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Collider Constraints applied toSimplified Models of Dark Matter fitted

to the Fermi-LAT gamma ray excessusing Bayesian Techniques

Guy Pitman

Department of Physics

University of Adelaide

This dissertation is submitted for the degree of

MPhil

October 2016

I would like to dedicate this thesis to my wife Julia who has enabled this undertaking by herpatience love and support

Declaration

I hereby declare that except where specific reference is made to the work of others thecontents of this dissertation are original and have not been submitted in whole or in partfor consideration for any other degree or qualification in this or any other university Thisdissertation is my own work and contains nothing which is the outcome of work done incollaboration with others except as specified in the text and Acknowledgements Thisdissertation contains fewer than 65000 words including appendices bibliography footnotestables and equations and has fewer than 150 figures

I certify that this work contains no material which has been accepted for the award of anyother degree or diploma in my name in any university or other tertiary institution and to thebest of my knowledge and belief contains no material previously published or written byanother person except where due reference has been made in the text In addition I certifythat no part of this work will in the future be used in a submission in my name for any otherdegree or diploma in any university or other tertiary institution without the prior approval ofthe University of Adelaide and where applicable any partner institution responsible for thejoint-award of this degree I give consent to this copy of my thesis when deposited in theUniversity Library being made available for loan and photocopying subject to the provisionsof the Copyright Act 1968

I acknowledge that copyright of published works contained within this thesis resides withthe copyright holder(s) of those works I also give permission for the digital version of mythesis to be made available on the web via the Universityrsquos digital research repository theLibrary Search and also through web search engines unless permission has been granted bythe University to restrict access for a period of time

Guy PitmanOctober 2016

Acknowledgements

And I would like to acknowledge the help and support of my supervisors Professor TonyWilliams and Dr Martin White as well as Assoc Professor Csaba Balasz who has assisted withinformation about a previous study Ankit Beniwal and Jinmian Li who assisted with runningMicrOmegas and LUXCalc I adapted the collider cuts programs originally developed bySky French and Martin White for my study

Contents

List of Figures xiii

List of Tables xv

1 Introduction 111 Motivation 312 Literature review 4

121 Simplified Models 4122 Collider Constraints 7

2 Review of Physics 921 Standard Model 9

211 Introduction 9212 Quantum Mechanics 9213 Field Theory 10214 Spin and Statistics 10215 Feynman Diagrams 11216 Gauge Symmetries and Quantum Electrodynamics (QED) 12217 The Standard Electroweak Model 13218 Higgs Mechanism 17219 Quantum Chromodynamics 222110 Full SM Lagrangian 23

22 Dark Matter 25221 Evidence for the existence of dark matter 25222 Searches for dark matter 30223 Possible signals of dark matter 30224 Gamma Ray Excess at the Centre of the Galaxy [65] 30

Contents vi

23 Background on ATLAS and CMS Experiments at the Large Hadron collider(LHC) 31231 ATLAS Experiment 32232 CMS Experiment 33

3 Fitting Models to the Observables 3531 Simplified Models Considered 3532 Observables 36

321 Dark Matter Abundance 36322 Gamma Rays from the Galactic Center 36323 Direct Detection - LUX 37

33 Calculations 39331 Mediator Decay 39332 Collider Cuts Analyses 42333 Description of Collider Cuts Analyses 43

4 Calculation Tools 5541 Summary 5542 FeynRules 5643 LUXCalc 5644 Multinest 5745 Madgraph 5846 Collider Cuts C++ Code 59

5 Majorana Model Results 6151 Bayesian Scans 6152 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 6453 Collider Constraints 67

531 Mediator Decay 67532 Collider Cuts Analyses 69

6 Real Scalar Model Results 7361 Bayesian Scans 7362 Best fit Gamma Ray Spectrum for the Real Scalar DM model 7663 Collider Constraints 77

631 Mediator Decay 77632 Collider Cuts Analyses 78

Contents vii

7 Real Vector Dark Matter Results 8171 Bayesian Scans 8172 Best fit Gamma Ray Spectrum for the Real Vector DM model 8473 Collider Constraints 84

731 Mediator Decay 84732 Collider Cuts Analyses 86

8 Conclusion 89

Bibliography 91

Appendix A Validation of Calculation Tools 97

Appendix B Branching ratio calculations for narrow width approximation 105B1 Code obtained from decayspy in Madgraph 105

List of Figures

21 Feynman Diagram of electron interacting with a muon 1122 Weak Interaction Vertices [48] 1523 Higgs Potential [49] 1824 Standard Model Particles and Forces [50] 2425 Bullet Cluster [52] 2526 Galaxy Rotation Curves [54] 2627 WMAP Cosmic Microwave Background Fluctuations [58] 2928 Dark Matter Interactions [60] 2929 Gamma Ray Excess from the Milky Way Center [75] 31210 ATLAS Experiment 31211 CMS Experiment 34

31 Main Feyman diagrams leading to the cross section for scalar decaying to apair of τ leptons 40

32 WidthmS vs mS 4033 WidthmS vs λb 4134 WidthmS vs λτ 4135 Main Feyman diagrams leading to the cross section for scalar decaying to a

pair of b quarks in the presence of at least one b quark 42

41 Calculation Tools 55

51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether 62

52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter 6353 Gamma Ray Spectrum 6454 Plots of log likelihoods by individual and combined constraints Masses in

GeV 6655 σ lowastBr(σ rarr ττ) versus Mass of Scalar 67

List of Figures ix

56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar 6857 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar 6858 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 6959 Excluded points from Collider Cuts and σBranching Ratio 70

61 Real Scalar Dark Matter - By Individual Constraint and All Together 7462 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter 7563 Gamma Ray Spectrum 7664 σ lowastBr(σ rarr ττ) versus Mass of Scalar 7765 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 7866 Excluded points from Collider Cuts and σBranching Ratio 80

71 Real Vector Dark Matter - By Individual Constraint and All Together 8272 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter 8373 Gamma Ray Spectrum 8474 σ lowastBr(σ rarr ττ) versus Mass of Scalar 8575 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 8676 Excluded points from Collider Cuts and σBranching Ratio 87

List of Tables

21 Quantum numbers of the Higgs field 1922 Weak Quantum numbers of Lepton and Quarks 21

31 Simplified Models 3532 95 CL by Signal Region 4433 Selection criteria common to all signal regions 4534 Selection criteria for signal regions A 4535 Selection criteria for signal regions C 4536 Signal Regions - Lepstop1 4837 Signal Regions Lepstop2 4938 Signal Regions 2bstop 5139 Signal Region ATLASmonobjet 52

51 Scanned Ranges 6152 Best Fit Parameters 64

61 Best Fit Parameters 76

71 Best Fit Parameters 84

A1 0 Leptons in the final state 98A2 1 Lepton in the Final state 100A3 2 Leptons in the final state 101A4 2b jets in the final state 102A5 Signal Efficiencies 90 CL on σ lim

exp[ f b] on pp gt tt +χχ 103

Chapter 1

Introduction

Dark matter (DM) was first postulated over 80 years ago when Swiss astronomer FritzZwicky observed a discrepancy between the amount of light emitted by a cluster of galaxiesand the total mass contained within the cluster inferred from the relative motion of thosegalaxies by a simple application of the theory of Newtonian gravitation The surprising resultof this observation was that the vast majority of the mass in the cluster did not emit lightwhich was contrary to the expectation that most of the mass would be carried by the starsSince that time further observations over a wide range of scales and experimental techniqueshave continued to point to the same result and refine it Some of these observations and otherevidence are discussed in section 22 We now know with certainty that in the entire Universeall of the matter we know about - stars planets gases and other cosmic objects such as blackholes can only account for less than 5 of the mass that we calculate to be there

A recent phenomenon that has received much attention is the significant deviation frombackground expectations of the Fermi Large Area Telescope(Fermi-LAT) gamma ray flux atthe galactic centre [1] A number of astrophysical explanations have been proposed includingmillisecond pulsars of supernova remnants [2] or burst-like continuous events at the galacticcentre but these are unresolved However it has also been noted that the observed Fermi-LATexcess is consistent with the annihilation of dark matter particles which would naturally beconcentrated at the Galactic centre in a manner consistent with the Navarro-Frenk-Whitedistribution of dark matter [3]

There are a number of other purely theoretical (particle physics) reasons to postulatethe existence of weakly interacting matter particles that could supply the missing mass andyet remain unobservable Weakly interacting massive particle (WIMPS) have been a majorfocus of Run I and ongoing Run II searches of the Large Hadron Collider (LHC) In spite

2

of substantial theoretical and experimental effort the microscopic properties of dark matterparticles are still unknown

One of the main problems with extracting these properties is the plethora of competingtheoretical models that provide dark matter candidates Another is the very weak interactionof these particles and the lack of experimental observables allowing one to discriminatebetween the competing models

A particularly fruitful approach is the simplified model approach which makes minimalassumptions about the DM candidates as extensions to the Standard Model interacting weaklywith ordinary matter through a mediator These can be coded as Lagrangian interaction termsusing tools such as FeynRules [4] and Calchep [5] and used to simulate Dark Matter relicdensities indirect detection expectations as well as direct detection expectations throughMicrOmegas [6] This in turn can be run under a Bayesian inference tool (see below)

Using a Bayesian inference tool such as MultiNest [7][8][9] one can scan the parameterspace of a particular model to find regions of highest probability for that model based onobservations of a range of phenomena including direct detection experiments such as ZEPLIN[10] XENON [11] DEAP [12] ARDM [13] DarkSide [14] PandaX [15] and LUX [16]indirect detection - eg Fermi-LAT gamma ray excess [17] Pamela-positron and antiprotonflux [18] and production at colliders MultiNest calculates the evidence value which is theaverage likelihood of the model weighted by prior probability and a posterior distributionwhich is the distribution of the parameter(s) after taking into account the observed databy scanning over a range of parameters based on prior probabilities This can be used tocompare models on a consistent basis The scans also provide information about the best fitparameters for the overall fit as well as for the seperate experimentsobservations

The simplified model approach has been taken in a number of papers to study the Fermi-LAT gamma ray excess (see literature review in section 12) and one in particular comparesthese models using a Bayesian inference tool which found that a Majorana fermion DMcandidate interacting via a scalar mediator with arbitrary couplings to third generationfermions showed the best fit [19] A subsequent paper [20] examined the most likelyparameter combinations that simultaneously fit the Planck relic density Fermi-LAT galacticcentre gamma ray AMS-02 positron flux and Pamela antiproton flux data

There has not as yet been a thorough study of the collider constraints on DM simplifiedmodels that fit the Fermi-LAT galactic centre excess In this thesis we reproduce the

11 Motivation 3

previous results of [20] using similar astrophysical constraints and then implement detailedsimulations of collider constraints on DM and mediator production

In Section 11 we review the motivation for this study and in Section 12 we considerrecent astronomical data which is consistent with the annihilation of DM particles andreview the literature and searches which attempt to explain it with a range of extensions tothe standard model

In Chapter 2 we review the Standard Model of particle physics (SM) the evidence fordark matter and the ATLAS and CMS experiments at the LHC

In Chapter 3 we describe the models considered in this paper the observables used inthe Bayesian scans and the collider experiments which provide limits against which theMagraph simulations of the three models can be compared

In Chapter 4 we review the calculation tools that have been used in this paper

In Chapters 5 6 and 7 we give the results of the Bayesian scans and collider cutsanalyses for each of the models and in Chapter 8 we summarise our conclusions

11 Motivation

The Fermi gamma ray space telescope has over the past few years detected a gamma raysignal from the inner few degrees around the galactic centre corresponding to a regionseveral hundred parsecs in radius The spectrum and angular distribution are compatible withthat from annihilating dark matter particles

A number of papers have recently attempted to explain the Gamma Ray excess in termsof extensions to the Standard Model taking into account constraints from ATLAS and CMScollider experiments direct detection of dark matter colliding off nucleii in well shieldeddetectors (eg LUX) as well as other indirect measurements (astrophysical observations)These papers take the approach of simplified models which describe the contact interactionsof a mediator beween DM and ordinary matter using an effective field theory approach(EFT) Until recently it has been difficult to discriminate between the results of these paperswhich look at different models and produce different statistical results A recent paper[19] Simplified Dark Matter Models Confront the Gamma Ray Excess used a Bayesianinference tool to compare a variety of simplified models using the Bayesian evidence values

12 Literature review 4

calculated for each model to find the most preferred explanation The motivation for thechosen models is discussed in section 121

A subsequent paper [20] Interpreting the Fermi-LAT gamma ray excess in the simplifiedframework studied the most likely model again using Bayesian inference to find the mostlikely parameters for the (most likely) model The data inputs (observables) to the scanwere the Planck measurement of the dark matter relic density [21] Fermi-LAT gamma rayflux data from the galactic center [17] cosmic positron flux data [22] cosmic anti-protonflux data [18] cosmic microwave background data and direct detection data from the LUXexperiment [16] The result of the scan was a posterior probability density function in theparameter space of the model

This paper extends the findings in [19] by adding collider constraints to the likelihoodestimations for the three models It is not a simple matter to add collider constraints becauseeach point in parameter space demands a significant amount of computing to calculate theexpected excess amount of observable particles corresponding to the particular experimentgiven the proposed simplified model A sampling method has been employed to calculatethe collider constraints using the probability of the parameters based on the astrophysicalobservables

12 Literature review

121 Simplified Models

A recent paper [23] summarised the requirements and properties that simplified models forLHC dark matter searches should have

The general principles are

bull Besides the Standard Model (SM) the Lagrangian should contain a DM candidate thatis absolutely stable or lives long enough to escape the LHC detector and a mediatorthat couples the two sectors

bull The Lagrangian should contain all terms that are renormalizable consistent withLorentz invariance SM symmetries and DM stability

bull Additional interactions should not violate exact and approximate global symmetries ofthe SM

12 Literature review 5

The examples of models that satisfy these requirements are

1 Scalar (or pseudo-scalar) s-channel mediator coupled to fermionic dark matter (Diracor Majorana)

2 Higgs portal DM where the DM is a scalar singlet or fermion singlet under the gaugesymmetries of the SM coupling to a scalar boson which mixes with the Higgs (Thefermion singlet is a specific realisation of 1 above)

3 DM is a mixture of an electroweak singlet and doublet as in the Minimal Supersym-metric SM (MSSM)

4 Vector (or Axial vector) s-channel mediator obtained by extending the SM gaugesymmetry by a new U(1)rsquo which is spontaneously broken such that the mediatoracquires mass

5 t-channel flavoured mediator (if the DM particle is a fermion χ the mediator can bea coloured scalar or vector particle φ -eg assuming a scalar mediator a couplingof the form φ χq requires either χ or φ to carry a flavor index to be consistent withminimal flavour violation (MFV) which postulates the flavour changing neutral currentstructure in the SM is preserved

Another recent paper [24] summarised the requirements and properties that simplifiedmodels for LHC dark matter searches should have The models studied in this paper arepossible candidates competing with Dirac fermion and complex scalar dark matter andmodels with vector and pseudo-scalar mediators Other models would have an elasticscattering cross section that will remain beyond the reach of direct detection experiments dueto the neutrino floor (the irreducible background produced by neutrino scattering) or arealready close to sensitivity to existing LUX and XENONIT experiments

A recent paper [25] Scalar Simplified Models for Dark Matter looks at Dirac fermiondark matter mediated by either a new scalar or pseudo-scalar to SM fermions This paperplaces bounds on the coupling combination gχgφ (coupling of DM to mediator times couplingof mediator to SM fermions) calculated from direct detection indirect detection and relicdensity by the mass of DM and mass of mediator Bounds are also calculated for colliderconstraints (mono-jet searches heavy-flavour searches and collider bounds on the 125 GeVHiggs are used to place limits on the monojet and heavy flavor channels which can betranslated into limits on the Higgs coupling to dark matter and experimental measurements

12 Literature review 6

of the Higgs width are used to constrain the addition of new channels to Higgs decay) Thepaper models the top loop in a full theory using the Monte Carlo generator MCFM [26]and extends the process to accommodate off-shell mediator production and decay to a darkmatter pair While this paper does not consider the Majorana fermion dark matter model theconstraints on the coupling product may have some relevence to the present study if only asan order of magnitude indicator

Constraining dark sectors with monojets and dijets [27] studies a simplified model withDirac fermion dark matter and an axial-vector mediator They use two classes of constraint-searches for DM production in events with large amounts of missing transverse energy inassociation with SM particles (eg monojet events) and direct constraints on the mediatorfrom monojet searches to show that for example where the choice of couplings of themediator to quarks and DM gA

q =gAχ=1 all mediator masses in the range 130 GeV ltMR lt3

TeV are excluded

The paper Constraining Dark Sectors at Colliders Beyond the Effective Theory Ap-proach [28] considers Higgs portal models with scalar pseudo-scalar vector and axial-vectormediators between the SM and the dark sector The paper takes account of loop processesand gluon fusion as well as quark-mediator interactions using MCFM Apart from mediatortype the models are characterised by free parameters of mediator width and mass darkmatter mass and the effective coupling combination gχgφ (coupling of DM to mediator timescoupling of mediator to SM fermions) The conclusion is that LHC searches can proberegions of the parameter space that are not able to be explored in direct detection experiments(since the latter measurements suffer from a velocity suppression of the WIMP-nucleoncross-section)

The paper Constraining the Fermi-LAT excess with multi-jet plus EmissT collider searches

[29] shows that models with a pseudo-scalar or scalar mediator are best constrained by multi-jet final states with Emiss

T The analysis was done using POWHEG-BOX-V2 at LO and usesM2

T variables together with multi-jet binning to exploit differences in signal and backgroundin the different regions of phase space to obtain a more stringent set of constraints on DMproduction This paper largely motivated the current work in that it showed that collidersearches that were not traditionally thought of as dark matter searches were neverthelessoften sensitive to dark matter produced in association with heavy quarks The combinationof searches used in this paper reflects this fact

12 Literature review 7

122 Collider Constraints

In order to constrain this model with collider constraints (direct detection limits) we considerthe following papers by the CMS and ATLAS collaborations which describe searches at theLHC

ATLAS Experiments

bull Search for dark matter in events with heavy quarks and missing transverse momentumin p p collisions with the ATLAS detector [30]

bull Search for direct third-generation squark pair production in final states with missingtransverse momentum and two b-jets in

radics= 8 TeV pp collisions with the ATLAS

detector[31]

bull Search for direct pair production of the top squark in all-hadronic final states inprotonndashproton collisions at

radics=8 TeV with the ATLAS detector [32]

bull Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

radic(s)=8TeV pp collisions using 21 f bminus1 of

ATLAS data [33]

bull Search for direct top squark pair production in final states with two leptons inradic

s = 8TeV pp collisions using 20 f bminus1 of ATLAS data [34]

bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

radics=8 TeV [35]

CMS Experiments

bull Searches for anomalous tt production in p p collisions atradic

s=8 TeV [36]

bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

radics=8 TeV [37]

bull Search for the production of dark matter in association with top-quark pairs in thesingle-lepton final state in proton-proton collisions at

radics = 8 TeV [38]

bull Search for new physics in monojet events in p p collisions atradic

s = 8 TeV(CMS) [39]

bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

s = 8 TeV [40]

12 Literature review 8

bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theSingle-lepton Final State in pp collisions at

radics=8 TeV [41]

bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

s=8 TeV [42]

bull Search for neutral MSSM Higgs bosons decaying into a pair of bottom quarks [43]

bull Search for neutral MSSM Higgs bosons decaying into a pair of τ leptons in p p collision[44]

bull Search for dark matter extra dimensions and unparticles in monojet events in proton-proton collisions at

radics=8 TeV [45]

In particular we have implemented Madgraph simulations of certain data analysesapplying to collider experiments described in papers [32][33][34][31][30][41] Theremaining models were not implemented in Madgraph because of inability to validate themodels against published results in the time available We developed C++ analysis code toimplement the collider cuts after showering with Pythia and basic analysis with Delpheswith output to root files The C++ code allowed multiple experimental cuts to be applied tothe same simulation output The C++ code was first checked against the results of the variousexperiments above and then the simulations were rerun using the three modelsrsquo Lagrangians(implemented in FeynRules) described in this paper

Chapter 2

Review of Physics

21 Standard Model

211 Introduction

The Standard Model of particle physics (SM) has sometimes been described as the theoryof Almost Everything The SM is a field theory which emerged in the second half of thetwentieth century from quantum mechanics and special relativity which themselves wereonly conceived after 1905 by Einsten Pauli Bohr Dirac and others In the first decades ofthe twenty first century the discovery of the Higgs boson [46][47] completed the pictureand confirmed the last previously unverified part of the SM The SM is a self consistenttheory that describes and classifies all known particles and their interactions (other thangravity) to an astonishing level of accuracy and detail However the theory is incomplete-it does not describe the complete theory of gravitation or account for DM (see below) ordark energy (the accelerating expansion of the universe) Related to these shortcomings aretheoretical deficiencies such as the asymmetry of matter and anti-matter the fact that the SMis inconsistent with general relativity (to the extent that one or both theories break down atspacetime singularities like the Big Bang and black hole event horizons)

212 Quantum Mechanics

Quantum Mechanics (QM) describes particles as probability waves which can be thoughtof as packets of energy which can be localised or spread through all space depending on how

21 Standard Model 10

accurately one measures the particlersquos momentum However QM is not itself consistent withspecial relativity It was realised too that because of relativity particles could be created anddestroyed and because of the uncertainty principle the theory should describe the probabilityof particles being created and destroyed on arbitrarily short timescales A quantum fieldtheory (QFT) is a way to describe this probability at every point of space and time

213 Field Theory

A field theory is described by a mathematical function of space and time called a LagrangianAs in classical mechanics where Lagrangians were introduced by Louis Lagrange at thebeginning of the nineteenth century to summarise dynamical systems the Lagrangian sum-marises the dynamics of space which is permeated by force fields The equations of motionarising from a Lagrangian are determined by finding the contours of the Lagrangian describedby stationary points (ie where the infinitesimal change in the Lagrangian is zero for a smallchange in field) Particles are generally described by singularities in the Lagrangian (wherethe values tend to infinity also called resonances)

214 Spin and Statistics

It was also realised that different types of particle needed to be represented by different fieldsin the Lagrangian Diracrsquos discovery that particles with spin 12 quantum numbers couldbe represented by a field obtained by effectively taking the square root of the relativisticequation (Klein Gordon equation) was probably the most significant advance in quantumfield theory It required the introduction of a matrix operators called the gamma matricesor Dirac matrices which helped explain the already known fact that spin half particles orfermions obey Fermi-Dirac statistics ie when two spin half particles are exchanged thesign of the wavefunction must change and hence they cannot occupy the same quantum stateunlike bosons with integer spin that can occupy the same quantum state

Other particles need to be represented by fields called vector pseudo-vector scalar (as inthe Klein Gordon equation) pseudo-scalar and complex fields A vector field is really fourfields one for each dimension of space plus time and is parameterised by the four gammamatrices γmicro which transform among themselves in a relativistically invariant way (Lorentzinvariance) A pseudo-vector field is a vector field which transforms in the same way but

21 Standard Model 11

with additional sign change under parity transformations (a simultaneous reversal of the threespatial directions) and is represented by γmicroγ5 A pseudo-scalar is a scalar which changes signunder parity transformations and a complex scalar field is represented by complex numbers

215 Feynman Diagrams

QFT took another giant leap forward when Richard Feynman realised that the propagationof particles described by the field could be described by expanding the so called S-matrixor scattering matrix derived from calculating the probability of initial states transitioning tofinal states The expansion is similar to a Taylor expansion in mathematics allowing for thespecial commuting properties of elements of the field The Feynman diagrams representingindividual terms in the expansion can be interpreted as interactions between the particlecomponents The S-Matrix (or more correctly the non trivial part called the T-Matrix) isderived using the interaction Hamiltonian of the field The Hamiltonian is an operator whoseeigenvalues or characteristic values give the energy quanta of the field The T-matrix canalso be described by a path integral which is an integral over all of quantum space weightedby the exponential of the Lagrangian This is completely equivalent to the Hamiltonianformulation and shows how the Lagrangian arises naturally out of the Hamiltonian (in factthe Lagrangian is the Legendre transformation of the Hamiltonian)

Figure 21 Feynman Diagram of electron interacting with a muon

γ

eminus

e+

micro+

microminus

The diagram Fig21 is a tree diagram (diagrams with closed loops indicate self in-teractions and are higher order terms in the expansion) and shows an electron (eminus) and ananti-muon (micro+) interacting by exchanging a photon

21 Standard Model 12

216 Gauge Symmetries and Quantum Electrodynamics (QED)

The Dirac equation with its gamma matrices ushered in the use of group theory to describethe transformation of fields (because the multiplication table of the gamma matrices isclosed and described by group theory) It also ushered in the idea of symmetries within theLagrangian- for example consider a Lagrangian

ψ(ipart minusm)ψ (21)

The Lagrangian is invariant under a global U(1) transformation which acts on the fieldsand hence the derivatives as follows

ψ rarr eiqαψ ψ rarr ψeminusiqα partmicroψ rarr eiqα

partmicroψ (22)

where qα is a global phase and α is a continuous parameter

A symmetry leads by Noetherrsquos theorem to a conserved current jmicro = qψγmicroψ and is thebasis of charge conservation since partmicro jmicro = 0 and q =

intd3x j0(x)

By promoting this transformation to a local one (ie α rarr α(x) where x = xmicro are thespace time co-ordinates) the transformations become

ψ rarr eiqα(x)ψ ψ rarr ψeminusiqα(x)partmicroψ rarr eiqα(x)(partmicroψ + iq(partmicroα(x))ψ) (23)

The derivative transformation introduces an extra term which would spoil the gaugeinvariance This can be solved by introducing a new field the photon field Amicro(x) whichinteracts with the fermion field ψ and transforms under a U(1) transformation

Amicro rarr Amicro minuspartmicroα(x) (24)

If we replace normal derivatives partmicro with covariant derivatives Dmicro = (partmicro + iqAmicro(x)) in theLagrangian then the extra term is cancelled and we can now see that the covariant derivativetransforms like ψ - under a local U(1) transform ie

Dmicro rarr Dmicroψprime = eiqα(x)Dmicroψ (25)

The additional term iqAmicro(x) can be interpreted in a geometric sense as a connectionwhich facilitates the parallel transport of vectors along a curve in a parallel consistent manner

21 Standard Model 13

We must add a gauge-invariant kinetic term to the Lagrangian to describe the dynamicsof the photon field This is given by the electromagnetic field strength tensor

Fmicroν = partmicroAν minuspartνAmicro (26)

The full Quantum Electrodynamics (QED) Lagrangian is then for electrons with mass mand charge -e

LQED = ψ(i Dminusm)ψ minus 14

Fmicroν(X)Fmicroν(x) (27)

This Lagrangian is Lorentz and U(1) guage invariant The interaction term is

Lint =+eψ Aψ = eψγmicro

ψAmicro = jmicro

EMAmicro (28)

where jmicro

EM is the electromagnetic four current

217 The Standard Electroweak Model

The Standard Electroweak (EW) Model was created by Glashow-Weinberg-Salam (also calledthe GWS model) and unifies the electomagnetic and weak interactions of the fundamentalparticles

The idea of gauge symmetry was extended to the idea of a non-abelian gauge symmetry(essentially the operators defining the symmetry do not commute and so are represented bymatrices- in this case the Pauli matrices for the SU(2) group) - the gauge group is the directproduct G = SU(2)

otimesU(1) It was known that weak interactions were mediated by Wplusmn

and Z0 particles where the superscripts are the signs of the electromagnetic charge Thesewere fairly massive but it was also known that through spontaneous symmetry breaking ofa continuous global non-abelian symmetry there would be one massive propagation modefor each spontaneously broken-symmetry generator and massless modes for the unbrokengenerators (the so-called Goldstone bosons)

This required an extra field to represent the gauge bosons called Bmicro (a vector fieldhaving the 4 spacetime degrees of freedom) With this identification the covariant derivativebecomes

Dmicro = partmicro minus igAmicro τ

2minus i

gprime

2Y Bmicro (29)

21 Standard Model 14

Here Y is the so called weak hypercharge of the field Bmicro which is a quantum numberdepending on particle (lepton or quark) g and gprime are two independent parameters in theGWS theory and the expression Amicro τ implies a sum over 3 components Aa

micro a=123 and thePauli matrices τa

This isnrsquot the full story because it had been known since 1956 when Chin Shuing Wuand her collaborators confirmed in an experiment prompted by Lee and Yang that nature isnot ambivalent to so-called parity It had been assumed that the laws of nature would notchange if all spatial coordinates were reversed (a parity transformation) The experimentinvolved the beta decay of cooled 60C atoms (more electrons were emitted in the backwardhemisphere with respect to spin direction of the atoms) This emission involved the weakinteraction and required that the Lagrangian for weak interactions contain terms which aremaximally parity violating- which turn out to be currents of the form

ψ(1minus γ5)γmicro

ψ (210)

The term

12(1minus γ

5) (211)

projects out a left-handed spinor and can be absorbed into the spinor to describe left-handedleptons (and as later discovered quarks) The right handed leptons are blind to the weakinteraction

The processes describing left-handed current interactions are shown in Fig 22

Analogously to isospin where the proton and neutron are considered two eigenstates ofthe nucleon we postulate that Bmicro is the gauge boson of an SU(2)transformation coupled toweak SU(2) doublets (

νe

eminus

)

(ud

) (212)

We may now write the weak SU(2) currents as eg

jimicro = (ν e)Lγmicro

τ i

2

e

)L (213)

21 Standard Model 15

Figure 22 Weak Interaction Vertices [48]

where τ i are the Pauli-spin matrices and the subscript L denotes the left-handed componentof the spinor projected out by Eqn211 The third current corresponding to i=3 is called theneutral current as it doesnrsquot change the charge of the particle involved in the interaction

We could postulate that the electromagnetic current is a linear combination of the leftand right elecromagnetic fields eL = 1

2(1minus γ5)e and eR = 12(1+ γ5)e

jemmicro = eLγmicroQeL + eRγmicroQeR (214)

where Q is the electromagnetic operator and eR is a singlet However this is not invariantunder SU(2)L To construct an invariant current we need to use the SU(2)L doublet 212The current which is an SU(2)L invariant U(1) current is

jYmicro = (ν e)LγmicroYL

e

)L+ eRγmicroYReR (215)

where hypercharges YL and YR are associated with the U(1)Y symmetry jYmicro is a linearcombination of the electromagnetic current and the weak neutral current thus explaining whythe hypercharge differs for the left and right-handed components since the electromagneticcurrent has the same charge for both components whilst the weak current exists only forleft-handed particles j3

micro and the third component of weak isospin T 3 allows us to calculate

21 Standard Model 16

the relationship Y = 2(QminusT 3) which can be seen by writing jYmicro as a linear combination ofj3micro (the weak neutral current that doesnrsquot change the charge of the particle involved in the

interaction) and jemmicro (using the convention that the jYmicro is multiplied by 1

2 to match the samefactor implicit in j3

micro ) Substituting

τ3 =

(1 00 minus1

)(216)

into equation 213 for i=3 and subtracting equation 215 and equating to 12 times equation

214

we get

eLγmicroQeL + eRγmicroQeR minus (νLγmicro

12

νL minus eLγmicro

12

eL) =12

eRγmicroYReR +12(ν e)LγmicroYL

e

)L (217)

from which we can read out

YR = 2QYL = 2Q+1 (218)

and T3(eR) = 0 T3(νL) =12 and T3(eL) =

12 The latter three identities are implied by

the fraction 12 inserted into the definition of equation 213

The Lagrangian kinetic terms of the fermions can then be written

L =minus14

FmicroνFmicroν minus 14

GmicroνGmicroν

+ sumgenerations

LL(i D)LL + lR(i D)lR + νR(i D)νR

+ sumgenerations

QL(i D)QL +UR(i D)UR + DR(i D)DR

(219)

LL are the left-handed lepton doublets QL are the left-handed quark doublets lRνR theright handed lepton singlets URDR the right handed quark singlets

The field strength tensors are given by

Fmicroν = partmicroAν minuspartνAmicro minus ig[Amicro Aν ] (220)

21 Standard Model 17

andGmicroν = partmicroBν minuspartνBmicro (221)

Bmicro is an abelian U(1) gauge field while Amicro is non-abelian and Fmicroν contains the com-mutator of the field through the action of the covariant derivative [Dmicro Dν ] This equationcontains no mass terms terms for the field Amicro for good reason- this would violate gaugeinvariance We know that the gauge bosons of the weak force have substantial mass and theway that this problem has been solved is through the Higgs mechanism

218 Higgs Mechanism

To solve the problem of getting gauge invariant mass terms three independent groupspublished papers in 1964 The mechanism was named after Peter Higgs To introduce theconcept consider a complex scalar field φ that permeates all space Consider a potential forthe Higgs field of the form

minusmicro2φ

daggerφ +λ (φ dagger

φ2) (222)

which has the form of a mexican hat Fig 23 This potential is still U(1) gauge invariant andcan be added to our Lagrangian together with kinetic+field strength terms to give

L = (Dmicroφ)dagger(Dmicroφ)minusmicro2φ

daggerφ +λ (φ dagger

φ2)minus 1

4FmicroνFmicroν (223)

It is easily seen that this is invariant to the transformations

Amicro rarr Amicro minuspartmicroη(x) (224)

φ(x)rarr eieη(x)φ(x) (225)

The minimum of this field is not at 0 but in a circular ring around 0 with a vacuum

expectation value(vev)radic

micro2

2λequiv vradic

2

We can parameterise φ as v+h(x)radic2

ei π

Fπ where h and π are referred to as the Higgs Bosonand Goldstone Boson respectively and are real scalar fields with no vevs Fπ is a constant

21 Standard Model 18

Figure 23 Higgs Potential [49]

Substituting this back into the Lagrangian 223 we get

minus14

FmicroνFmicroν minusevAmicropartmicro

π+e2v2

2AmicroAmicro +

12(partmicrohpart

microhminus2micro2h2)+

12

partmicroπpartmicro

π+(hπinteractions)(226)

This Lagrangian now seems to describe a theory with a photon Amicro of mass = ev a Higgsboson h with mass =

radic2micro and a massless Goldstone π

However there is a term mixing Amicro and part microπ in this expression and this shows Amicro and π

are not independent normal coordinates and we can not conclude that the third term is a massterm This π minusAmicro mixing can be removed by making a gauge transformation in equations224 and 225 making

φrarrv+h(x)radic2

ei π

Fπminusieη(x) (227)

and setting πrarr π

Fπminus eη(x) = 0 This choosing of a point on the circle for the vev is called

spontaneous symmetry breaking much like a vertical pencil falling and pointing horizontallyin the plane This choice of gauge is called unitary gauge The Goldstone π will thencompletely disappear from the theory and one says that the Goldstone has been eaten to givethe photon mass

21 Standard Model 19

This can be confirmed by counting degrees of freedom before and after spontaneoussymmetry breaking The original Lagrangian had four degrees of freedom- two from thereal massless vector field Amicro (massless photons have only two independent polarisationstates) and two from the complex scalar field (one each from the h and π fields) The newLagrangian has one degree of freedom from the h field and 3 degrees of freedom from themassive real vector field Amicro and the π field has disappeared

The actual Higgs field is postulated to be a complex scalar doublet with four degrees offreedom

Φ =

(φ+

φ0

)(228)

which transforms under SU(2) in the fundamental representation and also transforms underU(1) hypercharge with quantum numbers

Table 21 Quantum numbers of the Higgs field

T 3 Q Yφ+

12 1 1

φ0 minus12 1 0

We can parameterise the Higgs field in terms of deviations from the vacuum

Φ(x) =(

η1(x)+ iη2(x)v+σ(x)+ iη3(x)

) (229)

It can be shown that the three η fields can be transformed away by working in unitarygauge By analogy with the scalar Higgs field example above the classical action will beminimised for a constant value of the Higgs field Φ = Φ0 for which Φ

dagger0Φ0 = v2 This again

defines a range of minima and we can pick one arbitarily (this is equivalent to applying anSU(2)timesU(1) transform to the Higgs doublet transforming the top component to 0 and thebottom component to Φ0 = v) which corresponds to a vacuum of zero charge density andlt φ0 gt=+v

In this gauge we can write the Higgs doublet as

Φ =

(φ+

φ0

)rarr M

(0

v+ H(x)radic2

) (230)

where H(x) is a real Higgs field v is the vacuum expectation and M is an SU(2)timesU(1)matrix The matrix Mdagger converts the isospinor into a down form (an SU(2) rotation) followedby a U(1) rotation to a real number

21 Standard Model 20

If we consider the Higgs part of the Lagrangian

minus14(Fmicroν)

2 minus 14(Bmicroν)

2 +(DmicroΦ)dagger(DmicroΦ)minusλ (Φdagger

Φminus v2)2 (231)

Substituting from equation 230 into this and noting that

DmicroΦ = partmicroΦminus igW amicro τ

aΦminus 1

2ig

primeBmicroΦ (232)

We can express as

DmicroΦ = (partmicro minus i2

(gA3

micro +gprimeBmicro g(A1micro minusA2

micro)

g(A1micro +A2

micro) minusgA3micro +gprimeBmicro

))Φ equiv (partmicro minus i

2Amicro)Φ (233)

After some calculation the kinetic term is

(DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

14(v+

Hradic2)2[A 2]22 (234)

where the 22 subscript is the index in the matrix

If we defineWplusmn

micro =1radic2(A1

micro∓iA2micro) (235)

then [A 2]22 is given by

[A 2]22 =

(gprimeBmicro +gA3

micro

radic2gW+

microradic2gWminus

micro gprimeBmicro minusgA3micro

) (236)

We can now substitute this expression for [A 2]22 into equation 234 and get

(DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

14(v+

Hradic2)2(2g2Wminus

micro W+micro +(gprimeBmicro minusgA3micro)

2) (237)

This expression contains mass terms for three massive gauge bosons Wplusmn and the combi-nation of vector fields gprimeBmicro minusgA3

micro where note

21 Standard Model 21

Table 22 Weak Quantum numbers of Lepton and Quarks

T 3 Q YνL

12 0 -1

lminusL minus12 -1 -1

νR 0 0 0lminusR 0 -1 -2UL

12

23

13

DL minus12 minus1

313

UR 0 23

43

DR 0 minus13 minus2

3

Wminusmicro = (W+

micro )dagger equivW 1micro minus iW 2

micro (238)

Then the mass terms can be written

12

v2g2|Wmicro |2 +14

v2(gprimeBmicro minusgA3micro)

2 (239)

W+micro and W microminus can be associated with the W boson and its anti-particle and (minusgprimeBmicro +

gA3micro) with the Z Boson (after normalisation by

radicg2 +(gprime

)2) The combination gprimeA3micro +gBmicro

is orthogonal to Z and remains massless This can be associated with the Photon (afternormalisation) These predict mass terms for the W and Z bosons mW = vgradic

2and mZ =

vradic2

radicg2 +(gprime

)2It is again instructive to count the degrees of freedom before and after the Higgs mech-

anism Before we had a complex doublet Φ with four degrees of freedom one masslessB with two degrees of freedom and three massless W a fields each with two each givinga total of twelve degrees of freedom After spontaneous symmetry breaking we have onereal scalar Higgs field with one degree of freedom three massive weak bosons with nineand one massless photon with two again giving twelve degrees of freedom One says thescalar degrees of freedom have been eaten to give the Wplusmn and Z bosons their longitudinalcomponents (ie mass)

Table 22 shows T 3 the weak isospin quantum number (which describes how a particletransforms under SU(2)) Y the weak hypercharge (which describes how the particle trans-

21 Standard Model 22

forms under U(1)) and Q the electric charge of leptons and quarks The former two enterdirectly into the Lagrangian while the latter can be derived as a function of the former two

Strictly speaking the right handed neutrino which has no charge may not exist and is notpart of the standard model although some theories postulate a mass for this (hypothetical)particle

Writing gauge invariant couplings between ELeRφ with the vacuum expectation ofφ = v (Yukawa matrices λ

i ju and λ

i jd respectively for the up and down quarks) we get mass

terms for the quarks (and similarly for the leptons)

Mass terms for quarks minussumi j[(λi jd Qi

Lφd jR)+λ

i ju εab(Qi

L)aφlowastb u j

R +hc]

Mass terms for leptonsminussumi j[(λi jl Li

Lφ l jR)+λ

i jν εab(Li

L)aφlowastb ν

jR +hc]

Here hc is the hermitian conjugate of the first terms L and Q are lepton and quarkdoublets l and ν are the lepton singlet and corresponding neutrino d and u are singlet downand up quarks εab is the Levi-Civita (anti-symmetric) tensor

If we write the quark generations in a basis that diagonalises the Higgs couplings weget a unitary transformations Uu and Ud for the up and down quarks respectively which iscompounded in the boson current (Udagger

u Ud) called the CKM matrix where off-diagonal termsallow mixing between quark generations (this is also called the mass basis as opposed tothe flavour basis) A similar argument can give masses to right handed sterile neutrinos butthese can never be measured However a lepton analogue of the CKM matrix the PNMSmatrix allows left-handed neutrinos to mix between various generations and oscillate betweenflavours as they propagate through space allowing a calculation of mass differences

219 Quantum Chromodynamics

The final plank in SM theory tackling the strong force Quantum Chromodynamics wasproposed by Gell-Mann and Zweig in 1963 Three generations of quarks were proposedwith charges of 23 for uc and t and -13 for ds and b Initially SU(2) isospin was consideredtransforming u and d states but later irreducible representations of SU(3) were found to fit theexperimental data The need for an additional quantum number was motivated by the fact thatthe lightest excited state of the nucleon consisted of sss (the ∆++) To reconcile the baryon

21 Standard Model 23

spectrum with spin statistics colour was proposed Quarks transform under the fundamentalor 3 representation of colour symmetry and anti-quarks under 3 The inner product of 3and 3 is an invariant under SU(3) as is a totally anti-symmetric combination ε i jkqiq jqk (andthe anti-quark equivalent) It was discovered though deep inelastic scattering that quarksshowed asymptotic freedom (the binding force grew as the seperation grew) Non-abeliangauge theories showed this same property and colour was identified with the gauge quantumnumbers of the quarks The quanta of the gauge field are called gluons Because the forcesshow asymptotic freedom - ie the interaction becomes weaker at high energies and shorterdistances perturbation theory can be used (ie Feynman diagrams) at these energies but forlonger distances and lower energies lattice simulations must be done

2110 Full SM Lagrangian

The full SM can be written

L =minus14

BmicroνBmicroν minus 18

tr(FmicroνFmicroν)minus 12

tr(GmicroνGmicroν)

+ sumgenerations

(ν eL)σmicro iDmicro

(νL

eL

)+ eRσ

micro iDmicroeR + νRσmicro iDmicroνR +hc

+ sumgenerations

(u dL)σmicro iDmicro

(uL

dL

)+ uRσ

micro iDmicrouR + dRσmicro iDmicrodR +hc

minussumi j[(λ

i jl Li

Lφ l jR)+λ

i jν ε

ab(LiL)aφ

lowastb ν

jR +hc]

minussumi j[(λ

i jd Qi

Lφd jR)+λ

i ju ε

ab(QiL)aφ

lowastb u j

R +hc]

+ (Dmicroφ)dagger(Dmicroφ)minusmh[φφ minus v22]22v2

(240)

where σ micro are the extended Pauli matrices

(1 00 1

)

(0 11 0

)

(0 minusii 0

)

(1 00 minus1

)

The first line gives the gauge field strength terms the second the lepton dynamical termsthe third the quark dynamical terms the fourth the lepton (including neutrino) mass termsthe fifth the quark mass terms and the sixth the Higgs terms

The SU(3) gauge field is given by Gmicroν and which provides the colour quantum numbers

21 Standard Model 24

Figure 24 Standard Model Particles and Forces [50]

Note The covariant derivatives operating on the left-handed leptons contain only theU(1) and SU(2) gauge fields Bmicro Fmicro whereas those operating on left-handed quarks containU(1)SU(2) and SU(3) gauge fields Bmicro Fmicro Gmicro The covariant derivatives operating onright handed electrons and quarks are as above but do not contain the SU(2) gauge fieldsRight handed neutrinos have no gauge fields

The sums over i j above are over the different generations of leptons and quarks

The particles and forces that emerge from the SM are shown in Fig 24

22 Dark Matter 25

22 Dark Matter

221 Evidence for the existence of dark matter

2211 Bullet Cluster of galaxies

Most of the mass of the cluster as measured by a gravitational lensing map is concentratedaway from the centre of the object The visible matter (as measured by X-ray emission) ismostly present in the centre of the object The interpretation is that the two galaxy clustershave collided perpendicular to the line of sight with the mostly collisionless dark matterpassing straight through and the visible matter interacting and being trapped in the middleIt was estimated that the ratio of actual to visible matter was about 16 [51]

Figure 25 Bullet Cluster [52]

2212 Coma Cluster

The first evidence for dark matter came in the 1930s when Fritz Zwicky observed the Comacluster of galaxies over 1000 galaxies gravitationally bound to one another He observed thevelocities of individual galaxies in the cluster and estimated the amount of matter based onthe average galaxy He calculated that many of the galaxies should be able to escape fromthe cluster unless the mass was about 10 times higher than estimated

22 Dark Matter 26

2213 Rotation Curves [53]

Since those early observations many spiral galaxies have been observed with the rotationcurves calculated by doppler shift with the result that in all cases the outermost stars arerotating at rates fast enough to escape their host galaxies Indeed the rotation curves arealmost flat indicating large amounts of dark matter rotating with the galaxies and holdingthe galaxies together

Figure 26 Galaxy Rotation Curves [54]

2214 WIMPS MACHOS

The question is if the particles that make up most of the mass of the universe are not baryonswhat are they It is clear that the matter does not interact significantly with radiation so mustbe electrically neutral It has also not lost much of its kinetic energy sufficiently to relax intothe disks of galaxies like ordinary matter Massive particles may survive to the present ifthey carry a conserved additive or multiplicative quantum number This is so even if equalnumbers of particles and anti-particles can annihilate if the number densities become too lowto reduce the density further These particles have been dubbed Weakly interacting MassiveParticles (WIMPS) as they interact with ordinary matter only through a coupling strengthtypically encountered in the weak interaction They are predicted to have frozen out ofthe hot plasma and would give the correct relic density based solely on the weak interactioncross section (sometimes called the WIMP Miracle) Another possibility called MassiveCompact Halo Objects (MACHOS)- are brown dwarf stars Jupiter mass objects or remnantsfrom an earlier generation of star formation

22 Dark Matter 27

2215 MACHO Collaboration [55]

In order to estimate the mass of MACHOS a number of experiments have been conductedusing microlensing which makes use of the fact that if a massive object lies in the line ofsight of a much more distant star the light from the more distant star will be lensed and forma ring around the star It is extemely unlikely that the MACHO will pass directly through theline of sight but if there is a near miss the two images will be seperated by a small angleThe signature for microlensing is a time-symmetric brightening of a star as the MACHOpasses near the line of sight called a light curve One can obtain statistical information aboutthe MACHO masses from the shape and duration of the light curves and a large number ofstars (many million) must be monitored The MACHO Collaboration [55] have monitoredover 11 million stars and less than 20 events were detected The conclusion was that thiscould account for a small addition (perhaps up to 20 ) to the visible mass but not enoughto account for Dark Matter

2216 Big Bang Nucleosynthesis (BBN) [56]

Another source of evidence for DM comes from the theory of Nucleosynthesis in the fewseconds after the Big Bang In the first few minutes after the big bang the temperatures wereso high that all matter was dissociated but after about 3 minutes the temperature rapidlycooled to about 109 Kelvin Nucleosynthesis could begin when protons and neutrons couldstick together to form Deuterium (Heavy Hydrogen) and isotopes of Helium The ratio ofDeuterium to Hydrogen and Helium is very much a function of the baryon density and alsothe rate of expansion of the universe at the time of Nucleosynthesis There is a small windowwhen the density of Baryons is sufficient to account for the ratios of the various elementsThe rate of expansion of the universe is also a function of the total matter density The ratiosof these lighter elements have been calculated by spectral measurements with Helium some25 of the mass of the universe Deuterium 001 The theory of BBN accounts for theratios of all of the lighter elements if the ratio of Baryon mass to total mass is a few percentsuggesting that the missing matter is some exotic non-baryonic form

22 Dark Matter 28

2217 Cosmic Microwave Background [57]

The discovery of a uniform Microwave Background radiation by Penzias and Wilson in1965 was confirmation of the theory first proposed by Gamow and collaborators that theuniverse should be filled with black body radiation an afterglow of the big bang TheCOBE satelite launched in 1989 confirmed this black body radiation with temperature of2725+-002K with 95 confidence While extremely uniform it found tiny irregularitieswhich could be accounted for by irregularities in the density of the plasma at last scattering(gravitational redshifts or blueshifts called the Sachs Wolfe effect) and temperature andvelocity fluctuations in the plasma The anisotropy of the background has been furtheranalysed by WMAP (the Wilkinson Microwave Anisotropy Probe) launched in 2001 Thefluctuations in the density of the plasma are what account for the seeds of galaxy formationHowever detailed calculations have found that the fluctuations were too small to account forthe observed galaxy distribution without an electrically neutral gravitating matter in additionto ordinary matter that only became electrically neutral at the time of recombination [58]

In May 2009 the European Space Agency (ESA) launched the Planck sattelite a spacebased observatory to observe the universe at wavelengths between 03mm and 111 mmbroadly covering the far-infrared microwave and high frequency domains The missionrsquosmain goal was to study the cosmic microwave background across the whole sky at greatersensitivity and resolution than ever before Using the information gathered and the standardmodel of cosmology ESA was able to calculate cosmological parameters including the relicdensity of dark matter remaining after the Big Bang to a high degree of precision withinthe framework of the standard model of cosmology The standard model of cosmology canbe described by a relatively small number of parameters including the density of ordinarymatter dark matter and dark energy the speed of cosmic expansion at the present epoch (alsoknown as the Hubble constant) the geometry of the Universe and the relative amount ofthe primordial fluctuations embedded during inflation on different scales and their amplitude[59]

2218 LUX Experiment - Large Underground Xenon experiment [16]

The LUX experiment is conducted at the Sanford Underground research facility one mileunderground to minimise the background of cosmic rays The detector is a time projectionchamber allowing a 3D positioning of interactions occurring within its active volume LUX

22 Dark Matter 29

Figure 27 WMAP Cosmic Microwave Background Fluctuations [58]

Figure 28 Dark Matter Interactions [60]

uses as its target 368 kiliograms of liquefied ultra pure xenon which is a scintillator Interac-tions inside the xenon will create an amount of light proportional to the energy depositedThat light can be collected on arrays of light detectors sensitive to a single photon

Xenon is very pure and the amount of radiation originating within the target is limitedAlso being three times denser than water it can stop radiation originating from outside thedetector The 3D detector focuses on interactions near the center which is a quiet regionwhere rare dark matter interactions might be detected [16]

22 Dark Matter 30

222 Searches for dark matter

2221 Dark Matter Detection

Fig 28 illustrates the reactions studied under the different forms of experiment designed todetect DM

Direct detection experiments are mainly underground experiments with huge detectorsattempting to detect the recoil of dark matter off ordinary matter such as LUX [16] CDMS-II [61] XENON100 [11] COUPP [62] PICASSO [63] ZEPLIN [10] These involveinteractions between DM and SM particles with final states involving DM and SM particles

Indirect detection involves the search for the annihilation of DM particles in distantastrophysical sources where the DM content is expected to be large Example facilitiesinclude space based experiments (EGRET [64] PAMELA [18] FERMI-LAT [65]) balloonflights (ATIC [66] PEBS [67] PPB-BETS [68]) or atmospheric Cherenkov telescopes(HESS [69]MAGIC [70]) and neutrino telescopes (SuperKamiokande [71] IceCube [72]ANTARES [73] KM3NET [74]) Indirect Detection involves DM particles in the initial stateand SM particles in the final state

Collider detection involves SM particles in the initial state creating DM particles in thefinal state which can be detected by missing transverse energy and explained by variousmodels extending the SM

223 Possible signals of dark matter

224 Gamma Ray Excess at the Centre of the Galaxy [65]

The center of the galaxy is the most dense and active region and it is no surprise that itproduces intense gamma ray sources When astronomers subtract all known gamma raysources there remains an excess which extends out some 5000 light years from the centerHooper et al find that the annihilation of dark matter particles with a mass between 31 and40 Gev fits the excess to a remarkable degree based on Gamma Ray spectrum distributionand brightness

23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 31

Figure 29 Gamma Ray Excess from the Milky Way Center [75]

23 Background on ATLAS and CMS Experiments at theLarge Hadron collider (LHC)

The collider constraints in this paper are based on two of the experiments that are part of theLHC in CERN

Figure 210 ATLAS Experiment

The LHC accelerates proton beams in opposite directions around a 27 km ring of super-conducting magnets until they are made to collide at an interaction point inside one of the 4experiments placed in the ring An upgrade to the LHC has just been performed providing65 TeV per beam

23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 32

231 ATLAS Experiment

The ATLAS detector consists of concentric cylinders around the interaction point whichmeasure different aspects of the particles that are flung out of the collision point There are 4major parts the Inner Detector the Calorimeters the Muon spectrometer and the magnetsystems

2311 Inner Detector

The inner detector extends from a few centimeters to 12 meters from the interaction point andmeasures details of the charged particles and their momentum interacting with the materialIf particles seem to originate from a point other than the interaction point this may indicatethat the particle comes from the decay of a hadron with a bottom quark This is used to b-tagcertain particles

The inner detector consists of a pixel detector with three concentric layers and threeendcaps containing detecting silicon modules 250 microm thick The pixel detector has over 80million readout channels about 50 percent of the total channels for the experiment All thecomponents are radiation hardened to cope with the extremely high radiation in the proximityof the interaction point

The next layer is the semi-conductor layer (SCT) similar in concept to the pixel detectorbut with long narrow strips instead of pixels to cover a larger area It has 4 double layers ofsilicon strips with 63 million readout channels

The outermost component of the inner detector is the Transition Radiation Tracker It is acombination of a straw tracker and a transition radiation detector The detecting elementsare drift tubes (straws) each 4mm in diameter and up to 144 cm long Each straw is filledwith gas which becomes ionised when the radiation passes through The straws are held at-1500V with a thin wire down the center This drives negatively ionised atoms to the centercausing a pulse of electricity in the wire This causes a pattern of hit straws allowing the pathof the particles to be determined Between the straws materials with widely varying indicesof refraction cause transition radiation when particles pass through leaving much strongersignals in some straws Transition radiation is strongest for highly relativistic particles (thelightest particles- electrons and positrons) allowing strong signals to be associated with theseparticles

23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 33

2312 Calorimeters

The inner detector is surrounded by a solenoidal magnet of 2 Tesla (there is also an outertoroidal magnet) but surrounding the inner magnet are the calorimeters to measure theenergy of particles by absorbing them There is an inner electromagnetic calorimeter whichmeasures the angle between the detectorrsquos beam axis and the particle (or more preciselypseudo-rapidity) and its angle in the perpendicular plane to a high degree of accuracy Theabsorbing material is lead and stainless steel and the sampling material liquid argon Thehadron calorimeter measures the energy of particles that interact via the strong force Theenergy absorbing material is scintillating steel

2313 Muon Specrometer

The muon spectrometer surrounds the calorimeters and has a set of 1200 chambers tomeasure the tracks of outgoing muons with high precision using a magnetic field andtriggering chambers Very few particles of other types pass through the calorimeters and aresubsequently measured

2314 Magnets

The magnets are in two layers the inner solenoidal and outer toroidal which together causecharged particles to curve in the magetic field allowing their momentum to be determinedThe outer toroidal magnet consists of 8 very large air core superconducting magnets whichsupply the field in the muon spectrometer

232 CMS Experiment

The CMS Detector is similar in design to the ATLAS Detector with a single solenoidalmagnet which is the largest of its type ever built (length 13m and diameter 7m) and allowsthe tracker and calorimeter detectors to be placed inside the coil in a compact way Thestrong magnetic field gives a high resolution for measuring muons electrons and photons ahigh quality central tracking system and a hermetic hadron calorimeter designed to preventhadrons from escaping

23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 34

Figure 211 CMS Experiment

Chapter 3

Fitting Models to the Observables

31 Simplified Models Considered

In order to extend the findings of previous studies [19] to include collider constraints wefirst performed Bayesian scans [7] over three simplified models real scalar dark matter φ Majorana fermion χ and real vector Xmicro The purpose was to obtain the most probable regionsof each model then to use the resulting posterior samples to investigate whether the LHCexperiments have any sensitivity to the preferred regions

The three models couple to the mediator with interactions shown in the following table

Table 31 Simplified Models

Hypothesis real scalar DM Majorana fermion DM real vector DM

DM mediator int Lφ sup microφ mφ φ 2S Lχ sup iλχ

2 χγ5χS LX sup microX mX2 X microXmicroS

The interactions between the mediator and the standard fermions is assumed to be

LS sup f f S (31)

and in line with minimal flavour violation we only consider third generation fermions- ief = b tτ

For the purposes of these scans we consider the following observables

32 Observables 36

32 Observables

321 Dark Matter Abundance

We have assumed a single dark matter candidate which freezes out in the early universe witha central value as determined by Planck [21] of

ΩDMh2 = 01199plusmn 0031 (32)

h is the reduced hubble constant

The experimental uncertainty of the background relic density [59] is dominated by atheoretical uncertainty of 50 of the value of the scanned Ωh2 [76] We have added thetheoretical and the experimental uncertainties in quadrature

SD =radic(05Ωh2)2 + 00312 (33)

This gives a log likelihood of

minus05lowast (Ωh2 minus 1199)2

SD2 minus log(radic

2πSD) (34)

322 Gamma Rays from the Galactic Center

Assuming that the gamma ray flux excess from the galactic centre observed by Fermi-LAT iscaused by self annihilation of dark matter particles the differential flux

d2Φ

dEdΩ=

lt σv gt8πmχ

2 J(ψ)sumf

B fdN f

γ

dE(35)

has been calculated using both Micromegas v425 with a Navarro-Frenk-White (NFW) darkmatter galactic distribution function

ρ(r) = ρ0(rrs)

minusγ

(1+ rrs)3minusγ (36)

with γ = 126 and an angle of 5 to the galactic centre [19]

32 Observables 37

Here Φ is the flux dE is the differential energy lt σv gt the velocity averaged annihi-lation cross section of the dark matter across the galactic centre mχ the mass of the darkmatter particle B f =lt σv gt f σv is the annihilation fraction into the f f final state anddN f

γ dE is the energy distribution of photons produced in the annihilation channel with thefinal state f f The J factor is described below

The angle ψ to the galactic centre determines the normalisation of the gamma ray fluxthrough the factor

J(ψ) =int

losρ(l2 + r⊙2 minus2lr⊙cosψ)dl (37)

where ρ(r) is the dark matter density at distance r from the centre of the galaxy r⊙=85 kpcis the solar distance from the Galactic center ρ0 is set to reproduce local dark matter densityof 03GeVcm3

The central values for the gamma ray spectrum were taken from Fig 5 of [77] and thestandard deviations of individual flux points from the error bars on the graph In commonwith [78] we ignore possible segment to segment correlations

For the Fermi-LAT log likelihood we assumed each point was distributed with indepen-dent Gaussian likelihood with the experimental error for each data point given by the errorbar presented in Fig5 of [76] for each point We also assumed an equal theoretical errorfor each point which when added to the experimental error in quadrature gave an effectivelog likelihood for each point of minus05lowast (giminusdi)

2

2lowastσ2i

where gi are the calculated values and di theexperimental values and σi the experimental errors

323 Direct Detection - LUX

The LUX experiment [16] is the most constraining direct search experiment to date for awide range of WIMP models We have used LUXCalc [79] (modified to handle momentumdependent nucleon cross sections) to calculate the likelihood of the expected number ofsignal events for the Majorana fermion model

The likelihood function is taken as the Poisson distribution

L(mχ σ | N) = P(N | mχ σ) =(b+micro)Neminus(b+micro)

N (38)

32 Observables 38

where b= expected background of 064 and micro = micro(mχσ ) is the expected number of signalevents calculated for a given model

micro = MTint

infin

0dEφ(E)

dRdE

(E) (39)

where MT is the detector mass times time exposure and φ(E) is a global efficiency factorincorporating trigger efficiencies energy resolution and analysis cuts

The differential recoil rate of dark matter on nucleii as a function of recoil energy E

dRdE

=ρX

mχmA

intdvv f (v)

dσASI

dER (310)

where mA is the nucleon mass f (v) is the dark matter velocity distribution and

dσSIA

dER= Gχ(q2)

4micro2A

Emaxπ[Z f χ

p +(AminusZ) f χn ]

2F2A (q) (311)

where Emax = 2micro2Av2mA Gχ(q2) = q2

4m2χ

[24] is an adjustment for a momentum dependantcross section which applies only to the Majorana model

f χ

N =λχ

2m2SgSNN assuming that the relic density is the central value of 1199 We have

implemented a scaling of the direct search cross section in our code by multiplying gSNN by( Ω

Ω0)05 as a convenient method of allowing for the fact that the actual relic density enters

into the calculation of the cross section as a square

FA(q) is the nucleus form factor and

microA =mχmA

(mχ +mA)(312)

is the reduced WIMP-nucleon mass

The strength of the dark matter nucleon interaction between dark matter particles andthird generation quarks is

gSNN =2

27mN fT G sum

f=bt

λ f

m f (313)

where fT G = 1minus f NTuminus f N

Tdminus fTs and f N

Tu= 02 f N

Td= 026 fTs = 043 [20]

33 Calculations 39

For the real scalar and real vector models where the spin independent cross sections arenot momentum dependent we based the likelihood on the LUX 95 CL from [16] using acomplementary error function Li(d|mχ I) = 05lowastCEr f (dminusx(mχ )

σ) where x is the LUX limit

and d the calculated SI cross section We assumed a LUX error σ of 01lowastd lowastradic

2

33 Calculations

We coded the Lagrangians for the three models in Feynrules [4] Calculation of the expectedobservables (dark matter relic density Fermi-LAT gamma ray spectrum and LUX nucleonscattering amplitudes) were performed using Micromegas v425

331 Mediator Decay

A number of searches have previously been performed at the LHC which show sensitivity tothe decay of scalar mediators in the form of Higgs bosons (in beyond the standard modeltheories) to bottom quarks top quarks and τ leptons (all third generation fermions) Two inparticular [43 44] provide limits on the cross-sections for the production of the scalar timesthe branching ratios to brsquos and τrsquos In order to check these limits against each of the modelswe ran parameter scans in Madgraph at leading order over the top 900 points from the nestedsampling output from MultiNest [19]

The two processes were

1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to apair of τ leptons in pp collisions [44]

bull generate p p gt b b S where S is the scalar mediator

The process measures the decay of a scalar in the presence of b-jets that decay to τ+τminus

leptons Fig31 shows the main diagrams contributing to the process The leftmost diagramis a loop process and the centre and right are tree level however we only got Madgraph togenerated the middle diagram since we are interested in τ leptons and b quarks in the finalstate

The limit given in [44] only applies in the case of narrow resonances where narrow istaken to mean that the width of the resonance is less than 10 of its mass To test the validity

33 Calculations 40

Figure 31 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof τ leptons

of applying these limits to our models we show in Figures 32 to 34 the percentage widthof the resonance vs mS[GeV ] (pole mass of the scalar mediator) λb and λt (the couplingconstants to the b and t quarks) The probabilities are the posterior probabilities obtainedfrom MultiNest scans of the Majorana model varying the coupling constants and scalarmediator and calculating the decay width of the mediator The results are represented as 2dposterior probability heat maps in the central panels and 1d marginalised distributions on thex and y axes of each graph

16 18 20 22 24 26 28 30

log10(mS[GeV])

001

002

003

004

005

Widthm

S

00

04

08

12

0 100 200

Posterior Probability

Figure 32 WidthmS vs mS

The WidthmS was lt 01 at almost all points except where the mass of the scalar wasgreater than 1000 GeV λb was greater than 1 or λτ was greater than 1

This can be seen from the graphs in Figs 323334

33 Calculations 41

4 3 2 1 0

λb

001

002

003

004

005

WidthmS

000

015

030

045

0 100 200

Posterior Probability

Figure 33 WidthmS vs λb

5 4 3 2 1 0

λτ

001

002

003

004

005

WidthmS

000

015

030

045

0 100 200

Posterior Probability

Figure 34 WidthmS vs λτ

The decay widths are calculated in Madgraph at leading order and branching fractionscan be calculated to b t τ and χ (DM) The formulae for the calculations can be found indecayspy which is a subroutine of Madgraph which we have documented in Appendix BB1 Using these we calculated the branching ratio to τ+τminus so that the cross section timesbranching ratio can be compared to the limits in [80] shown in Chapters 56 and 7 in Figs5564 and 74 for each model

2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

This scan calculates the contributions of b-quark associated Higgs boson production inthe center and right diagrams of Fig 35 reproduced from [43] The scans did not includePythia and Delphes as only production cross sections were required and the producedparticles were in the final state and required no matching or merging

The Madgraph processes were

bull generate p p gt b S where S is the scalar mediator

bull add process p p gt b S j

bull add process p p gt b S

33 Calculations 42

Figure 35 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof b quarks in the presence of at least one b quark

bull add process p p gt b S j

The cross section times branching ratios to bb for mediator production in association withat least one b quark is shown for each model in Chapters 56 and 7 in Figs 5765 and 75

332 Collider Cuts Analyses

We generated two parameter scans with parameter combinations from the Multinest outputwhich were the basis of the plots of Fig 6 in [19] The plots in Fig 6 were marginalisedto two parameters each but the raw Multinest output contained 5 parameters plus the priorprobabilities of each combination normalised to total 10

The first scan was for χ produced with b quarks In order to force the mediator to decayto χ only we had χ in the final state and defined jets to include b quarks in Madgraph Thenwe generated the processes

bull generate p p gt χ χ j

bull add process p p gt χ χ j j

Jet matching was on

The second scan was for t quarks produced in the final state

bull generate p p gt χ χ tt

33 Calculations 43

No jet matching was required because we did not mix final state jet multiplicities at thematrix element level

The outputs from these two processes were normalised to 21 f bminus1 and combined

The results of these scans prior to collider cuts calculations are shown in Figs 5161and 71 These show the posterior probabilities of the various parameter combinations withdifferent constraints applied seperately and then together A more detailed commentary isgiven in each of the sections

We then sampled 900 points based on posterior probability from these scans and ran thesethrough Madgraph v425 then applied collider cuts with the C++ code using the limits fromthe following six ATLAS and CMS searches shown in Table32 and described below to findany exclusions

333 Description of Collider Cuts Analyses

In the following all masses and energies are in GeV and angles in radians unless specificallystated

3331 Lepstop0

Search for direct pair production of the top squark in all-hadronic final states in pro-tonndashproton collisions at

radics=8 TeV with the ATLAS detector[32]

This analysis searched for direct pair production of the supersymmetric partner to thetop quark using integrated luminosity of 201 f bminus1 The top squark was assumed to de-cay via t rarr t χ0

1 or t rarr bχ01 or t rarr bχ

plusmn1 rarr bW (lowast)χ1

0 where χ01 (χ

plusmn1 ) denotes the lightest

neutralino(chargino) in supersymmetric models The search targets a fully hadronic finalstate with four or more jets and large missing momentum The production of a mediator inassociation with top quarks followed by an invisible mediator decay to dark matter particleshas a similar final state to that considered in the ATLAS stop searches Thus we expect theATLAS reach to have sensitivity to the three dark matter models considered in this thesis

The collider cuts program mimicked the object reconstruction through jetlepton overlapremoval and selected signal electrons muons and τ particles with pT gt 10 GeV and

33 Calculations 44

Table 32 95 CL by Signal Region

Experiment Region Number

Lepstop0

SRA1 66SRA2 57SRA3 67SRA4 65SRC1 157SRC2 124SRC3 80

Lepstop1

SRtN1-1 857SRtN1-2 498SRtN1-3 389SRtN2 107SRtN3 85SRbC1 832SRbC2 195SRbC3 76

Lepstop2

L90 740L100 56L110 90L120 170

2bstop

SRA 857SRB 270SRA150 380SRA200 260SRA250 90SRA300 75SRA350 52

CMSTopDM1L SRA 1385

ATLASMonobjetSR1 1240SR2 790

33 Calculations 45

|η | lt 24724 and 247 respectively (for eminusmicrominusτ) Jets were selected with pT gt 20 GeVand |η |lt 45 b-jets were selected if they were tagged by random selection lt 70 and had|η |lt 25 and pT gt 20 GeV

These jets and leptons were further managed for overlap removal within ∆R lt 04 withnon b-jets removed if ∆R lt 02

The b-jets and non b-jets were further selected as signal jets if pT gt 35 GeV and |η |lt 28

These signal jets electrons and muons were finally binned according to the signal regionsin Tables 333435 below

Table 33 Selection criteria common to all signal regions

Trigger EmissT

Nlep 0b-tagged jets ⩾ 2

EmissT 150 GeV

|∆φ( jet pmissT )| gtπ5

mbminT gt175 GeV

Table 34 Selection criteria for signal regions A

SRA1 SRA2 SRA3 SRA4anti-kt R=04 jets ⩾ 6 pT gt 808035353535 GeV

m0b j j lt 225 GeV [50250] GeV

m1b j j lt 225 GeV [50400] GeV

min( jet i pmissT ) - gt50 GeV

τ veto yesEmiss

T gt150 GeV gt250 GeV gt300 GeV gt 350 GeV

Table 35 Selection criteria for signal regions C

SRC1 SRC2 SRC3anti-kt R=04 jets ⩾ 5 pT gt 8080353535 GeV

|∆φ(bb)| gt02 π

mbminT gt185 GeV gt200 GeV gt200 GeV

mbmaxT gt205 GeV gt290 GeV gt325 GeV

τ veto yesEmiss

T gt160 GeV gt160 GeV gt215 GeV

wherembmin

T =radic

2pbt Emiss

T [1minus cos∆φ(pbT pmiss

T )]gt 175 (314)

33 Calculations 46

and pbT is calculated for the b-tagged jet with the smallest ∆φ to the pmiss

T direction andmbmax

T is calculated with the pbT for the b-tagged jet with the largest ∆φ to the pmiss

T direction

m0b j j is obtained by first selecting the two jets with the highest b-tagging weight- from

the remaining jets the two jets closest in the η minusφ plane are combined to form a W bosoncandidate this candidate is then combined with the closest of the 2 b-tagged jets in the η minusφ

plane to form the first top candidate with mass m0b j j A second W boson candidate is formed

by repeating the procedure with the remaining jets this candidate is combined with thesecond of the selected b-tagged jets to form the second top candidate with mass m1

b j j Thecuts shown in Appendix1 Table A1 show the cumulative effect of the individual cuts appliedto the data the signal region cuts and the cumulative effects of individual cuts in SignalRegion C

3332 Lepstop1

Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

radics=8 TeV pp collisions using 21 f bminus1 of

ATLAS data[33]

The stop decay modes considered were those to a top quark and the lightest supersym-metric particle (LSP) as well as a bottom quark and the lightest chargino where the charginodecays to the LSP by emitting a W boson We should expect the three dark matter models inthis paper with couplings to t and b quarks to show some visibility in this search

The collider cuts program mimicked the object construction by selecting electrons andmuons with pT gt 10 GeV and |η |lt 247 and 24 respectively and jets with pT gt 20 GeVand |η |lt 10 b-jets with |η |lt 25 and pT gt 25 GeV were selected with random efficiency75 and mistag rate 2 (where non b-jets were mis-classified) We then removed any jetwithin ∆R = 02 of an electron and non overlapping jets with |η |lt 28 were used to removeany electrons and muons within ∆R = 04 of these while non-overlapping jets with |η |lt 25and pT gt 25 GeV were classified as signal jets Non overlapping electrons and muons withpT gt 25 GeV were classified as signal electrons and muons

Signal selection defined six regions to optimize sensitivity for different stop and LSPmasses as well as sensitivity to two considered stop decay scenarios Three signal regionslabelled SRbC 1-3 where bC is a moniker for b+Chargino and 3 regions labelled SRtN 1-3where tN is a moniker for t+Neutralino were optimised for the scenarios where t1 decaysto the respective particles indicated by the labels Increasing label numbers correspond toincreasingly strict selection criteria

33 Calculations 47

The dominant background in all signal regions arises from dileptonic tt events All threeSRtN selectons impose a requirement on the 3-jet mass m j j j of the hadronically decayingtop quark to specifically reject the tt background where both W bosons from the top quarkdecay leponically

For increasing stop mass and increasing difference between the stop and the LSP therequirents are tightened on Emiss

T on the ratio EmissT

radicHT where HT is the scalar sum of the

momenta of the four selected jets and also tightened on mT

To further reduce the dileptonic tt background for an event characterised by two one stepdecay chains a and b each producing a missing particle C use is made of the variable

mT 2 =min

pCTa + pC

T b = pmissT

[max(mTamtb)] (315)

where mTi is the transverse mass 1 of branch i for a given hypothetical allocation (pCTa pC

T b)

of the missing particle momenta Eqn 315 is extended for more general decay chains byamT 2 [81] which is an asymmetric form of mT 2 in which the missing particle is the W bosonin the branch for the lost lepton and the neutrino is the missing particle for the branch withthe observed charged lepton

∆φ( jet12 pmissT ) the azimuthal angle between leading or subleading jets and missing mo-

mentum is used to suppress the background from mostly multijet events with mis-measuredEmiss

T

Events with four or more jets are selected where the jets must satisfy |η | lt 25 and pT gt80604025 GeV respectively

These signal particles were then classified according to the cutflows analysis in Fig36below and compared with the paper in Appendix1 Table A2

3333 Lepstop2

Search for direct top squark pair production in final states with two leptons in p pcollisions at

radics=8TeV with the ATLAS detector[34]

Three different analysis strategies are used to search for t1 pair production Two targett1 rarr b+ χ

plusmn1 decay and the three body t1 rarr bW χ0

1 decay via an off-shell top quark whilst

1The transverse mass is defined as m2T = 2plep

T EmissT (1minus cos(∆φ)) of branch i where ∆φ is the azimuthal

angle between the lepton and the missing transverse momentum

33 Calculations 48

Table 36 Signal Regions - Lepstop1

Requirement SRtN2 SRtN3 SRbC1 SRbC2 SRbC3∆φ( jet1 pmiss

t )gt - 08 08 08 08∆φ( jet2 pmiss

T )gt 08 08 08 08 08Emiss

T [GeV ]gt 200 275 150 160 160Emiss

T radic

HT [GeV12 ]gt 13 11 7 8 8

mT [GeV ]gt 140 200 120 120 120me f f [GeV ]gt - - - 550 700amT 2[GeV ]gt 170 175 - 175 200mT

T 2[GeV ]gt - 80 - - -m j j j Yes Yes - - -

one targets t1 rarr t + χ01 to an on shell quark decay mode The analysis is arranged to look

at complementary mass splittings between χplusmn1 and χ0

1 smaller and larger than the W bosonmass The smaller splitting provides sensitivity to t1 three body decay

Object reconstruction was carried out by selecting baseline electronsmuons and τrsquos withpT gt 10 GeV and |η |lt 24724 and 247 respectively Baseline jets and b-jets were selectedwith pT gt 20 GeV and |η |lt 25 and b-jets with 70 efficiency Baseline jets were removedif within ∆R lt 02 of an electron and electrons and muons were removed if within ∆R lt 04of surviving jets Signal electrons required pT gt 10 GeV and signal jets pT gt 20 GeV and|η |lt 25

The analysis made use of the variable mT 2 [82] the stransverse mass used to mea-sure the masses of pair-produced semi-invisibly decaying heavy particles It is definedas mT 2(pT1 pT2qT ) =

minqT1+qT2=qT

max[mT (pT1qT1)mT (pT2qT2)] where mT indicatesthe transverse mass pT1 and pT2 are vectors and qT = qT1 + qT2 The minimisation isperformed over all possible decompositions of qT

Events were selected if they passed either a single-electron a single-muon a double-electron a double-muon or an electron-muon trigger Events are required to have exactlytwo opposite sign leptons and at least one of the leptons or muons must have a momentumgreater than 25 GeV and the invariant mass (rest mass) of the two leptons is required to begreater than 20 GeV In order to reduce the number of background events containing twoleptons produced by the on-shell decay of the Z boson the invariant mas must be outside711-111 GeV range

Two additional selections are applied to reduce the number of background events withhigh mT 2 arising from events with large Emiss

T due to mismeasured jets ∆φb lt 15 and∆φ gt 10 The quantity ∆φb is the azimuthal angle between pll

T b = pmissT + pl1

T +Pl2T The

33 Calculations 49

vector pllT b is the opposite of the vector sum of all the hadronic activity in the event For WW

and tt backgrounds it measures the transverse boost of the WW system and for the signalthe transverse boost of the chargino-chargino system The ∆φ variable is the azimuthal anglebetween the pmiss

T vector and the direction of the closest jet

By requiring mT 2 gt 40 GeV the tt background is the dominant background in both thesame-flavoured (SF) and different-flavoured (DF) events

Four signal regions are then defined as shown in Table 37 SR M90 has the loosestselection requiring MT 2gt 90 GeV and no additional requirement on jets and providessensitivity to scenarios with a small difference between the masses of the top-squark andchargino SR M110 and M120 have a loose selection on jets requiring two jets with pT

gt 20 GeV and mT 2 to be greater than 110 GeV and 120 GeV respectively They providesensitivity to scenarios with small to moderate difference between the invariant masses of thestop quark and the charginos SR M100 has a tighter jet slection and provides sensitivity toscenarios with a large mass difference between the both stop quark and the chargino and thestop quark and the neutralino

The analysis cut regions are summarised in Fig37 below

Table 37 Signal Regions Lepstop2

SR M90 M100 M110 M120pT leading lepton gt 25 GeV

∆φ(pmissT closest jet) gt10

∆φ(pmissT pll

T b) lt15mT 2 gt90 GeV gt100 GeV gt110 GeV gt120 GeV

pT leading jet no selection gt100 GeV gt20 GeV gt20 GeVpT second jet no selection gt50 GeV gt20 GeV gt20 GeV

To validate the code we ran the cuts shown in Appendix 1 Table A3 with the givenMadgraph simulation and compared to [83] It was not clear what baseline cuts had beenemployed in the ATLAS paper so our results were normalised to the 2 SF electrons beforecomparison Some of the ATLAS cut numbers were less than 3 and we did not produce anyevents passing these cuts but clearly the margin of error at this level was greater than thenumber of ATLAS events

33 Calculations 50

3334 2bstop

Search for direct third-generation squark pair production in final states with miss-ing transverse momentum and two b-jets in

radics= 8 TeV pp collisions with the ATLAS

detector[31]

Two sets of scenarios are considered In the first the coloured sbottom is the onlyproduction process decaying via b1 rarr bχ0

1 In the second the stop is the only productionprocess decaying via t1 rarr t χplusmn

1 and the lightest chargino (χplusmn1 ) decays via a virtual W boson

into the final state χ01 f f prime Both scenarios are characterised by the presence of two jets

resulting from the hadronisation of b quarks and large missing transverse momentum Theproduction of a mediator in association with top quarks followed by an invisible mediatordecay to dark matter particles has a similar final state as that considered in the ATLAS stopsearches Thus we expect the ATLAS reach to have sensitivity to the three dark mattermodels considered in this thesis

The collider cuts program mimicked object reconstruction by selecting baseline electronswith pT gt 7 GeV and |η |lt 247 and muons with pT gt 6 GeV and |η |lt 24 Baseline jetswere selected with pT gt 20 GeV and |η |lt 28 We then removed any jets within ∆R lt 02of an electron and any electron within ∆R lt 04 of surviving jets Signal electrons andmuons were required to have pT gt 25 GeV b-jets were selected with 60 tagging efficiencyand |η |lt 25 and pT gt 20 GeV

Two sets of signal regions are defined to provide sensitivity to the kinematic topologiesassociated with the different mass splittings between the sbottom or the stop and the neu-tralino In all cases the presence of at least one primary vertex (with at least five tracks withpT gt 04 GeV) is required Events are selected with Emiss

T gt 150 GeV and no electrons ormuons identified in the final state For signal region selections jets with |η | lt 28 are orderedaccording to their pT and 2 jets are required to be b-tagged

The variables are defined as follows

bull ∆φmin is the minimal azimuthal angle ∆φ between any of the three leading jets andpmiss

T

bull me f f (k) = sumki=1(p jet

T )i +EmissT where the index refers to the pT ordered list of jets

bull HT3 is defined as the scalar sum of the pT of the n leading pT ordered jets withoutincluding the 3 leading jets HT3 = sum

ni=4(p jet

T )i

bull mbb is the invariant mass of the two b-tagged jets in the event

33 Calculations 51

bull mCT is the contransverse mass [82] and is a kinematical variable that can be usedto measure the masses of pair produced semi-invisibly decaying particles For twoidentical decays of heavy particles into two invisible particles v1 and v2 and twoinvisible particles mCT is defined as m2

CT (v1v2) = [ET (v1)+ET (v2)]2 minus [pT (v1)minus

pT (v2)]2 where ET =

radicp2

T +m2 The the contransverse mass is invariant under equaland opposite boosts of the parent particles in the transverse plane and is bounded fromabove by an analytical expression of the particle masses For production of sbottompairs the bound is given by mmax

CT =m2(b)minusm2(χ0

1 )

m(b) and for tt events the bound is 135

GeV A similar equation can be written for stop pairs in term of the stop and charginomasses

A definition of the signal regions is given in the Table38

Table 38 Signal Regions 2bstop

Description SRA SRBEvent cleaning All signal regions

Lepton veto No emicro after overlap removal with pT gt7(6) GeV for e(micro)Emiss

T gt 150 GeV gt 250 GeVLeading jet pT ( j1) gt 130 GeV gt 150 GeVSecond jet pT ( j2) gt 50 GeV gt 30 GeVThird jet pT ( j3) veto if gt 50 GeV gt 30 GeV

∆φ(pmissT j1) - gt 25

b-tagging leading 2 jets(pT gt 50 GeV |η | lt 25)

2nd- and 3rd- leading jets(pT gt 30 GeV|η | lt 25)

∆φmin gt 04 gt 04Emiss

T me f f (k) EmissT me f f (2) gt 025 Emiss

T me f f (3) gt 04mCT gt 150200250300350 GeV -HT3 - lt 50 GeVmbb gt200 GeV -

The analysis cuts are summarised in Table A4 of Appendix 1

3335 ATLASMonobjet

Search for dark matter in events with heavy quarks and missing transverse momentumin pp collisions with the ATLAS detector[30]

Events were selected with high missing transverse momentum when produced in associa-tion with one or more jets identified as containing b quarks Final states with top quarks wereselected by requiring a high jet multiplicity and in some cases a single lepton The research

33 Calculations 52

studied effective contact operators in the form of a scalar operator Oscalar = sumqmqMN

lowastqqχχ

where MNlowast is the effective mass of the coupling and the quark and DM fields are denoted by

q and χ respectively These are the equivalent of the simplified models in this paper with themediator integrated out

Object reconstruction selected baseline electrons muons jets and τrsquos with pT gt 20 GeVand |η |lt 2525 and 45 respectively Signal jets were selected by removing any jets within∆R lt 02 of an electron and electrons and muons were selected by removing them if theywere within ∆R lt 04 of surviving jets b-jets were tagged with an efficiency of 60

Only signal regions SR1 and SR2 were analysed

The analysis split events into signal regions SR1 and SR2 according to the followingCriteria

Table 39 Signal Region ATLASmonobjet

Cut SR1 SR2Jet multiplicity n j 1minus2 3minus4

bminusjet multiplicity nb gt 0 (60 eff) gt 0 (60 eff)Lepton multiplicity nl 0 0

EmissT gt300 GeV gt200 GeV

Jet kinematics pb1T gt100 GeV pb1

T gt100 GeV p j2T gt100 (60) GeV

∆φ( j1EmissT ) gt 10 i = 12 gt 10 i = 14

Where p jiT (pbi

T ) represent the transverse momentum of the i-th jet b-tagged jet respec-tively If there is a second b-tagged jet the pT must be greater than 60 GeV

3336 CMSTop1L

Search for top-squark pair production in the single-lepton final state in pp collisionsat

radics=8 TeV[41]

This search was for pair production of top squarks in events with a single isolatedelectron or muon jets large missing transverse momentum and large transverse mass

(MT =radic

2EmissT pl

T (1minus cos(∆φ))) where plT is the leptonrsquos transverse momentum and ∆(φ)

is the difference between the azimuthal angles of the lepton and EmissT The 3 models

considered should have some visibility at certain parameter combinations in this regionThe search focuses on two decay modes of the top squark leading to the two processespp rarr t tlowast rarr tt χ0

1 χ01 rarr bbW+Wminusχ0

1 χ01 and pp rarr t tlowast rarr bbχ

+1 χ

minus1 rarr bbW+Wminusχ0

1 χ01 The

33 Calculations 53

lightest neutralino χ01 is considered the lightest supersymmetric particle and escapes without

detection The analysis is based on events where one of the W bosons decays leptonicallyand the other hadronically resulting in one isolated lepton and 4 jets two of which originatefrom b quarks The two neutralinos and the neutrino from the W decay can result in largemissing transverse momentom The kinematical end point of MT is lt the mass of the Wboson However for signal events the presence of LSPs in the final state allows MT to exceedthe mass of the W boson The search is therefore for events with large MT The dominantbackground with large MT arises from tt decays where W bosons decay leptonically withone of the leptons not being identified

To reduce the dominant tt background use was made of the MWT 2 variable defined as

the minimum mother particle mass compatible with the transverse momenta and mass shellconstraints [84]

Object reconstruction required electrons and muons to have pT gt 30 and 25 GeV respand |η |lt 25 and 21 respectively Baseline jets were required to have pT gt 30 and |η |lt 4and b-jets were tagged with an efficiency of 75 for |η |lt 24

Events were chosen if they contained one isolated electron or muon at least 4 jets and atleast one b-tagged jet and Emiss

T gt 100 GeV MT gt 160 GeV MWT 2 was required to be greater

than 200 GeV and the smallest azimuthal angle difference between baseline jets and EmissT

gt12

Chapter 4

Calculation Tools

41 Summary

Figure 41 Calculation Tools

The simplified models were implemented in Feynrules [4] and the model files importedinto MicrOmegas v425 which was used to calculate the log-likelihoods of the astrophysicalobservables MicrOmegas was modified to allow only s-channel decays in both relic densityand gamma ray calculations Direct detection constraints were easy to implement for thescalar and vector models by coding the analytical expression for the spin independent WIMP-nucleon scattering cross section but due to the momentum dependence of the WIMP-nucleon

42 FeynRules 55

scattering cross section of the Majorana fermion model LUXCalc v101 [79] was modifiedfor this model to implement momentum dependent cross sections [85] We used MicrOmegasv425 to calculate relic density values and to produce gamma ray spectra for comparisonwith the Fermi-LAT results Parameter scans were carried out using MultiNest v310 withthe parameter ranges given in Table 51 and using astrophysical constraints only Oncethese scans had converged a random sample of 900 points was chosen (weighted by theposterior probability) and then used to calculate collider constraints using the MadgraphMonte Carlo generator This included comparison with mediator production and decay limitsfrom the LHC experiments and the reproduction of searches for dark matter plus other finalstate objects which required dedicated C++ code that reproduces the cut and count analysesperformed by CMS and ATLAS

42 FeynRules

FeynRules is a Mathematicareg package that allows the calculation of Feynman rules inmomentum space for any QFT physics model The user needs to provide FeynRules withminimal information to describe the new model contained in the model-file This informationis then used to calculate a set of Fenman rules associated with the Lagrangian The Feynmanrules calculated by the code can then be used to implement new physics models into otherexisting tools such as Monte Carlo (MC) generators via a set of interfaces developed andmaintained by the MC authors

Feynrules can be run with a base model = SM (or MSSM) and user defined extensions tothe base model

43 LUXCalc

LUXCalc is a utility program for calculating and deriving WIMP-nucleon coupling limitsfrom the recent results of the LUX direct search dark matter experiment The program hasbeen used to derive limits on both spin-dependent and spin-independent WIMP-nucleoncouplings and it was found under standard astrophysical assumptions and common otherassumptions that LUX excludes the entire spin-dependent parameter space consistent witha dark matter interpretation of DAMArsquos anomalous signal and also found good agreementwith published LUX results [79]

We modified the program for the Majorana fermion model to implement momentumdependent cross -sections [85]

44 Multinest 56

44 Multinest

Multinest is an efficient Bayesian Inference Tool which uses a simultaneous ellipsoidalnested sampling method a Monte Carlo technique aimed at an efficient Bayesian Evidencecalculation with posterior inferences as a by product

Bayes theorem states that

Pr(θ |DH) =Pr(D|θ H)Pr(θ |H)

Pr(D|H) (41)

Here D is the data and H is the Hypothesis Pr(θ |DH) is the posterior probability ofthe parameters Pr(D|θ H) = L(θ) is the likelihood Pr(θ |H) is the prior and Pr(D|H) isthe evidence

The evidence Pr(D|H) =int

Pr(θ |DH)Pr(θ |H)d(θ) =int

L(θ)Pr(θ |H)d(θ) Thus ifwe define the prior volume dX = Pr(θ |H)d(θ) and

X(λ ) =int

L(θ)gtλ

Pr(θ |H)d(θ) (42)

where the integral extends over the region of parameter space within the iso likelihoodcontour L(θ) = λ then the evidence integral can be written

int 10 L (X)dX where L (X) the

inverse of equation 42 and is a monotonically decreasing function of X (the prior volumewhich is 1 for the total sample space and 0 for a single point)

Thus if one can evaluate likelihoods L (Xi) where 0 lt Xn lt lt X2 lt X1 lt X0 = 1the evidence can be estimated as sumLiwi where wi = 5(Ximinus1 minusXi+1) Ellipsoidal samplingapproximates the iso likelihood contour by a D dimensional covariance matrix of the currentset of active points New points are selected from the prior within this ellipsoidal bounduntil one is obtained that has a likelihood greater than the likelihood of the removed lowestlikelihood point This method is extended for multimodal sampling where there may bemore than one local maximum by clustering points that are well separated and constructingindividual ellipsoidal bounds for each cluster For more details consult [7]

The results of the analysis are expressed as a data file containing the sample probability-2loglikelihood followed by the parameters of the model for the particular sampling Theprobabilities are normalised to total 10

45 Madgraph 57

45 Madgraph

Madgraph is a general purpose matrix element calculator and Monte Carlo generator Basedon user-supplied Feynman rules (eg the output of Feynrules) the package calculates the tree-level matrix elements for any processes that the user specifies and generates samples of LHCevents including an accurate simulation of the proton collisions themselves (which requires aknowledge of the parton distribution functions that determine the relative proportion of eachparton inside the nucleus) Hadronisation of final state quarks is not possible in Madgraphand hence the code must be used in conjunction with Pythia which performs a parton shower

The Madgraph output was automatically showered by Pythia (Pythia describes partonradiation as successive parton emissions using Markov chain techniques based on Sudakovform factors and this description is only formally correct for the soft and collinear emissions)A Madgraph simulation is at the level of the fundamental particles (quarks and leptons)Madgraph generates a limited set of matrix element processes at LO representing hard (highenergy) emissions and passes these to Pythia for showering The final state partons in theMadgraph simulation must be matched to the Pythia final state partons to avoid doublecounting

The output from Pythia is automatically fed through Delphes where basic collider cutscan be performed (calculation of the numbers of particles and jets that would be observed atdifferent energies and locations within the collider) The output of the Delphes simulation is afile containing reconstructed events that can be analysed using the standard ROOT framework(the most popular data analysis package in particle physics) [86]

The parameters that gave the best fits to the direct [16] and indirect DM experiments [87]and relic density [59] are ranked by posterior probability from the Multinest [7] output

The parameters in the Madgraph simulations are stored in files in the cards sub directoryand in particular the run card param card and Delphes card record items relevent to thesimulation

In the run card the number of events the Beam 1 and 2 total energy in GeV (4000 each)and the type of jet matching were modified (to MLM type matching where the final statepartons in Madgraph are clustered according to the KT jet algorithm to find the equivalentparton shower history of the event there is a minimum cutoff scale for the momentum pT

given by the xqcut parameter) The maxjetflavour (=5 to include b quarks) and xqcut = 30GeV parameters were also modified

46 Collider Cuts C++ Code 58

The param card was modified to specify the masses of the new particles and the autodecay modes The Delphes card is overwritten with either the ATLAS or CMS defaultcards which record tracking efficiencies and isolation momentum and energy resolutionscalorimeter details jet finder and b and τ -tagging parameters However the isolation andtagging in this study were done by the collider cuts C++ code since multiple sets of cutswere applied to the same output from Pythia

When performing simulations of the three dark matter models it was necessary to definethe proton and jets to contain b quarks as this is not the default in Madgraph The processesgenerated included 1 and 2 jets

46 Collider Cuts C++ Code

Because of the volume of simulations required for the collider constraints we used C++code (originally developed by Martin White and Sky French and modified by me) that readsthe ROOT files and outputs the numbers of events that would have been observed under thechosen collider experiments had the model been true for the given parameters Thus the sameoutput files can be used for all of the collider cuts and hence no cuts were performed throughthe Delphes cards

In order to validate the C++ code we replicated the observed events with the papersdescribing the various collider searches (see Appendix)

Chapter 5

Majorana Model Results

51 Bayesian Scans

To build some intuition we first examined the constraining effect of each observable on itsown We ran separate scans for each constraint turned on and then for all the constraintstogether In Fig 51 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together The paramaters were scanned over the range in Table 51 below

Table 51 Scanned Ranges

Parameter mχ [GeV ] mS[GeV ] λt λb λτ

Range 0minus103 0minus103 10minus5 minus10 10minus5 minus10 10minus5 minus10

In Fig51 we can see that the preferred DM mass is centered around 35 GeV primarily dueto the influence of the Fermi-LAT gamma ray constraint whilst the relic density constraintallows a wide range of possible mediator and DM masses Direct detection (LUX) constraintssplit the mass regions into a top left and bottom right region and together these split theoverall mass parameter regions into two regions by mediator mass but with DM mass around35 GeV Later studies using a lower excess gamma ray spectrum tend to increase the preferredDM mass up to 50 GeV [76] and [88]

The marginalised posterior probabilities for all combinations of parameters taken twoat a time are shown in Fig52 These plots have all constraints (Gamma LUX and RelicDensity) applied together The idea of these graphs is to study the posterior probabilities ofcombinations of two paramaters taken at a time (the off-diagonal graphs) The diagonal plotsshow marginalised histograms of a single parameter at a time- this gives an indication of thespread of each parameter The plots show that mS and λb show a high degree of correlationfor mS gt 100 GeV but there is also a range of values of mS around 35 GeV which are

51 Bayesian Scans 60

1 0 1 2 3 4log10(mχ)[GeV]

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Figure 51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether

51 Bayesian Scans 61

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ms[Gev]5 4 3 2 1 0 1

log10(λt)5 4 3 2 1 0 1

log10(λb)5 4 3 2 1 0 1

log10(λτ)

Figure 52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter

52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 62

possible for λb around 001 λt and λτ show a wide range of possible values for two possibleranges of mS lt 35 GeV and greater than 100 GeV mχ shows a narrow range of possiblevalues around 35 GeV for all values of the other parameters

52 Best fit Gamma Ray Spectrum for the Majorana FermionDM model

We can see from the Fig51(a) scan that the maximum log-likelihood Fermi-LAT gammaray spectrum on its own shows values that centre around 35 GeV This is not changed byadding a relic density constraint or a direct detection (LUX) constraint

The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraintsis shown in Fig53 The parameters for this point were

Table 52 Best Fit Parameters

Parameter mχ [GeV ] mS[GeV ] λt λb λτ

Value 3332 49266 0322371 409990 0008106

10-1 100 101 102

E(GeV)

10

05

00

05

10

15

20

25

30

35

E2dφd

E(G

eVc

m2ss

r)

1e 6

Best fitData

Figure 53 Gamma Ray Spectrum

The relic density (Ωh2) at this point has a value of 0135 but the point is separated onlyslightly in posterior probability from points with a value of 01199 with mS lt 10 GeV (iethe posterior probability is almost the same differing by less than 1)

To better understand the interplay of the different constraints the three graphs in Fig54show the log likelihood surfaces and individual best fit points (best log likelihoods) for the γ

and Ω scans seperately and then combined The diagrams each choose a pair of couplings(fixing the remaining parameters) and display the 2d surfaces of minuslog(χ2) (proportional to

52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 63

the log likelihoods of the coupling combinations) obtained by varying the pair of couplingsThese graphs do not show marginalised posterior probabilities but rather are based on the-2log likelihood calculations directly from MicrOmegas for the given coupling combinationsWe have termed this value as χ2 for convenience and used the log of this value to give abetter range for the colours These log likelihood values are what drive the MultiNest searchof parameter space for the best parameter combinations

The graphs show a complex interplay of the different constraints resulting often in acompromise parameter combination The arrow on each graph points to the maximum point We have chosen a single representative set of couplings near the best overall fit point for allthe graphs

52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 64

00 05 10 15 20 25 30

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Figure 54 Plots of log likelihoods by individual and combined constraints Masses in GeV

53 Collider Constraints 65

53 Collider Constraints

531 Mediator Decay

1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof τ leptons in pp collisions [44] Refer also to Chapter 3 for details

We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

The results of the calculations are shown in the graph Fig55 and can be compared withthe limits in the table from the paper [44] in Fig 56 The black line is the observed ATLASlimit

Figure 55 σ lowastBr(σ rarr ττ) versus Mass of Scalar

0 200 400 600 800

mS[GeV]

10

5

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0(σ

(bbS

)lowastB

(Sgtττ

))[pb]

Observed LimitLikely PointsExcluded Points

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0 5 10 15 20 25 30 35 40 45

We can see that only one point here exceeded the limit and the points showed a widespread over σ(bbS)lowastBr(σ rarr ττ) by mS space which is consistent with the spread of pointson the MultiNest scan on the mS by λτ plane causing a spread of branching ratios to the τ

quark It may seem odd that only one point was excluded here but this point was sampledwith a very low probability having high coupling constants of 597 to the τ quark and 278 tothe t quark and only just exceeds the limit As such it can be explained as a random outlier

53 Collider Constraints 66

Figure 56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar

2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

This scan calculates the contributions of b-quark associated Higgs boson productionreproduced from [43] Refer to Chapter 3 for details

We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

The cross section times branching ratios to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig57 The black line is the CMSLimit

Figure 57 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar

0 200 400 600 800

mS[GeV]

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)lowastB

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bb))

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53 Collider Constraints 67

The results of this scan were compared to the limits in [89] with the plot shown inFig58

Figure 58 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

We can see that there are a number of exclusions across the range of mS which occuredfor high values of the coupling constants gt 1 for λbλt λtau - these can be seen also on Fig59The excluded parameter combinations represent less than 2 of the total sampled number

532 Collider Cuts Analyses

We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude and the results are shown against theposterior probabilities of the parameter combinations in Fig 59

The circles in Figure 59 show which points were excluded by the collider cuts as well asthe limits on cross sections times branching ratios to each of τ+τminus and bb

All the excluded points under the heading Collider Cuts showed exclusion by the Lepstop0experiment (Search for direct pair production of the top quark in all hadronic final states

53 Collider Constraints 68

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Figure 59 Excluded points from Collider Cuts and σBranching Ratio

53 Collider Constraints 69

[32]) Some of these same points were also excluded by Lepstop2 2bstop CMSTop andATLASMonobjet experiments There were expected events at other points but none weresignificant at the 95 exclusion level These significant exclusions occurred mainly wherethe mass of the mediator was over 100 GeV and higher values of the couplings to b and tquarks but there were a few points with mediator mass around 25 GeV The exclusions withmediator mass around 25 GeV were all accompanied by couplings to t quarks of order 10which is probably non-perturbative

Looking at the exclusions in Fig 59 we can see that these occur mainly in the regionof higher mediator mass gt 100 GeV and lt 400 GeV These exclusions come from theastrophysical constraints (Gamma) which show a strong correlation between λb and mS andhigh values of λt gt 1 However these mass regions do not overlap at the same couplingcombinations as the high likelihood region The green and white circles correspond to theexcluded points in Figs 57 and 58 respectively

The remaining regions of highest likelihood can be seen to be where 0001 λb lt 01accompanied by λt lt 0001 and mass of the mediator mS lt 10 GeV or 400 GeV lt mS lt1000GeV with mχ sim 35 GeV

The plot of λt by λb shows the significant regions do not overlap with the collider cutsand cross section constraints since the latter occur at high values of either coupling while thelikely region has low values of these couplings (also accompanied by low couplings to λτ )

Chapter 6

Real Scalar Model Results

61 Bayesian Scans

To build some intuition we first examined the constraining effect of each observable on itsown We ran seperate scans for each constraint turned on and then for all the constraintstogether In Figure 61 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together

In Figure 61 we can see that there are is a narrow region which dominates when allconstraints are taken into account This centers on mχ=9 GeV and a range of values for mS

from 35 GeV to 400 GeV The region is not the highest likelihood region for the Fermi-LATgamma ray spectrum on its own (as can be seen from the individual plot - there is a narrowband to the left of the highest likelihood band) nor for the LUX likelihood or relic densitybut represents a compromise between the three constraints

The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig62 The diagonal plots show marginalised histograms of a singleparameter at a time

61 Bayesian Scans 71

05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

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Figure 61 Real Scalar Dark Matter - By Individual Constraint and All Together

61 Bayesian Scans 72

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Figure 62 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter

62 Best fit Gamma Ray Spectrum for the Real Scalar DM model 73

62 Best fit Gamma Ray Spectrum for the Real Scalar DMmodel

We can see from Fig61 that the maximum log-likelihood Fermi-LAT Gamma ray spectrumon its own shows values of mχ that centre around 35 GeV However the LUX constraintin particular moves the best fit to a point around 9 GeV- the tight LUX constraint onlydisappears for low masses and at that point the proportion of τ+τminus annihilation needs toincrease to fit the constraints

The gamma ray spectrum at the points of best fit for Fermi-LAT including all constraintsis shown in Fig63 The parameters for these points were respectively and involve decays tomainly τ and b quarks respectively

Table 61 Best Fit Parameters

Parameter mχ [GeV ] mS[GeV ] λt λb λτ

Value 932 3526 000049 0002561 000781

10-1 100 101 102

E(GeV)

10

05

00

05

10

15

20

25

30

35

E2dφdE

(GeVc

m2ss

r)

1e 6

Best fitData

Figure 63 Gamma Ray Spectrum

This is not a particularly good fit and it occcurs at a relic density Ωh2 of 335 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

63 Collider Constraints 74

63 Collider Constraints

631 Mediator Decay

1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44]

We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

The results of the scan are shown in the graph Fig64 and can be compared with thelimits in the table from the paper [44] in Fig 56The black line is the observed ATLAS limit

Figure 64 σ lowastBr(σ rarr ττ) versus Mass of Scalar

0 200 400 600 800

mS[GeV]

8

6

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log 1

0(σ

(bbS

)lowastB

(Sgtττ

))[pb]

Observed LimitLikely PointsExcluded Points

050

100150200250300350

0 10 20 30 40 50 60

We can see that there is a narrow band in the region of interest in the σ(bbS)lowastB(S gt ττ)

by mS plane due to the high degree of correlation between mS and λτ seen in the likelihoodscan Values for mS gt 300Gev are excluded

2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinations

63 Collider Constraints 75

randomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig65 The black line is the CMSlimit

Figure 65 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

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050

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The results of this scan were compared to the limits in [89] with the plot shown inFig58

We can see a similar shape for the region of interest to that in Fig64 due to the highcorrelation between mS and λb over the region preferred by astrophysical constraints resultingin branching ratios highly correlated to ms Only one point is excluded with mS gt 500GeV

632 Collider Cuts Analyses

We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expected

63 Collider Constraints 76

with 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 66

We can see that collider cuts show some exclusions only for masses mχ below the highlikelihood region but these exclusions do not occur at the same λb and λt combinations asseen from the plot of these two couplings except for a small region at the highest mS andcouplings

All but one of the collider cut exclusions were excluded by the Lepstop0 experiment (thiswas excluded by 2bstop with high coupling to b quarks of 73) A number of the excludedpoints were also excluded by Lepstop2 2bstop CMSTopDM1L and ATLASMonobJet-(around 30 of the total for each)

63 Collider Constraints 77

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2

1

0

1

log 1

0(λ

t)

(c) λb by λt

5 4 3 2 1 0 1

log10(λb)

6

5

4

3

2

1

0

1

2

log 1

0(λ

τ)

(d) λb by λτ

5 4 3 2 1 0 1

log10(λt)

6

5

4

3

2

1

0

1

2

log 1

0(λ

τ)

(e) λt by λτ

5 4 3 2 1 0 1

log10(λb)

0

1

2

3

log 1

0(m

s)[GeV

]

(f) λb by mS

Figure 66 Excluded points from Collider Cuts and σBranching Ratio

Chapter 7

Real Vector Dark Matter Results

71 Bayesian Scans

In the Figure 71 we can see that the preferred DM mass is centered around 35 GeV primarilydue to the influence of the Fermi-LAT gamma ray constraint with relic density having a widerange of possible mediator and DM masses The result of combining all the constraints is todefine a small region around mχ = 8 GeV and mS = 20 GeV which dominates the posteriorprobability

The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig72 The diagonal plots show marginalised histograms of a singleparameter at a time

71 Bayesian Scans 79

05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

1

0

1

2

3

4

log 1

0(m

s)[GeV

]

(a) Gamma Only

05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

05

00

05

10

15

20

25

30

35

log 1

0(m

s)[GeV

]

(b) Relic Density

1 0 1 2 3 4log10(mχ)[GeV]

05

00

05

10

15

20

25

30

35

log 1

0(m

s)[GeV

]

(c) LUX

05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

05

00

05

10

15

20

25

30

35

log 1

0(m

s)[GeV

]

(d) All Constraints

Figure 71 Real Vector Dark Matter - By Individual Constraint and All Together

71 Bayesian Scans 80

00

05

10

15

20

25

30

log 1

0(m

χ)[GeV

]

00

05

10

15

20

25

30

ms[Gev

]

5

4

3

2

1

0

1

log 1

0(λ

t)

5

4

3

2

1

0

1

log 1

0(λ

b)

00 05 10 15 20 25 30

log10(mχ)[GeV]

5

4

3

2

1

0

1

log 1

0(λ

τ)

00 05 10 15 20 25 30

ms[Gev]5 4 3 2 1 0 1

log10(λt)5 4 3 2 1 0 1

log10(λb)5 4 3 2 1 0 1

log10(λτ)

Figure 72 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter

72 Best fit Gamma Ray Spectrum for the Real Vector DM model 81

72 Best fit Gamma Ray Spectrum for the Real Vector DMmodel

The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraints isshown in Fig73

Table 71 Best Fit Parameters

Parameter mχ [GeV ] mS[GeV ] λt λb λτ

Value 8447 20685 0000022 0000746 0002439

10-1 100 101 102

E(GeV)

10

05

00

05

10

15

20

25

30

35

E2dφdE

(GeVc

m2s

sr)

1e 6

Best fitData

Figure 73 Gamma Ray Spectrum

This is not a particularly good fit and it occcurs at a relic density Ωh2 of 025 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

73 Collider Constraints

731 Mediator Decay

1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44] (See Chapter 3 for more details)

We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

73 Collider Constraints 82

The results of the scan are shown in the graph Fig74 and can be compared with thelimits in the table from the paper [44] in Fig 56

Figure 74 σ lowastBr(σ rarr ττ) versus Mass of Scalar

0 200 400 600 800

mS[GeV]

8

6

4

2

0

2

log 1

0(σ

(bbS

)lowastB

(Sgtττ

))[pb]

Observed LimitLikely PointsExcluded Points

0100200300400500600700800

0 20 40 60 80 100120140

We can see that the region of highest likelihood is limited to mS lt 100 GeV and a narrowrange of couplings to λτ giving a narrow region of interest which falls well below the limit

2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig75 The black line is the CMSlimit

The results of this scan were compared to the limits in [89] with the plot shown in Fig58

We can see that again the region of interest for mS is below 100 GeV and the points fallwell below the limit

73 Collider Constraints 83

Figure 75 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

0 200 400 600 800

mS[GeV]

8

6

4

2

0

2

4

log

10(σ

(bS

+X

)lowastB

(Sgt

bb))

[pb]

Observed LimitLikely PointsExcluded Points

0100200300400500600700800

0 20 40 60 80 100120140

732 Collider Cuts Analyses

We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 76

We can see from Figure 76 that like the real scalar model the region of highest likelihoodhas λb and λt lt 001 while the excluded points from collider cuts and cross sections areabove 01 Most of the excluded points have mχ gt than the values in the region of highestlikelihood and all points are widely separated in parameter space from this region

Once again 89 of excluded points were excluded by Lepstop0 with many also beingexcluded by the other experiments (50 by Lepstop2 and 56 by CMSTopDM1L 35 byLepstop2 and 20 by ATLASMonobjet)

73 Collider Constraints 84

0 1 2 3

log10(mχ)[GeV]

0

1

2

3

log 1

0(m

s)[GeV

]Collider Cuts

σ lowastBr(σgt bS+X)

σ lowastBr(σgt ττ)

(a) mχ by mS

5 4 3 2 1 0 1

log10(λt)

0

1

2

3

log 1

0(m

s)[GeV

](b) λt by mS

5 4 3 2 1 0 1

log10(λb)

5

4

3

2

1

0

1

log 1

0(λ

t)

(c) λb by λt

5 4 3 2 1 0 1

log10(λb)

5

4

3

2

1

0

1

log 1

0(λ

τ)

(d) λb by λτ

5 4 3 2 1 0 1

log10(λt)

5

4

3

2

1

0

1

log 1

0(λ

τ)

(e) λt by λτ

5 4 3 2 1 0 1

log10(λb)

0

1

2

3

log 1

0(m

s)[GeV

]

(f) λb by mS

Figure 76 Excluded points from Collider Cuts and σBranching Ratio

Chapter 8

Conclusion

We first conducted Bayesian scans over mass and coupling parameters for three simplifiedmodels - Majorana fermion real scalar and real vector using direct detecion (LUX) indirectdetection (GCE γ rays) and relic density constraints to find the regions of highest likelihoodWe ran these scans one constraint at a time and then together to better understand the interplayof the different constraints on best fit and likely parameter regions

We also developed and tested C++ collider cut software specifically designed to mimickthe cuts applied in a number of ATLAS and CMS experiments that searched for Emiss

T inconnection with specific beyond the standard model (BSM) theories of Dark Matter

We then selected parameter combinations by randomly sampling the scanned resultsbased on the posterior probability distribution of the points and applied collider cuts andcross section times branching ratio calculations to these points to see if there would havebeen any sensitivity from the ATLAS and CMS experiments

We found that the Majorana fermion model showed the best overall fit to all the constraintswith relic density close to the experimental value of 01199 and a low χ2 error on the gammaspectrum fit The preferred mass mχ was around 35 GeV and mS ranging from lt10 GeV to500 GeV The smaller masses give a better fit for relic density and also are well separatedfrom current collider constraints mainly due the the fact that the exclusions occur mainlywith higher couplings to t quarks (greater than 1) which is consistent with the fact that manyof the experiments were targeting dark matter pair production associated with t quark decaysNevertheless there were some exclusions that did fall in the region sampled by randomselection from the posterior probability distribution

The preferred parameter regions were also consistent with the observation in [20] thatthe Majorana model showed the best overall fit with similar mass constraints We find thatthe couplings to the SM fermions should all be less that 001

86

The scalar and vector DM models give slightly less good fits especially to the currentrelic density estimate of 01199 but again the current collider constraints are well separatedfrom the best fit region and in fact none of the points randomly sampled from the posteriordistribution showed any exclusion The best fit parameter combination favours couplings toSM fermions of less than 001

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Appendix A

Validation of Calculation Tools

Lepstop0Search for direct pair production of the top squarkin all hadronic final states in proton=proton collisions atradic

s=8 TeV with the ATLAS detector [32]

Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1) mt1=600 GeV mn1=1 GeVWhere t1 is the top squark and n1 is the neutralino of the MSSM

94

Table A1 0 Leptons in the final state

Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 10000 460 460 100Emiss

T gt 130GeV 8574 395 451 88Leptonveto 6621 307 285 108Emiss

T gt 150GeV 6316 293 294 100Jet multiplicityand pT 1099 48 51 94|∆φ( jet pmiss

T )|gt π5 902 40 42 96gt= 2b jets 396 19 18 106τ veto 396 19 18 106mbmin

T gt 175GeV 270 13 13 105SRA1 79 4 4 104SRA2 70 3 3 93SRA3 67 3 3 97SRA4 52 2 2 91SRC exactly5 jets 1297 50 60 83SRC |∆φ( jet pmiss

T )|gt π5 1082 50 50 99SRC gt= 2b jets 430 20 20 97SRC τ veto 430 20 20 97SRC |∆φ(bb)|gt 02π 367 17 17 99SRC1 246 12 11 105SRC2 177 9 8 116SRC3 152 8 7 116

95

Lepstop1Search for direct top squark pair production infinal states with one isolated lepton jets and missing trans-verse momentum in

radics = 8 TeV pp collisions using 21 f bminus1

of ATLAS data[33]

Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1)(t1 gt b x+1 )(t1 gt b xminus1 )mt1=650 GeV mn1=1 GeV Where t1 is the top squark n1 is the neutralino xplusmn1 are thecharginos

96

Table A2 1 Lepton in the Final state

Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 100000 99990 99990 100Electron(= 1signal) 11558 11557 11785 984 jets(80604025) 5187 5186 6271 83gt= 1btag 4998 4997 5389 93Emiss

T gt 100GeV [allSRs] 4131 4131 4393 94Emiss

T radic

HT gt 5 [allSRs] 4026 4026 4286 94∆φ( jet2 pmiss

T )gt 08 [allSRs] 3565 3565 3817 93∆φ( jet1Pmiss

T )gt 08 [not SRtN2] 3446 3446 3723 93Emiss

T gt 200GeV (SRtN2) 2027 2027 2186 93Emiss

T radic

HT gt 13(SRtN2) 1156 1156 1172 99mT gt 140GeV (SRtN2) 839 839 974 86Emiss

T gt 275GeV (SRtN3) 948 948 965 98Emiss

T radic

HT gt 11(SRtN3) 932 932 937 99mT gt 200GeV (SRtN3) 456 456 566 81Emiss

T gt 150GeV (SRbC1minusSRbC3) 2803 2803 3080 91Emiss

T radic

HT gt 7(SRbC1minusSRbC3) 2742 2742 3017 91mT gt 12GeV (SRbC1minusSRbC3) 1917 1917 2347 82Emiss

T gt 160GeV (SRbC2SRbC3) 1836 1836 2257 81Emiss

T radic

HT gt 8(SRbC2SRbC3) 1788 1788 2175 82me f f gt 550GeV (SRbC2) 1741 1741 2042 85me f f gt 700GeV (SRbC3) 1075 1075 1398 77Muon(= 1signal) 11379 11378 11975 954 jets(80604025) 5264 5263 6046 87gt= 1btag 5101 5100 5711 89Emiss

T gt 100GeV [allSRs] 4199 4199 4770 88Emiss

T radic

HT gt 5 [all SRs] 4115 4115 4653 88∆φ( jet2 pmiss

T )gt 08 [all SRs] 3622 3622 4077 89∆φ( jet1 pmiss

T )gt 08 [not SRtN2] 3515 3515 3970 89Emiss

T gt 200GeV (SRtN2) 2136 2136 2534 84Emiss

T radic

HT gt 13(SRtN2) 1261 1261 1441 87mT gt 140GeV (SRtN2) 928 928 1033 90Emiss

T gt 275GeV (SRtN3) 1036 1036 1274 81Emiss

T radic

HT gt 11(SRtN3) 1018 1018 1235 82mT gt 200GeV (SRtN3) 497 497 612 81Emiss

T gt 150GeV (SRbC1minusSRbC3) 2891 2891 3380 86Emiss

T radic

HT gt 7(SRbC1minusSRbC3) 2821 2821 3290 86mT gt 120GeV (SRbC1minusSRbC3) 1977 1977 2290 86Emiss

T gt 160GeV (SRbC2SRbC3) 1913 1913 2220 86Emiss

T radic

HT gt 8(SRbC2SRbC3) 1863 1863 2152 87me f f gt 550GeV (SRbC2) 1826 1826 2116 86me f f gt 700GeV (SRbC3) 1131 1131 1452 78

97

Lepstop2Search for direct top squark pair production infinal states with two leptons in

radics =8 TeV pp collisions using

20 f bminus1 of ATLAS data[83][34]

Simulated in Madgraph with p p gt t1 t1(t1 gt b w+ n1)(t1 gt b wminusn1) mt1=180 GeV mn1=60 GeV Where t1 is the top squark n1 is the neutralino b1 is the bottom squark

Table A3 2 Leptons in the final state

Cut RawMe Scaled ATLAS MeATLAS PercentNoCuts 100000 324308 199000 1632BaselineCuts 3593 11652 7203 1622SF Signal Leptons 1699 5510 5510 1002OSSF Signal Leptons 1632 5293 5233 101mll gt 20Gev 1553 5037 5028 100leadinglepton pT 1479 4797 5048 95|mll minusmZ|gt 20Gev 1015 3292 3484 94∆φmin gt 10 850 2757 2160 128∆φb lt 15 680 2205 1757 126SRM90 [SF ] 13 42 60 70SRM120 [SF ] 0 0 19 0SRM100 +2 jets [SF ] 0 0 17 0SRM110 +2 jets [SF ] 0 0 1 02DF Signal Leptons 1737 5633 7164 792OSDF Signal Leptons 1670 5416 5396 100mll20Gev 1587 5147 5136 100leadinglepton pT 1501 4868 4931 99∆φmin gt 10 1247 4044 3122 130∆φb lt 15 1010 3276 2558 128SRM90 [SF ] 21 68 95 72SRM120 [SF ] 0 0 23 0SRM100 +2 jets [SF ] 0 0 19 0SRM110 +2 jets [SF ] 0 0 14 0

98

2bstopSearch for direct third-generation squark pair pro-duction in final states with missing transverse momentumand two b-jets[31]

Simulated in Madgraph with p p gt b1 b1(b1 gt b n1)(b1 gt b n1)

SRAmb1=500 GeV mn1=1 GeV SRBmb1=350 GeV mn1=320 GeVWhere b1 is the bottom squark and n1 is the neutralino

Table A4 2b jets in the final state

Cut RawMe Scaled ATLAS MeATLAS PercentSRA Nocuts 100000 173800 1738 100SRA MET gt 80 93407 162341 1606 101SRA Leptonveto 93302 162159 1505 108SRA MET gt 150 79449 138082 1323 104SRA Jet selection 10213 17750 119 149SRA mbb gt 200 8344 14502 96 151SRA mCT gt 150 6535 11358 82 139SRA mCT gt 200 5175 8994 67 134SRA mCT gt 250 3793 6592 51 129SRA mCT gt 300 2487 4322 35 123SRB NoCuts 100000 1624100 16241 100SRB leptonveto 11489 186593 4069 46SRB MET gt 250 2530 41090 757 54SRB Jet selection 63 1023 79 130SRB HT 3 lt 50 51 828 52 159

99

CMSTop1LSearch for the Production of Dark Matter in As-sociation with Top Quark Pairs in the Single-lepton FinalState in pp collisions[41]

Simulated in Madgraph with p p gt t t p1 p1

Where p1= Majorana fermion Table 3 shows Signal efficiencies and cross sections TheM scale was read from Figure 7 for 90 CL

Table A5 Signal Efficiencies 90 CL on σ limexp[ f b] on pp gt tt +χχ

Mχ[GeV ] My Signal Efficiency Quoted Efficiency My σ [ f b] Quoted σ [ f b]1 102 101 63 47

10 132 101 62 4650 143 120 50 38100 170 146 56 32200 185 173 69 27600 265 240 32 19

1000 320 276 41 17

Appendix B

Branching ratio calculations for narrowwidth approximation

B1 Code obtained from decayspy in Madgraph

Br(S rarr bb) = (minus24λ2b m2

b +6λ2b m2

s

radicminus4m2

bm2S +m4

S)16πm3S

Br(S rarr tt) = (6λ2t m2

S minus24λ2t m2

t

radicm4

S minus4ms2m2t )16πm3

S

Br(S rarr τ+

τminus) = (2λ

2τ m2

S minus8λ2τ m2

τ

radicm4

S minus4m2Sm2

τ)16πm3S

Br(S rarr χχ) = (2λ2χm2

S

radicm4

S minus4m2Sm2

χ)32πm3S

(B1)

Where

mS is the mass of the scalar mediator

mχ is the mass of the Dark Matter particle

mb is the mass of the b quark

mt is the mass of the t quark

mτ is the mass of the τ lepton

The coupling constants λ follow the same pattern

  • TITLE Collider Constraints applied to Simplified Models of Dark Matter fitted to the Fermi-LAT gamma ray excess using Bayesian Techniques
    • Dedication
    • Declaration
    • Acknowledgements
    • Contents
    • List of Figures
    • List of Tables
      • Chapter 1 Introduction
      • Chapter 2 Review of Physics
      • Chapter 3 Fitting Models to the Observables
      • Chapter 4 Calculation Tools
      • Chapter 5 Majorana Model Results
      • Chapter 6 Real Scalar Model Results
      • Chapter 7 Real Vector Dark Matter Results
      • Chapter 8 Conclusion
      • Bibliography
      • Appendix A Validation of Calculation Tools
      • Appendix B Branching ratio calculations for narrow width approximation

    I would like to dedicate this thesis to my wife Julia who has enabled this undertaking by herpatience love and support

    Declaration

    I hereby declare that except where specific reference is made to the work of others thecontents of this dissertation are original and have not been submitted in whole or in partfor consideration for any other degree or qualification in this or any other university Thisdissertation is my own work and contains nothing which is the outcome of work done incollaboration with others except as specified in the text and Acknowledgements Thisdissertation contains fewer than 65000 words including appendices bibliography footnotestables and equations and has fewer than 150 figures

    I certify that this work contains no material which has been accepted for the award of anyother degree or diploma in my name in any university or other tertiary institution and to thebest of my knowledge and belief contains no material previously published or written byanother person except where due reference has been made in the text In addition I certifythat no part of this work will in the future be used in a submission in my name for any otherdegree or diploma in any university or other tertiary institution without the prior approval ofthe University of Adelaide and where applicable any partner institution responsible for thejoint-award of this degree I give consent to this copy of my thesis when deposited in theUniversity Library being made available for loan and photocopying subject to the provisionsof the Copyright Act 1968

    I acknowledge that copyright of published works contained within this thesis resides withthe copyright holder(s) of those works I also give permission for the digital version of mythesis to be made available on the web via the Universityrsquos digital research repository theLibrary Search and also through web search engines unless permission has been granted bythe University to restrict access for a period of time

    Guy PitmanOctober 2016

    Acknowledgements

    And I would like to acknowledge the help and support of my supervisors Professor TonyWilliams and Dr Martin White as well as Assoc Professor Csaba Balasz who has assisted withinformation about a previous study Ankit Beniwal and Jinmian Li who assisted with runningMicrOmegas and LUXCalc I adapted the collider cuts programs originally developed bySky French and Martin White for my study

    Contents

    List of Figures xiii

    List of Tables xv

    1 Introduction 111 Motivation 312 Literature review 4

    121 Simplified Models 4122 Collider Constraints 7

    2 Review of Physics 921 Standard Model 9

    211 Introduction 9212 Quantum Mechanics 9213 Field Theory 10214 Spin and Statistics 10215 Feynman Diagrams 11216 Gauge Symmetries and Quantum Electrodynamics (QED) 12217 The Standard Electroweak Model 13218 Higgs Mechanism 17219 Quantum Chromodynamics 222110 Full SM Lagrangian 23

    22 Dark Matter 25221 Evidence for the existence of dark matter 25222 Searches for dark matter 30223 Possible signals of dark matter 30224 Gamma Ray Excess at the Centre of the Galaxy [65] 30

    Contents vi

    23 Background on ATLAS and CMS Experiments at the Large Hadron collider(LHC) 31231 ATLAS Experiment 32232 CMS Experiment 33

    3 Fitting Models to the Observables 3531 Simplified Models Considered 3532 Observables 36

    321 Dark Matter Abundance 36322 Gamma Rays from the Galactic Center 36323 Direct Detection - LUX 37

    33 Calculations 39331 Mediator Decay 39332 Collider Cuts Analyses 42333 Description of Collider Cuts Analyses 43

    4 Calculation Tools 5541 Summary 5542 FeynRules 5643 LUXCalc 5644 Multinest 5745 Madgraph 5846 Collider Cuts C++ Code 59

    5 Majorana Model Results 6151 Bayesian Scans 6152 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 6453 Collider Constraints 67

    531 Mediator Decay 67532 Collider Cuts Analyses 69

    6 Real Scalar Model Results 7361 Bayesian Scans 7362 Best fit Gamma Ray Spectrum for the Real Scalar DM model 7663 Collider Constraints 77

    631 Mediator Decay 77632 Collider Cuts Analyses 78

    Contents vii

    7 Real Vector Dark Matter Results 8171 Bayesian Scans 8172 Best fit Gamma Ray Spectrum for the Real Vector DM model 8473 Collider Constraints 84

    731 Mediator Decay 84732 Collider Cuts Analyses 86

    8 Conclusion 89

    Bibliography 91

    Appendix A Validation of Calculation Tools 97

    Appendix B Branching ratio calculations for narrow width approximation 105B1 Code obtained from decayspy in Madgraph 105

    List of Figures

    21 Feynman Diagram of electron interacting with a muon 1122 Weak Interaction Vertices [48] 1523 Higgs Potential [49] 1824 Standard Model Particles and Forces [50] 2425 Bullet Cluster [52] 2526 Galaxy Rotation Curves [54] 2627 WMAP Cosmic Microwave Background Fluctuations [58] 2928 Dark Matter Interactions [60] 2929 Gamma Ray Excess from the Milky Way Center [75] 31210 ATLAS Experiment 31211 CMS Experiment 34

    31 Main Feyman diagrams leading to the cross section for scalar decaying to apair of τ leptons 40

    32 WidthmS vs mS 4033 WidthmS vs λb 4134 WidthmS vs λτ 4135 Main Feyman diagrams leading to the cross section for scalar decaying to a

    pair of b quarks in the presence of at least one b quark 42

    41 Calculation Tools 55

    51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether 62

    52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter 6353 Gamma Ray Spectrum 6454 Plots of log likelihoods by individual and combined constraints Masses in

    GeV 6655 σ lowastBr(σ rarr ττ) versus Mass of Scalar 67

    List of Figures ix

    56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar 6857 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar 6858 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 6959 Excluded points from Collider Cuts and σBranching Ratio 70

    61 Real Scalar Dark Matter - By Individual Constraint and All Together 7462 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter 7563 Gamma Ray Spectrum 7664 σ lowastBr(σ rarr ττ) versus Mass of Scalar 7765 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 7866 Excluded points from Collider Cuts and σBranching Ratio 80

    71 Real Vector Dark Matter - By Individual Constraint and All Together 8272 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter 8373 Gamma Ray Spectrum 8474 σ lowastBr(σ rarr ττ) versus Mass of Scalar 8575 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 8676 Excluded points from Collider Cuts and σBranching Ratio 87

    List of Tables

    21 Quantum numbers of the Higgs field 1922 Weak Quantum numbers of Lepton and Quarks 21

    31 Simplified Models 3532 95 CL by Signal Region 4433 Selection criteria common to all signal regions 4534 Selection criteria for signal regions A 4535 Selection criteria for signal regions C 4536 Signal Regions - Lepstop1 4837 Signal Regions Lepstop2 4938 Signal Regions 2bstop 5139 Signal Region ATLASmonobjet 52

    51 Scanned Ranges 6152 Best Fit Parameters 64

    61 Best Fit Parameters 76

    71 Best Fit Parameters 84

    A1 0 Leptons in the final state 98A2 1 Lepton in the Final state 100A3 2 Leptons in the final state 101A4 2b jets in the final state 102A5 Signal Efficiencies 90 CL on σ lim

    exp[ f b] on pp gt tt +χχ 103

    Chapter 1

    Introduction

    Dark matter (DM) was first postulated over 80 years ago when Swiss astronomer FritzZwicky observed a discrepancy between the amount of light emitted by a cluster of galaxiesand the total mass contained within the cluster inferred from the relative motion of thosegalaxies by a simple application of the theory of Newtonian gravitation The surprising resultof this observation was that the vast majority of the mass in the cluster did not emit lightwhich was contrary to the expectation that most of the mass would be carried by the starsSince that time further observations over a wide range of scales and experimental techniqueshave continued to point to the same result and refine it Some of these observations and otherevidence are discussed in section 22 We now know with certainty that in the entire Universeall of the matter we know about - stars planets gases and other cosmic objects such as blackholes can only account for less than 5 of the mass that we calculate to be there

    A recent phenomenon that has received much attention is the significant deviation frombackground expectations of the Fermi Large Area Telescope(Fermi-LAT) gamma ray flux atthe galactic centre [1] A number of astrophysical explanations have been proposed includingmillisecond pulsars of supernova remnants [2] or burst-like continuous events at the galacticcentre but these are unresolved However it has also been noted that the observed Fermi-LATexcess is consistent with the annihilation of dark matter particles which would naturally beconcentrated at the Galactic centre in a manner consistent with the Navarro-Frenk-Whitedistribution of dark matter [3]

    There are a number of other purely theoretical (particle physics) reasons to postulatethe existence of weakly interacting matter particles that could supply the missing mass andyet remain unobservable Weakly interacting massive particle (WIMPS) have been a majorfocus of Run I and ongoing Run II searches of the Large Hadron Collider (LHC) In spite

    2

    of substantial theoretical and experimental effort the microscopic properties of dark matterparticles are still unknown

    One of the main problems with extracting these properties is the plethora of competingtheoretical models that provide dark matter candidates Another is the very weak interactionof these particles and the lack of experimental observables allowing one to discriminatebetween the competing models

    A particularly fruitful approach is the simplified model approach which makes minimalassumptions about the DM candidates as extensions to the Standard Model interacting weaklywith ordinary matter through a mediator These can be coded as Lagrangian interaction termsusing tools such as FeynRules [4] and Calchep [5] and used to simulate Dark Matter relicdensities indirect detection expectations as well as direct detection expectations throughMicrOmegas [6] This in turn can be run under a Bayesian inference tool (see below)

    Using a Bayesian inference tool such as MultiNest [7][8][9] one can scan the parameterspace of a particular model to find regions of highest probability for that model based onobservations of a range of phenomena including direct detection experiments such as ZEPLIN[10] XENON [11] DEAP [12] ARDM [13] DarkSide [14] PandaX [15] and LUX [16]indirect detection - eg Fermi-LAT gamma ray excess [17] Pamela-positron and antiprotonflux [18] and production at colliders MultiNest calculates the evidence value which is theaverage likelihood of the model weighted by prior probability and a posterior distributionwhich is the distribution of the parameter(s) after taking into account the observed databy scanning over a range of parameters based on prior probabilities This can be used tocompare models on a consistent basis The scans also provide information about the best fitparameters for the overall fit as well as for the seperate experimentsobservations

    The simplified model approach has been taken in a number of papers to study the Fermi-LAT gamma ray excess (see literature review in section 12) and one in particular comparesthese models using a Bayesian inference tool which found that a Majorana fermion DMcandidate interacting via a scalar mediator with arbitrary couplings to third generationfermions showed the best fit [19] A subsequent paper [20] examined the most likelyparameter combinations that simultaneously fit the Planck relic density Fermi-LAT galacticcentre gamma ray AMS-02 positron flux and Pamela antiproton flux data

    There has not as yet been a thorough study of the collider constraints on DM simplifiedmodels that fit the Fermi-LAT galactic centre excess In this thesis we reproduce the

    11 Motivation 3

    previous results of [20] using similar astrophysical constraints and then implement detailedsimulations of collider constraints on DM and mediator production

    In Section 11 we review the motivation for this study and in Section 12 we considerrecent astronomical data which is consistent with the annihilation of DM particles andreview the literature and searches which attempt to explain it with a range of extensions tothe standard model

    In Chapter 2 we review the Standard Model of particle physics (SM) the evidence fordark matter and the ATLAS and CMS experiments at the LHC

    In Chapter 3 we describe the models considered in this paper the observables used inthe Bayesian scans and the collider experiments which provide limits against which theMagraph simulations of the three models can be compared

    In Chapter 4 we review the calculation tools that have been used in this paper

    In Chapters 5 6 and 7 we give the results of the Bayesian scans and collider cutsanalyses for each of the models and in Chapter 8 we summarise our conclusions

    11 Motivation

    The Fermi gamma ray space telescope has over the past few years detected a gamma raysignal from the inner few degrees around the galactic centre corresponding to a regionseveral hundred parsecs in radius The spectrum and angular distribution are compatible withthat from annihilating dark matter particles

    A number of papers have recently attempted to explain the Gamma Ray excess in termsof extensions to the Standard Model taking into account constraints from ATLAS and CMScollider experiments direct detection of dark matter colliding off nucleii in well shieldeddetectors (eg LUX) as well as other indirect measurements (astrophysical observations)These papers take the approach of simplified models which describe the contact interactionsof a mediator beween DM and ordinary matter using an effective field theory approach(EFT) Until recently it has been difficult to discriminate between the results of these paperswhich look at different models and produce different statistical results A recent paper[19] Simplified Dark Matter Models Confront the Gamma Ray Excess used a Bayesianinference tool to compare a variety of simplified models using the Bayesian evidence values

    12 Literature review 4

    calculated for each model to find the most preferred explanation The motivation for thechosen models is discussed in section 121

    A subsequent paper [20] Interpreting the Fermi-LAT gamma ray excess in the simplifiedframework studied the most likely model again using Bayesian inference to find the mostlikely parameters for the (most likely) model The data inputs (observables) to the scanwere the Planck measurement of the dark matter relic density [21] Fermi-LAT gamma rayflux data from the galactic center [17] cosmic positron flux data [22] cosmic anti-protonflux data [18] cosmic microwave background data and direct detection data from the LUXexperiment [16] The result of the scan was a posterior probability density function in theparameter space of the model

    This paper extends the findings in [19] by adding collider constraints to the likelihoodestimations for the three models It is not a simple matter to add collider constraints becauseeach point in parameter space demands a significant amount of computing to calculate theexpected excess amount of observable particles corresponding to the particular experimentgiven the proposed simplified model A sampling method has been employed to calculatethe collider constraints using the probability of the parameters based on the astrophysicalobservables

    12 Literature review

    121 Simplified Models

    A recent paper [23] summarised the requirements and properties that simplified models forLHC dark matter searches should have

    The general principles are

    bull Besides the Standard Model (SM) the Lagrangian should contain a DM candidate thatis absolutely stable or lives long enough to escape the LHC detector and a mediatorthat couples the two sectors

    bull The Lagrangian should contain all terms that are renormalizable consistent withLorentz invariance SM symmetries and DM stability

    bull Additional interactions should not violate exact and approximate global symmetries ofthe SM

    12 Literature review 5

    The examples of models that satisfy these requirements are

    1 Scalar (or pseudo-scalar) s-channel mediator coupled to fermionic dark matter (Diracor Majorana)

    2 Higgs portal DM where the DM is a scalar singlet or fermion singlet under the gaugesymmetries of the SM coupling to a scalar boson which mixes with the Higgs (Thefermion singlet is a specific realisation of 1 above)

    3 DM is a mixture of an electroweak singlet and doublet as in the Minimal Supersym-metric SM (MSSM)

    4 Vector (or Axial vector) s-channel mediator obtained by extending the SM gaugesymmetry by a new U(1)rsquo which is spontaneously broken such that the mediatoracquires mass

    5 t-channel flavoured mediator (if the DM particle is a fermion χ the mediator can bea coloured scalar or vector particle φ -eg assuming a scalar mediator a couplingof the form φ χq requires either χ or φ to carry a flavor index to be consistent withminimal flavour violation (MFV) which postulates the flavour changing neutral currentstructure in the SM is preserved

    Another recent paper [24] summarised the requirements and properties that simplifiedmodels for LHC dark matter searches should have The models studied in this paper arepossible candidates competing with Dirac fermion and complex scalar dark matter andmodels with vector and pseudo-scalar mediators Other models would have an elasticscattering cross section that will remain beyond the reach of direct detection experiments dueto the neutrino floor (the irreducible background produced by neutrino scattering) or arealready close to sensitivity to existing LUX and XENONIT experiments

    A recent paper [25] Scalar Simplified Models for Dark Matter looks at Dirac fermiondark matter mediated by either a new scalar or pseudo-scalar to SM fermions This paperplaces bounds on the coupling combination gχgφ (coupling of DM to mediator times couplingof mediator to SM fermions) calculated from direct detection indirect detection and relicdensity by the mass of DM and mass of mediator Bounds are also calculated for colliderconstraints (mono-jet searches heavy-flavour searches and collider bounds on the 125 GeVHiggs are used to place limits on the monojet and heavy flavor channels which can betranslated into limits on the Higgs coupling to dark matter and experimental measurements

    12 Literature review 6

    of the Higgs width are used to constrain the addition of new channels to Higgs decay) Thepaper models the top loop in a full theory using the Monte Carlo generator MCFM [26]and extends the process to accommodate off-shell mediator production and decay to a darkmatter pair While this paper does not consider the Majorana fermion dark matter model theconstraints on the coupling product may have some relevence to the present study if only asan order of magnitude indicator

    Constraining dark sectors with monojets and dijets [27] studies a simplified model withDirac fermion dark matter and an axial-vector mediator They use two classes of constraint-searches for DM production in events with large amounts of missing transverse energy inassociation with SM particles (eg monojet events) and direct constraints on the mediatorfrom monojet searches to show that for example where the choice of couplings of themediator to quarks and DM gA

    q =gAχ=1 all mediator masses in the range 130 GeV ltMR lt3

    TeV are excluded

    The paper Constraining Dark Sectors at Colliders Beyond the Effective Theory Ap-proach [28] considers Higgs portal models with scalar pseudo-scalar vector and axial-vectormediators between the SM and the dark sector The paper takes account of loop processesand gluon fusion as well as quark-mediator interactions using MCFM Apart from mediatortype the models are characterised by free parameters of mediator width and mass darkmatter mass and the effective coupling combination gχgφ (coupling of DM to mediator timescoupling of mediator to SM fermions) The conclusion is that LHC searches can proberegions of the parameter space that are not able to be explored in direct detection experiments(since the latter measurements suffer from a velocity suppression of the WIMP-nucleoncross-section)

    The paper Constraining the Fermi-LAT excess with multi-jet plus EmissT collider searches

    [29] shows that models with a pseudo-scalar or scalar mediator are best constrained by multi-jet final states with Emiss

    T The analysis was done using POWHEG-BOX-V2 at LO and usesM2

    T variables together with multi-jet binning to exploit differences in signal and backgroundin the different regions of phase space to obtain a more stringent set of constraints on DMproduction This paper largely motivated the current work in that it showed that collidersearches that were not traditionally thought of as dark matter searches were neverthelessoften sensitive to dark matter produced in association with heavy quarks The combinationof searches used in this paper reflects this fact

    12 Literature review 7

    122 Collider Constraints

    In order to constrain this model with collider constraints (direct detection limits) we considerthe following papers by the CMS and ATLAS collaborations which describe searches at theLHC

    ATLAS Experiments

    bull Search for dark matter in events with heavy quarks and missing transverse momentumin p p collisions with the ATLAS detector [30]

    bull Search for direct third-generation squark pair production in final states with missingtransverse momentum and two b-jets in

    radics= 8 TeV pp collisions with the ATLAS

    detector[31]

    bull Search for direct pair production of the top squark in all-hadronic final states inprotonndashproton collisions at

    radics=8 TeV with the ATLAS detector [32]

    bull Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

    radic(s)=8TeV pp collisions using 21 f bminus1 of

    ATLAS data [33]

    bull Search for direct top squark pair production in final states with two leptons inradic

    s = 8TeV pp collisions using 20 f bminus1 of ATLAS data [34]

    bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

    radics=8 TeV [35]

    CMS Experiments

    bull Searches for anomalous tt production in p p collisions atradic

    s=8 TeV [36]

    bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

    radics=8 TeV [37]

    bull Search for the production of dark matter in association with top-quark pairs in thesingle-lepton final state in proton-proton collisions at

    radics = 8 TeV [38]

    bull Search for new physics in monojet events in p p collisions atradic

    s = 8 TeV(CMS) [39]

    bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

    s = 8 TeV [40]

    12 Literature review 8

    bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theSingle-lepton Final State in pp collisions at

    radics=8 TeV [41]

    bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

    s=8 TeV [42]

    bull Search for neutral MSSM Higgs bosons decaying into a pair of bottom quarks [43]

    bull Search for neutral MSSM Higgs bosons decaying into a pair of τ leptons in p p collision[44]

    bull Search for dark matter extra dimensions and unparticles in monojet events in proton-proton collisions at

    radics=8 TeV [45]

    In particular we have implemented Madgraph simulations of certain data analysesapplying to collider experiments described in papers [32][33][34][31][30][41] Theremaining models were not implemented in Madgraph because of inability to validate themodels against published results in the time available We developed C++ analysis code toimplement the collider cuts after showering with Pythia and basic analysis with Delpheswith output to root files The C++ code allowed multiple experimental cuts to be applied tothe same simulation output The C++ code was first checked against the results of the variousexperiments above and then the simulations were rerun using the three modelsrsquo Lagrangians(implemented in FeynRules) described in this paper

    Chapter 2

    Review of Physics

    21 Standard Model

    211 Introduction

    The Standard Model of particle physics (SM) has sometimes been described as the theoryof Almost Everything The SM is a field theory which emerged in the second half of thetwentieth century from quantum mechanics and special relativity which themselves wereonly conceived after 1905 by Einsten Pauli Bohr Dirac and others In the first decades ofthe twenty first century the discovery of the Higgs boson [46][47] completed the pictureand confirmed the last previously unverified part of the SM The SM is a self consistenttheory that describes and classifies all known particles and their interactions (other thangravity) to an astonishing level of accuracy and detail However the theory is incomplete-it does not describe the complete theory of gravitation or account for DM (see below) ordark energy (the accelerating expansion of the universe) Related to these shortcomings aretheoretical deficiencies such as the asymmetry of matter and anti-matter the fact that the SMis inconsistent with general relativity (to the extent that one or both theories break down atspacetime singularities like the Big Bang and black hole event horizons)

    212 Quantum Mechanics

    Quantum Mechanics (QM) describes particles as probability waves which can be thoughtof as packets of energy which can be localised or spread through all space depending on how

    21 Standard Model 10

    accurately one measures the particlersquos momentum However QM is not itself consistent withspecial relativity It was realised too that because of relativity particles could be created anddestroyed and because of the uncertainty principle the theory should describe the probabilityof particles being created and destroyed on arbitrarily short timescales A quantum fieldtheory (QFT) is a way to describe this probability at every point of space and time

    213 Field Theory

    A field theory is described by a mathematical function of space and time called a LagrangianAs in classical mechanics where Lagrangians were introduced by Louis Lagrange at thebeginning of the nineteenth century to summarise dynamical systems the Lagrangian sum-marises the dynamics of space which is permeated by force fields The equations of motionarising from a Lagrangian are determined by finding the contours of the Lagrangian describedby stationary points (ie where the infinitesimal change in the Lagrangian is zero for a smallchange in field) Particles are generally described by singularities in the Lagrangian (wherethe values tend to infinity also called resonances)

    214 Spin and Statistics

    It was also realised that different types of particle needed to be represented by different fieldsin the Lagrangian Diracrsquos discovery that particles with spin 12 quantum numbers couldbe represented by a field obtained by effectively taking the square root of the relativisticequation (Klein Gordon equation) was probably the most significant advance in quantumfield theory It required the introduction of a matrix operators called the gamma matricesor Dirac matrices which helped explain the already known fact that spin half particles orfermions obey Fermi-Dirac statistics ie when two spin half particles are exchanged thesign of the wavefunction must change and hence they cannot occupy the same quantum stateunlike bosons with integer spin that can occupy the same quantum state

    Other particles need to be represented by fields called vector pseudo-vector scalar (as inthe Klein Gordon equation) pseudo-scalar and complex fields A vector field is really fourfields one for each dimension of space plus time and is parameterised by the four gammamatrices γmicro which transform among themselves in a relativistically invariant way (Lorentzinvariance) A pseudo-vector field is a vector field which transforms in the same way but

    21 Standard Model 11

    with additional sign change under parity transformations (a simultaneous reversal of the threespatial directions) and is represented by γmicroγ5 A pseudo-scalar is a scalar which changes signunder parity transformations and a complex scalar field is represented by complex numbers

    215 Feynman Diagrams

    QFT took another giant leap forward when Richard Feynman realised that the propagationof particles described by the field could be described by expanding the so called S-matrixor scattering matrix derived from calculating the probability of initial states transitioning tofinal states The expansion is similar to a Taylor expansion in mathematics allowing for thespecial commuting properties of elements of the field The Feynman diagrams representingindividual terms in the expansion can be interpreted as interactions between the particlecomponents The S-Matrix (or more correctly the non trivial part called the T-Matrix) isderived using the interaction Hamiltonian of the field The Hamiltonian is an operator whoseeigenvalues or characteristic values give the energy quanta of the field The T-matrix canalso be described by a path integral which is an integral over all of quantum space weightedby the exponential of the Lagrangian This is completely equivalent to the Hamiltonianformulation and shows how the Lagrangian arises naturally out of the Hamiltonian (in factthe Lagrangian is the Legendre transformation of the Hamiltonian)

    Figure 21 Feynman Diagram of electron interacting with a muon

    γ

    eminus

    e+

    micro+

    microminus

    The diagram Fig21 is a tree diagram (diagrams with closed loops indicate self in-teractions and are higher order terms in the expansion) and shows an electron (eminus) and ananti-muon (micro+) interacting by exchanging a photon

    21 Standard Model 12

    216 Gauge Symmetries and Quantum Electrodynamics (QED)

    The Dirac equation with its gamma matrices ushered in the use of group theory to describethe transformation of fields (because the multiplication table of the gamma matrices isclosed and described by group theory) It also ushered in the idea of symmetries within theLagrangian- for example consider a Lagrangian

    ψ(ipart minusm)ψ (21)

    The Lagrangian is invariant under a global U(1) transformation which acts on the fieldsand hence the derivatives as follows

    ψ rarr eiqαψ ψ rarr ψeminusiqα partmicroψ rarr eiqα

    partmicroψ (22)

    where qα is a global phase and α is a continuous parameter

    A symmetry leads by Noetherrsquos theorem to a conserved current jmicro = qψγmicroψ and is thebasis of charge conservation since partmicro jmicro = 0 and q =

    intd3x j0(x)

    By promoting this transformation to a local one (ie α rarr α(x) where x = xmicro are thespace time co-ordinates) the transformations become

    ψ rarr eiqα(x)ψ ψ rarr ψeminusiqα(x)partmicroψ rarr eiqα(x)(partmicroψ + iq(partmicroα(x))ψ) (23)

    The derivative transformation introduces an extra term which would spoil the gaugeinvariance This can be solved by introducing a new field the photon field Amicro(x) whichinteracts with the fermion field ψ and transforms under a U(1) transformation

    Amicro rarr Amicro minuspartmicroα(x) (24)

    If we replace normal derivatives partmicro with covariant derivatives Dmicro = (partmicro + iqAmicro(x)) in theLagrangian then the extra term is cancelled and we can now see that the covariant derivativetransforms like ψ - under a local U(1) transform ie

    Dmicro rarr Dmicroψprime = eiqα(x)Dmicroψ (25)

    The additional term iqAmicro(x) can be interpreted in a geometric sense as a connectionwhich facilitates the parallel transport of vectors along a curve in a parallel consistent manner

    21 Standard Model 13

    We must add a gauge-invariant kinetic term to the Lagrangian to describe the dynamicsof the photon field This is given by the electromagnetic field strength tensor

    Fmicroν = partmicroAν minuspartνAmicro (26)

    The full Quantum Electrodynamics (QED) Lagrangian is then for electrons with mass mand charge -e

    LQED = ψ(i Dminusm)ψ minus 14

    Fmicroν(X)Fmicroν(x) (27)

    This Lagrangian is Lorentz and U(1) guage invariant The interaction term is

    Lint =+eψ Aψ = eψγmicro

    ψAmicro = jmicro

    EMAmicro (28)

    where jmicro

    EM is the electromagnetic four current

    217 The Standard Electroweak Model

    The Standard Electroweak (EW) Model was created by Glashow-Weinberg-Salam (also calledthe GWS model) and unifies the electomagnetic and weak interactions of the fundamentalparticles

    The idea of gauge symmetry was extended to the idea of a non-abelian gauge symmetry(essentially the operators defining the symmetry do not commute and so are represented bymatrices- in this case the Pauli matrices for the SU(2) group) - the gauge group is the directproduct G = SU(2)

    otimesU(1) It was known that weak interactions were mediated by Wplusmn

    and Z0 particles where the superscripts are the signs of the electromagnetic charge Thesewere fairly massive but it was also known that through spontaneous symmetry breaking ofa continuous global non-abelian symmetry there would be one massive propagation modefor each spontaneously broken-symmetry generator and massless modes for the unbrokengenerators (the so-called Goldstone bosons)

    This required an extra field to represent the gauge bosons called Bmicro (a vector fieldhaving the 4 spacetime degrees of freedom) With this identification the covariant derivativebecomes

    Dmicro = partmicro minus igAmicro τ

    2minus i

    gprime

    2Y Bmicro (29)

    21 Standard Model 14

    Here Y is the so called weak hypercharge of the field Bmicro which is a quantum numberdepending on particle (lepton or quark) g and gprime are two independent parameters in theGWS theory and the expression Amicro τ implies a sum over 3 components Aa

    micro a=123 and thePauli matrices τa

    This isnrsquot the full story because it had been known since 1956 when Chin Shuing Wuand her collaborators confirmed in an experiment prompted by Lee and Yang that nature isnot ambivalent to so-called parity It had been assumed that the laws of nature would notchange if all spatial coordinates were reversed (a parity transformation) The experimentinvolved the beta decay of cooled 60C atoms (more electrons were emitted in the backwardhemisphere with respect to spin direction of the atoms) This emission involved the weakinteraction and required that the Lagrangian for weak interactions contain terms which aremaximally parity violating- which turn out to be currents of the form

    ψ(1minus γ5)γmicro

    ψ (210)

    The term

    12(1minus γ

    5) (211)

    projects out a left-handed spinor and can be absorbed into the spinor to describe left-handedleptons (and as later discovered quarks) The right handed leptons are blind to the weakinteraction

    The processes describing left-handed current interactions are shown in Fig 22

    Analogously to isospin where the proton and neutron are considered two eigenstates ofthe nucleon we postulate that Bmicro is the gauge boson of an SU(2)transformation coupled toweak SU(2) doublets (

    νe

    eminus

    )

    (ud

    ) (212)

    We may now write the weak SU(2) currents as eg

    jimicro = (ν e)Lγmicro

    τ i

    2

    e

    )L (213)

    21 Standard Model 15

    Figure 22 Weak Interaction Vertices [48]

    where τ i are the Pauli-spin matrices and the subscript L denotes the left-handed componentof the spinor projected out by Eqn211 The third current corresponding to i=3 is called theneutral current as it doesnrsquot change the charge of the particle involved in the interaction

    We could postulate that the electromagnetic current is a linear combination of the leftand right elecromagnetic fields eL = 1

    2(1minus γ5)e and eR = 12(1+ γ5)e

    jemmicro = eLγmicroQeL + eRγmicroQeR (214)

    where Q is the electromagnetic operator and eR is a singlet However this is not invariantunder SU(2)L To construct an invariant current we need to use the SU(2)L doublet 212The current which is an SU(2)L invariant U(1) current is

    jYmicro = (ν e)LγmicroYL

    e

    )L+ eRγmicroYReR (215)

    where hypercharges YL and YR are associated with the U(1)Y symmetry jYmicro is a linearcombination of the electromagnetic current and the weak neutral current thus explaining whythe hypercharge differs for the left and right-handed components since the electromagneticcurrent has the same charge for both components whilst the weak current exists only forleft-handed particles j3

    micro and the third component of weak isospin T 3 allows us to calculate

    21 Standard Model 16

    the relationship Y = 2(QminusT 3) which can be seen by writing jYmicro as a linear combination ofj3micro (the weak neutral current that doesnrsquot change the charge of the particle involved in the

    interaction) and jemmicro (using the convention that the jYmicro is multiplied by 1

    2 to match the samefactor implicit in j3

    micro ) Substituting

    τ3 =

    (1 00 minus1

    )(216)

    into equation 213 for i=3 and subtracting equation 215 and equating to 12 times equation

    214

    we get

    eLγmicroQeL + eRγmicroQeR minus (νLγmicro

    12

    νL minus eLγmicro

    12

    eL) =12

    eRγmicroYReR +12(ν e)LγmicroYL

    e

    )L (217)

    from which we can read out

    YR = 2QYL = 2Q+1 (218)

    and T3(eR) = 0 T3(νL) =12 and T3(eL) =

    12 The latter three identities are implied by

    the fraction 12 inserted into the definition of equation 213

    The Lagrangian kinetic terms of the fermions can then be written

    L =minus14

    FmicroνFmicroν minus 14

    GmicroνGmicroν

    + sumgenerations

    LL(i D)LL + lR(i D)lR + νR(i D)νR

    + sumgenerations

    QL(i D)QL +UR(i D)UR + DR(i D)DR

    (219)

    LL are the left-handed lepton doublets QL are the left-handed quark doublets lRνR theright handed lepton singlets URDR the right handed quark singlets

    The field strength tensors are given by

    Fmicroν = partmicroAν minuspartνAmicro minus ig[Amicro Aν ] (220)

    21 Standard Model 17

    andGmicroν = partmicroBν minuspartνBmicro (221)

    Bmicro is an abelian U(1) gauge field while Amicro is non-abelian and Fmicroν contains the com-mutator of the field through the action of the covariant derivative [Dmicro Dν ] This equationcontains no mass terms terms for the field Amicro for good reason- this would violate gaugeinvariance We know that the gauge bosons of the weak force have substantial mass and theway that this problem has been solved is through the Higgs mechanism

    218 Higgs Mechanism

    To solve the problem of getting gauge invariant mass terms three independent groupspublished papers in 1964 The mechanism was named after Peter Higgs To introduce theconcept consider a complex scalar field φ that permeates all space Consider a potential forthe Higgs field of the form

    minusmicro2φ

    daggerφ +λ (φ dagger

    φ2) (222)

    which has the form of a mexican hat Fig 23 This potential is still U(1) gauge invariant andcan be added to our Lagrangian together with kinetic+field strength terms to give

    L = (Dmicroφ)dagger(Dmicroφ)minusmicro2φ

    daggerφ +λ (φ dagger

    φ2)minus 1

    4FmicroνFmicroν (223)

    It is easily seen that this is invariant to the transformations

    Amicro rarr Amicro minuspartmicroη(x) (224)

    φ(x)rarr eieη(x)φ(x) (225)

    The minimum of this field is not at 0 but in a circular ring around 0 with a vacuum

    expectation value(vev)radic

    micro2

    2λequiv vradic

    2

    We can parameterise φ as v+h(x)radic2

    ei π

    Fπ where h and π are referred to as the Higgs Bosonand Goldstone Boson respectively and are real scalar fields with no vevs Fπ is a constant

    21 Standard Model 18

    Figure 23 Higgs Potential [49]

    Substituting this back into the Lagrangian 223 we get

    minus14

    FmicroνFmicroν minusevAmicropartmicro

    π+e2v2

    2AmicroAmicro +

    12(partmicrohpart

    microhminus2micro2h2)+

    12

    partmicroπpartmicro

    π+(hπinteractions)(226)

    This Lagrangian now seems to describe a theory with a photon Amicro of mass = ev a Higgsboson h with mass =

    radic2micro and a massless Goldstone π

    However there is a term mixing Amicro and part microπ in this expression and this shows Amicro and π

    are not independent normal coordinates and we can not conclude that the third term is a massterm This π minusAmicro mixing can be removed by making a gauge transformation in equations224 and 225 making

    φrarrv+h(x)radic2

    ei π

    Fπminusieη(x) (227)

    and setting πrarr π

    Fπminus eη(x) = 0 This choosing of a point on the circle for the vev is called

    spontaneous symmetry breaking much like a vertical pencil falling and pointing horizontallyin the plane This choice of gauge is called unitary gauge The Goldstone π will thencompletely disappear from the theory and one says that the Goldstone has been eaten to givethe photon mass

    21 Standard Model 19

    This can be confirmed by counting degrees of freedom before and after spontaneoussymmetry breaking The original Lagrangian had four degrees of freedom- two from thereal massless vector field Amicro (massless photons have only two independent polarisationstates) and two from the complex scalar field (one each from the h and π fields) The newLagrangian has one degree of freedom from the h field and 3 degrees of freedom from themassive real vector field Amicro and the π field has disappeared

    The actual Higgs field is postulated to be a complex scalar doublet with four degrees offreedom

    Φ =

    (φ+

    φ0

    )(228)

    which transforms under SU(2) in the fundamental representation and also transforms underU(1) hypercharge with quantum numbers

    Table 21 Quantum numbers of the Higgs field

    T 3 Q Yφ+

    12 1 1

    φ0 minus12 1 0

    We can parameterise the Higgs field in terms of deviations from the vacuum

    Φ(x) =(

    η1(x)+ iη2(x)v+σ(x)+ iη3(x)

    ) (229)

    It can be shown that the three η fields can be transformed away by working in unitarygauge By analogy with the scalar Higgs field example above the classical action will beminimised for a constant value of the Higgs field Φ = Φ0 for which Φ

    dagger0Φ0 = v2 This again

    defines a range of minima and we can pick one arbitarily (this is equivalent to applying anSU(2)timesU(1) transform to the Higgs doublet transforming the top component to 0 and thebottom component to Φ0 = v) which corresponds to a vacuum of zero charge density andlt φ0 gt=+v

    In this gauge we can write the Higgs doublet as

    Φ =

    (φ+

    φ0

    )rarr M

    (0

    v+ H(x)radic2

    ) (230)

    where H(x) is a real Higgs field v is the vacuum expectation and M is an SU(2)timesU(1)matrix The matrix Mdagger converts the isospinor into a down form (an SU(2) rotation) followedby a U(1) rotation to a real number

    21 Standard Model 20

    If we consider the Higgs part of the Lagrangian

    minus14(Fmicroν)

    2 minus 14(Bmicroν)

    2 +(DmicroΦ)dagger(DmicroΦ)minusλ (Φdagger

    Φminus v2)2 (231)

    Substituting from equation 230 into this and noting that

    DmicroΦ = partmicroΦminus igW amicro τ

    aΦminus 1

    2ig

    primeBmicroΦ (232)

    We can express as

    DmicroΦ = (partmicro minus i2

    (gA3

    micro +gprimeBmicro g(A1micro minusA2

    micro)

    g(A1micro +A2

    micro) minusgA3micro +gprimeBmicro

    ))Φ equiv (partmicro minus i

    2Amicro)Φ (233)

    After some calculation the kinetic term is

    (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

    14(v+

    Hradic2)2[A 2]22 (234)

    where the 22 subscript is the index in the matrix

    If we defineWplusmn

    micro =1radic2(A1

    micro∓iA2micro) (235)

    then [A 2]22 is given by

    [A 2]22 =

    (gprimeBmicro +gA3

    micro

    radic2gW+

    microradic2gWminus

    micro gprimeBmicro minusgA3micro

    ) (236)

    We can now substitute this expression for [A 2]22 into equation 234 and get

    (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

    14(v+

    Hradic2)2(2g2Wminus

    micro W+micro +(gprimeBmicro minusgA3micro)

    2) (237)

    This expression contains mass terms for three massive gauge bosons Wplusmn and the combi-nation of vector fields gprimeBmicro minusgA3

    micro where note

    21 Standard Model 21

    Table 22 Weak Quantum numbers of Lepton and Quarks

    T 3 Q YνL

    12 0 -1

    lminusL minus12 -1 -1

    νR 0 0 0lminusR 0 -1 -2UL

    12

    23

    13

    DL minus12 minus1

    313

    UR 0 23

    43

    DR 0 minus13 minus2

    3

    Wminusmicro = (W+

    micro )dagger equivW 1micro minus iW 2

    micro (238)

    Then the mass terms can be written

    12

    v2g2|Wmicro |2 +14

    v2(gprimeBmicro minusgA3micro)

    2 (239)

    W+micro and W microminus can be associated with the W boson and its anti-particle and (minusgprimeBmicro +

    gA3micro) with the Z Boson (after normalisation by

    radicg2 +(gprime

    )2) The combination gprimeA3micro +gBmicro

    is orthogonal to Z and remains massless This can be associated with the Photon (afternormalisation) These predict mass terms for the W and Z bosons mW = vgradic

    2and mZ =

    vradic2

    radicg2 +(gprime

    )2It is again instructive to count the degrees of freedom before and after the Higgs mech-

    anism Before we had a complex doublet Φ with four degrees of freedom one masslessB with two degrees of freedom and three massless W a fields each with two each givinga total of twelve degrees of freedom After spontaneous symmetry breaking we have onereal scalar Higgs field with one degree of freedom three massive weak bosons with nineand one massless photon with two again giving twelve degrees of freedom One says thescalar degrees of freedom have been eaten to give the Wplusmn and Z bosons their longitudinalcomponents (ie mass)

    Table 22 shows T 3 the weak isospin quantum number (which describes how a particletransforms under SU(2)) Y the weak hypercharge (which describes how the particle trans-

    21 Standard Model 22

    forms under U(1)) and Q the electric charge of leptons and quarks The former two enterdirectly into the Lagrangian while the latter can be derived as a function of the former two

    Strictly speaking the right handed neutrino which has no charge may not exist and is notpart of the standard model although some theories postulate a mass for this (hypothetical)particle

    Writing gauge invariant couplings between ELeRφ with the vacuum expectation ofφ = v (Yukawa matrices λ

    i ju and λ

    i jd respectively for the up and down quarks) we get mass

    terms for the quarks (and similarly for the leptons)

    Mass terms for quarks minussumi j[(λi jd Qi

    Lφd jR)+λ

    i ju εab(Qi

    L)aφlowastb u j

    R +hc]

    Mass terms for leptonsminussumi j[(λi jl Li

    Lφ l jR)+λ

    i jν εab(Li

    L)aφlowastb ν

    jR +hc]

    Here hc is the hermitian conjugate of the first terms L and Q are lepton and quarkdoublets l and ν are the lepton singlet and corresponding neutrino d and u are singlet downand up quarks εab is the Levi-Civita (anti-symmetric) tensor

    If we write the quark generations in a basis that diagonalises the Higgs couplings weget a unitary transformations Uu and Ud for the up and down quarks respectively which iscompounded in the boson current (Udagger

    u Ud) called the CKM matrix where off-diagonal termsallow mixing between quark generations (this is also called the mass basis as opposed tothe flavour basis) A similar argument can give masses to right handed sterile neutrinos butthese can never be measured However a lepton analogue of the CKM matrix the PNMSmatrix allows left-handed neutrinos to mix between various generations and oscillate betweenflavours as they propagate through space allowing a calculation of mass differences

    219 Quantum Chromodynamics

    The final plank in SM theory tackling the strong force Quantum Chromodynamics wasproposed by Gell-Mann and Zweig in 1963 Three generations of quarks were proposedwith charges of 23 for uc and t and -13 for ds and b Initially SU(2) isospin was consideredtransforming u and d states but later irreducible representations of SU(3) were found to fit theexperimental data The need for an additional quantum number was motivated by the fact thatthe lightest excited state of the nucleon consisted of sss (the ∆++) To reconcile the baryon

    21 Standard Model 23

    spectrum with spin statistics colour was proposed Quarks transform under the fundamentalor 3 representation of colour symmetry and anti-quarks under 3 The inner product of 3and 3 is an invariant under SU(3) as is a totally anti-symmetric combination ε i jkqiq jqk (andthe anti-quark equivalent) It was discovered though deep inelastic scattering that quarksshowed asymptotic freedom (the binding force grew as the seperation grew) Non-abeliangauge theories showed this same property and colour was identified with the gauge quantumnumbers of the quarks The quanta of the gauge field are called gluons Because the forcesshow asymptotic freedom - ie the interaction becomes weaker at high energies and shorterdistances perturbation theory can be used (ie Feynman diagrams) at these energies but forlonger distances and lower energies lattice simulations must be done

    2110 Full SM Lagrangian

    The full SM can be written

    L =minus14

    BmicroνBmicroν minus 18

    tr(FmicroνFmicroν)minus 12

    tr(GmicroνGmicroν)

    + sumgenerations

    (ν eL)σmicro iDmicro

    (νL

    eL

    )+ eRσ

    micro iDmicroeR + νRσmicro iDmicroνR +hc

    + sumgenerations

    (u dL)σmicro iDmicro

    (uL

    dL

    )+ uRσ

    micro iDmicrouR + dRσmicro iDmicrodR +hc

    minussumi j[(λ

    i jl Li

    Lφ l jR)+λ

    i jν ε

    ab(LiL)aφ

    lowastb ν

    jR +hc]

    minussumi j[(λ

    i jd Qi

    Lφd jR)+λ

    i ju ε

    ab(QiL)aφ

    lowastb u j

    R +hc]

    + (Dmicroφ)dagger(Dmicroφ)minusmh[φφ minus v22]22v2

    (240)

    where σ micro are the extended Pauli matrices

    (1 00 1

    )

    (0 11 0

    )

    (0 minusii 0

    )

    (1 00 minus1

    )

    The first line gives the gauge field strength terms the second the lepton dynamical termsthe third the quark dynamical terms the fourth the lepton (including neutrino) mass termsthe fifth the quark mass terms and the sixth the Higgs terms

    The SU(3) gauge field is given by Gmicroν and which provides the colour quantum numbers

    21 Standard Model 24

    Figure 24 Standard Model Particles and Forces [50]

    Note The covariant derivatives operating on the left-handed leptons contain only theU(1) and SU(2) gauge fields Bmicro Fmicro whereas those operating on left-handed quarks containU(1)SU(2) and SU(3) gauge fields Bmicro Fmicro Gmicro The covariant derivatives operating onright handed electrons and quarks are as above but do not contain the SU(2) gauge fieldsRight handed neutrinos have no gauge fields

    The sums over i j above are over the different generations of leptons and quarks

    The particles and forces that emerge from the SM are shown in Fig 24

    22 Dark Matter 25

    22 Dark Matter

    221 Evidence for the existence of dark matter

    2211 Bullet Cluster of galaxies

    Most of the mass of the cluster as measured by a gravitational lensing map is concentratedaway from the centre of the object The visible matter (as measured by X-ray emission) ismostly present in the centre of the object The interpretation is that the two galaxy clustershave collided perpendicular to the line of sight with the mostly collisionless dark matterpassing straight through and the visible matter interacting and being trapped in the middleIt was estimated that the ratio of actual to visible matter was about 16 [51]

    Figure 25 Bullet Cluster [52]

    2212 Coma Cluster

    The first evidence for dark matter came in the 1930s when Fritz Zwicky observed the Comacluster of galaxies over 1000 galaxies gravitationally bound to one another He observed thevelocities of individual galaxies in the cluster and estimated the amount of matter based onthe average galaxy He calculated that many of the galaxies should be able to escape fromthe cluster unless the mass was about 10 times higher than estimated

    22 Dark Matter 26

    2213 Rotation Curves [53]

    Since those early observations many spiral galaxies have been observed with the rotationcurves calculated by doppler shift with the result that in all cases the outermost stars arerotating at rates fast enough to escape their host galaxies Indeed the rotation curves arealmost flat indicating large amounts of dark matter rotating with the galaxies and holdingthe galaxies together

    Figure 26 Galaxy Rotation Curves [54]

    2214 WIMPS MACHOS

    The question is if the particles that make up most of the mass of the universe are not baryonswhat are they It is clear that the matter does not interact significantly with radiation so mustbe electrically neutral It has also not lost much of its kinetic energy sufficiently to relax intothe disks of galaxies like ordinary matter Massive particles may survive to the present ifthey carry a conserved additive or multiplicative quantum number This is so even if equalnumbers of particles and anti-particles can annihilate if the number densities become too lowto reduce the density further These particles have been dubbed Weakly interacting MassiveParticles (WIMPS) as they interact with ordinary matter only through a coupling strengthtypically encountered in the weak interaction They are predicted to have frozen out ofthe hot plasma and would give the correct relic density based solely on the weak interactioncross section (sometimes called the WIMP Miracle) Another possibility called MassiveCompact Halo Objects (MACHOS)- are brown dwarf stars Jupiter mass objects or remnantsfrom an earlier generation of star formation

    22 Dark Matter 27

    2215 MACHO Collaboration [55]

    In order to estimate the mass of MACHOS a number of experiments have been conductedusing microlensing which makes use of the fact that if a massive object lies in the line ofsight of a much more distant star the light from the more distant star will be lensed and forma ring around the star It is extemely unlikely that the MACHO will pass directly through theline of sight but if there is a near miss the two images will be seperated by a small angleThe signature for microlensing is a time-symmetric brightening of a star as the MACHOpasses near the line of sight called a light curve One can obtain statistical information aboutthe MACHO masses from the shape and duration of the light curves and a large number ofstars (many million) must be monitored The MACHO Collaboration [55] have monitoredover 11 million stars and less than 20 events were detected The conclusion was that thiscould account for a small addition (perhaps up to 20 ) to the visible mass but not enoughto account for Dark Matter

    2216 Big Bang Nucleosynthesis (BBN) [56]

    Another source of evidence for DM comes from the theory of Nucleosynthesis in the fewseconds after the Big Bang In the first few minutes after the big bang the temperatures wereso high that all matter was dissociated but after about 3 minutes the temperature rapidlycooled to about 109 Kelvin Nucleosynthesis could begin when protons and neutrons couldstick together to form Deuterium (Heavy Hydrogen) and isotopes of Helium The ratio ofDeuterium to Hydrogen and Helium is very much a function of the baryon density and alsothe rate of expansion of the universe at the time of Nucleosynthesis There is a small windowwhen the density of Baryons is sufficient to account for the ratios of the various elementsThe rate of expansion of the universe is also a function of the total matter density The ratiosof these lighter elements have been calculated by spectral measurements with Helium some25 of the mass of the universe Deuterium 001 The theory of BBN accounts for theratios of all of the lighter elements if the ratio of Baryon mass to total mass is a few percentsuggesting that the missing matter is some exotic non-baryonic form

    22 Dark Matter 28

    2217 Cosmic Microwave Background [57]

    The discovery of a uniform Microwave Background radiation by Penzias and Wilson in1965 was confirmation of the theory first proposed by Gamow and collaborators that theuniverse should be filled with black body radiation an afterglow of the big bang TheCOBE satelite launched in 1989 confirmed this black body radiation with temperature of2725+-002K with 95 confidence While extremely uniform it found tiny irregularitieswhich could be accounted for by irregularities in the density of the plasma at last scattering(gravitational redshifts or blueshifts called the Sachs Wolfe effect) and temperature andvelocity fluctuations in the plasma The anisotropy of the background has been furtheranalysed by WMAP (the Wilkinson Microwave Anisotropy Probe) launched in 2001 Thefluctuations in the density of the plasma are what account for the seeds of galaxy formationHowever detailed calculations have found that the fluctuations were too small to account forthe observed galaxy distribution without an electrically neutral gravitating matter in additionto ordinary matter that only became electrically neutral at the time of recombination [58]

    In May 2009 the European Space Agency (ESA) launched the Planck sattelite a spacebased observatory to observe the universe at wavelengths between 03mm and 111 mmbroadly covering the far-infrared microwave and high frequency domains The missionrsquosmain goal was to study the cosmic microwave background across the whole sky at greatersensitivity and resolution than ever before Using the information gathered and the standardmodel of cosmology ESA was able to calculate cosmological parameters including the relicdensity of dark matter remaining after the Big Bang to a high degree of precision withinthe framework of the standard model of cosmology The standard model of cosmology canbe described by a relatively small number of parameters including the density of ordinarymatter dark matter and dark energy the speed of cosmic expansion at the present epoch (alsoknown as the Hubble constant) the geometry of the Universe and the relative amount ofthe primordial fluctuations embedded during inflation on different scales and their amplitude[59]

    2218 LUX Experiment - Large Underground Xenon experiment [16]

    The LUX experiment is conducted at the Sanford Underground research facility one mileunderground to minimise the background of cosmic rays The detector is a time projectionchamber allowing a 3D positioning of interactions occurring within its active volume LUX

    22 Dark Matter 29

    Figure 27 WMAP Cosmic Microwave Background Fluctuations [58]

    Figure 28 Dark Matter Interactions [60]

    uses as its target 368 kiliograms of liquefied ultra pure xenon which is a scintillator Interac-tions inside the xenon will create an amount of light proportional to the energy depositedThat light can be collected on arrays of light detectors sensitive to a single photon

    Xenon is very pure and the amount of radiation originating within the target is limitedAlso being three times denser than water it can stop radiation originating from outside thedetector The 3D detector focuses on interactions near the center which is a quiet regionwhere rare dark matter interactions might be detected [16]

    22 Dark Matter 30

    222 Searches for dark matter

    2221 Dark Matter Detection

    Fig 28 illustrates the reactions studied under the different forms of experiment designed todetect DM

    Direct detection experiments are mainly underground experiments with huge detectorsattempting to detect the recoil of dark matter off ordinary matter such as LUX [16] CDMS-II [61] XENON100 [11] COUPP [62] PICASSO [63] ZEPLIN [10] These involveinteractions between DM and SM particles with final states involving DM and SM particles

    Indirect detection involves the search for the annihilation of DM particles in distantastrophysical sources where the DM content is expected to be large Example facilitiesinclude space based experiments (EGRET [64] PAMELA [18] FERMI-LAT [65]) balloonflights (ATIC [66] PEBS [67] PPB-BETS [68]) or atmospheric Cherenkov telescopes(HESS [69]MAGIC [70]) and neutrino telescopes (SuperKamiokande [71] IceCube [72]ANTARES [73] KM3NET [74]) Indirect Detection involves DM particles in the initial stateand SM particles in the final state

    Collider detection involves SM particles in the initial state creating DM particles in thefinal state which can be detected by missing transverse energy and explained by variousmodels extending the SM

    223 Possible signals of dark matter

    224 Gamma Ray Excess at the Centre of the Galaxy [65]

    The center of the galaxy is the most dense and active region and it is no surprise that itproduces intense gamma ray sources When astronomers subtract all known gamma raysources there remains an excess which extends out some 5000 light years from the centerHooper et al find that the annihilation of dark matter particles with a mass between 31 and40 Gev fits the excess to a remarkable degree based on Gamma Ray spectrum distributionand brightness

    23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 31

    Figure 29 Gamma Ray Excess from the Milky Way Center [75]

    23 Background on ATLAS and CMS Experiments at theLarge Hadron collider (LHC)

    The collider constraints in this paper are based on two of the experiments that are part of theLHC in CERN

    Figure 210 ATLAS Experiment

    The LHC accelerates proton beams in opposite directions around a 27 km ring of super-conducting magnets until they are made to collide at an interaction point inside one of the 4experiments placed in the ring An upgrade to the LHC has just been performed providing65 TeV per beam

    23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 32

    231 ATLAS Experiment

    The ATLAS detector consists of concentric cylinders around the interaction point whichmeasure different aspects of the particles that are flung out of the collision point There are 4major parts the Inner Detector the Calorimeters the Muon spectrometer and the magnetsystems

    2311 Inner Detector

    The inner detector extends from a few centimeters to 12 meters from the interaction point andmeasures details of the charged particles and their momentum interacting with the materialIf particles seem to originate from a point other than the interaction point this may indicatethat the particle comes from the decay of a hadron with a bottom quark This is used to b-tagcertain particles

    The inner detector consists of a pixel detector with three concentric layers and threeendcaps containing detecting silicon modules 250 microm thick The pixel detector has over 80million readout channels about 50 percent of the total channels for the experiment All thecomponents are radiation hardened to cope with the extremely high radiation in the proximityof the interaction point

    The next layer is the semi-conductor layer (SCT) similar in concept to the pixel detectorbut with long narrow strips instead of pixels to cover a larger area It has 4 double layers ofsilicon strips with 63 million readout channels

    The outermost component of the inner detector is the Transition Radiation Tracker It is acombination of a straw tracker and a transition radiation detector The detecting elementsare drift tubes (straws) each 4mm in diameter and up to 144 cm long Each straw is filledwith gas which becomes ionised when the radiation passes through The straws are held at-1500V with a thin wire down the center This drives negatively ionised atoms to the centercausing a pulse of electricity in the wire This causes a pattern of hit straws allowing the pathof the particles to be determined Between the straws materials with widely varying indicesof refraction cause transition radiation when particles pass through leaving much strongersignals in some straws Transition radiation is strongest for highly relativistic particles (thelightest particles- electrons and positrons) allowing strong signals to be associated with theseparticles

    23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 33

    2312 Calorimeters

    The inner detector is surrounded by a solenoidal magnet of 2 Tesla (there is also an outertoroidal magnet) but surrounding the inner magnet are the calorimeters to measure theenergy of particles by absorbing them There is an inner electromagnetic calorimeter whichmeasures the angle between the detectorrsquos beam axis and the particle (or more preciselypseudo-rapidity) and its angle in the perpendicular plane to a high degree of accuracy Theabsorbing material is lead and stainless steel and the sampling material liquid argon Thehadron calorimeter measures the energy of particles that interact via the strong force Theenergy absorbing material is scintillating steel

    2313 Muon Specrometer

    The muon spectrometer surrounds the calorimeters and has a set of 1200 chambers tomeasure the tracks of outgoing muons with high precision using a magnetic field andtriggering chambers Very few particles of other types pass through the calorimeters and aresubsequently measured

    2314 Magnets

    The magnets are in two layers the inner solenoidal and outer toroidal which together causecharged particles to curve in the magetic field allowing their momentum to be determinedThe outer toroidal magnet consists of 8 very large air core superconducting magnets whichsupply the field in the muon spectrometer

    232 CMS Experiment

    The CMS Detector is similar in design to the ATLAS Detector with a single solenoidalmagnet which is the largest of its type ever built (length 13m and diameter 7m) and allowsthe tracker and calorimeter detectors to be placed inside the coil in a compact way Thestrong magnetic field gives a high resolution for measuring muons electrons and photons ahigh quality central tracking system and a hermetic hadron calorimeter designed to preventhadrons from escaping

    23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 34

    Figure 211 CMS Experiment

    Chapter 3

    Fitting Models to the Observables

    31 Simplified Models Considered

    In order to extend the findings of previous studies [19] to include collider constraints wefirst performed Bayesian scans [7] over three simplified models real scalar dark matter φ Majorana fermion χ and real vector Xmicro The purpose was to obtain the most probable regionsof each model then to use the resulting posterior samples to investigate whether the LHCexperiments have any sensitivity to the preferred regions

    The three models couple to the mediator with interactions shown in the following table

    Table 31 Simplified Models

    Hypothesis real scalar DM Majorana fermion DM real vector DM

    DM mediator int Lφ sup microφ mφ φ 2S Lχ sup iλχ

    2 χγ5χS LX sup microX mX2 X microXmicroS

    The interactions between the mediator and the standard fermions is assumed to be

    LS sup f f S (31)

    and in line with minimal flavour violation we only consider third generation fermions- ief = b tτ

    For the purposes of these scans we consider the following observables

    32 Observables 36

    32 Observables

    321 Dark Matter Abundance

    We have assumed a single dark matter candidate which freezes out in the early universe witha central value as determined by Planck [21] of

    ΩDMh2 = 01199plusmn 0031 (32)

    h is the reduced hubble constant

    The experimental uncertainty of the background relic density [59] is dominated by atheoretical uncertainty of 50 of the value of the scanned Ωh2 [76] We have added thetheoretical and the experimental uncertainties in quadrature

    SD =radic(05Ωh2)2 + 00312 (33)

    This gives a log likelihood of

    minus05lowast (Ωh2 minus 1199)2

    SD2 minus log(radic

    2πSD) (34)

    322 Gamma Rays from the Galactic Center

    Assuming that the gamma ray flux excess from the galactic centre observed by Fermi-LAT iscaused by self annihilation of dark matter particles the differential flux

    d2Φ

    dEdΩ=

    lt σv gt8πmχ

    2 J(ψ)sumf

    B fdN f

    γ

    dE(35)

    has been calculated using both Micromegas v425 with a Navarro-Frenk-White (NFW) darkmatter galactic distribution function

    ρ(r) = ρ0(rrs)

    minusγ

    (1+ rrs)3minusγ (36)

    with γ = 126 and an angle of 5 to the galactic centre [19]

    32 Observables 37

    Here Φ is the flux dE is the differential energy lt σv gt the velocity averaged annihi-lation cross section of the dark matter across the galactic centre mχ the mass of the darkmatter particle B f =lt σv gt f σv is the annihilation fraction into the f f final state anddN f

    γ dE is the energy distribution of photons produced in the annihilation channel with thefinal state f f The J factor is described below

    The angle ψ to the galactic centre determines the normalisation of the gamma ray fluxthrough the factor

    J(ψ) =int

    losρ(l2 + r⊙2 minus2lr⊙cosψ)dl (37)

    where ρ(r) is the dark matter density at distance r from the centre of the galaxy r⊙=85 kpcis the solar distance from the Galactic center ρ0 is set to reproduce local dark matter densityof 03GeVcm3

    The central values for the gamma ray spectrum were taken from Fig 5 of [77] and thestandard deviations of individual flux points from the error bars on the graph In commonwith [78] we ignore possible segment to segment correlations

    For the Fermi-LAT log likelihood we assumed each point was distributed with indepen-dent Gaussian likelihood with the experimental error for each data point given by the errorbar presented in Fig5 of [76] for each point We also assumed an equal theoretical errorfor each point which when added to the experimental error in quadrature gave an effectivelog likelihood for each point of minus05lowast (giminusdi)

    2

    2lowastσ2i

    where gi are the calculated values and di theexperimental values and σi the experimental errors

    323 Direct Detection - LUX

    The LUX experiment [16] is the most constraining direct search experiment to date for awide range of WIMP models We have used LUXCalc [79] (modified to handle momentumdependent nucleon cross sections) to calculate the likelihood of the expected number ofsignal events for the Majorana fermion model

    The likelihood function is taken as the Poisson distribution

    L(mχ σ | N) = P(N | mχ σ) =(b+micro)Neminus(b+micro)

    N (38)

    32 Observables 38

    where b= expected background of 064 and micro = micro(mχσ ) is the expected number of signalevents calculated for a given model

    micro = MTint

    infin

    0dEφ(E)

    dRdE

    (E) (39)

    where MT is the detector mass times time exposure and φ(E) is a global efficiency factorincorporating trigger efficiencies energy resolution and analysis cuts

    The differential recoil rate of dark matter on nucleii as a function of recoil energy E

    dRdE

    =ρX

    mχmA

    intdvv f (v)

    dσASI

    dER (310)

    where mA is the nucleon mass f (v) is the dark matter velocity distribution and

    dσSIA

    dER= Gχ(q2)

    4micro2A

    Emaxπ[Z f χ

    p +(AminusZ) f χn ]

    2F2A (q) (311)

    where Emax = 2micro2Av2mA Gχ(q2) = q2

    4m2χ

    [24] is an adjustment for a momentum dependantcross section which applies only to the Majorana model

    f χ

    N =λχ

    2m2SgSNN assuming that the relic density is the central value of 1199 We have

    implemented a scaling of the direct search cross section in our code by multiplying gSNN by( Ω

    Ω0)05 as a convenient method of allowing for the fact that the actual relic density enters

    into the calculation of the cross section as a square

    FA(q) is the nucleus form factor and

    microA =mχmA

    (mχ +mA)(312)

    is the reduced WIMP-nucleon mass

    The strength of the dark matter nucleon interaction between dark matter particles andthird generation quarks is

    gSNN =2

    27mN fT G sum

    f=bt

    λ f

    m f (313)

    where fT G = 1minus f NTuminus f N

    Tdminus fTs and f N

    Tu= 02 f N

    Td= 026 fTs = 043 [20]

    33 Calculations 39

    For the real scalar and real vector models where the spin independent cross sections arenot momentum dependent we based the likelihood on the LUX 95 CL from [16] using acomplementary error function Li(d|mχ I) = 05lowastCEr f (dminusx(mχ )

    σ) where x is the LUX limit

    and d the calculated SI cross section We assumed a LUX error σ of 01lowastd lowastradic

    2

    33 Calculations

    We coded the Lagrangians for the three models in Feynrules [4] Calculation of the expectedobservables (dark matter relic density Fermi-LAT gamma ray spectrum and LUX nucleonscattering amplitudes) were performed using Micromegas v425

    331 Mediator Decay

    A number of searches have previously been performed at the LHC which show sensitivity tothe decay of scalar mediators in the form of Higgs bosons (in beyond the standard modeltheories) to bottom quarks top quarks and τ leptons (all third generation fermions) Two inparticular [43 44] provide limits on the cross-sections for the production of the scalar timesthe branching ratios to brsquos and τrsquos In order to check these limits against each of the modelswe ran parameter scans in Madgraph at leading order over the top 900 points from the nestedsampling output from MultiNest [19]

    The two processes were

    1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to apair of τ leptons in pp collisions [44]

    bull generate p p gt b b S where S is the scalar mediator

    The process measures the decay of a scalar in the presence of b-jets that decay to τ+τminus

    leptons Fig31 shows the main diagrams contributing to the process The leftmost diagramis a loop process and the centre and right are tree level however we only got Madgraph togenerated the middle diagram since we are interested in τ leptons and b quarks in the finalstate

    The limit given in [44] only applies in the case of narrow resonances where narrow istaken to mean that the width of the resonance is less than 10 of its mass To test the validity

    33 Calculations 40

    Figure 31 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof τ leptons

    of applying these limits to our models we show in Figures 32 to 34 the percentage widthof the resonance vs mS[GeV ] (pole mass of the scalar mediator) λb and λt (the couplingconstants to the b and t quarks) The probabilities are the posterior probabilities obtainedfrom MultiNest scans of the Majorana model varying the coupling constants and scalarmediator and calculating the decay width of the mediator The results are represented as 2dposterior probability heat maps in the central panels and 1d marginalised distributions on thex and y axes of each graph

    16 18 20 22 24 26 28 30

    log10(mS[GeV])

    001

    002

    003

    004

    005

    Widthm

    S

    00

    04

    08

    12

    0 100 200

    Posterior Probability

    Figure 32 WidthmS vs mS

    The WidthmS was lt 01 at almost all points except where the mass of the scalar wasgreater than 1000 GeV λb was greater than 1 or λτ was greater than 1

    This can be seen from the graphs in Figs 323334

    33 Calculations 41

    4 3 2 1 0

    λb

    001

    002

    003

    004

    005

    WidthmS

    000

    015

    030

    045

    0 100 200

    Posterior Probability

    Figure 33 WidthmS vs λb

    5 4 3 2 1 0

    λτ

    001

    002

    003

    004

    005

    WidthmS

    000

    015

    030

    045

    0 100 200

    Posterior Probability

    Figure 34 WidthmS vs λτ

    The decay widths are calculated in Madgraph at leading order and branching fractionscan be calculated to b t τ and χ (DM) The formulae for the calculations can be found indecayspy which is a subroutine of Madgraph which we have documented in Appendix BB1 Using these we calculated the branching ratio to τ+τminus so that the cross section timesbranching ratio can be compared to the limits in [80] shown in Chapters 56 and 7 in Figs5564 and 74 for each model

    2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

    This scan calculates the contributions of b-quark associated Higgs boson production inthe center and right diagrams of Fig 35 reproduced from [43] The scans did not includePythia and Delphes as only production cross sections were required and the producedparticles were in the final state and required no matching or merging

    The Madgraph processes were

    bull generate p p gt b S where S is the scalar mediator

    bull add process p p gt b S j

    bull add process p p gt b S

    33 Calculations 42

    Figure 35 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof b quarks in the presence of at least one b quark

    bull add process p p gt b S j

    The cross section times branching ratios to bb for mediator production in association withat least one b quark is shown for each model in Chapters 56 and 7 in Figs 5765 and 75

    332 Collider Cuts Analyses

    We generated two parameter scans with parameter combinations from the Multinest outputwhich were the basis of the plots of Fig 6 in [19] The plots in Fig 6 were marginalisedto two parameters each but the raw Multinest output contained 5 parameters plus the priorprobabilities of each combination normalised to total 10

    The first scan was for χ produced with b quarks In order to force the mediator to decayto χ only we had χ in the final state and defined jets to include b quarks in Madgraph Thenwe generated the processes

    bull generate p p gt χ χ j

    bull add process p p gt χ χ j j

    Jet matching was on

    The second scan was for t quarks produced in the final state

    bull generate p p gt χ χ tt

    33 Calculations 43

    No jet matching was required because we did not mix final state jet multiplicities at thematrix element level

    The outputs from these two processes were normalised to 21 f bminus1 and combined

    The results of these scans prior to collider cuts calculations are shown in Figs 5161and 71 These show the posterior probabilities of the various parameter combinations withdifferent constraints applied seperately and then together A more detailed commentary isgiven in each of the sections

    We then sampled 900 points based on posterior probability from these scans and ran thesethrough Madgraph v425 then applied collider cuts with the C++ code using the limits fromthe following six ATLAS and CMS searches shown in Table32 and described below to findany exclusions

    333 Description of Collider Cuts Analyses

    In the following all masses and energies are in GeV and angles in radians unless specificallystated

    3331 Lepstop0

    Search for direct pair production of the top squark in all-hadronic final states in pro-tonndashproton collisions at

    radics=8 TeV with the ATLAS detector[32]

    This analysis searched for direct pair production of the supersymmetric partner to thetop quark using integrated luminosity of 201 f bminus1 The top squark was assumed to de-cay via t rarr t χ0

    1 or t rarr bχ01 or t rarr bχ

    plusmn1 rarr bW (lowast)χ1

    0 where χ01 (χ

    plusmn1 ) denotes the lightest

    neutralino(chargino) in supersymmetric models The search targets a fully hadronic finalstate with four or more jets and large missing momentum The production of a mediator inassociation with top quarks followed by an invisible mediator decay to dark matter particleshas a similar final state to that considered in the ATLAS stop searches Thus we expect theATLAS reach to have sensitivity to the three dark matter models considered in this thesis

    The collider cuts program mimicked the object reconstruction through jetlepton overlapremoval and selected signal electrons muons and τ particles with pT gt 10 GeV and

    33 Calculations 44

    Table 32 95 CL by Signal Region

    Experiment Region Number

    Lepstop0

    SRA1 66SRA2 57SRA3 67SRA4 65SRC1 157SRC2 124SRC3 80

    Lepstop1

    SRtN1-1 857SRtN1-2 498SRtN1-3 389SRtN2 107SRtN3 85SRbC1 832SRbC2 195SRbC3 76

    Lepstop2

    L90 740L100 56L110 90L120 170

    2bstop

    SRA 857SRB 270SRA150 380SRA200 260SRA250 90SRA300 75SRA350 52

    CMSTopDM1L SRA 1385

    ATLASMonobjetSR1 1240SR2 790

    33 Calculations 45

    |η | lt 24724 and 247 respectively (for eminusmicrominusτ) Jets were selected with pT gt 20 GeVand |η |lt 45 b-jets were selected if they were tagged by random selection lt 70 and had|η |lt 25 and pT gt 20 GeV

    These jets and leptons were further managed for overlap removal within ∆R lt 04 withnon b-jets removed if ∆R lt 02

    The b-jets and non b-jets were further selected as signal jets if pT gt 35 GeV and |η |lt 28

    These signal jets electrons and muons were finally binned according to the signal regionsin Tables 333435 below

    Table 33 Selection criteria common to all signal regions

    Trigger EmissT

    Nlep 0b-tagged jets ⩾ 2

    EmissT 150 GeV

    |∆φ( jet pmissT )| gtπ5

    mbminT gt175 GeV

    Table 34 Selection criteria for signal regions A

    SRA1 SRA2 SRA3 SRA4anti-kt R=04 jets ⩾ 6 pT gt 808035353535 GeV

    m0b j j lt 225 GeV [50250] GeV

    m1b j j lt 225 GeV [50400] GeV

    min( jet i pmissT ) - gt50 GeV

    τ veto yesEmiss

    T gt150 GeV gt250 GeV gt300 GeV gt 350 GeV

    Table 35 Selection criteria for signal regions C

    SRC1 SRC2 SRC3anti-kt R=04 jets ⩾ 5 pT gt 8080353535 GeV

    |∆φ(bb)| gt02 π

    mbminT gt185 GeV gt200 GeV gt200 GeV

    mbmaxT gt205 GeV gt290 GeV gt325 GeV

    τ veto yesEmiss

    T gt160 GeV gt160 GeV gt215 GeV

    wherembmin

    T =radic

    2pbt Emiss

    T [1minus cos∆φ(pbT pmiss

    T )]gt 175 (314)

    33 Calculations 46

    and pbT is calculated for the b-tagged jet with the smallest ∆φ to the pmiss

    T direction andmbmax

    T is calculated with the pbT for the b-tagged jet with the largest ∆φ to the pmiss

    T direction

    m0b j j is obtained by first selecting the two jets with the highest b-tagging weight- from

    the remaining jets the two jets closest in the η minusφ plane are combined to form a W bosoncandidate this candidate is then combined with the closest of the 2 b-tagged jets in the η minusφ

    plane to form the first top candidate with mass m0b j j A second W boson candidate is formed

    by repeating the procedure with the remaining jets this candidate is combined with thesecond of the selected b-tagged jets to form the second top candidate with mass m1

    b j j Thecuts shown in Appendix1 Table A1 show the cumulative effect of the individual cuts appliedto the data the signal region cuts and the cumulative effects of individual cuts in SignalRegion C

    3332 Lepstop1

    Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

    radics=8 TeV pp collisions using 21 f bminus1 of

    ATLAS data[33]

    The stop decay modes considered were those to a top quark and the lightest supersym-metric particle (LSP) as well as a bottom quark and the lightest chargino where the charginodecays to the LSP by emitting a W boson We should expect the three dark matter models inthis paper with couplings to t and b quarks to show some visibility in this search

    The collider cuts program mimicked the object construction by selecting electrons andmuons with pT gt 10 GeV and |η |lt 247 and 24 respectively and jets with pT gt 20 GeVand |η |lt 10 b-jets with |η |lt 25 and pT gt 25 GeV were selected with random efficiency75 and mistag rate 2 (where non b-jets were mis-classified) We then removed any jetwithin ∆R = 02 of an electron and non overlapping jets with |η |lt 28 were used to removeany electrons and muons within ∆R = 04 of these while non-overlapping jets with |η |lt 25and pT gt 25 GeV were classified as signal jets Non overlapping electrons and muons withpT gt 25 GeV were classified as signal electrons and muons

    Signal selection defined six regions to optimize sensitivity for different stop and LSPmasses as well as sensitivity to two considered stop decay scenarios Three signal regionslabelled SRbC 1-3 where bC is a moniker for b+Chargino and 3 regions labelled SRtN 1-3where tN is a moniker for t+Neutralino were optimised for the scenarios where t1 decaysto the respective particles indicated by the labels Increasing label numbers correspond toincreasingly strict selection criteria

    33 Calculations 47

    The dominant background in all signal regions arises from dileptonic tt events All threeSRtN selectons impose a requirement on the 3-jet mass m j j j of the hadronically decayingtop quark to specifically reject the tt background where both W bosons from the top quarkdecay leponically

    For increasing stop mass and increasing difference between the stop and the LSP therequirents are tightened on Emiss

    T on the ratio EmissT

    radicHT where HT is the scalar sum of the

    momenta of the four selected jets and also tightened on mT

    To further reduce the dileptonic tt background for an event characterised by two one stepdecay chains a and b each producing a missing particle C use is made of the variable

    mT 2 =min

    pCTa + pC

    T b = pmissT

    [max(mTamtb)] (315)

    where mTi is the transverse mass 1 of branch i for a given hypothetical allocation (pCTa pC

    T b)

    of the missing particle momenta Eqn 315 is extended for more general decay chains byamT 2 [81] which is an asymmetric form of mT 2 in which the missing particle is the W bosonin the branch for the lost lepton and the neutrino is the missing particle for the branch withthe observed charged lepton

    ∆φ( jet12 pmissT ) the azimuthal angle between leading or subleading jets and missing mo-

    mentum is used to suppress the background from mostly multijet events with mis-measuredEmiss

    T

    Events with four or more jets are selected where the jets must satisfy |η | lt 25 and pT gt80604025 GeV respectively

    These signal particles were then classified according to the cutflows analysis in Fig36below and compared with the paper in Appendix1 Table A2

    3333 Lepstop2

    Search for direct top squark pair production in final states with two leptons in p pcollisions at

    radics=8TeV with the ATLAS detector[34]

    Three different analysis strategies are used to search for t1 pair production Two targett1 rarr b+ χ

    plusmn1 decay and the three body t1 rarr bW χ0

    1 decay via an off-shell top quark whilst

    1The transverse mass is defined as m2T = 2plep

    T EmissT (1minus cos(∆φ)) of branch i where ∆φ is the azimuthal

    angle between the lepton and the missing transverse momentum

    33 Calculations 48

    Table 36 Signal Regions - Lepstop1

    Requirement SRtN2 SRtN3 SRbC1 SRbC2 SRbC3∆φ( jet1 pmiss

    t )gt - 08 08 08 08∆φ( jet2 pmiss

    T )gt 08 08 08 08 08Emiss

    T [GeV ]gt 200 275 150 160 160Emiss

    T radic

    HT [GeV12 ]gt 13 11 7 8 8

    mT [GeV ]gt 140 200 120 120 120me f f [GeV ]gt - - - 550 700amT 2[GeV ]gt 170 175 - 175 200mT

    T 2[GeV ]gt - 80 - - -m j j j Yes Yes - - -

    one targets t1 rarr t + χ01 to an on shell quark decay mode The analysis is arranged to look

    at complementary mass splittings between χplusmn1 and χ0

    1 smaller and larger than the W bosonmass The smaller splitting provides sensitivity to t1 three body decay

    Object reconstruction was carried out by selecting baseline electronsmuons and τrsquos withpT gt 10 GeV and |η |lt 24724 and 247 respectively Baseline jets and b-jets were selectedwith pT gt 20 GeV and |η |lt 25 and b-jets with 70 efficiency Baseline jets were removedif within ∆R lt 02 of an electron and electrons and muons were removed if within ∆R lt 04of surviving jets Signal electrons required pT gt 10 GeV and signal jets pT gt 20 GeV and|η |lt 25

    The analysis made use of the variable mT 2 [82] the stransverse mass used to mea-sure the masses of pair-produced semi-invisibly decaying heavy particles It is definedas mT 2(pT1 pT2qT ) =

    minqT1+qT2=qT

    max[mT (pT1qT1)mT (pT2qT2)] where mT indicatesthe transverse mass pT1 and pT2 are vectors and qT = qT1 + qT2 The minimisation isperformed over all possible decompositions of qT

    Events were selected if they passed either a single-electron a single-muon a double-electron a double-muon or an electron-muon trigger Events are required to have exactlytwo opposite sign leptons and at least one of the leptons or muons must have a momentumgreater than 25 GeV and the invariant mass (rest mass) of the two leptons is required to begreater than 20 GeV In order to reduce the number of background events containing twoleptons produced by the on-shell decay of the Z boson the invariant mas must be outside711-111 GeV range

    Two additional selections are applied to reduce the number of background events withhigh mT 2 arising from events with large Emiss

    T due to mismeasured jets ∆φb lt 15 and∆φ gt 10 The quantity ∆φb is the azimuthal angle between pll

    T b = pmissT + pl1

    T +Pl2T The

    33 Calculations 49

    vector pllT b is the opposite of the vector sum of all the hadronic activity in the event For WW

    and tt backgrounds it measures the transverse boost of the WW system and for the signalthe transverse boost of the chargino-chargino system The ∆φ variable is the azimuthal anglebetween the pmiss

    T vector and the direction of the closest jet

    By requiring mT 2 gt 40 GeV the tt background is the dominant background in both thesame-flavoured (SF) and different-flavoured (DF) events

    Four signal regions are then defined as shown in Table 37 SR M90 has the loosestselection requiring MT 2gt 90 GeV and no additional requirement on jets and providessensitivity to scenarios with a small difference between the masses of the top-squark andchargino SR M110 and M120 have a loose selection on jets requiring two jets with pT

    gt 20 GeV and mT 2 to be greater than 110 GeV and 120 GeV respectively They providesensitivity to scenarios with small to moderate difference between the invariant masses of thestop quark and the charginos SR M100 has a tighter jet slection and provides sensitivity toscenarios with a large mass difference between the both stop quark and the chargino and thestop quark and the neutralino

    The analysis cut regions are summarised in Fig37 below

    Table 37 Signal Regions Lepstop2

    SR M90 M100 M110 M120pT leading lepton gt 25 GeV

    ∆φ(pmissT closest jet) gt10

    ∆φ(pmissT pll

    T b) lt15mT 2 gt90 GeV gt100 GeV gt110 GeV gt120 GeV

    pT leading jet no selection gt100 GeV gt20 GeV gt20 GeVpT second jet no selection gt50 GeV gt20 GeV gt20 GeV

    To validate the code we ran the cuts shown in Appendix 1 Table A3 with the givenMadgraph simulation and compared to [83] It was not clear what baseline cuts had beenemployed in the ATLAS paper so our results were normalised to the 2 SF electrons beforecomparison Some of the ATLAS cut numbers were less than 3 and we did not produce anyevents passing these cuts but clearly the margin of error at this level was greater than thenumber of ATLAS events

    33 Calculations 50

    3334 2bstop

    Search for direct third-generation squark pair production in final states with miss-ing transverse momentum and two b-jets in

    radics= 8 TeV pp collisions with the ATLAS

    detector[31]

    Two sets of scenarios are considered In the first the coloured sbottom is the onlyproduction process decaying via b1 rarr bχ0

    1 In the second the stop is the only productionprocess decaying via t1 rarr t χplusmn

    1 and the lightest chargino (χplusmn1 ) decays via a virtual W boson

    into the final state χ01 f f prime Both scenarios are characterised by the presence of two jets

    resulting from the hadronisation of b quarks and large missing transverse momentum Theproduction of a mediator in association with top quarks followed by an invisible mediatordecay to dark matter particles has a similar final state as that considered in the ATLAS stopsearches Thus we expect the ATLAS reach to have sensitivity to the three dark mattermodels considered in this thesis

    The collider cuts program mimicked object reconstruction by selecting baseline electronswith pT gt 7 GeV and |η |lt 247 and muons with pT gt 6 GeV and |η |lt 24 Baseline jetswere selected with pT gt 20 GeV and |η |lt 28 We then removed any jets within ∆R lt 02of an electron and any electron within ∆R lt 04 of surviving jets Signal electrons andmuons were required to have pT gt 25 GeV b-jets were selected with 60 tagging efficiencyand |η |lt 25 and pT gt 20 GeV

    Two sets of signal regions are defined to provide sensitivity to the kinematic topologiesassociated with the different mass splittings between the sbottom or the stop and the neu-tralino In all cases the presence of at least one primary vertex (with at least five tracks withpT gt 04 GeV) is required Events are selected with Emiss

    T gt 150 GeV and no electrons ormuons identified in the final state For signal region selections jets with |η | lt 28 are orderedaccording to their pT and 2 jets are required to be b-tagged

    The variables are defined as follows

    bull ∆φmin is the minimal azimuthal angle ∆φ between any of the three leading jets andpmiss

    T

    bull me f f (k) = sumki=1(p jet

    T )i +EmissT where the index refers to the pT ordered list of jets

    bull HT3 is defined as the scalar sum of the pT of the n leading pT ordered jets withoutincluding the 3 leading jets HT3 = sum

    ni=4(p jet

    T )i

    bull mbb is the invariant mass of the two b-tagged jets in the event

    33 Calculations 51

    bull mCT is the contransverse mass [82] and is a kinematical variable that can be usedto measure the masses of pair produced semi-invisibly decaying particles For twoidentical decays of heavy particles into two invisible particles v1 and v2 and twoinvisible particles mCT is defined as m2

    CT (v1v2) = [ET (v1)+ET (v2)]2 minus [pT (v1)minus

    pT (v2)]2 where ET =

    radicp2

    T +m2 The the contransverse mass is invariant under equaland opposite boosts of the parent particles in the transverse plane and is bounded fromabove by an analytical expression of the particle masses For production of sbottompairs the bound is given by mmax

    CT =m2(b)minusm2(χ0

    1 )

    m(b) and for tt events the bound is 135

    GeV A similar equation can be written for stop pairs in term of the stop and charginomasses

    A definition of the signal regions is given in the Table38

    Table 38 Signal Regions 2bstop

    Description SRA SRBEvent cleaning All signal regions

    Lepton veto No emicro after overlap removal with pT gt7(6) GeV for e(micro)Emiss

    T gt 150 GeV gt 250 GeVLeading jet pT ( j1) gt 130 GeV gt 150 GeVSecond jet pT ( j2) gt 50 GeV gt 30 GeVThird jet pT ( j3) veto if gt 50 GeV gt 30 GeV

    ∆φ(pmissT j1) - gt 25

    b-tagging leading 2 jets(pT gt 50 GeV |η | lt 25)

    2nd- and 3rd- leading jets(pT gt 30 GeV|η | lt 25)

    ∆φmin gt 04 gt 04Emiss

    T me f f (k) EmissT me f f (2) gt 025 Emiss

    T me f f (3) gt 04mCT gt 150200250300350 GeV -HT3 - lt 50 GeVmbb gt200 GeV -

    The analysis cuts are summarised in Table A4 of Appendix 1

    3335 ATLASMonobjet

    Search for dark matter in events with heavy quarks and missing transverse momentumin pp collisions with the ATLAS detector[30]

    Events were selected with high missing transverse momentum when produced in associa-tion with one or more jets identified as containing b quarks Final states with top quarks wereselected by requiring a high jet multiplicity and in some cases a single lepton The research

    33 Calculations 52

    studied effective contact operators in the form of a scalar operator Oscalar = sumqmqMN

    lowastqqχχ

    where MNlowast is the effective mass of the coupling and the quark and DM fields are denoted by

    q and χ respectively These are the equivalent of the simplified models in this paper with themediator integrated out

    Object reconstruction selected baseline electrons muons jets and τrsquos with pT gt 20 GeVand |η |lt 2525 and 45 respectively Signal jets were selected by removing any jets within∆R lt 02 of an electron and electrons and muons were selected by removing them if theywere within ∆R lt 04 of surviving jets b-jets were tagged with an efficiency of 60

    Only signal regions SR1 and SR2 were analysed

    The analysis split events into signal regions SR1 and SR2 according to the followingCriteria

    Table 39 Signal Region ATLASmonobjet

    Cut SR1 SR2Jet multiplicity n j 1minus2 3minus4

    bminusjet multiplicity nb gt 0 (60 eff) gt 0 (60 eff)Lepton multiplicity nl 0 0

    EmissT gt300 GeV gt200 GeV

    Jet kinematics pb1T gt100 GeV pb1

    T gt100 GeV p j2T gt100 (60) GeV

    ∆φ( j1EmissT ) gt 10 i = 12 gt 10 i = 14

    Where p jiT (pbi

    T ) represent the transverse momentum of the i-th jet b-tagged jet respec-tively If there is a second b-tagged jet the pT must be greater than 60 GeV

    3336 CMSTop1L

    Search for top-squark pair production in the single-lepton final state in pp collisionsat

    radics=8 TeV[41]

    This search was for pair production of top squarks in events with a single isolatedelectron or muon jets large missing transverse momentum and large transverse mass

    (MT =radic

    2EmissT pl

    T (1minus cos(∆φ))) where plT is the leptonrsquos transverse momentum and ∆(φ)

    is the difference between the azimuthal angles of the lepton and EmissT The 3 models

    considered should have some visibility at certain parameter combinations in this regionThe search focuses on two decay modes of the top squark leading to the two processespp rarr t tlowast rarr tt χ0

    1 χ01 rarr bbW+Wminusχ0

    1 χ01 and pp rarr t tlowast rarr bbχ

    +1 χ

    minus1 rarr bbW+Wminusχ0

    1 χ01 The

    33 Calculations 53

    lightest neutralino χ01 is considered the lightest supersymmetric particle and escapes without

    detection The analysis is based on events where one of the W bosons decays leptonicallyand the other hadronically resulting in one isolated lepton and 4 jets two of which originatefrom b quarks The two neutralinos and the neutrino from the W decay can result in largemissing transverse momentom The kinematical end point of MT is lt the mass of the Wboson However for signal events the presence of LSPs in the final state allows MT to exceedthe mass of the W boson The search is therefore for events with large MT The dominantbackground with large MT arises from tt decays where W bosons decay leptonically withone of the leptons not being identified

    To reduce the dominant tt background use was made of the MWT 2 variable defined as

    the minimum mother particle mass compatible with the transverse momenta and mass shellconstraints [84]

    Object reconstruction required electrons and muons to have pT gt 30 and 25 GeV respand |η |lt 25 and 21 respectively Baseline jets were required to have pT gt 30 and |η |lt 4and b-jets were tagged with an efficiency of 75 for |η |lt 24

    Events were chosen if they contained one isolated electron or muon at least 4 jets and atleast one b-tagged jet and Emiss

    T gt 100 GeV MT gt 160 GeV MWT 2 was required to be greater

    than 200 GeV and the smallest azimuthal angle difference between baseline jets and EmissT

    gt12

    Chapter 4

    Calculation Tools

    41 Summary

    Figure 41 Calculation Tools

    The simplified models were implemented in Feynrules [4] and the model files importedinto MicrOmegas v425 which was used to calculate the log-likelihoods of the astrophysicalobservables MicrOmegas was modified to allow only s-channel decays in both relic densityand gamma ray calculations Direct detection constraints were easy to implement for thescalar and vector models by coding the analytical expression for the spin independent WIMP-nucleon scattering cross section but due to the momentum dependence of the WIMP-nucleon

    42 FeynRules 55

    scattering cross section of the Majorana fermion model LUXCalc v101 [79] was modifiedfor this model to implement momentum dependent cross sections [85] We used MicrOmegasv425 to calculate relic density values and to produce gamma ray spectra for comparisonwith the Fermi-LAT results Parameter scans were carried out using MultiNest v310 withthe parameter ranges given in Table 51 and using astrophysical constraints only Oncethese scans had converged a random sample of 900 points was chosen (weighted by theposterior probability) and then used to calculate collider constraints using the MadgraphMonte Carlo generator This included comparison with mediator production and decay limitsfrom the LHC experiments and the reproduction of searches for dark matter plus other finalstate objects which required dedicated C++ code that reproduces the cut and count analysesperformed by CMS and ATLAS

    42 FeynRules

    FeynRules is a Mathematicareg package that allows the calculation of Feynman rules inmomentum space for any QFT physics model The user needs to provide FeynRules withminimal information to describe the new model contained in the model-file This informationis then used to calculate a set of Fenman rules associated with the Lagrangian The Feynmanrules calculated by the code can then be used to implement new physics models into otherexisting tools such as Monte Carlo (MC) generators via a set of interfaces developed andmaintained by the MC authors

    Feynrules can be run with a base model = SM (or MSSM) and user defined extensions tothe base model

    43 LUXCalc

    LUXCalc is a utility program for calculating and deriving WIMP-nucleon coupling limitsfrom the recent results of the LUX direct search dark matter experiment The program hasbeen used to derive limits on both spin-dependent and spin-independent WIMP-nucleoncouplings and it was found under standard astrophysical assumptions and common otherassumptions that LUX excludes the entire spin-dependent parameter space consistent witha dark matter interpretation of DAMArsquos anomalous signal and also found good agreementwith published LUX results [79]

    We modified the program for the Majorana fermion model to implement momentumdependent cross -sections [85]

    44 Multinest 56

    44 Multinest

    Multinest is an efficient Bayesian Inference Tool which uses a simultaneous ellipsoidalnested sampling method a Monte Carlo technique aimed at an efficient Bayesian Evidencecalculation with posterior inferences as a by product

    Bayes theorem states that

    Pr(θ |DH) =Pr(D|θ H)Pr(θ |H)

    Pr(D|H) (41)

    Here D is the data and H is the Hypothesis Pr(θ |DH) is the posterior probability ofthe parameters Pr(D|θ H) = L(θ) is the likelihood Pr(θ |H) is the prior and Pr(D|H) isthe evidence

    The evidence Pr(D|H) =int

    Pr(θ |DH)Pr(θ |H)d(θ) =int

    L(θ)Pr(θ |H)d(θ) Thus ifwe define the prior volume dX = Pr(θ |H)d(θ) and

    X(λ ) =int

    L(θ)gtλ

    Pr(θ |H)d(θ) (42)

    where the integral extends over the region of parameter space within the iso likelihoodcontour L(θ) = λ then the evidence integral can be written

    int 10 L (X)dX where L (X) the

    inverse of equation 42 and is a monotonically decreasing function of X (the prior volumewhich is 1 for the total sample space and 0 for a single point)

    Thus if one can evaluate likelihoods L (Xi) where 0 lt Xn lt lt X2 lt X1 lt X0 = 1the evidence can be estimated as sumLiwi where wi = 5(Ximinus1 minusXi+1) Ellipsoidal samplingapproximates the iso likelihood contour by a D dimensional covariance matrix of the currentset of active points New points are selected from the prior within this ellipsoidal bounduntil one is obtained that has a likelihood greater than the likelihood of the removed lowestlikelihood point This method is extended for multimodal sampling where there may bemore than one local maximum by clustering points that are well separated and constructingindividual ellipsoidal bounds for each cluster For more details consult [7]

    The results of the analysis are expressed as a data file containing the sample probability-2loglikelihood followed by the parameters of the model for the particular sampling Theprobabilities are normalised to total 10

    45 Madgraph 57

    45 Madgraph

    Madgraph is a general purpose matrix element calculator and Monte Carlo generator Basedon user-supplied Feynman rules (eg the output of Feynrules) the package calculates the tree-level matrix elements for any processes that the user specifies and generates samples of LHCevents including an accurate simulation of the proton collisions themselves (which requires aknowledge of the parton distribution functions that determine the relative proportion of eachparton inside the nucleus) Hadronisation of final state quarks is not possible in Madgraphand hence the code must be used in conjunction with Pythia which performs a parton shower

    The Madgraph output was automatically showered by Pythia (Pythia describes partonradiation as successive parton emissions using Markov chain techniques based on Sudakovform factors and this description is only formally correct for the soft and collinear emissions)A Madgraph simulation is at the level of the fundamental particles (quarks and leptons)Madgraph generates a limited set of matrix element processes at LO representing hard (highenergy) emissions and passes these to Pythia for showering The final state partons in theMadgraph simulation must be matched to the Pythia final state partons to avoid doublecounting

    The output from Pythia is automatically fed through Delphes where basic collider cutscan be performed (calculation of the numbers of particles and jets that would be observed atdifferent energies and locations within the collider) The output of the Delphes simulation is afile containing reconstructed events that can be analysed using the standard ROOT framework(the most popular data analysis package in particle physics) [86]

    The parameters that gave the best fits to the direct [16] and indirect DM experiments [87]and relic density [59] are ranked by posterior probability from the Multinest [7] output

    The parameters in the Madgraph simulations are stored in files in the cards sub directoryand in particular the run card param card and Delphes card record items relevent to thesimulation

    In the run card the number of events the Beam 1 and 2 total energy in GeV (4000 each)and the type of jet matching were modified (to MLM type matching where the final statepartons in Madgraph are clustered according to the KT jet algorithm to find the equivalentparton shower history of the event there is a minimum cutoff scale for the momentum pT

    given by the xqcut parameter) The maxjetflavour (=5 to include b quarks) and xqcut = 30GeV parameters were also modified

    46 Collider Cuts C++ Code 58

    The param card was modified to specify the masses of the new particles and the autodecay modes The Delphes card is overwritten with either the ATLAS or CMS defaultcards which record tracking efficiencies and isolation momentum and energy resolutionscalorimeter details jet finder and b and τ -tagging parameters However the isolation andtagging in this study were done by the collider cuts C++ code since multiple sets of cutswere applied to the same output from Pythia

    When performing simulations of the three dark matter models it was necessary to definethe proton and jets to contain b quarks as this is not the default in Madgraph The processesgenerated included 1 and 2 jets

    46 Collider Cuts C++ Code

    Because of the volume of simulations required for the collider constraints we used C++code (originally developed by Martin White and Sky French and modified by me) that readsthe ROOT files and outputs the numbers of events that would have been observed under thechosen collider experiments had the model been true for the given parameters Thus the sameoutput files can be used for all of the collider cuts and hence no cuts were performed throughthe Delphes cards

    In order to validate the C++ code we replicated the observed events with the papersdescribing the various collider searches (see Appendix)

    Chapter 5

    Majorana Model Results

    51 Bayesian Scans

    To build some intuition we first examined the constraining effect of each observable on itsown We ran separate scans for each constraint turned on and then for all the constraintstogether In Fig 51 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together The paramaters were scanned over the range in Table 51 below

    Table 51 Scanned Ranges

    Parameter mχ [GeV ] mS[GeV ] λt λb λτ

    Range 0minus103 0minus103 10minus5 minus10 10minus5 minus10 10minus5 minus10

    In Fig51 we can see that the preferred DM mass is centered around 35 GeV primarily dueto the influence of the Fermi-LAT gamma ray constraint whilst the relic density constraintallows a wide range of possible mediator and DM masses Direct detection (LUX) constraintssplit the mass regions into a top left and bottom right region and together these split theoverall mass parameter regions into two regions by mediator mass but with DM mass around35 GeV Later studies using a lower excess gamma ray spectrum tend to increase the preferredDM mass up to 50 GeV [76] and [88]

    The marginalised posterior probabilities for all combinations of parameters taken twoat a time are shown in Fig52 These plots have all constraints (Gamma LUX and RelicDensity) applied together The idea of these graphs is to study the posterior probabilities ofcombinations of two paramaters taken at a time (the off-diagonal graphs) The diagonal plotsshow marginalised histograms of a single parameter at a time- this gives an indication of thespread of each parameter The plots show that mS and λb show a high degree of correlationfor mS gt 100 GeV but there is also a range of values of mS around 35 GeV which are

    51 Bayesian Scans 60

    1 0 1 2 3 4log10(mχ)[GeV]

    1

    0

    1

    2

    3

    4

    log 1

    0(m

    s)[GeV

    ]

    (a) Gamma Only

    1 0 1 2 3 4log10(mχ)[GeV]

    1

    0

    1

    2

    3

    4

    log 1

    0(m

    s)[GeV

    ]

    (b) Relic Density

    1 0 1 2 3 4log10(mχ)[GeV]

    1

    0

    1

    2

    3

    4

    log 1

    0(m

    s)[GeV

    ]

    (c) LUX

    05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

    05

    00

    05

    10

    15

    20

    25

    30

    35

    log 1

    0(m

    s)[GeV

    ]

    (d) All Constraints

    Figure 51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether

    51 Bayesian Scans 61

    00

    05

    10

    15

    20

    25

    30

    log 1

    0(m

    χ)[GeV

    ]

    00

    05

    10

    15

    20

    25

    30

    ms[Gev

    ]

    5

    4

    3

    2

    1

    0

    1

    log 1

    0(λ

    t)

    5

    4

    3

    2

    1

    0

    1

    log 1

    0(λ

    b)

    00 05 10 15 20 25 30

    log10(mχ)[GeV]

    5

    4

    3

    2

    1

    0

    1

    log 1

    0(λ

    τ)

    00 05 10 15 20 25 30

    ms[Gev]5 4 3 2 1 0 1

    log10(λt)5 4 3 2 1 0 1

    log10(λb)5 4 3 2 1 0 1

    log10(λτ)

    Figure 52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter

    52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 62

    possible for λb around 001 λt and λτ show a wide range of possible values for two possibleranges of mS lt 35 GeV and greater than 100 GeV mχ shows a narrow range of possiblevalues around 35 GeV for all values of the other parameters

    52 Best fit Gamma Ray Spectrum for the Majorana FermionDM model

    We can see from the Fig51(a) scan that the maximum log-likelihood Fermi-LAT gammaray spectrum on its own shows values that centre around 35 GeV This is not changed byadding a relic density constraint or a direct detection (LUX) constraint

    The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraintsis shown in Fig53 The parameters for this point were

    Table 52 Best Fit Parameters

    Parameter mχ [GeV ] mS[GeV ] λt λb λτ

    Value 3332 49266 0322371 409990 0008106

    10-1 100 101 102

    E(GeV)

    10

    05

    00

    05

    10

    15

    20

    25

    30

    35

    E2dφd

    E(G

    eVc

    m2ss

    r)

    1e 6

    Best fitData

    Figure 53 Gamma Ray Spectrum

    The relic density (Ωh2) at this point has a value of 0135 but the point is separated onlyslightly in posterior probability from points with a value of 01199 with mS lt 10 GeV (iethe posterior probability is almost the same differing by less than 1)

    To better understand the interplay of the different constraints the three graphs in Fig54show the log likelihood surfaces and individual best fit points (best log likelihoods) for the γ

    and Ω scans seperately and then combined The diagrams each choose a pair of couplings(fixing the remaining parameters) and display the 2d surfaces of minuslog(χ2) (proportional to

    52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 63

    the log likelihoods of the coupling combinations) obtained by varying the pair of couplingsThese graphs do not show marginalised posterior probabilities but rather are based on the-2log likelihood calculations directly from MicrOmegas for the given coupling combinationsWe have termed this value as χ2 for convenience and used the log of this value to give abetter range for the colours These log likelihood values are what drive the MultiNest searchof parameter space for the best parameter combinations

    The graphs show a complex interplay of the different constraints resulting often in acompromise parameter combination The arrow on each graph points to the maximum point We have chosen a single representative set of couplings near the best overall fit point for allthe graphs

    52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 64

    00 05 10 15 20 25 30

    log10(mχ)

    00

    05

    10

    15

    20

    25

    30

    log

    10(m

    S)

    Max

    minuslog10(χ2(Γ)) λt = 487 λτ = 024 λb = 0344

    16

    14

    12

    10

    8

    6

    4

    2

    0

    γ Maximum at mχ=416 GeV mS=2188 GeV

    00 05 10 15 20 25 30

    log10(mχ)

    00

    05

    10

    15

    20

    25

    30

    log

    10(m

    S)

    Max

    minuslog10(χ2(Omega)) λt = 487 λτ = 024 λb = 0344

    28

    24

    20

    16

    12

    08

    04

    00

    04

    Ω Maximum at mχ=363 GeV mS=1659 GeV

    00 05 10 15 20 25 30

    log10(mχ)

    00

    05

    10

    15

    20

    25

    30

    log

    10(m

    S)

    Max

    minuslog10(χ2(Both)) λt = 487 λτ = 024 λb = 0344

    16

    14

    12

    10

    8

    6

    4

    2

    0

    Both Maximum at mχ=8317 GeV mS=2884GeV The best fit point for these couplings maybe above below or between the individual bestfit points but will on average be between thepoints

    Figure 54 Plots of log likelihoods by individual and combined constraints Masses in GeV

    53 Collider Constraints 65

    53 Collider Constraints

    531 Mediator Decay

    1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof τ leptons in pp collisions [44] Refer also to Chapter 3 for details

    We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

    The results of the calculations are shown in the graph Fig55 and can be compared withthe limits in the table from the paper [44] in Fig 56 The black line is the observed ATLASlimit

    Figure 55 σ lowastBr(σ rarr ττ) versus Mass of Scalar

    0 200 400 600 800

    mS[GeV]

    10

    5

    0

    log 1

    0(σ

    (bbS

    )lowastB

    (Sgtττ

    ))[pb]

    Observed LimitLikely PointsExcluded Points

    0

    20

    40

    60

    80

    100

    120

    0 5 10 15 20 25 30 35 40 45

    We can see that only one point here exceeded the limit and the points showed a widespread over σ(bbS)lowastBr(σ rarr ττ) by mS space which is consistent with the spread of pointson the MultiNest scan on the mS by λτ plane causing a spread of branching ratios to the τ

    quark It may seem odd that only one point was excluded here but this point was sampledwith a very low probability having high coupling constants of 597 to the τ quark and 278 tothe t quark and only just exceeds the limit As such it can be explained as a random outlier

    53 Collider Constraints 66

    Figure 56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar

    2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

    This scan calculates the contributions of b-quark associated Higgs boson productionreproduced from [43] Refer to Chapter 3 for details

    We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

    The cross section times branching ratios to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig57 The black line is the CMSLimit

    Figure 57 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar

    0 200 400 600 800

    mS[GeV]

    15

    10

    5

    0

    5

    log

    10(σ

    (bS

    +X

    )lowastB

    (Sgt

    bb))

    [pb]

    Observed LimitLikely PointsExcluded Points

    0

    20

    40

    60

    80

    100

    120

    0 50 100 150 200 250

    53 Collider Constraints 67

    The results of this scan were compared to the limits in [89] with the plot shown inFig58

    Figure 58 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

    We can see that there are a number of exclusions across the range of mS which occuredfor high values of the coupling constants gt 1 for λbλt λtau - these can be seen also on Fig59The excluded parameter combinations represent less than 2 of the total sampled number

    532 Collider Cuts Analyses

    We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude and the results are shown against theposterior probabilities of the parameter combinations in Fig 59

    The circles in Figure 59 show which points were excluded by the collider cuts as well asthe limits on cross sections times branching ratios to each of τ+τminus and bb

    All the excluded points under the heading Collider Cuts showed exclusion by the Lepstop0experiment (Search for direct pair production of the top quark in all hadronic final states

    53 Collider Constraints 68

    0 1 2 3

    log10(mχ)[GeV]

    0

    1

    2

    3

    log 1

    0(m

    s)[GeV

    ]Collider Cuts

    σ lowastBr(σgt bS+X)

    σ lowastBr(σgt ττ)

    (a) mχ by mS

    6 5 4 3 2 1 0 1 2

    log10(λt)

    0

    1

    2

    3

    log 1

    0(m

    s)[GeV

    ](b) λt by mS

    5 4 3 2 1 0 1

    log10(λb)

    6

    5

    4

    3

    2

    1

    0

    1

    2

    log 1

    0(λ

    t)

    (c) λb by λt

    5 4 3 2 1 0 1

    log10(λb)

    6

    5

    4

    3

    2

    1

    0

    1

    2

    log 1

    0(λ

    τ)

    (d) λb by λτ

    6 5 4 3 2 1 0 1 2

    log10(λt)

    6

    5

    4

    3

    2

    1

    0

    1

    2

    log 1

    0(λ

    τ)

    (e) λt by λτ

    5 4 3 2 1 0 1

    log10(λb)

    0

    1

    2

    3

    log 1

    0(m

    s)[GeV

    ]

    (f) λb by mS

    Figure 59 Excluded points from Collider Cuts and σBranching Ratio

    53 Collider Constraints 69

    [32]) Some of these same points were also excluded by Lepstop2 2bstop CMSTop andATLASMonobjet experiments There were expected events at other points but none weresignificant at the 95 exclusion level These significant exclusions occurred mainly wherethe mass of the mediator was over 100 GeV and higher values of the couplings to b and tquarks but there were a few points with mediator mass around 25 GeV The exclusions withmediator mass around 25 GeV were all accompanied by couplings to t quarks of order 10which is probably non-perturbative

    Looking at the exclusions in Fig 59 we can see that these occur mainly in the regionof higher mediator mass gt 100 GeV and lt 400 GeV These exclusions come from theastrophysical constraints (Gamma) which show a strong correlation between λb and mS andhigh values of λt gt 1 However these mass regions do not overlap at the same couplingcombinations as the high likelihood region The green and white circles correspond to theexcluded points in Figs 57 and 58 respectively

    The remaining regions of highest likelihood can be seen to be where 0001 λb lt 01accompanied by λt lt 0001 and mass of the mediator mS lt 10 GeV or 400 GeV lt mS lt1000GeV with mχ sim 35 GeV

    The plot of λt by λb shows the significant regions do not overlap with the collider cutsand cross section constraints since the latter occur at high values of either coupling while thelikely region has low values of these couplings (also accompanied by low couplings to λτ )

    Chapter 6

    Real Scalar Model Results

    61 Bayesian Scans

    To build some intuition we first examined the constraining effect of each observable on itsown We ran seperate scans for each constraint turned on and then for all the constraintstogether In Figure 61 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together

    In Figure 61 we can see that there are is a narrow region which dominates when allconstraints are taken into account This centers on mχ=9 GeV and a range of values for mS

    from 35 GeV to 400 GeV The region is not the highest likelihood region for the Fermi-LATgamma ray spectrum on its own (as can be seen from the individual plot - there is a narrowband to the left of the highest likelihood band) nor for the LUX likelihood or relic densitybut represents a compromise between the three constraints

    The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig62 The diagonal plots show marginalised histograms of a singleparameter at a time

    61 Bayesian Scans 71

    05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

    05

    00

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    log 1

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    s)[GeV

    ]

    (a) Gamma Only

    05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

    05

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    log 1

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    s)[GeV

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    05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

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    s)[GeV

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    05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

    05

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    s)[GeV

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    (d) All Constraints

    Figure 61 Real Scalar Dark Matter - By Individual Constraint and All Together

    61 Bayesian Scans 72

    00

    05

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    log 1

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    χ)[GeV

    ]

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    ]

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    t)

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    b)

    00 05 10 15 20 25 30

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    0(λ

    τ)

    00 05 10 15 20 25 30

    ms[Gev]5 4 3 2 1 0 1

    log10(λt)5 4 3 2 1 0 1

    log10(λb)5 4 3 2 1 0 1

    log10(λτ)

    Figure 62 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter

    62 Best fit Gamma Ray Spectrum for the Real Scalar DM model 73

    62 Best fit Gamma Ray Spectrum for the Real Scalar DMmodel

    We can see from Fig61 that the maximum log-likelihood Fermi-LAT Gamma ray spectrumon its own shows values of mχ that centre around 35 GeV However the LUX constraintin particular moves the best fit to a point around 9 GeV- the tight LUX constraint onlydisappears for low masses and at that point the proportion of τ+τminus annihilation needs toincrease to fit the constraints

    The gamma ray spectrum at the points of best fit for Fermi-LAT including all constraintsis shown in Fig63 The parameters for these points were respectively and involve decays tomainly τ and b quarks respectively

    Table 61 Best Fit Parameters

    Parameter mχ [GeV ] mS[GeV ] λt λb λτ

    Value 932 3526 000049 0002561 000781

    10-1 100 101 102

    E(GeV)

    10

    05

    00

    05

    10

    15

    20

    25

    30

    35

    E2dφdE

    (GeVc

    m2ss

    r)

    1e 6

    Best fitData

    Figure 63 Gamma Ray Spectrum

    This is not a particularly good fit and it occcurs at a relic density Ωh2 of 335 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

    63 Collider Constraints 74

    63 Collider Constraints

    631 Mediator Decay

    1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44]

    We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

    The results of the scan are shown in the graph Fig64 and can be compared with thelimits in the table from the paper [44] in Fig 56The black line is the observed ATLAS limit

    Figure 64 σ lowastBr(σ rarr ττ) versus Mass of Scalar

    0 200 400 600 800

    mS[GeV]

    8

    6

    4

    2

    0

    2

    4

    log 1

    0(σ

    (bbS

    )lowastB

    (Sgtττ

    ))[pb]

    Observed LimitLikely PointsExcluded Points

    050

    100150200250300350

    0 10 20 30 40 50 60

    We can see that there is a narrow band in the region of interest in the σ(bbS)lowastB(S gt ττ)

    by mS plane due to the high degree of correlation between mS and λτ seen in the likelihoodscan Values for mS gt 300Gev are excluded

    2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

    We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinations

    63 Collider Constraints 75

    randomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

    The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig65 The black line is the CMSlimit

    Figure 65 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

    0 200 400 600 800

    mS[GeV]

    8

    6

    4

    2

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    2

    4

    log

    10(σ

    (bS

    +X

    )lowastB

    (Sgt

    bb))

    [pb]

    Observed LimitLikely PointsExcluded Points

    050

    100150200250300350

    0 10 20 30 40 50 60

    The results of this scan were compared to the limits in [89] with the plot shown inFig58

    We can see a similar shape for the region of interest to that in Fig64 due to the highcorrelation between mS and λb over the region preferred by astrophysical constraints resultingin branching ratios highly correlated to ms Only one point is excluded with mS gt 500GeV

    632 Collider Cuts Analyses

    We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expected

    63 Collider Constraints 76

    with 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 66

    We can see that collider cuts show some exclusions only for masses mχ below the highlikelihood region but these exclusions do not occur at the same λb and λt combinations asseen from the plot of these two couplings except for a small region at the highest mS andcouplings

    All but one of the collider cut exclusions were excluded by the Lepstop0 experiment (thiswas excluded by 2bstop with high coupling to b quarks of 73) A number of the excludedpoints were also excluded by Lepstop2 2bstop CMSTopDM1L and ATLASMonobJet-(around 30 of the total for each)

    63 Collider Constraints 77

    0 1 2 3

    log10(mχ)[GeV]

    0

    1

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    s)[GeV

    ]Collider Cuts

    σ lowastBr(σgt bS+X)

    σ lowastBr(σgt ττ)

    (a) mχ by mS

    5 4 3 2 1 0 1

    log10(λt)

    0

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    s)[GeV

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    t)

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    τ)

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    τ)

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    s)[GeV

    ]

    (f) λb by mS

    Figure 66 Excluded points from Collider Cuts and σBranching Ratio

    Chapter 7

    Real Vector Dark Matter Results

    71 Bayesian Scans

    In the Figure 71 we can see that the preferred DM mass is centered around 35 GeV primarilydue to the influence of the Fermi-LAT gamma ray constraint with relic density having a widerange of possible mediator and DM masses The result of combining all the constraints is todefine a small region around mχ = 8 GeV and mS = 20 GeV which dominates the posteriorprobability

    The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig72 The diagonal plots show marginalised histograms of a singleparameter at a time

    71 Bayesian Scans 79

    05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

    1

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    s)[GeV

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    Figure 71 Real Vector Dark Matter - By Individual Constraint and All Together

    71 Bayesian Scans 80

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    b)

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    τ)

    00 05 10 15 20 25 30

    ms[Gev]5 4 3 2 1 0 1

    log10(λt)5 4 3 2 1 0 1

    log10(λb)5 4 3 2 1 0 1

    log10(λτ)

    Figure 72 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter

    72 Best fit Gamma Ray Spectrum for the Real Vector DM model 81

    72 Best fit Gamma Ray Spectrum for the Real Vector DMmodel

    The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraints isshown in Fig73

    Table 71 Best Fit Parameters

    Parameter mχ [GeV ] mS[GeV ] λt λb λτ

    Value 8447 20685 0000022 0000746 0002439

    10-1 100 101 102

    E(GeV)

    10

    05

    00

    05

    10

    15

    20

    25

    30

    35

    E2dφdE

    (GeVc

    m2s

    sr)

    1e 6

    Best fitData

    Figure 73 Gamma Ray Spectrum

    This is not a particularly good fit and it occcurs at a relic density Ωh2 of 025 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

    73 Collider Constraints

    731 Mediator Decay

    1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44] (See Chapter 3 for more details)

    We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

    73 Collider Constraints 82

    The results of the scan are shown in the graph Fig74 and can be compared with thelimits in the table from the paper [44] in Fig 56

    Figure 74 σ lowastBr(σ rarr ττ) versus Mass of Scalar

    0 200 400 600 800

    mS[GeV]

    8

    6

    4

    2

    0

    2

    log 1

    0(σ

    (bbS

    )lowastB

    (Sgtττ

    ))[pb]

    Observed LimitLikely PointsExcluded Points

    0100200300400500600700800

    0 20 40 60 80 100120140

    We can see that the region of highest likelihood is limited to mS lt 100 GeV and a narrowrange of couplings to λτ giving a narrow region of interest which falls well below the limit

    2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

    We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

    The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig75 The black line is the CMSlimit

    The results of this scan were compared to the limits in [89] with the plot shown in Fig58

    We can see that again the region of interest for mS is below 100 GeV and the points fallwell below the limit

    73 Collider Constraints 83

    Figure 75 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

    0 200 400 600 800

    mS[GeV]

    8

    6

    4

    2

    0

    2

    4

    log

    10(σ

    (bS

    +X

    )lowastB

    (Sgt

    bb))

    [pb]

    Observed LimitLikely PointsExcluded Points

    0100200300400500600700800

    0 20 40 60 80 100120140

    732 Collider Cuts Analyses

    We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 76

    We can see from Figure 76 that like the real scalar model the region of highest likelihoodhas λb and λt lt 001 while the excluded points from collider cuts and cross sections areabove 01 Most of the excluded points have mχ gt than the values in the region of highestlikelihood and all points are widely separated in parameter space from this region

    Once again 89 of excluded points were excluded by Lepstop0 with many also beingexcluded by the other experiments (50 by Lepstop2 and 56 by CMSTopDM1L 35 byLepstop2 and 20 by ATLASMonobjet)

    73 Collider Constraints 84

    0 1 2 3

    log10(mχ)[GeV]

    0

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    s)[GeV

    ]Collider Cuts

    σ lowastBr(σgt bS+X)

    σ lowastBr(σgt ττ)

    (a) mχ by mS

    5 4 3 2 1 0 1

    log10(λt)

    0

    1

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    s)[GeV

    ](b) λt by mS

    5 4 3 2 1 0 1

    log10(λb)

    5

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    1

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    t)

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    5 4 3 2 1 0 1

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    5

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    τ)

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    5 4 3 2 1 0 1

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    5

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    τ)

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    5 4 3 2 1 0 1

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    0

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    s)[GeV

    ]

    (f) λb by mS

    Figure 76 Excluded points from Collider Cuts and σBranching Ratio

    Chapter 8

    Conclusion

    We first conducted Bayesian scans over mass and coupling parameters for three simplifiedmodels - Majorana fermion real scalar and real vector using direct detecion (LUX) indirectdetection (GCE γ rays) and relic density constraints to find the regions of highest likelihoodWe ran these scans one constraint at a time and then together to better understand the interplayof the different constraints on best fit and likely parameter regions

    We also developed and tested C++ collider cut software specifically designed to mimickthe cuts applied in a number of ATLAS and CMS experiments that searched for Emiss

    T inconnection with specific beyond the standard model (BSM) theories of Dark Matter

    We then selected parameter combinations by randomly sampling the scanned resultsbased on the posterior probability distribution of the points and applied collider cuts andcross section times branching ratio calculations to these points to see if there would havebeen any sensitivity from the ATLAS and CMS experiments

    We found that the Majorana fermion model showed the best overall fit to all the constraintswith relic density close to the experimental value of 01199 and a low χ2 error on the gammaspectrum fit The preferred mass mχ was around 35 GeV and mS ranging from lt10 GeV to500 GeV The smaller masses give a better fit for relic density and also are well separatedfrom current collider constraints mainly due the the fact that the exclusions occur mainlywith higher couplings to t quarks (greater than 1) which is consistent with the fact that manyof the experiments were targeting dark matter pair production associated with t quark decaysNevertheless there were some exclusions that did fall in the region sampled by randomselection from the posterior probability distribution

    The preferred parameter regions were also consistent with the observation in [20] thatthe Majorana model showed the best overall fit with similar mass constraints We find thatthe couplings to the SM fermions should all be less that 001

    86

    The scalar and vector DM models give slightly less good fits especially to the currentrelic density estimate of 01199 but again the current collider constraints are well separatedfrom the best fit region and in fact none of the points randomly sampled from the posteriordistribution showed any exclusion The best fit parameter combination favours couplings toSM fermions of less than 001

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    [55] KGriest et al Macho collaboration search for baryonic dark matter via gravitationalmicrolensing arXiv astro-ph95066016 1995

    [56] Richard H Cyburt Brian D Fields Keith A Olive and Tsung-Han Yeh Big bangnucleosynthesis 2015 RevModPhys 88(015004) 2016

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    [58] The Wilkinson Microwave Anisotropy Probe (WMAP) Wmap skymap httpcosmologyberkeleyeduEducationCosmologyEssaysimagesWMAP_skymapjpg2005

    [59] P A R Ade et al Planck 2013 results xvi cosmological parameters arXiv13035076v3 2013

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    [79] Christopher Savage Andre Scaffidi Martin White and Anthony G Williams Luxlikelihood and limits on spin-independent and spin-dependent wimp couplings withluxcalc PhysRev D 92(103519) 2015

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    [87] Jonathan Feng Mpik website http1bpblogspotcom-U0ltb81JltQUXuBvRV4BbIAAAAAAAAE4kK3x0lQ50d4As1600direct+indirect+collider+imagepng 2005

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    Appendix A

    Validation of Calculation Tools

    Lepstop0Search for direct pair production of the top squarkin all hadronic final states in proton=proton collisions atradic

    s=8 TeV with the ATLAS detector [32]

    Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1) mt1=600 GeV mn1=1 GeVWhere t1 is the top squark and n1 is the neutralino of the MSSM

    94

    Table A1 0 Leptons in the final state

    Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 10000 460 460 100Emiss

    T gt 130GeV 8574 395 451 88Leptonveto 6621 307 285 108Emiss

    T gt 150GeV 6316 293 294 100Jet multiplicityand pT 1099 48 51 94|∆φ( jet pmiss

    T )|gt π5 902 40 42 96gt= 2b jets 396 19 18 106τ veto 396 19 18 106mbmin

    T gt 175GeV 270 13 13 105SRA1 79 4 4 104SRA2 70 3 3 93SRA3 67 3 3 97SRA4 52 2 2 91SRC exactly5 jets 1297 50 60 83SRC |∆φ( jet pmiss

    T )|gt π5 1082 50 50 99SRC gt= 2b jets 430 20 20 97SRC τ veto 430 20 20 97SRC |∆φ(bb)|gt 02π 367 17 17 99SRC1 246 12 11 105SRC2 177 9 8 116SRC3 152 8 7 116

    95

    Lepstop1Search for direct top squark pair production infinal states with one isolated lepton jets and missing trans-verse momentum in

    radics = 8 TeV pp collisions using 21 f bminus1

    of ATLAS data[33]

    Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1)(t1 gt b x+1 )(t1 gt b xminus1 )mt1=650 GeV mn1=1 GeV Where t1 is the top squark n1 is the neutralino xplusmn1 are thecharginos

    96

    Table A2 1 Lepton in the Final state

    Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 100000 99990 99990 100Electron(= 1signal) 11558 11557 11785 984 jets(80604025) 5187 5186 6271 83gt= 1btag 4998 4997 5389 93Emiss

    T gt 100GeV [allSRs] 4131 4131 4393 94Emiss

    T radic

    HT gt 5 [allSRs] 4026 4026 4286 94∆φ( jet2 pmiss

    T )gt 08 [allSRs] 3565 3565 3817 93∆φ( jet1Pmiss

    T )gt 08 [not SRtN2] 3446 3446 3723 93Emiss

    T gt 200GeV (SRtN2) 2027 2027 2186 93Emiss

    T radic

    HT gt 13(SRtN2) 1156 1156 1172 99mT gt 140GeV (SRtN2) 839 839 974 86Emiss

    T gt 275GeV (SRtN3) 948 948 965 98Emiss

    T radic

    HT gt 11(SRtN3) 932 932 937 99mT gt 200GeV (SRtN3) 456 456 566 81Emiss

    T gt 150GeV (SRbC1minusSRbC3) 2803 2803 3080 91Emiss

    T radic

    HT gt 7(SRbC1minusSRbC3) 2742 2742 3017 91mT gt 12GeV (SRbC1minusSRbC3) 1917 1917 2347 82Emiss

    T gt 160GeV (SRbC2SRbC3) 1836 1836 2257 81Emiss

    T radic

    HT gt 8(SRbC2SRbC3) 1788 1788 2175 82me f f gt 550GeV (SRbC2) 1741 1741 2042 85me f f gt 700GeV (SRbC3) 1075 1075 1398 77Muon(= 1signal) 11379 11378 11975 954 jets(80604025) 5264 5263 6046 87gt= 1btag 5101 5100 5711 89Emiss

    T gt 100GeV [allSRs] 4199 4199 4770 88Emiss

    T radic

    HT gt 5 [all SRs] 4115 4115 4653 88∆φ( jet2 pmiss

    T )gt 08 [all SRs] 3622 3622 4077 89∆φ( jet1 pmiss

    T )gt 08 [not SRtN2] 3515 3515 3970 89Emiss

    T gt 200GeV (SRtN2) 2136 2136 2534 84Emiss

    T radic

    HT gt 13(SRtN2) 1261 1261 1441 87mT gt 140GeV (SRtN2) 928 928 1033 90Emiss

    T gt 275GeV (SRtN3) 1036 1036 1274 81Emiss

    T radic

    HT gt 11(SRtN3) 1018 1018 1235 82mT gt 200GeV (SRtN3) 497 497 612 81Emiss

    T gt 150GeV (SRbC1minusSRbC3) 2891 2891 3380 86Emiss

    T radic

    HT gt 7(SRbC1minusSRbC3) 2821 2821 3290 86mT gt 120GeV (SRbC1minusSRbC3) 1977 1977 2290 86Emiss

    T gt 160GeV (SRbC2SRbC3) 1913 1913 2220 86Emiss

    T radic

    HT gt 8(SRbC2SRbC3) 1863 1863 2152 87me f f gt 550GeV (SRbC2) 1826 1826 2116 86me f f gt 700GeV (SRbC3) 1131 1131 1452 78

    97

    Lepstop2Search for direct top squark pair production infinal states with two leptons in

    radics =8 TeV pp collisions using

    20 f bminus1 of ATLAS data[83][34]

    Simulated in Madgraph with p p gt t1 t1(t1 gt b w+ n1)(t1 gt b wminusn1) mt1=180 GeV mn1=60 GeV Where t1 is the top squark n1 is the neutralino b1 is the bottom squark

    Table A3 2 Leptons in the final state

    Cut RawMe Scaled ATLAS MeATLAS PercentNoCuts 100000 324308 199000 1632BaselineCuts 3593 11652 7203 1622SF Signal Leptons 1699 5510 5510 1002OSSF Signal Leptons 1632 5293 5233 101mll gt 20Gev 1553 5037 5028 100leadinglepton pT 1479 4797 5048 95|mll minusmZ|gt 20Gev 1015 3292 3484 94∆φmin gt 10 850 2757 2160 128∆φb lt 15 680 2205 1757 126SRM90 [SF ] 13 42 60 70SRM120 [SF ] 0 0 19 0SRM100 +2 jets [SF ] 0 0 17 0SRM110 +2 jets [SF ] 0 0 1 02DF Signal Leptons 1737 5633 7164 792OSDF Signal Leptons 1670 5416 5396 100mll20Gev 1587 5147 5136 100leadinglepton pT 1501 4868 4931 99∆φmin gt 10 1247 4044 3122 130∆φb lt 15 1010 3276 2558 128SRM90 [SF ] 21 68 95 72SRM120 [SF ] 0 0 23 0SRM100 +2 jets [SF ] 0 0 19 0SRM110 +2 jets [SF ] 0 0 14 0

    98

    2bstopSearch for direct third-generation squark pair pro-duction in final states with missing transverse momentumand two b-jets[31]

    Simulated in Madgraph with p p gt b1 b1(b1 gt b n1)(b1 gt b n1)

    SRAmb1=500 GeV mn1=1 GeV SRBmb1=350 GeV mn1=320 GeVWhere b1 is the bottom squark and n1 is the neutralino

    Table A4 2b jets in the final state

    Cut RawMe Scaled ATLAS MeATLAS PercentSRA Nocuts 100000 173800 1738 100SRA MET gt 80 93407 162341 1606 101SRA Leptonveto 93302 162159 1505 108SRA MET gt 150 79449 138082 1323 104SRA Jet selection 10213 17750 119 149SRA mbb gt 200 8344 14502 96 151SRA mCT gt 150 6535 11358 82 139SRA mCT gt 200 5175 8994 67 134SRA mCT gt 250 3793 6592 51 129SRA mCT gt 300 2487 4322 35 123SRB NoCuts 100000 1624100 16241 100SRB leptonveto 11489 186593 4069 46SRB MET gt 250 2530 41090 757 54SRB Jet selection 63 1023 79 130SRB HT 3 lt 50 51 828 52 159

    99

    CMSTop1LSearch for the Production of Dark Matter in As-sociation with Top Quark Pairs in the Single-lepton FinalState in pp collisions[41]

    Simulated in Madgraph with p p gt t t p1 p1

    Where p1= Majorana fermion Table 3 shows Signal efficiencies and cross sections TheM scale was read from Figure 7 for 90 CL

    Table A5 Signal Efficiencies 90 CL on σ limexp[ f b] on pp gt tt +χχ

    Mχ[GeV ] My Signal Efficiency Quoted Efficiency My σ [ f b] Quoted σ [ f b]1 102 101 63 47

    10 132 101 62 4650 143 120 50 38100 170 146 56 32200 185 173 69 27600 265 240 32 19

    1000 320 276 41 17

    Appendix B

    Branching ratio calculations for narrowwidth approximation

    B1 Code obtained from decayspy in Madgraph

    Br(S rarr bb) = (minus24λ2b m2

    b +6λ2b m2

    s

    radicminus4m2

    bm2S +m4

    S)16πm3S

    Br(S rarr tt) = (6λ2t m2

    S minus24λ2t m2

    t

    radicm4

    S minus4ms2m2t )16πm3

    S

    Br(S rarr τ+

    τminus) = (2λ

    2τ m2

    S minus8λ2τ m2

    τ

    radicm4

    S minus4m2Sm2

    τ)16πm3S

    Br(S rarr χχ) = (2λ2χm2

    S

    radicm4

    S minus4m2Sm2

    χ)32πm3S

    (B1)

    Where

    mS is the mass of the scalar mediator

    mχ is the mass of the Dark Matter particle

    mb is the mass of the b quark

    mt is the mass of the t quark

    mτ is the mass of the τ lepton

    The coupling constants λ follow the same pattern

    • TITLE Collider Constraints applied to Simplified Models of Dark Matter fitted to the Fermi-LAT gamma ray excess using Bayesian Techniques
      • Dedication
      • Declaration
      • Acknowledgements
      • Contents
      • List of Figures
      • List of Tables
        • Chapter 1 Introduction
        • Chapter 2 Review of Physics
        • Chapter 3 Fitting Models to the Observables
        • Chapter 4 Calculation Tools
        • Chapter 5 Majorana Model Results
        • Chapter 6 Real Scalar Model Results
        • Chapter 7 Real Vector Dark Matter Results
        • Chapter 8 Conclusion
        • Bibliography
        • Appendix A Validation of Calculation Tools
        • Appendix B Branching ratio calculations for narrow width approximation

      Declaration

      I hereby declare that except where specific reference is made to the work of others thecontents of this dissertation are original and have not been submitted in whole or in partfor consideration for any other degree or qualification in this or any other university Thisdissertation is my own work and contains nothing which is the outcome of work done incollaboration with others except as specified in the text and Acknowledgements Thisdissertation contains fewer than 65000 words including appendices bibliography footnotestables and equations and has fewer than 150 figures

      I certify that this work contains no material which has been accepted for the award of anyother degree or diploma in my name in any university or other tertiary institution and to thebest of my knowledge and belief contains no material previously published or written byanother person except where due reference has been made in the text In addition I certifythat no part of this work will in the future be used in a submission in my name for any otherdegree or diploma in any university or other tertiary institution without the prior approval ofthe University of Adelaide and where applicable any partner institution responsible for thejoint-award of this degree I give consent to this copy of my thesis when deposited in theUniversity Library being made available for loan and photocopying subject to the provisionsof the Copyright Act 1968

      I acknowledge that copyright of published works contained within this thesis resides withthe copyright holder(s) of those works I also give permission for the digital version of mythesis to be made available on the web via the Universityrsquos digital research repository theLibrary Search and also through web search engines unless permission has been granted bythe University to restrict access for a period of time

      Guy PitmanOctober 2016

      Acknowledgements

      And I would like to acknowledge the help and support of my supervisors Professor TonyWilliams and Dr Martin White as well as Assoc Professor Csaba Balasz who has assisted withinformation about a previous study Ankit Beniwal and Jinmian Li who assisted with runningMicrOmegas and LUXCalc I adapted the collider cuts programs originally developed bySky French and Martin White for my study

      Contents

      List of Figures xiii

      List of Tables xv

      1 Introduction 111 Motivation 312 Literature review 4

      121 Simplified Models 4122 Collider Constraints 7

      2 Review of Physics 921 Standard Model 9

      211 Introduction 9212 Quantum Mechanics 9213 Field Theory 10214 Spin and Statistics 10215 Feynman Diagrams 11216 Gauge Symmetries and Quantum Electrodynamics (QED) 12217 The Standard Electroweak Model 13218 Higgs Mechanism 17219 Quantum Chromodynamics 222110 Full SM Lagrangian 23

      22 Dark Matter 25221 Evidence for the existence of dark matter 25222 Searches for dark matter 30223 Possible signals of dark matter 30224 Gamma Ray Excess at the Centre of the Galaxy [65] 30

      Contents vi

      23 Background on ATLAS and CMS Experiments at the Large Hadron collider(LHC) 31231 ATLAS Experiment 32232 CMS Experiment 33

      3 Fitting Models to the Observables 3531 Simplified Models Considered 3532 Observables 36

      321 Dark Matter Abundance 36322 Gamma Rays from the Galactic Center 36323 Direct Detection - LUX 37

      33 Calculations 39331 Mediator Decay 39332 Collider Cuts Analyses 42333 Description of Collider Cuts Analyses 43

      4 Calculation Tools 5541 Summary 5542 FeynRules 5643 LUXCalc 5644 Multinest 5745 Madgraph 5846 Collider Cuts C++ Code 59

      5 Majorana Model Results 6151 Bayesian Scans 6152 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 6453 Collider Constraints 67

      531 Mediator Decay 67532 Collider Cuts Analyses 69

      6 Real Scalar Model Results 7361 Bayesian Scans 7362 Best fit Gamma Ray Spectrum for the Real Scalar DM model 7663 Collider Constraints 77

      631 Mediator Decay 77632 Collider Cuts Analyses 78

      Contents vii

      7 Real Vector Dark Matter Results 8171 Bayesian Scans 8172 Best fit Gamma Ray Spectrum for the Real Vector DM model 8473 Collider Constraints 84

      731 Mediator Decay 84732 Collider Cuts Analyses 86

      8 Conclusion 89

      Bibliography 91

      Appendix A Validation of Calculation Tools 97

      Appendix B Branching ratio calculations for narrow width approximation 105B1 Code obtained from decayspy in Madgraph 105

      List of Figures

      21 Feynman Diagram of electron interacting with a muon 1122 Weak Interaction Vertices [48] 1523 Higgs Potential [49] 1824 Standard Model Particles and Forces [50] 2425 Bullet Cluster [52] 2526 Galaxy Rotation Curves [54] 2627 WMAP Cosmic Microwave Background Fluctuations [58] 2928 Dark Matter Interactions [60] 2929 Gamma Ray Excess from the Milky Way Center [75] 31210 ATLAS Experiment 31211 CMS Experiment 34

      31 Main Feyman diagrams leading to the cross section for scalar decaying to apair of τ leptons 40

      32 WidthmS vs mS 4033 WidthmS vs λb 4134 WidthmS vs λτ 4135 Main Feyman diagrams leading to the cross section for scalar decaying to a

      pair of b quarks in the presence of at least one b quark 42

      41 Calculation Tools 55

      51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether 62

      52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter 6353 Gamma Ray Spectrum 6454 Plots of log likelihoods by individual and combined constraints Masses in

      GeV 6655 σ lowastBr(σ rarr ττ) versus Mass of Scalar 67

      List of Figures ix

      56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar 6857 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar 6858 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 6959 Excluded points from Collider Cuts and σBranching Ratio 70

      61 Real Scalar Dark Matter - By Individual Constraint and All Together 7462 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter 7563 Gamma Ray Spectrum 7664 σ lowastBr(σ rarr ττ) versus Mass of Scalar 7765 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 7866 Excluded points from Collider Cuts and σBranching Ratio 80

      71 Real Vector Dark Matter - By Individual Constraint and All Together 8272 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter 8373 Gamma Ray Spectrum 8474 σ lowastBr(σ rarr ττ) versus Mass of Scalar 8575 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 8676 Excluded points from Collider Cuts and σBranching Ratio 87

      List of Tables

      21 Quantum numbers of the Higgs field 1922 Weak Quantum numbers of Lepton and Quarks 21

      31 Simplified Models 3532 95 CL by Signal Region 4433 Selection criteria common to all signal regions 4534 Selection criteria for signal regions A 4535 Selection criteria for signal regions C 4536 Signal Regions - Lepstop1 4837 Signal Regions Lepstop2 4938 Signal Regions 2bstop 5139 Signal Region ATLASmonobjet 52

      51 Scanned Ranges 6152 Best Fit Parameters 64

      61 Best Fit Parameters 76

      71 Best Fit Parameters 84

      A1 0 Leptons in the final state 98A2 1 Lepton in the Final state 100A3 2 Leptons in the final state 101A4 2b jets in the final state 102A5 Signal Efficiencies 90 CL on σ lim

      exp[ f b] on pp gt tt +χχ 103

      Chapter 1

      Introduction

      Dark matter (DM) was first postulated over 80 years ago when Swiss astronomer FritzZwicky observed a discrepancy between the amount of light emitted by a cluster of galaxiesand the total mass contained within the cluster inferred from the relative motion of thosegalaxies by a simple application of the theory of Newtonian gravitation The surprising resultof this observation was that the vast majority of the mass in the cluster did not emit lightwhich was contrary to the expectation that most of the mass would be carried by the starsSince that time further observations over a wide range of scales and experimental techniqueshave continued to point to the same result and refine it Some of these observations and otherevidence are discussed in section 22 We now know with certainty that in the entire Universeall of the matter we know about - stars planets gases and other cosmic objects such as blackholes can only account for less than 5 of the mass that we calculate to be there

      A recent phenomenon that has received much attention is the significant deviation frombackground expectations of the Fermi Large Area Telescope(Fermi-LAT) gamma ray flux atthe galactic centre [1] A number of astrophysical explanations have been proposed includingmillisecond pulsars of supernova remnants [2] or burst-like continuous events at the galacticcentre but these are unresolved However it has also been noted that the observed Fermi-LATexcess is consistent with the annihilation of dark matter particles which would naturally beconcentrated at the Galactic centre in a manner consistent with the Navarro-Frenk-Whitedistribution of dark matter [3]

      There are a number of other purely theoretical (particle physics) reasons to postulatethe existence of weakly interacting matter particles that could supply the missing mass andyet remain unobservable Weakly interacting massive particle (WIMPS) have been a majorfocus of Run I and ongoing Run II searches of the Large Hadron Collider (LHC) In spite

      2

      of substantial theoretical and experimental effort the microscopic properties of dark matterparticles are still unknown

      One of the main problems with extracting these properties is the plethora of competingtheoretical models that provide dark matter candidates Another is the very weak interactionof these particles and the lack of experimental observables allowing one to discriminatebetween the competing models

      A particularly fruitful approach is the simplified model approach which makes minimalassumptions about the DM candidates as extensions to the Standard Model interacting weaklywith ordinary matter through a mediator These can be coded as Lagrangian interaction termsusing tools such as FeynRules [4] and Calchep [5] and used to simulate Dark Matter relicdensities indirect detection expectations as well as direct detection expectations throughMicrOmegas [6] This in turn can be run under a Bayesian inference tool (see below)

      Using a Bayesian inference tool such as MultiNest [7][8][9] one can scan the parameterspace of a particular model to find regions of highest probability for that model based onobservations of a range of phenomena including direct detection experiments such as ZEPLIN[10] XENON [11] DEAP [12] ARDM [13] DarkSide [14] PandaX [15] and LUX [16]indirect detection - eg Fermi-LAT gamma ray excess [17] Pamela-positron and antiprotonflux [18] and production at colliders MultiNest calculates the evidence value which is theaverage likelihood of the model weighted by prior probability and a posterior distributionwhich is the distribution of the parameter(s) after taking into account the observed databy scanning over a range of parameters based on prior probabilities This can be used tocompare models on a consistent basis The scans also provide information about the best fitparameters for the overall fit as well as for the seperate experimentsobservations

      The simplified model approach has been taken in a number of papers to study the Fermi-LAT gamma ray excess (see literature review in section 12) and one in particular comparesthese models using a Bayesian inference tool which found that a Majorana fermion DMcandidate interacting via a scalar mediator with arbitrary couplings to third generationfermions showed the best fit [19] A subsequent paper [20] examined the most likelyparameter combinations that simultaneously fit the Planck relic density Fermi-LAT galacticcentre gamma ray AMS-02 positron flux and Pamela antiproton flux data

      There has not as yet been a thorough study of the collider constraints on DM simplifiedmodels that fit the Fermi-LAT galactic centre excess In this thesis we reproduce the

      11 Motivation 3

      previous results of [20] using similar astrophysical constraints and then implement detailedsimulations of collider constraints on DM and mediator production

      In Section 11 we review the motivation for this study and in Section 12 we considerrecent astronomical data which is consistent with the annihilation of DM particles andreview the literature and searches which attempt to explain it with a range of extensions tothe standard model

      In Chapter 2 we review the Standard Model of particle physics (SM) the evidence fordark matter and the ATLAS and CMS experiments at the LHC

      In Chapter 3 we describe the models considered in this paper the observables used inthe Bayesian scans and the collider experiments which provide limits against which theMagraph simulations of the three models can be compared

      In Chapter 4 we review the calculation tools that have been used in this paper

      In Chapters 5 6 and 7 we give the results of the Bayesian scans and collider cutsanalyses for each of the models and in Chapter 8 we summarise our conclusions

      11 Motivation

      The Fermi gamma ray space telescope has over the past few years detected a gamma raysignal from the inner few degrees around the galactic centre corresponding to a regionseveral hundred parsecs in radius The spectrum and angular distribution are compatible withthat from annihilating dark matter particles

      A number of papers have recently attempted to explain the Gamma Ray excess in termsof extensions to the Standard Model taking into account constraints from ATLAS and CMScollider experiments direct detection of dark matter colliding off nucleii in well shieldeddetectors (eg LUX) as well as other indirect measurements (astrophysical observations)These papers take the approach of simplified models which describe the contact interactionsof a mediator beween DM and ordinary matter using an effective field theory approach(EFT) Until recently it has been difficult to discriminate between the results of these paperswhich look at different models and produce different statistical results A recent paper[19] Simplified Dark Matter Models Confront the Gamma Ray Excess used a Bayesianinference tool to compare a variety of simplified models using the Bayesian evidence values

      12 Literature review 4

      calculated for each model to find the most preferred explanation The motivation for thechosen models is discussed in section 121

      A subsequent paper [20] Interpreting the Fermi-LAT gamma ray excess in the simplifiedframework studied the most likely model again using Bayesian inference to find the mostlikely parameters for the (most likely) model The data inputs (observables) to the scanwere the Planck measurement of the dark matter relic density [21] Fermi-LAT gamma rayflux data from the galactic center [17] cosmic positron flux data [22] cosmic anti-protonflux data [18] cosmic microwave background data and direct detection data from the LUXexperiment [16] The result of the scan was a posterior probability density function in theparameter space of the model

      This paper extends the findings in [19] by adding collider constraints to the likelihoodestimations for the three models It is not a simple matter to add collider constraints becauseeach point in parameter space demands a significant amount of computing to calculate theexpected excess amount of observable particles corresponding to the particular experimentgiven the proposed simplified model A sampling method has been employed to calculatethe collider constraints using the probability of the parameters based on the astrophysicalobservables

      12 Literature review

      121 Simplified Models

      A recent paper [23] summarised the requirements and properties that simplified models forLHC dark matter searches should have

      The general principles are

      bull Besides the Standard Model (SM) the Lagrangian should contain a DM candidate thatis absolutely stable or lives long enough to escape the LHC detector and a mediatorthat couples the two sectors

      bull The Lagrangian should contain all terms that are renormalizable consistent withLorentz invariance SM symmetries and DM stability

      bull Additional interactions should not violate exact and approximate global symmetries ofthe SM

      12 Literature review 5

      The examples of models that satisfy these requirements are

      1 Scalar (or pseudo-scalar) s-channel mediator coupled to fermionic dark matter (Diracor Majorana)

      2 Higgs portal DM where the DM is a scalar singlet or fermion singlet under the gaugesymmetries of the SM coupling to a scalar boson which mixes with the Higgs (Thefermion singlet is a specific realisation of 1 above)

      3 DM is a mixture of an electroweak singlet and doublet as in the Minimal Supersym-metric SM (MSSM)

      4 Vector (or Axial vector) s-channel mediator obtained by extending the SM gaugesymmetry by a new U(1)rsquo which is spontaneously broken such that the mediatoracquires mass

      5 t-channel flavoured mediator (if the DM particle is a fermion χ the mediator can bea coloured scalar or vector particle φ -eg assuming a scalar mediator a couplingof the form φ χq requires either χ or φ to carry a flavor index to be consistent withminimal flavour violation (MFV) which postulates the flavour changing neutral currentstructure in the SM is preserved

      Another recent paper [24] summarised the requirements and properties that simplifiedmodels for LHC dark matter searches should have The models studied in this paper arepossible candidates competing with Dirac fermion and complex scalar dark matter andmodels with vector and pseudo-scalar mediators Other models would have an elasticscattering cross section that will remain beyond the reach of direct detection experiments dueto the neutrino floor (the irreducible background produced by neutrino scattering) or arealready close to sensitivity to existing LUX and XENONIT experiments

      A recent paper [25] Scalar Simplified Models for Dark Matter looks at Dirac fermiondark matter mediated by either a new scalar or pseudo-scalar to SM fermions This paperplaces bounds on the coupling combination gχgφ (coupling of DM to mediator times couplingof mediator to SM fermions) calculated from direct detection indirect detection and relicdensity by the mass of DM and mass of mediator Bounds are also calculated for colliderconstraints (mono-jet searches heavy-flavour searches and collider bounds on the 125 GeVHiggs are used to place limits on the monojet and heavy flavor channels which can betranslated into limits on the Higgs coupling to dark matter and experimental measurements

      12 Literature review 6

      of the Higgs width are used to constrain the addition of new channels to Higgs decay) Thepaper models the top loop in a full theory using the Monte Carlo generator MCFM [26]and extends the process to accommodate off-shell mediator production and decay to a darkmatter pair While this paper does not consider the Majorana fermion dark matter model theconstraints on the coupling product may have some relevence to the present study if only asan order of magnitude indicator

      Constraining dark sectors with monojets and dijets [27] studies a simplified model withDirac fermion dark matter and an axial-vector mediator They use two classes of constraint-searches for DM production in events with large amounts of missing transverse energy inassociation with SM particles (eg monojet events) and direct constraints on the mediatorfrom monojet searches to show that for example where the choice of couplings of themediator to quarks and DM gA

      q =gAχ=1 all mediator masses in the range 130 GeV ltMR lt3

      TeV are excluded

      The paper Constraining Dark Sectors at Colliders Beyond the Effective Theory Ap-proach [28] considers Higgs portal models with scalar pseudo-scalar vector and axial-vectormediators between the SM and the dark sector The paper takes account of loop processesand gluon fusion as well as quark-mediator interactions using MCFM Apart from mediatortype the models are characterised by free parameters of mediator width and mass darkmatter mass and the effective coupling combination gχgφ (coupling of DM to mediator timescoupling of mediator to SM fermions) The conclusion is that LHC searches can proberegions of the parameter space that are not able to be explored in direct detection experiments(since the latter measurements suffer from a velocity suppression of the WIMP-nucleoncross-section)

      The paper Constraining the Fermi-LAT excess with multi-jet plus EmissT collider searches

      [29] shows that models with a pseudo-scalar or scalar mediator are best constrained by multi-jet final states with Emiss

      T The analysis was done using POWHEG-BOX-V2 at LO and usesM2

      T variables together with multi-jet binning to exploit differences in signal and backgroundin the different regions of phase space to obtain a more stringent set of constraints on DMproduction This paper largely motivated the current work in that it showed that collidersearches that were not traditionally thought of as dark matter searches were neverthelessoften sensitive to dark matter produced in association with heavy quarks The combinationof searches used in this paper reflects this fact

      12 Literature review 7

      122 Collider Constraints

      In order to constrain this model with collider constraints (direct detection limits) we considerthe following papers by the CMS and ATLAS collaborations which describe searches at theLHC

      ATLAS Experiments

      bull Search for dark matter in events with heavy quarks and missing transverse momentumin p p collisions with the ATLAS detector [30]

      bull Search for direct third-generation squark pair production in final states with missingtransverse momentum and two b-jets in

      radics= 8 TeV pp collisions with the ATLAS

      detector[31]

      bull Search for direct pair production of the top squark in all-hadronic final states inprotonndashproton collisions at

      radics=8 TeV with the ATLAS detector [32]

      bull Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

      radic(s)=8TeV pp collisions using 21 f bminus1 of

      ATLAS data [33]

      bull Search for direct top squark pair production in final states with two leptons inradic

      s = 8TeV pp collisions using 20 f bminus1 of ATLAS data [34]

      bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

      radics=8 TeV [35]

      CMS Experiments

      bull Searches for anomalous tt production in p p collisions atradic

      s=8 TeV [36]

      bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

      radics=8 TeV [37]

      bull Search for the production of dark matter in association with top-quark pairs in thesingle-lepton final state in proton-proton collisions at

      radics = 8 TeV [38]

      bull Search for new physics in monojet events in p p collisions atradic

      s = 8 TeV(CMS) [39]

      bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

      s = 8 TeV [40]

      12 Literature review 8

      bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theSingle-lepton Final State in pp collisions at

      radics=8 TeV [41]

      bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

      s=8 TeV [42]

      bull Search for neutral MSSM Higgs bosons decaying into a pair of bottom quarks [43]

      bull Search for neutral MSSM Higgs bosons decaying into a pair of τ leptons in p p collision[44]

      bull Search for dark matter extra dimensions and unparticles in monojet events in proton-proton collisions at

      radics=8 TeV [45]

      In particular we have implemented Madgraph simulations of certain data analysesapplying to collider experiments described in papers [32][33][34][31][30][41] Theremaining models were not implemented in Madgraph because of inability to validate themodels against published results in the time available We developed C++ analysis code toimplement the collider cuts after showering with Pythia and basic analysis with Delpheswith output to root files The C++ code allowed multiple experimental cuts to be applied tothe same simulation output The C++ code was first checked against the results of the variousexperiments above and then the simulations were rerun using the three modelsrsquo Lagrangians(implemented in FeynRules) described in this paper

      Chapter 2

      Review of Physics

      21 Standard Model

      211 Introduction

      The Standard Model of particle physics (SM) has sometimes been described as the theoryof Almost Everything The SM is a field theory which emerged in the second half of thetwentieth century from quantum mechanics and special relativity which themselves wereonly conceived after 1905 by Einsten Pauli Bohr Dirac and others In the first decades ofthe twenty first century the discovery of the Higgs boson [46][47] completed the pictureand confirmed the last previously unverified part of the SM The SM is a self consistenttheory that describes and classifies all known particles and their interactions (other thangravity) to an astonishing level of accuracy and detail However the theory is incomplete-it does not describe the complete theory of gravitation or account for DM (see below) ordark energy (the accelerating expansion of the universe) Related to these shortcomings aretheoretical deficiencies such as the asymmetry of matter and anti-matter the fact that the SMis inconsistent with general relativity (to the extent that one or both theories break down atspacetime singularities like the Big Bang and black hole event horizons)

      212 Quantum Mechanics

      Quantum Mechanics (QM) describes particles as probability waves which can be thoughtof as packets of energy which can be localised or spread through all space depending on how

      21 Standard Model 10

      accurately one measures the particlersquos momentum However QM is not itself consistent withspecial relativity It was realised too that because of relativity particles could be created anddestroyed and because of the uncertainty principle the theory should describe the probabilityof particles being created and destroyed on arbitrarily short timescales A quantum fieldtheory (QFT) is a way to describe this probability at every point of space and time

      213 Field Theory

      A field theory is described by a mathematical function of space and time called a LagrangianAs in classical mechanics where Lagrangians were introduced by Louis Lagrange at thebeginning of the nineteenth century to summarise dynamical systems the Lagrangian sum-marises the dynamics of space which is permeated by force fields The equations of motionarising from a Lagrangian are determined by finding the contours of the Lagrangian describedby stationary points (ie where the infinitesimal change in the Lagrangian is zero for a smallchange in field) Particles are generally described by singularities in the Lagrangian (wherethe values tend to infinity also called resonances)

      214 Spin and Statistics

      It was also realised that different types of particle needed to be represented by different fieldsin the Lagrangian Diracrsquos discovery that particles with spin 12 quantum numbers couldbe represented by a field obtained by effectively taking the square root of the relativisticequation (Klein Gordon equation) was probably the most significant advance in quantumfield theory It required the introduction of a matrix operators called the gamma matricesor Dirac matrices which helped explain the already known fact that spin half particles orfermions obey Fermi-Dirac statistics ie when two spin half particles are exchanged thesign of the wavefunction must change and hence they cannot occupy the same quantum stateunlike bosons with integer spin that can occupy the same quantum state

      Other particles need to be represented by fields called vector pseudo-vector scalar (as inthe Klein Gordon equation) pseudo-scalar and complex fields A vector field is really fourfields one for each dimension of space plus time and is parameterised by the four gammamatrices γmicro which transform among themselves in a relativistically invariant way (Lorentzinvariance) A pseudo-vector field is a vector field which transforms in the same way but

      21 Standard Model 11

      with additional sign change under parity transformations (a simultaneous reversal of the threespatial directions) and is represented by γmicroγ5 A pseudo-scalar is a scalar which changes signunder parity transformations and a complex scalar field is represented by complex numbers

      215 Feynman Diagrams

      QFT took another giant leap forward when Richard Feynman realised that the propagationof particles described by the field could be described by expanding the so called S-matrixor scattering matrix derived from calculating the probability of initial states transitioning tofinal states The expansion is similar to a Taylor expansion in mathematics allowing for thespecial commuting properties of elements of the field The Feynman diagrams representingindividual terms in the expansion can be interpreted as interactions between the particlecomponents The S-Matrix (or more correctly the non trivial part called the T-Matrix) isderived using the interaction Hamiltonian of the field The Hamiltonian is an operator whoseeigenvalues or characteristic values give the energy quanta of the field The T-matrix canalso be described by a path integral which is an integral over all of quantum space weightedby the exponential of the Lagrangian This is completely equivalent to the Hamiltonianformulation and shows how the Lagrangian arises naturally out of the Hamiltonian (in factthe Lagrangian is the Legendre transformation of the Hamiltonian)

      Figure 21 Feynman Diagram of electron interacting with a muon

      γ

      eminus

      e+

      micro+

      microminus

      The diagram Fig21 is a tree diagram (diagrams with closed loops indicate self in-teractions and are higher order terms in the expansion) and shows an electron (eminus) and ananti-muon (micro+) interacting by exchanging a photon

      21 Standard Model 12

      216 Gauge Symmetries and Quantum Electrodynamics (QED)

      The Dirac equation with its gamma matrices ushered in the use of group theory to describethe transformation of fields (because the multiplication table of the gamma matrices isclosed and described by group theory) It also ushered in the idea of symmetries within theLagrangian- for example consider a Lagrangian

      ψ(ipart minusm)ψ (21)

      The Lagrangian is invariant under a global U(1) transformation which acts on the fieldsand hence the derivatives as follows

      ψ rarr eiqαψ ψ rarr ψeminusiqα partmicroψ rarr eiqα

      partmicroψ (22)

      where qα is a global phase and α is a continuous parameter

      A symmetry leads by Noetherrsquos theorem to a conserved current jmicro = qψγmicroψ and is thebasis of charge conservation since partmicro jmicro = 0 and q =

      intd3x j0(x)

      By promoting this transformation to a local one (ie α rarr α(x) where x = xmicro are thespace time co-ordinates) the transformations become

      ψ rarr eiqα(x)ψ ψ rarr ψeminusiqα(x)partmicroψ rarr eiqα(x)(partmicroψ + iq(partmicroα(x))ψ) (23)

      The derivative transformation introduces an extra term which would spoil the gaugeinvariance This can be solved by introducing a new field the photon field Amicro(x) whichinteracts with the fermion field ψ and transforms under a U(1) transformation

      Amicro rarr Amicro minuspartmicroα(x) (24)

      If we replace normal derivatives partmicro with covariant derivatives Dmicro = (partmicro + iqAmicro(x)) in theLagrangian then the extra term is cancelled and we can now see that the covariant derivativetransforms like ψ - under a local U(1) transform ie

      Dmicro rarr Dmicroψprime = eiqα(x)Dmicroψ (25)

      The additional term iqAmicro(x) can be interpreted in a geometric sense as a connectionwhich facilitates the parallel transport of vectors along a curve in a parallel consistent manner

      21 Standard Model 13

      We must add a gauge-invariant kinetic term to the Lagrangian to describe the dynamicsof the photon field This is given by the electromagnetic field strength tensor

      Fmicroν = partmicroAν minuspartνAmicro (26)

      The full Quantum Electrodynamics (QED) Lagrangian is then for electrons with mass mand charge -e

      LQED = ψ(i Dminusm)ψ minus 14

      Fmicroν(X)Fmicroν(x) (27)

      This Lagrangian is Lorentz and U(1) guage invariant The interaction term is

      Lint =+eψ Aψ = eψγmicro

      ψAmicro = jmicro

      EMAmicro (28)

      where jmicro

      EM is the electromagnetic four current

      217 The Standard Electroweak Model

      The Standard Electroweak (EW) Model was created by Glashow-Weinberg-Salam (also calledthe GWS model) and unifies the electomagnetic and weak interactions of the fundamentalparticles

      The idea of gauge symmetry was extended to the idea of a non-abelian gauge symmetry(essentially the operators defining the symmetry do not commute and so are represented bymatrices- in this case the Pauli matrices for the SU(2) group) - the gauge group is the directproduct G = SU(2)

      otimesU(1) It was known that weak interactions were mediated by Wplusmn

      and Z0 particles where the superscripts are the signs of the electromagnetic charge Thesewere fairly massive but it was also known that through spontaneous symmetry breaking ofa continuous global non-abelian symmetry there would be one massive propagation modefor each spontaneously broken-symmetry generator and massless modes for the unbrokengenerators (the so-called Goldstone bosons)

      This required an extra field to represent the gauge bosons called Bmicro (a vector fieldhaving the 4 spacetime degrees of freedom) With this identification the covariant derivativebecomes

      Dmicro = partmicro minus igAmicro τ

      2minus i

      gprime

      2Y Bmicro (29)

      21 Standard Model 14

      Here Y is the so called weak hypercharge of the field Bmicro which is a quantum numberdepending on particle (lepton or quark) g and gprime are two independent parameters in theGWS theory and the expression Amicro τ implies a sum over 3 components Aa

      micro a=123 and thePauli matrices τa

      This isnrsquot the full story because it had been known since 1956 when Chin Shuing Wuand her collaborators confirmed in an experiment prompted by Lee and Yang that nature isnot ambivalent to so-called parity It had been assumed that the laws of nature would notchange if all spatial coordinates were reversed (a parity transformation) The experimentinvolved the beta decay of cooled 60C atoms (more electrons were emitted in the backwardhemisphere with respect to spin direction of the atoms) This emission involved the weakinteraction and required that the Lagrangian for weak interactions contain terms which aremaximally parity violating- which turn out to be currents of the form

      ψ(1minus γ5)γmicro

      ψ (210)

      The term

      12(1minus γ

      5) (211)

      projects out a left-handed spinor and can be absorbed into the spinor to describe left-handedleptons (and as later discovered quarks) The right handed leptons are blind to the weakinteraction

      The processes describing left-handed current interactions are shown in Fig 22

      Analogously to isospin where the proton and neutron are considered two eigenstates ofthe nucleon we postulate that Bmicro is the gauge boson of an SU(2)transformation coupled toweak SU(2) doublets (

      νe

      eminus

      )

      (ud

      ) (212)

      We may now write the weak SU(2) currents as eg

      jimicro = (ν e)Lγmicro

      τ i

      2

      e

      )L (213)

      21 Standard Model 15

      Figure 22 Weak Interaction Vertices [48]

      where τ i are the Pauli-spin matrices and the subscript L denotes the left-handed componentof the spinor projected out by Eqn211 The third current corresponding to i=3 is called theneutral current as it doesnrsquot change the charge of the particle involved in the interaction

      We could postulate that the electromagnetic current is a linear combination of the leftand right elecromagnetic fields eL = 1

      2(1minus γ5)e and eR = 12(1+ γ5)e

      jemmicro = eLγmicroQeL + eRγmicroQeR (214)

      where Q is the electromagnetic operator and eR is a singlet However this is not invariantunder SU(2)L To construct an invariant current we need to use the SU(2)L doublet 212The current which is an SU(2)L invariant U(1) current is

      jYmicro = (ν e)LγmicroYL

      e

      )L+ eRγmicroYReR (215)

      where hypercharges YL and YR are associated with the U(1)Y symmetry jYmicro is a linearcombination of the electromagnetic current and the weak neutral current thus explaining whythe hypercharge differs for the left and right-handed components since the electromagneticcurrent has the same charge for both components whilst the weak current exists only forleft-handed particles j3

      micro and the third component of weak isospin T 3 allows us to calculate

      21 Standard Model 16

      the relationship Y = 2(QminusT 3) which can be seen by writing jYmicro as a linear combination ofj3micro (the weak neutral current that doesnrsquot change the charge of the particle involved in the

      interaction) and jemmicro (using the convention that the jYmicro is multiplied by 1

      2 to match the samefactor implicit in j3

      micro ) Substituting

      τ3 =

      (1 00 minus1

      )(216)

      into equation 213 for i=3 and subtracting equation 215 and equating to 12 times equation

      214

      we get

      eLγmicroQeL + eRγmicroQeR minus (νLγmicro

      12

      νL minus eLγmicro

      12

      eL) =12

      eRγmicroYReR +12(ν e)LγmicroYL

      e

      )L (217)

      from which we can read out

      YR = 2QYL = 2Q+1 (218)

      and T3(eR) = 0 T3(νL) =12 and T3(eL) =

      12 The latter three identities are implied by

      the fraction 12 inserted into the definition of equation 213

      The Lagrangian kinetic terms of the fermions can then be written

      L =minus14

      FmicroνFmicroν minus 14

      GmicroνGmicroν

      + sumgenerations

      LL(i D)LL + lR(i D)lR + νR(i D)νR

      + sumgenerations

      QL(i D)QL +UR(i D)UR + DR(i D)DR

      (219)

      LL are the left-handed lepton doublets QL are the left-handed quark doublets lRνR theright handed lepton singlets URDR the right handed quark singlets

      The field strength tensors are given by

      Fmicroν = partmicroAν minuspartνAmicro minus ig[Amicro Aν ] (220)

      21 Standard Model 17

      andGmicroν = partmicroBν minuspartνBmicro (221)

      Bmicro is an abelian U(1) gauge field while Amicro is non-abelian and Fmicroν contains the com-mutator of the field through the action of the covariant derivative [Dmicro Dν ] This equationcontains no mass terms terms for the field Amicro for good reason- this would violate gaugeinvariance We know that the gauge bosons of the weak force have substantial mass and theway that this problem has been solved is through the Higgs mechanism

      218 Higgs Mechanism

      To solve the problem of getting gauge invariant mass terms three independent groupspublished papers in 1964 The mechanism was named after Peter Higgs To introduce theconcept consider a complex scalar field φ that permeates all space Consider a potential forthe Higgs field of the form

      minusmicro2φ

      daggerφ +λ (φ dagger

      φ2) (222)

      which has the form of a mexican hat Fig 23 This potential is still U(1) gauge invariant andcan be added to our Lagrangian together with kinetic+field strength terms to give

      L = (Dmicroφ)dagger(Dmicroφ)minusmicro2φ

      daggerφ +λ (φ dagger

      φ2)minus 1

      4FmicroνFmicroν (223)

      It is easily seen that this is invariant to the transformations

      Amicro rarr Amicro minuspartmicroη(x) (224)

      φ(x)rarr eieη(x)φ(x) (225)

      The minimum of this field is not at 0 but in a circular ring around 0 with a vacuum

      expectation value(vev)radic

      micro2

      2λequiv vradic

      2

      We can parameterise φ as v+h(x)radic2

      ei π

      Fπ where h and π are referred to as the Higgs Bosonand Goldstone Boson respectively and are real scalar fields with no vevs Fπ is a constant

      21 Standard Model 18

      Figure 23 Higgs Potential [49]

      Substituting this back into the Lagrangian 223 we get

      minus14

      FmicroνFmicroν minusevAmicropartmicro

      π+e2v2

      2AmicroAmicro +

      12(partmicrohpart

      microhminus2micro2h2)+

      12

      partmicroπpartmicro

      π+(hπinteractions)(226)

      This Lagrangian now seems to describe a theory with a photon Amicro of mass = ev a Higgsboson h with mass =

      radic2micro and a massless Goldstone π

      However there is a term mixing Amicro and part microπ in this expression and this shows Amicro and π

      are not independent normal coordinates and we can not conclude that the third term is a massterm This π minusAmicro mixing can be removed by making a gauge transformation in equations224 and 225 making

      φrarrv+h(x)radic2

      ei π

      Fπminusieη(x) (227)

      and setting πrarr π

      Fπminus eη(x) = 0 This choosing of a point on the circle for the vev is called

      spontaneous symmetry breaking much like a vertical pencil falling and pointing horizontallyin the plane This choice of gauge is called unitary gauge The Goldstone π will thencompletely disappear from the theory and one says that the Goldstone has been eaten to givethe photon mass

      21 Standard Model 19

      This can be confirmed by counting degrees of freedom before and after spontaneoussymmetry breaking The original Lagrangian had four degrees of freedom- two from thereal massless vector field Amicro (massless photons have only two independent polarisationstates) and two from the complex scalar field (one each from the h and π fields) The newLagrangian has one degree of freedom from the h field and 3 degrees of freedom from themassive real vector field Amicro and the π field has disappeared

      The actual Higgs field is postulated to be a complex scalar doublet with four degrees offreedom

      Φ =

      (φ+

      φ0

      )(228)

      which transforms under SU(2) in the fundamental representation and also transforms underU(1) hypercharge with quantum numbers

      Table 21 Quantum numbers of the Higgs field

      T 3 Q Yφ+

      12 1 1

      φ0 minus12 1 0

      We can parameterise the Higgs field in terms of deviations from the vacuum

      Φ(x) =(

      η1(x)+ iη2(x)v+σ(x)+ iη3(x)

      ) (229)

      It can be shown that the three η fields can be transformed away by working in unitarygauge By analogy with the scalar Higgs field example above the classical action will beminimised for a constant value of the Higgs field Φ = Φ0 for which Φ

      dagger0Φ0 = v2 This again

      defines a range of minima and we can pick one arbitarily (this is equivalent to applying anSU(2)timesU(1) transform to the Higgs doublet transforming the top component to 0 and thebottom component to Φ0 = v) which corresponds to a vacuum of zero charge density andlt φ0 gt=+v

      In this gauge we can write the Higgs doublet as

      Φ =

      (φ+

      φ0

      )rarr M

      (0

      v+ H(x)radic2

      ) (230)

      where H(x) is a real Higgs field v is the vacuum expectation and M is an SU(2)timesU(1)matrix The matrix Mdagger converts the isospinor into a down form (an SU(2) rotation) followedby a U(1) rotation to a real number

      21 Standard Model 20

      If we consider the Higgs part of the Lagrangian

      minus14(Fmicroν)

      2 minus 14(Bmicroν)

      2 +(DmicroΦ)dagger(DmicroΦ)minusλ (Φdagger

      Φminus v2)2 (231)

      Substituting from equation 230 into this and noting that

      DmicroΦ = partmicroΦminus igW amicro τ

      aΦminus 1

      2ig

      primeBmicroΦ (232)

      We can express as

      DmicroΦ = (partmicro minus i2

      (gA3

      micro +gprimeBmicro g(A1micro minusA2

      micro)

      g(A1micro +A2

      micro) minusgA3micro +gprimeBmicro

      ))Φ equiv (partmicro minus i

      2Amicro)Φ (233)

      After some calculation the kinetic term is

      (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

      14(v+

      Hradic2)2[A 2]22 (234)

      where the 22 subscript is the index in the matrix

      If we defineWplusmn

      micro =1radic2(A1

      micro∓iA2micro) (235)

      then [A 2]22 is given by

      [A 2]22 =

      (gprimeBmicro +gA3

      micro

      radic2gW+

      microradic2gWminus

      micro gprimeBmicro minusgA3micro

      ) (236)

      We can now substitute this expression for [A 2]22 into equation 234 and get

      (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

      14(v+

      Hradic2)2(2g2Wminus

      micro W+micro +(gprimeBmicro minusgA3micro)

      2) (237)

      This expression contains mass terms for three massive gauge bosons Wplusmn and the combi-nation of vector fields gprimeBmicro minusgA3

      micro where note

      21 Standard Model 21

      Table 22 Weak Quantum numbers of Lepton and Quarks

      T 3 Q YνL

      12 0 -1

      lminusL minus12 -1 -1

      νR 0 0 0lminusR 0 -1 -2UL

      12

      23

      13

      DL minus12 minus1

      313

      UR 0 23

      43

      DR 0 minus13 minus2

      3

      Wminusmicro = (W+

      micro )dagger equivW 1micro minus iW 2

      micro (238)

      Then the mass terms can be written

      12

      v2g2|Wmicro |2 +14

      v2(gprimeBmicro minusgA3micro)

      2 (239)

      W+micro and W microminus can be associated with the W boson and its anti-particle and (minusgprimeBmicro +

      gA3micro) with the Z Boson (after normalisation by

      radicg2 +(gprime

      )2) The combination gprimeA3micro +gBmicro

      is orthogonal to Z and remains massless This can be associated with the Photon (afternormalisation) These predict mass terms for the W and Z bosons mW = vgradic

      2and mZ =

      vradic2

      radicg2 +(gprime

      )2It is again instructive to count the degrees of freedom before and after the Higgs mech-

      anism Before we had a complex doublet Φ with four degrees of freedom one masslessB with two degrees of freedom and three massless W a fields each with two each givinga total of twelve degrees of freedom After spontaneous symmetry breaking we have onereal scalar Higgs field with one degree of freedom three massive weak bosons with nineand one massless photon with two again giving twelve degrees of freedom One says thescalar degrees of freedom have been eaten to give the Wplusmn and Z bosons their longitudinalcomponents (ie mass)

      Table 22 shows T 3 the weak isospin quantum number (which describes how a particletransforms under SU(2)) Y the weak hypercharge (which describes how the particle trans-

      21 Standard Model 22

      forms under U(1)) and Q the electric charge of leptons and quarks The former two enterdirectly into the Lagrangian while the latter can be derived as a function of the former two

      Strictly speaking the right handed neutrino which has no charge may not exist and is notpart of the standard model although some theories postulate a mass for this (hypothetical)particle

      Writing gauge invariant couplings between ELeRφ with the vacuum expectation ofφ = v (Yukawa matrices λ

      i ju and λ

      i jd respectively for the up and down quarks) we get mass

      terms for the quarks (and similarly for the leptons)

      Mass terms for quarks minussumi j[(λi jd Qi

      Lφd jR)+λ

      i ju εab(Qi

      L)aφlowastb u j

      R +hc]

      Mass terms for leptonsminussumi j[(λi jl Li

      Lφ l jR)+λ

      i jν εab(Li

      L)aφlowastb ν

      jR +hc]

      Here hc is the hermitian conjugate of the first terms L and Q are lepton and quarkdoublets l and ν are the lepton singlet and corresponding neutrino d and u are singlet downand up quarks εab is the Levi-Civita (anti-symmetric) tensor

      If we write the quark generations in a basis that diagonalises the Higgs couplings weget a unitary transformations Uu and Ud for the up and down quarks respectively which iscompounded in the boson current (Udagger

      u Ud) called the CKM matrix where off-diagonal termsallow mixing between quark generations (this is also called the mass basis as opposed tothe flavour basis) A similar argument can give masses to right handed sterile neutrinos butthese can never be measured However a lepton analogue of the CKM matrix the PNMSmatrix allows left-handed neutrinos to mix between various generations and oscillate betweenflavours as they propagate through space allowing a calculation of mass differences

      219 Quantum Chromodynamics

      The final plank in SM theory tackling the strong force Quantum Chromodynamics wasproposed by Gell-Mann and Zweig in 1963 Three generations of quarks were proposedwith charges of 23 for uc and t and -13 for ds and b Initially SU(2) isospin was consideredtransforming u and d states but later irreducible representations of SU(3) were found to fit theexperimental data The need for an additional quantum number was motivated by the fact thatthe lightest excited state of the nucleon consisted of sss (the ∆++) To reconcile the baryon

      21 Standard Model 23

      spectrum with spin statistics colour was proposed Quarks transform under the fundamentalor 3 representation of colour symmetry and anti-quarks under 3 The inner product of 3and 3 is an invariant under SU(3) as is a totally anti-symmetric combination ε i jkqiq jqk (andthe anti-quark equivalent) It was discovered though deep inelastic scattering that quarksshowed asymptotic freedom (the binding force grew as the seperation grew) Non-abeliangauge theories showed this same property and colour was identified with the gauge quantumnumbers of the quarks The quanta of the gauge field are called gluons Because the forcesshow asymptotic freedom - ie the interaction becomes weaker at high energies and shorterdistances perturbation theory can be used (ie Feynman diagrams) at these energies but forlonger distances and lower energies lattice simulations must be done

      2110 Full SM Lagrangian

      The full SM can be written

      L =minus14

      BmicroνBmicroν minus 18

      tr(FmicroνFmicroν)minus 12

      tr(GmicroνGmicroν)

      + sumgenerations

      (ν eL)σmicro iDmicro

      (νL

      eL

      )+ eRσ

      micro iDmicroeR + νRσmicro iDmicroνR +hc

      + sumgenerations

      (u dL)σmicro iDmicro

      (uL

      dL

      )+ uRσ

      micro iDmicrouR + dRσmicro iDmicrodR +hc

      minussumi j[(λ

      i jl Li

      Lφ l jR)+λ

      i jν ε

      ab(LiL)aφ

      lowastb ν

      jR +hc]

      minussumi j[(λ

      i jd Qi

      Lφd jR)+λ

      i ju ε

      ab(QiL)aφ

      lowastb u j

      R +hc]

      + (Dmicroφ)dagger(Dmicroφ)minusmh[φφ minus v22]22v2

      (240)

      where σ micro are the extended Pauli matrices

      (1 00 1

      )

      (0 11 0

      )

      (0 minusii 0

      )

      (1 00 minus1

      )

      The first line gives the gauge field strength terms the second the lepton dynamical termsthe third the quark dynamical terms the fourth the lepton (including neutrino) mass termsthe fifth the quark mass terms and the sixth the Higgs terms

      The SU(3) gauge field is given by Gmicroν and which provides the colour quantum numbers

      21 Standard Model 24

      Figure 24 Standard Model Particles and Forces [50]

      Note The covariant derivatives operating on the left-handed leptons contain only theU(1) and SU(2) gauge fields Bmicro Fmicro whereas those operating on left-handed quarks containU(1)SU(2) and SU(3) gauge fields Bmicro Fmicro Gmicro The covariant derivatives operating onright handed electrons and quarks are as above but do not contain the SU(2) gauge fieldsRight handed neutrinos have no gauge fields

      The sums over i j above are over the different generations of leptons and quarks

      The particles and forces that emerge from the SM are shown in Fig 24

      22 Dark Matter 25

      22 Dark Matter

      221 Evidence for the existence of dark matter

      2211 Bullet Cluster of galaxies

      Most of the mass of the cluster as measured by a gravitational lensing map is concentratedaway from the centre of the object The visible matter (as measured by X-ray emission) ismostly present in the centre of the object The interpretation is that the two galaxy clustershave collided perpendicular to the line of sight with the mostly collisionless dark matterpassing straight through and the visible matter interacting and being trapped in the middleIt was estimated that the ratio of actual to visible matter was about 16 [51]

      Figure 25 Bullet Cluster [52]

      2212 Coma Cluster

      The first evidence for dark matter came in the 1930s when Fritz Zwicky observed the Comacluster of galaxies over 1000 galaxies gravitationally bound to one another He observed thevelocities of individual galaxies in the cluster and estimated the amount of matter based onthe average galaxy He calculated that many of the galaxies should be able to escape fromthe cluster unless the mass was about 10 times higher than estimated

      22 Dark Matter 26

      2213 Rotation Curves [53]

      Since those early observations many spiral galaxies have been observed with the rotationcurves calculated by doppler shift with the result that in all cases the outermost stars arerotating at rates fast enough to escape their host galaxies Indeed the rotation curves arealmost flat indicating large amounts of dark matter rotating with the galaxies and holdingthe galaxies together

      Figure 26 Galaxy Rotation Curves [54]

      2214 WIMPS MACHOS

      The question is if the particles that make up most of the mass of the universe are not baryonswhat are they It is clear that the matter does not interact significantly with radiation so mustbe electrically neutral It has also not lost much of its kinetic energy sufficiently to relax intothe disks of galaxies like ordinary matter Massive particles may survive to the present ifthey carry a conserved additive or multiplicative quantum number This is so even if equalnumbers of particles and anti-particles can annihilate if the number densities become too lowto reduce the density further These particles have been dubbed Weakly interacting MassiveParticles (WIMPS) as they interact with ordinary matter only through a coupling strengthtypically encountered in the weak interaction They are predicted to have frozen out ofthe hot plasma and would give the correct relic density based solely on the weak interactioncross section (sometimes called the WIMP Miracle) Another possibility called MassiveCompact Halo Objects (MACHOS)- are brown dwarf stars Jupiter mass objects or remnantsfrom an earlier generation of star formation

      22 Dark Matter 27

      2215 MACHO Collaboration [55]

      In order to estimate the mass of MACHOS a number of experiments have been conductedusing microlensing which makes use of the fact that if a massive object lies in the line ofsight of a much more distant star the light from the more distant star will be lensed and forma ring around the star It is extemely unlikely that the MACHO will pass directly through theline of sight but if there is a near miss the two images will be seperated by a small angleThe signature for microlensing is a time-symmetric brightening of a star as the MACHOpasses near the line of sight called a light curve One can obtain statistical information aboutthe MACHO masses from the shape and duration of the light curves and a large number ofstars (many million) must be monitored The MACHO Collaboration [55] have monitoredover 11 million stars and less than 20 events were detected The conclusion was that thiscould account for a small addition (perhaps up to 20 ) to the visible mass but not enoughto account for Dark Matter

      2216 Big Bang Nucleosynthesis (BBN) [56]

      Another source of evidence for DM comes from the theory of Nucleosynthesis in the fewseconds after the Big Bang In the first few minutes after the big bang the temperatures wereso high that all matter was dissociated but after about 3 minutes the temperature rapidlycooled to about 109 Kelvin Nucleosynthesis could begin when protons and neutrons couldstick together to form Deuterium (Heavy Hydrogen) and isotopes of Helium The ratio ofDeuterium to Hydrogen and Helium is very much a function of the baryon density and alsothe rate of expansion of the universe at the time of Nucleosynthesis There is a small windowwhen the density of Baryons is sufficient to account for the ratios of the various elementsThe rate of expansion of the universe is also a function of the total matter density The ratiosof these lighter elements have been calculated by spectral measurements with Helium some25 of the mass of the universe Deuterium 001 The theory of BBN accounts for theratios of all of the lighter elements if the ratio of Baryon mass to total mass is a few percentsuggesting that the missing matter is some exotic non-baryonic form

      22 Dark Matter 28

      2217 Cosmic Microwave Background [57]

      The discovery of a uniform Microwave Background radiation by Penzias and Wilson in1965 was confirmation of the theory first proposed by Gamow and collaborators that theuniverse should be filled with black body radiation an afterglow of the big bang TheCOBE satelite launched in 1989 confirmed this black body radiation with temperature of2725+-002K with 95 confidence While extremely uniform it found tiny irregularitieswhich could be accounted for by irregularities in the density of the plasma at last scattering(gravitational redshifts or blueshifts called the Sachs Wolfe effect) and temperature andvelocity fluctuations in the plasma The anisotropy of the background has been furtheranalysed by WMAP (the Wilkinson Microwave Anisotropy Probe) launched in 2001 Thefluctuations in the density of the plasma are what account for the seeds of galaxy formationHowever detailed calculations have found that the fluctuations were too small to account forthe observed galaxy distribution without an electrically neutral gravitating matter in additionto ordinary matter that only became electrically neutral at the time of recombination [58]

      In May 2009 the European Space Agency (ESA) launched the Planck sattelite a spacebased observatory to observe the universe at wavelengths between 03mm and 111 mmbroadly covering the far-infrared microwave and high frequency domains The missionrsquosmain goal was to study the cosmic microwave background across the whole sky at greatersensitivity and resolution than ever before Using the information gathered and the standardmodel of cosmology ESA was able to calculate cosmological parameters including the relicdensity of dark matter remaining after the Big Bang to a high degree of precision withinthe framework of the standard model of cosmology The standard model of cosmology canbe described by a relatively small number of parameters including the density of ordinarymatter dark matter and dark energy the speed of cosmic expansion at the present epoch (alsoknown as the Hubble constant) the geometry of the Universe and the relative amount ofthe primordial fluctuations embedded during inflation on different scales and their amplitude[59]

      2218 LUX Experiment - Large Underground Xenon experiment [16]

      The LUX experiment is conducted at the Sanford Underground research facility one mileunderground to minimise the background of cosmic rays The detector is a time projectionchamber allowing a 3D positioning of interactions occurring within its active volume LUX

      22 Dark Matter 29

      Figure 27 WMAP Cosmic Microwave Background Fluctuations [58]

      Figure 28 Dark Matter Interactions [60]

      uses as its target 368 kiliograms of liquefied ultra pure xenon which is a scintillator Interac-tions inside the xenon will create an amount of light proportional to the energy depositedThat light can be collected on arrays of light detectors sensitive to a single photon

      Xenon is very pure and the amount of radiation originating within the target is limitedAlso being three times denser than water it can stop radiation originating from outside thedetector The 3D detector focuses on interactions near the center which is a quiet regionwhere rare dark matter interactions might be detected [16]

      22 Dark Matter 30

      222 Searches for dark matter

      2221 Dark Matter Detection

      Fig 28 illustrates the reactions studied under the different forms of experiment designed todetect DM

      Direct detection experiments are mainly underground experiments with huge detectorsattempting to detect the recoil of dark matter off ordinary matter such as LUX [16] CDMS-II [61] XENON100 [11] COUPP [62] PICASSO [63] ZEPLIN [10] These involveinteractions between DM and SM particles with final states involving DM and SM particles

      Indirect detection involves the search for the annihilation of DM particles in distantastrophysical sources where the DM content is expected to be large Example facilitiesinclude space based experiments (EGRET [64] PAMELA [18] FERMI-LAT [65]) balloonflights (ATIC [66] PEBS [67] PPB-BETS [68]) or atmospheric Cherenkov telescopes(HESS [69]MAGIC [70]) and neutrino telescopes (SuperKamiokande [71] IceCube [72]ANTARES [73] KM3NET [74]) Indirect Detection involves DM particles in the initial stateand SM particles in the final state

      Collider detection involves SM particles in the initial state creating DM particles in thefinal state which can be detected by missing transverse energy and explained by variousmodels extending the SM

      223 Possible signals of dark matter

      224 Gamma Ray Excess at the Centre of the Galaxy [65]

      The center of the galaxy is the most dense and active region and it is no surprise that itproduces intense gamma ray sources When astronomers subtract all known gamma raysources there remains an excess which extends out some 5000 light years from the centerHooper et al find that the annihilation of dark matter particles with a mass between 31 and40 Gev fits the excess to a remarkable degree based on Gamma Ray spectrum distributionand brightness

      23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 31

      Figure 29 Gamma Ray Excess from the Milky Way Center [75]

      23 Background on ATLAS and CMS Experiments at theLarge Hadron collider (LHC)

      The collider constraints in this paper are based on two of the experiments that are part of theLHC in CERN

      Figure 210 ATLAS Experiment

      The LHC accelerates proton beams in opposite directions around a 27 km ring of super-conducting magnets until they are made to collide at an interaction point inside one of the 4experiments placed in the ring An upgrade to the LHC has just been performed providing65 TeV per beam

      23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 32

      231 ATLAS Experiment

      The ATLAS detector consists of concentric cylinders around the interaction point whichmeasure different aspects of the particles that are flung out of the collision point There are 4major parts the Inner Detector the Calorimeters the Muon spectrometer and the magnetsystems

      2311 Inner Detector

      The inner detector extends from a few centimeters to 12 meters from the interaction point andmeasures details of the charged particles and their momentum interacting with the materialIf particles seem to originate from a point other than the interaction point this may indicatethat the particle comes from the decay of a hadron with a bottom quark This is used to b-tagcertain particles

      The inner detector consists of a pixel detector with three concentric layers and threeendcaps containing detecting silicon modules 250 microm thick The pixel detector has over 80million readout channels about 50 percent of the total channels for the experiment All thecomponents are radiation hardened to cope with the extremely high radiation in the proximityof the interaction point

      The next layer is the semi-conductor layer (SCT) similar in concept to the pixel detectorbut with long narrow strips instead of pixels to cover a larger area It has 4 double layers ofsilicon strips with 63 million readout channels

      The outermost component of the inner detector is the Transition Radiation Tracker It is acombination of a straw tracker and a transition radiation detector The detecting elementsare drift tubes (straws) each 4mm in diameter and up to 144 cm long Each straw is filledwith gas which becomes ionised when the radiation passes through The straws are held at-1500V with a thin wire down the center This drives negatively ionised atoms to the centercausing a pulse of electricity in the wire This causes a pattern of hit straws allowing the pathof the particles to be determined Between the straws materials with widely varying indicesof refraction cause transition radiation when particles pass through leaving much strongersignals in some straws Transition radiation is strongest for highly relativistic particles (thelightest particles- electrons and positrons) allowing strong signals to be associated with theseparticles

      23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 33

      2312 Calorimeters

      The inner detector is surrounded by a solenoidal magnet of 2 Tesla (there is also an outertoroidal magnet) but surrounding the inner magnet are the calorimeters to measure theenergy of particles by absorbing them There is an inner electromagnetic calorimeter whichmeasures the angle between the detectorrsquos beam axis and the particle (or more preciselypseudo-rapidity) and its angle in the perpendicular plane to a high degree of accuracy Theabsorbing material is lead and stainless steel and the sampling material liquid argon Thehadron calorimeter measures the energy of particles that interact via the strong force Theenergy absorbing material is scintillating steel

      2313 Muon Specrometer

      The muon spectrometer surrounds the calorimeters and has a set of 1200 chambers tomeasure the tracks of outgoing muons with high precision using a magnetic field andtriggering chambers Very few particles of other types pass through the calorimeters and aresubsequently measured

      2314 Magnets

      The magnets are in two layers the inner solenoidal and outer toroidal which together causecharged particles to curve in the magetic field allowing their momentum to be determinedThe outer toroidal magnet consists of 8 very large air core superconducting magnets whichsupply the field in the muon spectrometer

      232 CMS Experiment

      The CMS Detector is similar in design to the ATLAS Detector with a single solenoidalmagnet which is the largest of its type ever built (length 13m and diameter 7m) and allowsthe tracker and calorimeter detectors to be placed inside the coil in a compact way Thestrong magnetic field gives a high resolution for measuring muons electrons and photons ahigh quality central tracking system and a hermetic hadron calorimeter designed to preventhadrons from escaping

      23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 34

      Figure 211 CMS Experiment

      Chapter 3

      Fitting Models to the Observables

      31 Simplified Models Considered

      In order to extend the findings of previous studies [19] to include collider constraints wefirst performed Bayesian scans [7] over three simplified models real scalar dark matter φ Majorana fermion χ and real vector Xmicro The purpose was to obtain the most probable regionsof each model then to use the resulting posterior samples to investigate whether the LHCexperiments have any sensitivity to the preferred regions

      The three models couple to the mediator with interactions shown in the following table

      Table 31 Simplified Models

      Hypothesis real scalar DM Majorana fermion DM real vector DM

      DM mediator int Lφ sup microφ mφ φ 2S Lχ sup iλχ

      2 χγ5χS LX sup microX mX2 X microXmicroS

      The interactions between the mediator and the standard fermions is assumed to be

      LS sup f f S (31)

      and in line with minimal flavour violation we only consider third generation fermions- ief = b tτ

      For the purposes of these scans we consider the following observables

      32 Observables 36

      32 Observables

      321 Dark Matter Abundance

      We have assumed a single dark matter candidate which freezes out in the early universe witha central value as determined by Planck [21] of

      ΩDMh2 = 01199plusmn 0031 (32)

      h is the reduced hubble constant

      The experimental uncertainty of the background relic density [59] is dominated by atheoretical uncertainty of 50 of the value of the scanned Ωh2 [76] We have added thetheoretical and the experimental uncertainties in quadrature

      SD =radic(05Ωh2)2 + 00312 (33)

      This gives a log likelihood of

      minus05lowast (Ωh2 minus 1199)2

      SD2 minus log(radic

      2πSD) (34)

      322 Gamma Rays from the Galactic Center

      Assuming that the gamma ray flux excess from the galactic centre observed by Fermi-LAT iscaused by self annihilation of dark matter particles the differential flux

      d2Φ

      dEdΩ=

      lt σv gt8πmχ

      2 J(ψ)sumf

      B fdN f

      γ

      dE(35)

      has been calculated using both Micromegas v425 with a Navarro-Frenk-White (NFW) darkmatter galactic distribution function

      ρ(r) = ρ0(rrs)

      minusγ

      (1+ rrs)3minusγ (36)

      with γ = 126 and an angle of 5 to the galactic centre [19]

      32 Observables 37

      Here Φ is the flux dE is the differential energy lt σv gt the velocity averaged annihi-lation cross section of the dark matter across the galactic centre mχ the mass of the darkmatter particle B f =lt σv gt f σv is the annihilation fraction into the f f final state anddN f

      γ dE is the energy distribution of photons produced in the annihilation channel with thefinal state f f The J factor is described below

      The angle ψ to the galactic centre determines the normalisation of the gamma ray fluxthrough the factor

      J(ψ) =int

      losρ(l2 + r⊙2 minus2lr⊙cosψ)dl (37)

      where ρ(r) is the dark matter density at distance r from the centre of the galaxy r⊙=85 kpcis the solar distance from the Galactic center ρ0 is set to reproduce local dark matter densityof 03GeVcm3

      The central values for the gamma ray spectrum were taken from Fig 5 of [77] and thestandard deviations of individual flux points from the error bars on the graph In commonwith [78] we ignore possible segment to segment correlations

      For the Fermi-LAT log likelihood we assumed each point was distributed with indepen-dent Gaussian likelihood with the experimental error for each data point given by the errorbar presented in Fig5 of [76] for each point We also assumed an equal theoretical errorfor each point which when added to the experimental error in quadrature gave an effectivelog likelihood for each point of minus05lowast (giminusdi)

      2

      2lowastσ2i

      where gi are the calculated values and di theexperimental values and σi the experimental errors

      323 Direct Detection - LUX

      The LUX experiment [16] is the most constraining direct search experiment to date for awide range of WIMP models We have used LUXCalc [79] (modified to handle momentumdependent nucleon cross sections) to calculate the likelihood of the expected number ofsignal events for the Majorana fermion model

      The likelihood function is taken as the Poisson distribution

      L(mχ σ | N) = P(N | mχ σ) =(b+micro)Neminus(b+micro)

      N (38)

      32 Observables 38

      where b= expected background of 064 and micro = micro(mχσ ) is the expected number of signalevents calculated for a given model

      micro = MTint

      infin

      0dEφ(E)

      dRdE

      (E) (39)

      where MT is the detector mass times time exposure and φ(E) is a global efficiency factorincorporating trigger efficiencies energy resolution and analysis cuts

      The differential recoil rate of dark matter on nucleii as a function of recoil energy E

      dRdE

      =ρX

      mχmA

      intdvv f (v)

      dσASI

      dER (310)

      where mA is the nucleon mass f (v) is the dark matter velocity distribution and

      dσSIA

      dER= Gχ(q2)

      4micro2A

      Emaxπ[Z f χ

      p +(AminusZ) f χn ]

      2F2A (q) (311)

      where Emax = 2micro2Av2mA Gχ(q2) = q2

      4m2χ

      [24] is an adjustment for a momentum dependantcross section which applies only to the Majorana model

      f χ

      N =λχ

      2m2SgSNN assuming that the relic density is the central value of 1199 We have

      implemented a scaling of the direct search cross section in our code by multiplying gSNN by( Ω

      Ω0)05 as a convenient method of allowing for the fact that the actual relic density enters

      into the calculation of the cross section as a square

      FA(q) is the nucleus form factor and

      microA =mχmA

      (mχ +mA)(312)

      is the reduced WIMP-nucleon mass

      The strength of the dark matter nucleon interaction between dark matter particles andthird generation quarks is

      gSNN =2

      27mN fT G sum

      f=bt

      λ f

      m f (313)

      where fT G = 1minus f NTuminus f N

      Tdminus fTs and f N

      Tu= 02 f N

      Td= 026 fTs = 043 [20]

      33 Calculations 39

      For the real scalar and real vector models where the spin independent cross sections arenot momentum dependent we based the likelihood on the LUX 95 CL from [16] using acomplementary error function Li(d|mχ I) = 05lowastCEr f (dminusx(mχ )

      σ) where x is the LUX limit

      and d the calculated SI cross section We assumed a LUX error σ of 01lowastd lowastradic

      2

      33 Calculations

      We coded the Lagrangians for the three models in Feynrules [4] Calculation of the expectedobservables (dark matter relic density Fermi-LAT gamma ray spectrum and LUX nucleonscattering amplitudes) were performed using Micromegas v425

      331 Mediator Decay

      A number of searches have previously been performed at the LHC which show sensitivity tothe decay of scalar mediators in the form of Higgs bosons (in beyond the standard modeltheories) to bottom quarks top quarks and τ leptons (all third generation fermions) Two inparticular [43 44] provide limits on the cross-sections for the production of the scalar timesthe branching ratios to brsquos and τrsquos In order to check these limits against each of the modelswe ran parameter scans in Madgraph at leading order over the top 900 points from the nestedsampling output from MultiNest [19]

      The two processes were

      1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to apair of τ leptons in pp collisions [44]

      bull generate p p gt b b S where S is the scalar mediator

      The process measures the decay of a scalar in the presence of b-jets that decay to τ+τminus

      leptons Fig31 shows the main diagrams contributing to the process The leftmost diagramis a loop process and the centre and right are tree level however we only got Madgraph togenerated the middle diagram since we are interested in τ leptons and b quarks in the finalstate

      The limit given in [44] only applies in the case of narrow resonances where narrow istaken to mean that the width of the resonance is less than 10 of its mass To test the validity

      33 Calculations 40

      Figure 31 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof τ leptons

      of applying these limits to our models we show in Figures 32 to 34 the percentage widthof the resonance vs mS[GeV ] (pole mass of the scalar mediator) λb and λt (the couplingconstants to the b and t quarks) The probabilities are the posterior probabilities obtainedfrom MultiNest scans of the Majorana model varying the coupling constants and scalarmediator and calculating the decay width of the mediator The results are represented as 2dposterior probability heat maps in the central panels and 1d marginalised distributions on thex and y axes of each graph

      16 18 20 22 24 26 28 30

      log10(mS[GeV])

      001

      002

      003

      004

      005

      Widthm

      S

      00

      04

      08

      12

      0 100 200

      Posterior Probability

      Figure 32 WidthmS vs mS

      The WidthmS was lt 01 at almost all points except where the mass of the scalar wasgreater than 1000 GeV λb was greater than 1 or λτ was greater than 1

      This can be seen from the graphs in Figs 323334

      33 Calculations 41

      4 3 2 1 0

      λb

      001

      002

      003

      004

      005

      WidthmS

      000

      015

      030

      045

      0 100 200

      Posterior Probability

      Figure 33 WidthmS vs λb

      5 4 3 2 1 0

      λτ

      001

      002

      003

      004

      005

      WidthmS

      000

      015

      030

      045

      0 100 200

      Posterior Probability

      Figure 34 WidthmS vs λτ

      The decay widths are calculated in Madgraph at leading order and branching fractionscan be calculated to b t τ and χ (DM) The formulae for the calculations can be found indecayspy which is a subroutine of Madgraph which we have documented in Appendix BB1 Using these we calculated the branching ratio to τ+τminus so that the cross section timesbranching ratio can be compared to the limits in [80] shown in Chapters 56 and 7 in Figs5564 and 74 for each model

      2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

      This scan calculates the contributions of b-quark associated Higgs boson production inthe center and right diagrams of Fig 35 reproduced from [43] The scans did not includePythia and Delphes as only production cross sections were required and the producedparticles were in the final state and required no matching or merging

      The Madgraph processes were

      bull generate p p gt b S where S is the scalar mediator

      bull add process p p gt b S j

      bull add process p p gt b S

      33 Calculations 42

      Figure 35 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof b quarks in the presence of at least one b quark

      bull add process p p gt b S j

      The cross section times branching ratios to bb for mediator production in association withat least one b quark is shown for each model in Chapters 56 and 7 in Figs 5765 and 75

      332 Collider Cuts Analyses

      We generated two parameter scans with parameter combinations from the Multinest outputwhich were the basis of the plots of Fig 6 in [19] The plots in Fig 6 were marginalisedto two parameters each but the raw Multinest output contained 5 parameters plus the priorprobabilities of each combination normalised to total 10

      The first scan was for χ produced with b quarks In order to force the mediator to decayto χ only we had χ in the final state and defined jets to include b quarks in Madgraph Thenwe generated the processes

      bull generate p p gt χ χ j

      bull add process p p gt χ χ j j

      Jet matching was on

      The second scan was for t quarks produced in the final state

      bull generate p p gt χ χ tt

      33 Calculations 43

      No jet matching was required because we did not mix final state jet multiplicities at thematrix element level

      The outputs from these two processes were normalised to 21 f bminus1 and combined

      The results of these scans prior to collider cuts calculations are shown in Figs 5161and 71 These show the posterior probabilities of the various parameter combinations withdifferent constraints applied seperately and then together A more detailed commentary isgiven in each of the sections

      We then sampled 900 points based on posterior probability from these scans and ran thesethrough Madgraph v425 then applied collider cuts with the C++ code using the limits fromthe following six ATLAS and CMS searches shown in Table32 and described below to findany exclusions

      333 Description of Collider Cuts Analyses

      In the following all masses and energies are in GeV and angles in radians unless specificallystated

      3331 Lepstop0

      Search for direct pair production of the top squark in all-hadronic final states in pro-tonndashproton collisions at

      radics=8 TeV with the ATLAS detector[32]

      This analysis searched for direct pair production of the supersymmetric partner to thetop quark using integrated luminosity of 201 f bminus1 The top squark was assumed to de-cay via t rarr t χ0

      1 or t rarr bχ01 or t rarr bχ

      plusmn1 rarr bW (lowast)χ1

      0 where χ01 (χ

      plusmn1 ) denotes the lightest

      neutralino(chargino) in supersymmetric models The search targets a fully hadronic finalstate with four or more jets and large missing momentum The production of a mediator inassociation with top quarks followed by an invisible mediator decay to dark matter particleshas a similar final state to that considered in the ATLAS stop searches Thus we expect theATLAS reach to have sensitivity to the three dark matter models considered in this thesis

      The collider cuts program mimicked the object reconstruction through jetlepton overlapremoval and selected signal electrons muons and τ particles with pT gt 10 GeV and

      33 Calculations 44

      Table 32 95 CL by Signal Region

      Experiment Region Number

      Lepstop0

      SRA1 66SRA2 57SRA3 67SRA4 65SRC1 157SRC2 124SRC3 80

      Lepstop1

      SRtN1-1 857SRtN1-2 498SRtN1-3 389SRtN2 107SRtN3 85SRbC1 832SRbC2 195SRbC3 76

      Lepstop2

      L90 740L100 56L110 90L120 170

      2bstop

      SRA 857SRB 270SRA150 380SRA200 260SRA250 90SRA300 75SRA350 52

      CMSTopDM1L SRA 1385

      ATLASMonobjetSR1 1240SR2 790

      33 Calculations 45

      |η | lt 24724 and 247 respectively (for eminusmicrominusτ) Jets were selected with pT gt 20 GeVand |η |lt 45 b-jets were selected if they were tagged by random selection lt 70 and had|η |lt 25 and pT gt 20 GeV

      These jets and leptons were further managed for overlap removal within ∆R lt 04 withnon b-jets removed if ∆R lt 02

      The b-jets and non b-jets were further selected as signal jets if pT gt 35 GeV and |η |lt 28

      These signal jets electrons and muons were finally binned according to the signal regionsin Tables 333435 below

      Table 33 Selection criteria common to all signal regions

      Trigger EmissT

      Nlep 0b-tagged jets ⩾ 2

      EmissT 150 GeV

      |∆φ( jet pmissT )| gtπ5

      mbminT gt175 GeV

      Table 34 Selection criteria for signal regions A

      SRA1 SRA2 SRA3 SRA4anti-kt R=04 jets ⩾ 6 pT gt 808035353535 GeV

      m0b j j lt 225 GeV [50250] GeV

      m1b j j lt 225 GeV [50400] GeV

      min( jet i pmissT ) - gt50 GeV

      τ veto yesEmiss

      T gt150 GeV gt250 GeV gt300 GeV gt 350 GeV

      Table 35 Selection criteria for signal regions C

      SRC1 SRC2 SRC3anti-kt R=04 jets ⩾ 5 pT gt 8080353535 GeV

      |∆φ(bb)| gt02 π

      mbminT gt185 GeV gt200 GeV gt200 GeV

      mbmaxT gt205 GeV gt290 GeV gt325 GeV

      τ veto yesEmiss

      T gt160 GeV gt160 GeV gt215 GeV

      wherembmin

      T =radic

      2pbt Emiss

      T [1minus cos∆φ(pbT pmiss

      T )]gt 175 (314)

      33 Calculations 46

      and pbT is calculated for the b-tagged jet with the smallest ∆φ to the pmiss

      T direction andmbmax

      T is calculated with the pbT for the b-tagged jet with the largest ∆φ to the pmiss

      T direction

      m0b j j is obtained by first selecting the two jets with the highest b-tagging weight- from

      the remaining jets the two jets closest in the η minusφ plane are combined to form a W bosoncandidate this candidate is then combined with the closest of the 2 b-tagged jets in the η minusφ

      plane to form the first top candidate with mass m0b j j A second W boson candidate is formed

      by repeating the procedure with the remaining jets this candidate is combined with thesecond of the selected b-tagged jets to form the second top candidate with mass m1

      b j j Thecuts shown in Appendix1 Table A1 show the cumulative effect of the individual cuts appliedto the data the signal region cuts and the cumulative effects of individual cuts in SignalRegion C

      3332 Lepstop1

      Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

      radics=8 TeV pp collisions using 21 f bminus1 of

      ATLAS data[33]

      The stop decay modes considered were those to a top quark and the lightest supersym-metric particle (LSP) as well as a bottom quark and the lightest chargino where the charginodecays to the LSP by emitting a W boson We should expect the three dark matter models inthis paper with couplings to t and b quarks to show some visibility in this search

      The collider cuts program mimicked the object construction by selecting electrons andmuons with pT gt 10 GeV and |η |lt 247 and 24 respectively and jets with pT gt 20 GeVand |η |lt 10 b-jets with |η |lt 25 and pT gt 25 GeV were selected with random efficiency75 and mistag rate 2 (where non b-jets were mis-classified) We then removed any jetwithin ∆R = 02 of an electron and non overlapping jets with |η |lt 28 were used to removeany electrons and muons within ∆R = 04 of these while non-overlapping jets with |η |lt 25and pT gt 25 GeV were classified as signal jets Non overlapping electrons and muons withpT gt 25 GeV were classified as signal electrons and muons

      Signal selection defined six regions to optimize sensitivity for different stop and LSPmasses as well as sensitivity to two considered stop decay scenarios Three signal regionslabelled SRbC 1-3 where bC is a moniker for b+Chargino and 3 regions labelled SRtN 1-3where tN is a moniker for t+Neutralino were optimised for the scenarios where t1 decaysto the respective particles indicated by the labels Increasing label numbers correspond toincreasingly strict selection criteria

      33 Calculations 47

      The dominant background in all signal regions arises from dileptonic tt events All threeSRtN selectons impose a requirement on the 3-jet mass m j j j of the hadronically decayingtop quark to specifically reject the tt background where both W bosons from the top quarkdecay leponically

      For increasing stop mass and increasing difference between the stop and the LSP therequirents are tightened on Emiss

      T on the ratio EmissT

      radicHT where HT is the scalar sum of the

      momenta of the four selected jets and also tightened on mT

      To further reduce the dileptonic tt background for an event characterised by two one stepdecay chains a and b each producing a missing particle C use is made of the variable

      mT 2 =min

      pCTa + pC

      T b = pmissT

      [max(mTamtb)] (315)

      where mTi is the transverse mass 1 of branch i for a given hypothetical allocation (pCTa pC

      T b)

      of the missing particle momenta Eqn 315 is extended for more general decay chains byamT 2 [81] which is an asymmetric form of mT 2 in which the missing particle is the W bosonin the branch for the lost lepton and the neutrino is the missing particle for the branch withthe observed charged lepton

      ∆φ( jet12 pmissT ) the azimuthal angle between leading or subleading jets and missing mo-

      mentum is used to suppress the background from mostly multijet events with mis-measuredEmiss

      T

      Events with four or more jets are selected where the jets must satisfy |η | lt 25 and pT gt80604025 GeV respectively

      These signal particles were then classified according to the cutflows analysis in Fig36below and compared with the paper in Appendix1 Table A2

      3333 Lepstop2

      Search for direct top squark pair production in final states with two leptons in p pcollisions at

      radics=8TeV with the ATLAS detector[34]

      Three different analysis strategies are used to search for t1 pair production Two targett1 rarr b+ χ

      plusmn1 decay and the three body t1 rarr bW χ0

      1 decay via an off-shell top quark whilst

      1The transverse mass is defined as m2T = 2plep

      T EmissT (1minus cos(∆φ)) of branch i where ∆φ is the azimuthal

      angle between the lepton and the missing transverse momentum

      33 Calculations 48

      Table 36 Signal Regions - Lepstop1

      Requirement SRtN2 SRtN3 SRbC1 SRbC2 SRbC3∆φ( jet1 pmiss

      t )gt - 08 08 08 08∆φ( jet2 pmiss

      T )gt 08 08 08 08 08Emiss

      T [GeV ]gt 200 275 150 160 160Emiss

      T radic

      HT [GeV12 ]gt 13 11 7 8 8

      mT [GeV ]gt 140 200 120 120 120me f f [GeV ]gt - - - 550 700amT 2[GeV ]gt 170 175 - 175 200mT

      T 2[GeV ]gt - 80 - - -m j j j Yes Yes - - -

      one targets t1 rarr t + χ01 to an on shell quark decay mode The analysis is arranged to look

      at complementary mass splittings between χplusmn1 and χ0

      1 smaller and larger than the W bosonmass The smaller splitting provides sensitivity to t1 three body decay

      Object reconstruction was carried out by selecting baseline electronsmuons and τrsquos withpT gt 10 GeV and |η |lt 24724 and 247 respectively Baseline jets and b-jets were selectedwith pT gt 20 GeV and |η |lt 25 and b-jets with 70 efficiency Baseline jets were removedif within ∆R lt 02 of an electron and electrons and muons were removed if within ∆R lt 04of surviving jets Signal electrons required pT gt 10 GeV and signal jets pT gt 20 GeV and|η |lt 25

      The analysis made use of the variable mT 2 [82] the stransverse mass used to mea-sure the masses of pair-produced semi-invisibly decaying heavy particles It is definedas mT 2(pT1 pT2qT ) =

      minqT1+qT2=qT

      max[mT (pT1qT1)mT (pT2qT2)] where mT indicatesthe transverse mass pT1 and pT2 are vectors and qT = qT1 + qT2 The minimisation isperformed over all possible decompositions of qT

      Events were selected if they passed either a single-electron a single-muon a double-electron a double-muon or an electron-muon trigger Events are required to have exactlytwo opposite sign leptons and at least one of the leptons or muons must have a momentumgreater than 25 GeV and the invariant mass (rest mass) of the two leptons is required to begreater than 20 GeV In order to reduce the number of background events containing twoleptons produced by the on-shell decay of the Z boson the invariant mas must be outside711-111 GeV range

      Two additional selections are applied to reduce the number of background events withhigh mT 2 arising from events with large Emiss

      T due to mismeasured jets ∆φb lt 15 and∆φ gt 10 The quantity ∆φb is the azimuthal angle between pll

      T b = pmissT + pl1

      T +Pl2T The

      33 Calculations 49

      vector pllT b is the opposite of the vector sum of all the hadronic activity in the event For WW

      and tt backgrounds it measures the transverse boost of the WW system and for the signalthe transverse boost of the chargino-chargino system The ∆φ variable is the azimuthal anglebetween the pmiss

      T vector and the direction of the closest jet

      By requiring mT 2 gt 40 GeV the tt background is the dominant background in both thesame-flavoured (SF) and different-flavoured (DF) events

      Four signal regions are then defined as shown in Table 37 SR M90 has the loosestselection requiring MT 2gt 90 GeV and no additional requirement on jets and providessensitivity to scenarios with a small difference between the masses of the top-squark andchargino SR M110 and M120 have a loose selection on jets requiring two jets with pT

      gt 20 GeV and mT 2 to be greater than 110 GeV and 120 GeV respectively They providesensitivity to scenarios with small to moderate difference between the invariant masses of thestop quark and the charginos SR M100 has a tighter jet slection and provides sensitivity toscenarios with a large mass difference between the both stop quark and the chargino and thestop quark and the neutralino

      The analysis cut regions are summarised in Fig37 below

      Table 37 Signal Regions Lepstop2

      SR M90 M100 M110 M120pT leading lepton gt 25 GeV

      ∆φ(pmissT closest jet) gt10

      ∆φ(pmissT pll

      T b) lt15mT 2 gt90 GeV gt100 GeV gt110 GeV gt120 GeV

      pT leading jet no selection gt100 GeV gt20 GeV gt20 GeVpT second jet no selection gt50 GeV gt20 GeV gt20 GeV

      To validate the code we ran the cuts shown in Appendix 1 Table A3 with the givenMadgraph simulation and compared to [83] It was not clear what baseline cuts had beenemployed in the ATLAS paper so our results were normalised to the 2 SF electrons beforecomparison Some of the ATLAS cut numbers were less than 3 and we did not produce anyevents passing these cuts but clearly the margin of error at this level was greater than thenumber of ATLAS events

      33 Calculations 50

      3334 2bstop

      Search for direct third-generation squark pair production in final states with miss-ing transverse momentum and two b-jets in

      radics= 8 TeV pp collisions with the ATLAS

      detector[31]

      Two sets of scenarios are considered In the first the coloured sbottom is the onlyproduction process decaying via b1 rarr bχ0

      1 In the second the stop is the only productionprocess decaying via t1 rarr t χplusmn

      1 and the lightest chargino (χplusmn1 ) decays via a virtual W boson

      into the final state χ01 f f prime Both scenarios are characterised by the presence of two jets

      resulting from the hadronisation of b quarks and large missing transverse momentum Theproduction of a mediator in association with top quarks followed by an invisible mediatordecay to dark matter particles has a similar final state as that considered in the ATLAS stopsearches Thus we expect the ATLAS reach to have sensitivity to the three dark mattermodels considered in this thesis

      The collider cuts program mimicked object reconstruction by selecting baseline electronswith pT gt 7 GeV and |η |lt 247 and muons with pT gt 6 GeV and |η |lt 24 Baseline jetswere selected with pT gt 20 GeV and |η |lt 28 We then removed any jets within ∆R lt 02of an electron and any electron within ∆R lt 04 of surviving jets Signal electrons andmuons were required to have pT gt 25 GeV b-jets were selected with 60 tagging efficiencyand |η |lt 25 and pT gt 20 GeV

      Two sets of signal regions are defined to provide sensitivity to the kinematic topologiesassociated with the different mass splittings between the sbottom or the stop and the neu-tralino In all cases the presence of at least one primary vertex (with at least five tracks withpT gt 04 GeV) is required Events are selected with Emiss

      T gt 150 GeV and no electrons ormuons identified in the final state For signal region selections jets with |η | lt 28 are orderedaccording to their pT and 2 jets are required to be b-tagged

      The variables are defined as follows

      bull ∆φmin is the minimal azimuthal angle ∆φ between any of the three leading jets andpmiss

      T

      bull me f f (k) = sumki=1(p jet

      T )i +EmissT where the index refers to the pT ordered list of jets

      bull HT3 is defined as the scalar sum of the pT of the n leading pT ordered jets withoutincluding the 3 leading jets HT3 = sum

      ni=4(p jet

      T )i

      bull mbb is the invariant mass of the two b-tagged jets in the event

      33 Calculations 51

      bull mCT is the contransverse mass [82] and is a kinematical variable that can be usedto measure the masses of pair produced semi-invisibly decaying particles For twoidentical decays of heavy particles into two invisible particles v1 and v2 and twoinvisible particles mCT is defined as m2

      CT (v1v2) = [ET (v1)+ET (v2)]2 minus [pT (v1)minus

      pT (v2)]2 where ET =

      radicp2

      T +m2 The the contransverse mass is invariant under equaland opposite boosts of the parent particles in the transverse plane and is bounded fromabove by an analytical expression of the particle masses For production of sbottompairs the bound is given by mmax

      CT =m2(b)minusm2(χ0

      1 )

      m(b) and for tt events the bound is 135

      GeV A similar equation can be written for stop pairs in term of the stop and charginomasses

      A definition of the signal regions is given in the Table38

      Table 38 Signal Regions 2bstop

      Description SRA SRBEvent cleaning All signal regions

      Lepton veto No emicro after overlap removal with pT gt7(6) GeV for e(micro)Emiss

      T gt 150 GeV gt 250 GeVLeading jet pT ( j1) gt 130 GeV gt 150 GeVSecond jet pT ( j2) gt 50 GeV gt 30 GeVThird jet pT ( j3) veto if gt 50 GeV gt 30 GeV

      ∆φ(pmissT j1) - gt 25

      b-tagging leading 2 jets(pT gt 50 GeV |η | lt 25)

      2nd- and 3rd- leading jets(pT gt 30 GeV|η | lt 25)

      ∆φmin gt 04 gt 04Emiss

      T me f f (k) EmissT me f f (2) gt 025 Emiss

      T me f f (3) gt 04mCT gt 150200250300350 GeV -HT3 - lt 50 GeVmbb gt200 GeV -

      The analysis cuts are summarised in Table A4 of Appendix 1

      3335 ATLASMonobjet

      Search for dark matter in events with heavy quarks and missing transverse momentumin pp collisions with the ATLAS detector[30]

      Events were selected with high missing transverse momentum when produced in associa-tion with one or more jets identified as containing b quarks Final states with top quarks wereselected by requiring a high jet multiplicity and in some cases a single lepton The research

      33 Calculations 52

      studied effective contact operators in the form of a scalar operator Oscalar = sumqmqMN

      lowastqqχχ

      where MNlowast is the effective mass of the coupling and the quark and DM fields are denoted by

      q and χ respectively These are the equivalent of the simplified models in this paper with themediator integrated out

      Object reconstruction selected baseline electrons muons jets and τrsquos with pT gt 20 GeVand |η |lt 2525 and 45 respectively Signal jets were selected by removing any jets within∆R lt 02 of an electron and electrons and muons were selected by removing them if theywere within ∆R lt 04 of surviving jets b-jets were tagged with an efficiency of 60

      Only signal regions SR1 and SR2 were analysed

      The analysis split events into signal regions SR1 and SR2 according to the followingCriteria

      Table 39 Signal Region ATLASmonobjet

      Cut SR1 SR2Jet multiplicity n j 1minus2 3minus4

      bminusjet multiplicity nb gt 0 (60 eff) gt 0 (60 eff)Lepton multiplicity nl 0 0

      EmissT gt300 GeV gt200 GeV

      Jet kinematics pb1T gt100 GeV pb1

      T gt100 GeV p j2T gt100 (60) GeV

      ∆φ( j1EmissT ) gt 10 i = 12 gt 10 i = 14

      Where p jiT (pbi

      T ) represent the transverse momentum of the i-th jet b-tagged jet respec-tively If there is a second b-tagged jet the pT must be greater than 60 GeV

      3336 CMSTop1L

      Search for top-squark pair production in the single-lepton final state in pp collisionsat

      radics=8 TeV[41]

      This search was for pair production of top squarks in events with a single isolatedelectron or muon jets large missing transverse momentum and large transverse mass

      (MT =radic

      2EmissT pl

      T (1minus cos(∆φ))) where plT is the leptonrsquos transverse momentum and ∆(φ)

      is the difference between the azimuthal angles of the lepton and EmissT The 3 models

      considered should have some visibility at certain parameter combinations in this regionThe search focuses on two decay modes of the top squark leading to the two processespp rarr t tlowast rarr tt χ0

      1 χ01 rarr bbW+Wminusχ0

      1 χ01 and pp rarr t tlowast rarr bbχ

      +1 χ

      minus1 rarr bbW+Wminusχ0

      1 χ01 The

      33 Calculations 53

      lightest neutralino χ01 is considered the lightest supersymmetric particle and escapes without

      detection The analysis is based on events where one of the W bosons decays leptonicallyand the other hadronically resulting in one isolated lepton and 4 jets two of which originatefrom b quarks The two neutralinos and the neutrino from the W decay can result in largemissing transverse momentom The kinematical end point of MT is lt the mass of the Wboson However for signal events the presence of LSPs in the final state allows MT to exceedthe mass of the W boson The search is therefore for events with large MT The dominantbackground with large MT arises from tt decays where W bosons decay leptonically withone of the leptons not being identified

      To reduce the dominant tt background use was made of the MWT 2 variable defined as

      the minimum mother particle mass compatible with the transverse momenta and mass shellconstraints [84]

      Object reconstruction required electrons and muons to have pT gt 30 and 25 GeV respand |η |lt 25 and 21 respectively Baseline jets were required to have pT gt 30 and |η |lt 4and b-jets were tagged with an efficiency of 75 for |η |lt 24

      Events were chosen if they contained one isolated electron or muon at least 4 jets and atleast one b-tagged jet and Emiss

      T gt 100 GeV MT gt 160 GeV MWT 2 was required to be greater

      than 200 GeV and the smallest azimuthal angle difference between baseline jets and EmissT

      gt12

      Chapter 4

      Calculation Tools

      41 Summary

      Figure 41 Calculation Tools

      The simplified models were implemented in Feynrules [4] and the model files importedinto MicrOmegas v425 which was used to calculate the log-likelihoods of the astrophysicalobservables MicrOmegas was modified to allow only s-channel decays in both relic densityand gamma ray calculations Direct detection constraints were easy to implement for thescalar and vector models by coding the analytical expression for the spin independent WIMP-nucleon scattering cross section but due to the momentum dependence of the WIMP-nucleon

      42 FeynRules 55

      scattering cross section of the Majorana fermion model LUXCalc v101 [79] was modifiedfor this model to implement momentum dependent cross sections [85] We used MicrOmegasv425 to calculate relic density values and to produce gamma ray spectra for comparisonwith the Fermi-LAT results Parameter scans were carried out using MultiNest v310 withthe parameter ranges given in Table 51 and using astrophysical constraints only Oncethese scans had converged a random sample of 900 points was chosen (weighted by theposterior probability) and then used to calculate collider constraints using the MadgraphMonte Carlo generator This included comparison with mediator production and decay limitsfrom the LHC experiments and the reproduction of searches for dark matter plus other finalstate objects which required dedicated C++ code that reproduces the cut and count analysesperformed by CMS and ATLAS

      42 FeynRules

      FeynRules is a Mathematicareg package that allows the calculation of Feynman rules inmomentum space for any QFT physics model The user needs to provide FeynRules withminimal information to describe the new model contained in the model-file This informationis then used to calculate a set of Fenman rules associated with the Lagrangian The Feynmanrules calculated by the code can then be used to implement new physics models into otherexisting tools such as Monte Carlo (MC) generators via a set of interfaces developed andmaintained by the MC authors

      Feynrules can be run with a base model = SM (or MSSM) and user defined extensions tothe base model

      43 LUXCalc

      LUXCalc is a utility program for calculating and deriving WIMP-nucleon coupling limitsfrom the recent results of the LUX direct search dark matter experiment The program hasbeen used to derive limits on both spin-dependent and spin-independent WIMP-nucleoncouplings and it was found under standard astrophysical assumptions and common otherassumptions that LUX excludes the entire spin-dependent parameter space consistent witha dark matter interpretation of DAMArsquos anomalous signal and also found good agreementwith published LUX results [79]

      We modified the program for the Majorana fermion model to implement momentumdependent cross -sections [85]

      44 Multinest 56

      44 Multinest

      Multinest is an efficient Bayesian Inference Tool which uses a simultaneous ellipsoidalnested sampling method a Monte Carlo technique aimed at an efficient Bayesian Evidencecalculation with posterior inferences as a by product

      Bayes theorem states that

      Pr(θ |DH) =Pr(D|θ H)Pr(θ |H)

      Pr(D|H) (41)

      Here D is the data and H is the Hypothesis Pr(θ |DH) is the posterior probability ofthe parameters Pr(D|θ H) = L(θ) is the likelihood Pr(θ |H) is the prior and Pr(D|H) isthe evidence

      The evidence Pr(D|H) =int

      Pr(θ |DH)Pr(θ |H)d(θ) =int

      L(θ)Pr(θ |H)d(θ) Thus ifwe define the prior volume dX = Pr(θ |H)d(θ) and

      X(λ ) =int

      L(θ)gtλ

      Pr(θ |H)d(θ) (42)

      where the integral extends over the region of parameter space within the iso likelihoodcontour L(θ) = λ then the evidence integral can be written

      int 10 L (X)dX where L (X) the

      inverse of equation 42 and is a monotonically decreasing function of X (the prior volumewhich is 1 for the total sample space and 0 for a single point)

      Thus if one can evaluate likelihoods L (Xi) where 0 lt Xn lt lt X2 lt X1 lt X0 = 1the evidence can be estimated as sumLiwi where wi = 5(Ximinus1 minusXi+1) Ellipsoidal samplingapproximates the iso likelihood contour by a D dimensional covariance matrix of the currentset of active points New points are selected from the prior within this ellipsoidal bounduntil one is obtained that has a likelihood greater than the likelihood of the removed lowestlikelihood point This method is extended for multimodal sampling where there may bemore than one local maximum by clustering points that are well separated and constructingindividual ellipsoidal bounds for each cluster For more details consult [7]

      The results of the analysis are expressed as a data file containing the sample probability-2loglikelihood followed by the parameters of the model for the particular sampling Theprobabilities are normalised to total 10

      45 Madgraph 57

      45 Madgraph

      Madgraph is a general purpose matrix element calculator and Monte Carlo generator Basedon user-supplied Feynman rules (eg the output of Feynrules) the package calculates the tree-level matrix elements for any processes that the user specifies and generates samples of LHCevents including an accurate simulation of the proton collisions themselves (which requires aknowledge of the parton distribution functions that determine the relative proportion of eachparton inside the nucleus) Hadronisation of final state quarks is not possible in Madgraphand hence the code must be used in conjunction with Pythia which performs a parton shower

      The Madgraph output was automatically showered by Pythia (Pythia describes partonradiation as successive parton emissions using Markov chain techniques based on Sudakovform factors and this description is only formally correct for the soft and collinear emissions)A Madgraph simulation is at the level of the fundamental particles (quarks and leptons)Madgraph generates a limited set of matrix element processes at LO representing hard (highenergy) emissions and passes these to Pythia for showering The final state partons in theMadgraph simulation must be matched to the Pythia final state partons to avoid doublecounting

      The output from Pythia is automatically fed through Delphes where basic collider cutscan be performed (calculation of the numbers of particles and jets that would be observed atdifferent energies and locations within the collider) The output of the Delphes simulation is afile containing reconstructed events that can be analysed using the standard ROOT framework(the most popular data analysis package in particle physics) [86]

      The parameters that gave the best fits to the direct [16] and indirect DM experiments [87]and relic density [59] are ranked by posterior probability from the Multinest [7] output

      The parameters in the Madgraph simulations are stored in files in the cards sub directoryand in particular the run card param card and Delphes card record items relevent to thesimulation

      In the run card the number of events the Beam 1 and 2 total energy in GeV (4000 each)and the type of jet matching were modified (to MLM type matching where the final statepartons in Madgraph are clustered according to the KT jet algorithm to find the equivalentparton shower history of the event there is a minimum cutoff scale for the momentum pT

      given by the xqcut parameter) The maxjetflavour (=5 to include b quarks) and xqcut = 30GeV parameters were also modified

      46 Collider Cuts C++ Code 58

      The param card was modified to specify the masses of the new particles and the autodecay modes The Delphes card is overwritten with either the ATLAS or CMS defaultcards which record tracking efficiencies and isolation momentum and energy resolutionscalorimeter details jet finder and b and τ -tagging parameters However the isolation andtagging in this study were done by the collider cuts C++ code since multiple sets of cutswere applied to the same output from Pythia

      When performing simulations of the three dark matter models it was necessary to definethe proton and jets to contain b quarks as this is not the default in Madgraph The processesgenerated included 1 and 2 jets

      46 Collider Cuts C++ Code

      Because of the volume of simulations required for the collider constraints we used C++code (originally developed by Martin White and Sky French and modified by me) that readsthe ROOT files and outputs the numbers of events that would have been observed under thechosen collider experiments had the model been true for the given parameters Thus the sameoutput files can be used for all of the collider cuts and hence no cuts were performed throughthe Delphes cards

      In order to validate the C++ code we replicated the observed events with the papersdescribing the various collider searches (see Appendix)

      Chapter 5

      Majorana Model Results

      51 Bayesian Scans

      To build some intuition we first examined the constraining effect of each observable on itsown We ran separate scans for each constraint turned on and then for all the constraintstogether In Fig 51 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together The paramaters were scanned over the range in Table 51 below

      Table 51 Scanned Ranges

      Parameter mχ [GeV ] mS[GeV ] λt λb λτ

      Range 0minus103 0minus103 10minus5 minus10 10minus5 minus10 10minus5 minus10

      In Fig51 we can see that the preferred DM mass is centered around 35 GeV primarily dueto the influence of the Fermi-LAT gamma ray constraint whilst the relic density constraintallows a wide range of possible mediator and DM masses Direct detection (LUX) constraintssplit the mass regions into a top left and bottom right region and together these split theoverall mass parameter regions into two regions by mediator mass but with DM mass around35 GeV Later studies using a lower excess gamma ray spectrum tend to increase the preferredDM mass up to 50 GeV [76] and [88]

      The marginalised posterior probabilities for all combinations of parameters taken twoat a time are shown in Fig52 These plots have all constraints (Gamma LUX and RelicDensity) applied together The idea of these graphs is to study the posterior probabilities ofcombinations of two paramaters taken at a time (the off-diagonal graphs) The diagonal plotsshow marginalised histograms of a single parameter at a time- this gives an indication of thespread of each parameter The plots show that mS and λb show a high degree of correlationfor mS gt 100 GeV but there is also a range of values of mS around 35 GeV which are

      51 Bayesian Scans 60

      1 0 1 2 3 4log10(mχ)[GeV]

      1

      0

      1

      2

      3

      4

      log 1

      0(m

      s)[GeV

      ]

      (a) Gamma Only

      1 0 1 2 3 4log10(mχ)[GeV]

      1

      0

      1

      2

      3

      4

      log 1

      0(m

      s)[GeV

      ]

      (b) Relic Density

      1 0 1 2 3 4log10(mχ)[GeV]

      1

      0

      1

      2

      3

      4

      log 1

      0(m

      s)[GeV

      ]

      (c) LUX

      05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

      05

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      s)[GeV

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      (d) All Constraints

      Figure 51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether

      51 Bayesian Scans 61

      00

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      χ)[GeV

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      t)

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      b)

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      τ)

      00 05 10 15 20 25 30

      ms[Gev]5 4 3 2 1 0 1

      log10(λt)5 4 3 2 1 0 1

      log10(λb)5 4 3 2 1 0 1

      log10(λτ)

      Figure 52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter

      52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 62

      possible for λb around 001 λt and λτ show a wide range of possible values for two possibleranges of mS lt 35 GeV and greater than 100 GeV mχ shows a narrow range of possiblevalues around 35 GeV for all values of the other parameters

      52 Best fit Gamma Ray Spectrum for the Majorana FermionDM model

      We can see from the Fig51(a) scan that the maximum log-likelihood Fermi-LAT gammaray spectrum on its own shows values that centre around 35 GeV This is not changed byadding a relic density constraint or a direct detection (LUX) constraint

      The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraintsis shown in Fig53 The parameters for this point were

      Table 52 Best Fit Parameters

      Parameter mχ [GeV ] mS[GeV ] λt λb λτ

      Value 3332 49266 0322371 409990 0008106

      10-1 100 101 102

      E(GeV)

      10

      05

      00

      05

      10

      15

      20

      25

      30

      35

      E2dφd

      E(G

      eVc

      m2ss

      r)

      1e 6

      Best fitData

      Figure 53 Gamma Ray Spectrum

      The relic density (Ωh2) at this point has a value of 0135 but the point is separated onlyslightly in posterior probability from points with a value of 01199 with mS lt 10 GeV (iethe posterior probability is almost the same differing by less than 1)

      To better understand the interplay of the different constraints the three graphs in Fig54show the log likelihood surfaces and individual best fit points (best log likelihoods) for the γ

      and Ω scans seperately and then combined The diagrams each choose a pair of couplings(fixing the remaining parameters) and display the 2d surfaces of minuslog(χ2) (proportional to

      52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 63

      the log likelihoods of the coupling combinations) obtained by varying the pair of couplingsThese graphs do not show marginalised posterior probabilities but rather are based on the-2log likelihood calculations directly from MicrOmegas for the given coupling combinationsWe have termed this value as χ2 for convenience and used the log of this value to give abetter range for the colours These log likelihood values are what drive the MultiNest searchof parameter space for the best parameter combinations

      The graphs show a complex interplay of the different constraints resulting often in acompromise parameter combination The arrow on each graph points to the maximum point We have chosen a single representative set of couplings near the best overall fit point for allthe graphs

      52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 64

      00 05 10 15 20 25 30

      log10(mχ)

      00

      05

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      10(m

      S)

      Max

      minuslog10(χ2(Γ)) λt = 487 λτ = 024 λb = 0344

      16

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      00 05 10 15 20 25 30

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      S)

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      minuslog10(χ2(Omega)) λt = 487 λτ = 024 λb = 0344

      28

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      00 05 10 15 20 25 30

      log10(mχ)

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      S)

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      minuslog10(χ2(Both)) λt = 487 λτ = 024 λb = 0344

      16

      14

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      6

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      0

      Both Maximum at mχ=8317 GeV mS=2884GeV The best fit point for these couplings maybe above below or between the individual bestfit points but will on average be between thepoints

      Figure 54 Plots of log likelihoods by individual and combined constraints Masses in GeV

      53 Collider Constraints 65

      53 Collider Constraints

      531 Mediator Decay

      1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof τ leptons in pp collisions [44] Refer also to Chapter 3 for details

      We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

      The results of the calculations are shown in the graph Fig55 and can be compared withthe limits in the table from the paper [44] in Fig 56 The black line is the observed ATLASlimit

      Figure 55 σ lowastBr(σ rarr ττ) versus Mass of Scalar

      0 200 400 600 800

      mS[GeV]

      10

      5

      0

      log 1

      0(σ

      (bbS

      )lowastB

      (Sgtττ

      ))[pb]

      Observed LimitLikely PointsExcluded Points

      0

      20

      40

      60

      80

      100

      120

      0 5 10 15 20 25 30 35 40 45

      We can see that only one point here exceeded the limit and the points showed a widespread over σ(bbS)lowastBr(σ rarr ττ) by mS space which is consistent with the spread of pointson the MultiNest scan on the mS by λτ plane causing a spread of branching ratios to the τ

      quark It may seem odd that only one point was excluded here but this point was sampledwith a very low probability having high coupling constants of 597 to the τ quark and 278 tothe t quark and only just exceeds the limit As such it can be explained as a random outlier

      53 Collider Constraints 66

      Figure 56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar

      2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

      This scan calculates the contributions of b-quark associated Higgs boson productionreproduced from [43] Refer to Chapter 3 for details

      We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

      The cross section times branching ratios to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig57 The black line is the CMSLimit

      Figure 57 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar

      0 200 400 600 800

      mS[GeV]

      15

      10

      5

      0

      5

      log

      10(σ

      (bS

      +X

      )lowastB

      (Sgt

      bb))

      [pb]

      Observed LimitLikely PointsExcluded Points

      0

      20

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      120

      0 50 100 150 200 250

      53 Collider Constraints 67

      The results of this scan were compared to the limits in [89] with the plot shown inFig58

      Figure 58 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

      We can see that there are a number of exclusions across the range of mS which occuredfor high values of the coupling constants gt 1 for λbλt λtau - these can be seen also on Fig59The excluded parameter combinations represent less than 2 of the total sampled number

      532 Collider Cuts Analyses

      We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude and the results are shown against theposterior probabilities of the parameter combinations in Fig 59

      The circles in Figure 59 show which points were excluded by the collider cuts as well asthe limits on cross sections times branching ratios to each of τ+τminus and bb

      All the excluded points under the heading Collider Cuts showed exclusion by the Lepstop0experiment (Search for direct pair production of the top quark in all hadronic final states

      53 Collider Constraints 68

      0 1 2 3

      log10(mχ)[GeV]

      0

      1

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      3

      log 1

      0(m

      s)[GeV

      ]Collider Cuts

      σ lowastBr(σgt bS+X)

      σ lowastBr(σgt ττ)

      (a) mχ by mS

      6 5 4 3 2 1 0 1 2

      log10(λt)

      0

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      s)[GeV

      ](b) λt by mS

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      s)[GeV

      ]

      (f) λb by mS

      Figure 59 Excluded points from Collider Cuts and σBranching Ratio

      53 Collider Constraints 69

      [32]) Some of these same points were also excluded by Lepstop2 2bstop CMSTop andATLASMonobjet experiments There were expected events at other points but none weresignificant at the 95 exclusion level These significant exclusions occurred mainly wherethe mass of the mediator was over 100 GeV and higher values of the couplings to b and tquarks but there were a few points with mediator mass around 25 GeV The exclusions withmediator mass around 25 GeV were all accompanied by couplings to t quarks of order 10which is probably non-perturbative

      Looking at the exclusions in Fig 59 we can see that these occur mainly in the regionof higher mediator mass gt 100 GeV and lt 400 GeV These exclusions come from theastrophysical constraints (Gamma) which show a strong correlation between λb and mS andhigh values of λt gt 1 However these mass regions do not overlap at the same couplingcombinations as the high likelihood region The green and white circles correspond to theexcluded points in Figs 57 and 58 respectively

      The remaining regions of highest likelihood can be seen to be where 0001 λb lt 01accompanied by λt lt 0001 and mass of the mediator mS lt 10 GeV or 400 GeV lt mS lt1000GeV with mχ sim 35 GeV

      The plot of λt by λb shows the significant regions do not overlap with the collider cutsand cross section constraints since the latter occur at high values of either coupling while thelikely region has low values of these couplings (also accompanied by low couplings to λτ )

      Chapter 6

      Real Scalar Model Results

      61 Bayesian Scans

      To build some intuition we first examined the constraining effect of each observable on itsown We ran seperate scans for each constraint turned on and then for all the constraintstogether In Figure 61 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together

      In Figure 61 we can see that there are is a narrow region which dominates when allconstraints are taken into account This centers on mχ=9 GeV and a range of values for mS

      from 35 GeV to 400 GeV The region is not the highest likelihood region for the Fermi-LATgamma ray spectrum on its own (as can be seen from the individual plot - there is a narrowband to the left of the highest likelihood band) nor for the LUX likelihood or relic densitybut represents a compromise between the three constraints

      The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig62 The diagonal plots show marginalised histograms of a singleparameter at a time

      61 Bayesian Scans 71

      05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

      05

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      s)[GeV

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      05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

      05

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      05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

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      05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

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      (d) All Constraints

      Figure 61 Real Scalar Dark Matter - By Individual Constraint and All Together

      61 Bayesian Scans 72

      00

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      log10(λb)5 4 3 2 1 0 1

      log10(λτ)

      Figure 62 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter

      62 Best fit Gamma Ray Spectrum for the Real Scalar DM model 73

      62 Best fit Gamma Ray Spectrum for the Real Scalar DMmodel

      We can see from Fig61 that the maximum log-likelihood Fermi-LAT Gamma ray spectrumon its own shows values of mχ that centre around 35 GeV However the LUX constraintin particular moves the best fit to a point around 9 GeV- the tight LUX constraint onlydisappears for low masses and at that point the proportion of τ+τminus annihilation needs toincrease to fit the constraints

      The gamma ray spectrum at the points of best fit for Fermi-LAT including all constraintsis shown in Fig63 The parameters for these points were respectively and involve decays tomainly τ and b quarks respectively

      Table 61 Best Fit Parameters

      Parameter mχ [GeV ] mS[GeV ] λt λb λτ

      Value 932 3526 000049 0002561 000781

      10-1 100 101 102

      E(GeV)

      10

      05

      00

      05

      10

      15

      20

      25

      30

      35

      E2dφdE

      (GeVc

      m2ss

      r)

      1e 6

      Best fitData

      Figure 63 Gamma Ray Spectrum

      This is not a particularly good fit and it occcurs at a relic density Ωh2 of 335 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

      63 Collider Constraints 74

      63 Collider Constraints

      631 Mediator Decay

      1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44]

      We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

      The results of the scan are shown in the graph Fig64 and can be compared with thelimits in the table from the paper [44] in Fig 56The black line is the observed ATLAS limit

      Figure 64 σ lowastBr(σ rarr ττ) versus Mass of Scalar

      0 200 400 600 800

      mS[GeV]

      8

      6

      4

      2

      0

      2

      4

      log 1

      0(σ

      (bbS

      )lowastB

      (Sgtττ

      ))[pb]

      Observed LimitLikely PointsExcluded Points

      050

      100150200250300350

      0 10 20 30 40 50 60

      We can see that there is a narrow band in the region of interest in the σ(bbS)lowastB(S gt ττ)

      by mS plane due to the high degree of correlation between mS and λτ seen in the likelihoodscan Values for mS gt 300Gev are excluded

      2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

      We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinations

      63 Collider Constraints 75

      randomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

      The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig65 The black line is the CMSlimit

      Figure 65 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

      0 200 400 600 800

      mS[GeV]

      8

      6

      4

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      4

      log

      10(σ

      (bS

      +X

      )lowastB

      (Sgt

      bb))

      [pb]

      Observed LimitLikely PointsExcluded Points

      050

      100150200250300350

      0 10 20 30 40 50 60

      The results of this scan were compared to the limits in [89] with the plot shown inFig58

      We can see a similar shape for the region of interest to that in Fig64 due to the highcorrelation between mS and λb over the region preferred by astrophysical constraints resultingin branching ratios highly correlated to ms Only one point is excluded with mS gt 500GeV

      632 Collider Cuts Analyses

      We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expected

      63 Collider Constraints 76

      with 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 66

      We can see that collider cuts show some exclusions only for masses mχ below the highlikelihood region but these exclusions do not occur at the same λb and λt combinations asseen from the plot of these two couplings except for a small region at the highest mS andcouplings

      All but one of the collider cut exclusions were excluded by the Lepstop0 experiment (thiswas excluded by 2bstop with high coupling to b quarks of 73) A number of the excludedpoints were also excluded by Lepstop2 2bstop CMSTopDM1L and ATLASMonobJet-(around 30 of the total for each)

      63 Collider Constraints 77

      0 1 2 3

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      s)[GeV

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      σ lowastBr(σgt bS+X)

      σ lowastBr(σgt ττ)

      (a) mχ by mS

      5 4 3 2 1 0 1

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      s)[GeV

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      s)[GeV

      ]

      (f) λb by mS

      Figure 66 Excluded points from Collider Cuts and σBranching Ratio

      Chapter 7

      Real Vector Dark Matter Results

      71 Bayesian Scans

      In the Figure 71 we can see that the preferred DM mass is centered around 35 GeV primarilydue to the influence of the Fermi-LAT gamma ray constraint with relic density having a widerange of possible mediator and DM masses The result of combining all the constraints is todefine a small region around mχ = 8 GeV and mS = 20 GeV which dominates the posteriorprobability

      The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig72 The diagonal plots show marginalised histograms of a singleparameter at a time

      71 Bayesian Scans 79

      05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

      1

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      s)[GeV

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      05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

      05

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      1 0 1 2 3 4log10(mχ)[GeV]

      05

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      05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

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      (d) All Constraints

      Figure 71 Real Vector Dark Matter - By Individual Constraint and All Together

      71 Bayesian Scans 80

      00

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      τ)

      00 05 10 15 20 25 30

      ms[Gev]5 4 3 2 1 0 1

      log10(λt)5 4 3 2 1 0 1

      log10(λb)5 4 3 2 1 0 1

      log10(λτ)

      Figure 72 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter

      72 Best fit Gamma Ray Spectrum for the Real Vector DM model 81

      72 Best fit Gamma Ray Spectrum for the Real Vector DMmodel

      The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraints isshown in Fig73

      Table 71 Best Fit Parameters

      Parameter mχ [GeV ] mS[GeV ] λt λb λτ

      Value 8447 20685 0000022 0000746 0002439

      10-1 100 101 102

      E(GeV)

      10

      05

      00

      05

      10

      15

      20

      25

      30

      35

      E2dφdE

      (GeVc

      m2s

      sr)

      1e 6

      Best fitData

      Figure 73 Gamma Ray Spectrum

      This is not a particularly good fit and it occcurs at a relic density Ωh2 of 025 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

      73 Collider Constraints

      731 Mediator Decay

      1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44] (See Chapter 3 for more details)

      We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

      73 Collider Constraints 82

      The results of the scan are shown in the graph Fig74 and can be compared with thelimits in the table from the paper [44] in Fig 56

      Figure 74 σ lowastBr(σ rarr ττ) versus Mass of Scalar

      0 200 400 600 800

      mS[GeV]

      8

      6

      4

      2

      0

      2

      log 1

      0(σ

      (bbS

      )lowastB

      (Sgtττ

      ))[pb]

      Observed LimitLikely PointsExcluded Points

      0100200300400500600700800

      0 20 40 60 80 100120140

      We can see that the region of highest likelihood is limited to mS lt 100 GeV and a narrowrange of couplings to λτ giving a narrow region of interest which falls well below the limit

      2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

      We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

      The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig75 The black line is the CMSlimit

      The results of this scan were compared to the limits in [89] with the plot shown in Fig58

      We can see that again the region of interest for mS is below 100 GeV and the points fallwell below the limit

      73 Collider Constraints 83

      Figure 75 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

      0 200 400 600 800

      mS[GeV]

      8

      6

      4

      2

      0

      2

      4

      log

      10(σ

      (bS

      +X

      )lowastB

      (Sgt

      bb))

      [pb]

      Observed LimitLikely PointsExcluded Points

      0100200300400500600700800

      0 20 40 60 80 100120140

      732 Collider Cuts Analyses

      We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 76

      We can see from Figure 76 that like the real scalar model the region of highest likelihoodhas λb and λt lt 001 while the excluded points from collider cuts and cross sections areabove 01 Most of the excluded points have mχ gt than the values in the region of highestlikelihood and all points are widely separated in parameter space from this region

      Once again 89 of excluded points were excluded by Lepstop0 with many also beingexcluded by the other experiments (50 by Lepstop2 and 56 by CMSTopDM1L 35 byLepstop2 and 20 by ATLASMonobjet)

      73 Collider Constraints 84

      0 1 2 3

      log10(mχ)[GeV]

      0

      1

      2

      3

      log 1

      0(m

      s)[GeV

      ]Collider Cuts

      σ lowastBr(σgt bS+X)

      σ lowastBr(σgt ττ)

      (a) mχ by mS

      5 4 3 2 1 0 1

      log10(λt)

      0

      1

      2

      3

      log 1

      0(m

      s)[GeV

      ](b) λt by mS

      5 4 3 2 1 0 1

      log10(λb)

      5

      4

      3

      2

      1

      0

      1

      log 1

      0(λ

      t)

      (c) λb by λt

      5 4 3 2 1 0 1

      log10(λb)

      5

      4

      3

      2

      1

      0

      1

      log 1

      0(λ

      τ)

      (d) λb by λτ

      5 4 3 2 1 0 1

      log10(λt)

      5

      4

      3

      2

      1

      0

      1

      log 1

      0(λ

      τ)

      (e) λt by λτ

      5 4 3 2 1 0 1

      log10(λb)

      0

      1

      2

      3

      log 1

      0(m

      s)[GeV

      ]

      (f) λb by mS

      Figure 76 Excluded points from Collider Cuts and σBranching Ratio

      Chapter 8

      Conclusion

      We first conducted Bayesian scans over mass and coupling parameters for three simplifiedmodels - Majorana fermion real scalar and real vector using direct detecion (LUX) indirectdetection (GCE γ rays) and relic density constraints to find the regions of highest likelihoodWe ran these scans one constraint at a time and then together to better understand the interplayof the different constraints on best fit and likely parameter regions

      We also developed and tested C++ collider cut software specifically designed to mimickthe cuts applied in a number of ATLAS and CMS experiments that searched for Emiss

      T inconnection with specific beyond the standard model (BSM) theories of Dark Matter

      We then selected parameter combinations by randomly sampling the scanned resultsbased on the posterior probability distribution of the points and applied collider cuts andcross section times branching ratio calculations to these points to see if there would havebeen any sensitivity from the ATLAS and CMS experiments

      We found that the Majorana fermion model showed the best overall fit to all the constraintswith relic density close to the experimental value of 01199 and a low χ2 error on the gammaspectrum fit The preferred mass mχ was around 35 GeV and mS ranging from lt10 GeV to500 GeV The smaller masses give a better fit for relic density and also are well separatedfrom current collider constraints mainly due the the fact that the exclusions occur mainlywith higher couplings to t quarks (greater than 1) which is consistent with the fact that manyof the experiments were targeting dark matter pair production associated with t quark decaysNevertheless there were some exclusions that did fall in the region sampled by randomselection from the posterior probability distribution

      The preferred parameter regions were also consistent with the observation in [20] thatthe Majorana model showed the best overall fit with similar mass constraints We find thatthe couplings to the SM fermions should all be less that 001

      86

      The scalar and vector DM models give slightly less good fits especially to the currentrelic density estimate of 01199 but again the current collider constraints are well separatedfrom the best fit region and in fact none of the points randomly sampled from the posteriordistribution showed any exclusion The best fit parameter combination favours couplings toSM fermions of less than 001

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      Appendix A

      Validation of Calculation Tools

      Lepstop0Search for direct pair production of the top squarkin all hadronic final states in proton=proton collisions atradic

      s=8 TeV with the ATLAS detector [32]

      Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1) mt1=600 GeV mn1=1 GeVWhere t1 is the top squark and n1 is the neutralino of the MSSM

      94

      Table A1 0 Leptons in the final state

      Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 10000 460 460 100Emiss

      T gt 130GeV 8574 395 451 88Leptonveto 6621 307 285 108Emiss

      T gt 150GeV 6316 293 294 100Jet multiplicityand pT 1099 48 51 94|∆φ( jet pmiss

      T )|gt π5 902 40 42 96gt= 2b jets 396 19 18 106τ veto 396 19 18 106mbmin

      T gt 175GeV 270 13 13 105SRA1 79 4 4 104SRA2 70 3 3 93SRA3 67 3 3 97SRA4 52 2 2 91SRC exactly5 jets 1297 50 60 83SRC |∆φ( jet pmiss

      T )|gt π5 1082 50 50 99SRC gt= 2b jets 430 20 20 97SRC τ veto 430 20 20 97SRC |∆φ(bb)|gt 02π 367 17 17 99SRC1 246 12 11 105SRC2 177 9 8 116SRC3 152 8 7 116

      95

      Lepstop1Search for direct top squark pair production infinal states with one isolated lepton jets and missing trans-verse momentum in

      radics = 8 TeV pp collisions using 21 f bminus1

      of ATLAS data[33]

      Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1)(t1 gt b x+1 )(t1 gt b xminus1 )mt1=650 GeV mn1=1 GeV Where t1 is the top squark n1 is the neutralino xplusmn1 are thecharginos

      96

      Table A2 1 Lepton in the Final state

      Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 100000 99990 99990 100Electron(= 1signal) 11558 11557 11785 984 jets(80604025) 5187 5186 6271 83gt= 1btag 4998 4997 5389 93Emiss

      T gt 100GeV [allSRs] 4131 4131 4393 94Emiss

      T radic

      HT gt 5 [allSRs] 4026 4026 4286 94∆φ( jet2 pmiss

      T )gt 08 [allSRs] 3565 3565 3817 93∆φ( jet1Pmiss

      T )gt 08 [not SRtN2] 3446 3446 3723 93Emiss

      T gt 200GeV (SRtN2) 2027 2027 2186 93Emiss

      T radic

      HT gt 13(SRtN2) 1156 1156 1172 99mT gt 140GeV (SRtN2) 839 839 974 86Emiss

      T gt 275GeV (SRtN3) 948 948 965 98Emiss

      T radic

      HT gt 11(SRtN3) 932 932 937 99mT gt 200GeV (SRtN3) 456 456 566 81Emiss

      T gt 150GeV (SRbC1minusSRbC3) 2803 2803 3080 91Emiss

      T radic

      HT gt 7(SRbC1minusSRbC3) 2742 2742 3017 91mT gt 12GeV (SRbC1minusSRbC3) 1917 1917 2347 82Emiss

      T gt 160GeV (SRbC2SRbC3) 1836 1836 2257 81Emiss

      T radic

      HT gt 8(SRbC2SRbC3) 1788 1788 2175 82me f f gt 550GeV (SRbC2) 1741 1741 2042 85me f f gt 700GeV (SRbC3) 1075 1075 1398 77Muon(= 1signal) 11379 11378 11975 954 jets(80604025) 5264 5263 6046 87gt= 1btag 5101 5100 5711 89Emiss

      T gt 100GeV [allSRs] 4199 4199 4770 88Emiss

      T radic

      HT gt 5 [all SRs] 4115 4115 4653 88∆φ( jet2 pmiss

      T )gt 08 [all SRs] 3622 3622 4077 89∆φ( jet1 pmiss

      T )gt 08 [not SRtN2] 3515 3515 3970 89Emiss

      T gt 200GeV (SRtN2) 2136 2136 2534 84Emiss

      T radic

      HT gt 13(SRtN2) 1261 1261 1441 87mT gt 140GeV (SRtN2) 928 928 1033 90Emiss

      T gt 275GeV (SRtN3) 1036 1036 1274 81Emiss

      T radic

      HT gt 11(SRtN3) 1018 1018 1235 82mT gt 200GeV (SRtN3) 497 497 612 81Emiss

      T gt 150GeV (SRbC1minusSRbC3) 2891 2891 3380 86Emiss

      T radic

      HT gt 7(SRbC1minusSRbC3) 2821 2821 3290 86mT gt 120GeV (SRbC1minusSRbC3) 1977 1977 2290 86Emiss

      T gt 160GeV (SRbC2SRbC3) 1913 1913 2220 86Emiss

      T radic

      HT gt 8(SRbC2SRbC3) 1863 1863 2152 87me f f gt 550GeV (SRbC2) 1826 1826 2116 86me f f gt 700GeV (SRbC3) 1131 1131 1452 78

      97

      Lepstop2Search for direct top squark pair production infinal states with two leptons in

      radics =8 TeV pp collisions using

      20 f bminus1 of ATLAS data[83][34]

      Simulated in Madgraph with p p gt t1 t1(t1 gt b w+ n1)(t1 gt b wminusn1) mt1=180 GeV mn1=60 GeV Where t1 is the top squark n1 is the neutralino b1 is the bottom squark

      Table A3 2 Leptons in the final state

      Cut RawMe Scaled ATLAS MeATLAS PercentNoCuts 100000 324308 199000 1632BaselineCuts 3593 11652 7203 1622SF Signal Leptons 1699 5510 5510 1002OSSF Signal Leptons 1632 5293 5233 101mll gt 20Gev 1553 5037 5028 100leadinglepton pT 1479 4797 5048 95|mll minusmZ|gt 20Gev 1015 3292 3484 94∆φmin gt 10 850 2757 2160 128∆φb lt 15 680 2205 1757 126SRM90 [SF ] 13 42 60 70SRM120 [SF ] 0 0 19 0SRM100 +2 jets [SF ] 0 0 17 0SRM110 +2 jets [SF ] 0 0 1 02DF Signal Leptons 1737 5633 7164 792OSDF Signal Leptons 1670 5416 5396 100mll20Gev 1587 5147 5136 100leadinglepton pT 1501 4868 4931 99∆φmin gt 10 1247 4044 3122 130∆φb lt 15 1010 3276 2558 128SRM90 [SF ] 21 68 95 72SRM120 [SF ] 0 0 23 0SRM100 +2 jets [SF ] 0 0 19 0SRM110 +2 jets [SF ] 0 0 14 0

      98

      2bstopSearch for direct third-generation squark pair pro-duction in final states with missing transverse momentumand two b-jets[31]

      Simulated in Madgraph with p p gt b1 b1(b1 gt b n1)(b1 gt b n1)

      SRAmb1=500 GeV mn1=1 GeV SRBmb1=350 GeV mn1=320 GeVWhere b1 is the bottom squark and n1 is the neutralino

      Table A4 2b jets in the final state

      Cut RawMe Scaled ATLAS MeATLAS PercentSRA Nocuts 100000 173800 1738 100SRA MET gt 80 93407 162341 1606 101SRA Leptonveto 93302 162159 1505 108SRA MET gt 150 79449 138082 1323 104SRA Jet selection 10213 17750 119 149SRA mbb gt 200 8344 14502 96 151SRA mCT gt 150 6535 11358 82 139SRA mCT gt 200 5175 8994 67 134SRA mCT gt 250 3793 6592 51 129SRA mCT gt 300 2487 4322 35 123SRB NoCuts 100000 1624100 16241 100SRB leptonveto 11489 186593 4069 46SRB MET gt 250 2530 41090 757 54SRB Jet selection 63 1023 79 130SRB HT 3 lt 50 51 828 52 159

      99

      CMSTop1LSearch for the Production of Dark Matter in As-sociation with Top Quark Pairs in the Single-lepton FinalState in pp collisions[41]

      Simulated in Madgraph with p p gt t t p1 p1

      Where p1= Majorana fermion Table 3 shows Signal efficiencies and cross sections TheM scale was read from Figure 7 for 90 CL

      Table A5 Signal Efficiencies 90 CL on σ limexp[ f b] on pp gt tt +χχ

      Mχ[GeV ] My Signal Efficiency Quoted Efficiency My σ [ f b] Quoted σ [ f b]1 102 101 63 47

      10 132 101 62 4650 143 120 50 38100 170 146 56 32200 185 173 69 27600 265 240 32 19

      1000 320 276 41 17

      Appendix B

      Branching ratio calculations for narrowwidth approximation

      B1 Code obtained from decayspy in Madgraph

      Br(S rarr bb) = (minus24λ2b m2

      b +6λ2b m2

      s

      radicminus4m2

      bm2S +m4

      S)16πm3S

      Br(S rarr tt) = (6λ2t m2

      S minus24λ2t m2

      t

      radicm4

      S minus4ms2m2t )16πm3

      S

      Br(S rarr τ+

      τminus) = (2λ

      2τ m2

      S minus8λ2τ m2

      τ

      radicm4

      S minus4m2Sm2

      τ)16πm3S

      Br(S rarr χχ) = (2λ2χm2

      S

      radicm4

      S minus4m2Sm2

      χ)32πm3S

      (B1)

      Where

      mS is the mass of the scalar mediator

      mχ is the mass of the Dark Matter particle

      mb is the mass of the b quark

      mt is the mass of the t quark

      mτ is the mass of the τ lepton

      The coupling constants λ follow the same pattern

      • TITLE Collider Constraints applied to Simplified Models of Dark Matter fitted to the Fermi-LAT gamma ray excess using Bayesian Techniques
        • Dedication
        • Declaration
        • Acknowledgements
        • Contents
        • List of Figures
        • List of Tables
          • Chapter 1 Introduction
          • Chapter 2 Review of Physics
          • Chapter 3 Fitting Models to the Observables
          • Chapter 4 Calculation Tools
          • Chapter 5 Majorana Model Results
          • Chapter 6 Real Scalar Model Results
          • Chapter 7 Real Vector Dark Matter Results
          • Chapter 8 Conclusion
          • Bibliography
          • Appendix A Validation of Calculation Tools
          • Appendix B Branching ratio calculations for narrow width approximation

        Acknowledgements

        And I would like to acknowledge the help and support of my supervisors Professor TonyWilliams and Dr Martin White as well as Assoc Professor Csaba Balasz who has assisted withinformation about a previous study Ankit Beniwal and Jinmian Li who assisted with runningMicrOmegas and LUXCalc I adapted the collider cuts programs originally developed bySky French and Martin White for my study

        Contents

        List of Figures xiii

        List of Tables xv

        1 Introduction 111 Motivation 312 Literature review 4

        121 Simplified Models 4122 Collider Constraints 7

        2 Review of Physics 921 Standard Model 9

        211 Introduction 9212 Quantum Mechanics 9213 Field Theory 10214 Spin and Statistics 10215 Feynman Diagrams 11216 Gauge Symmetries and Quantum Electrodynamics (QED) 12217 The Standard Electroweak Model 13218 Higgs Mechanism 17219 Quantum Chromodynamics 222110 Full SM Lagrangian 23

        22 Dark Matter 25221 Evidence for the existence of dark matter 25222 Searches for dark matter 30223 Possible signals of dark matter 30224 Gamma Ray Excess at the Centre of the Galaxy [65] 30

        Contents vi

        23 Background on ATLAS and CMS Experiments at the Large Hadron collider(LHC) 31231 ATLAS Experiment 32232 CMS Experiment 33

        3 Fitting Models to the Observables 3531 Simplified Models Considered 3532 Observables 36

        321 Dark Matter Abundance 36322 Gamma Rays from the Galactic Center 36323 Direct Detection - LUX 37

        33 Calculations 39331 Mediator Decay 39332 Collider Cuts Analyses 42333 Description of Collider Cuts Analyses 43

        4 Calculation Tools 5541 Summary 5542 FeynRules 5643 LUXCalc 5644 Multinest 5745 Madgraph 5846 Collider Cuts C++ Code 59

        5 Majorana Model Results 6151 Bayesian Scans 6152 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 6453 Collider Constraints 67

        531 Mediator Decay 67532 Collider Cuts Analyses 69

        6 Real Scalar Model Results 7361 Bayesian Scans 7362 Best fit Gamma Ray Spectrum for the Real Scalar DM model 7663 Collider Constraints 77

        631 Mediator Decay 77632 Collider Cuts Analyses 78

        Contents vii

        7 Real Vector Dark Matter Results 8171 Bayesian Scans 8172 Best fit Gamma Ray Spectrum for the Real Vector DM model 8473 Collider Constraints 84

        731 Mediator Decay 84732 Collider Cuts Analyses 86

        8 Conclusion 89

        Bibliography 91

        Appendix A Validation of Calculation Tools 97

        Appendix B Branching ratio calculations for narrow width approximation 105B1 Code obtained from decayspy in Madgraph 105

        List of Figures

        21 Feynman Diagram of electron interacting with a muon 1122 Weak Interaction Vertices [48] 1523 Higgs Potential [49] 1824 Standard Model Particles and Forces [50] 2425 Bullet Cluster [52] 2526 Galaxy Rotation Curves [54] 2627 WMAP Cosmic Microwave Background Fluctuations [58] 2928 Dark Matter Interactions [60] 2929 Gamma Ray Excess from the Milky Way Center [75] 31210 ATLAS Experiment 31211 CMS Experiment 34

        31 Main Feyman diagrams leading to the cross section for scalar decaying to apair of τ leptons 40

        32 WidthmS vs mS 4033 WidthmS vs λb 4134 WidthmS vs λτ 4135 Main Feyman diagrams leading to the cross section for scalar decaying to a

        pair of b quarks in the presence of at least one b quark 42

        41 Calculation Tools 55

        51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether 62

        52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter 6353 Gamma Ray Spectrum 6454 Plots of log likelihoods by individual and combined constraints Masses in

        GeV 6655 σ lowastBr(σ rarr ττ) versus Mass of Scalar 67

        List of Figures ix

        56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar 6857 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar 6858 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 6959 Excluded points from Collider Cuts and σBranching Ratio 70

        61 Real Scalar Dark Matter - By Individual Constraint and All Together 7462 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter 7563 Gamma Ray Spectrum 7664 σ lowastBr(σ rarr ττ) versus Mass of Scalar 7765 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 7866 Excluded points from Collider Cuts and σBranching Ratio 80

        71 Real Vector Dark Matter - By Individual Constraint and All Together 8272 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter 8373 Gamma Ray Spectrum 8474 σ lowastBr(σ rarr ττ) versus Mass of Scalar 8575 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 8676 Excluded points from Collider Cuts and σBranching Ratio 87

        List of Tables

        21 Quantum numbers of the Higgs field 1922 Weak Quantum numbers of Lepton and Quarks 21

        31 Simplified Models 3532 95 CL by Signal Region 4433 Selection criteria common to all signal regions 4534 Selection criteria for signal regions A 4535 Selection criteria for signal regions C 4536 Signal Regions - Lepstop1 4837 Signal Regions Lepstop2 4938 Signal Regions 2bstop 5139 Signal Region ATLASmonobjet 52

        51 Scanned Ranges 6152 Best Fit Parameters 64

        61 Best Fit Parameters 76

        71 Best Fit Parameters 84

        A1 0 Leptons in the final state 98A2 1 Lepton in the Final state 100A3 2 Leptons in the final state 101A4 2b jets in the final state 102A5 Signal Efficiencies 90 CL on σ lim

        exp[ f b] on pp gt tt +χχ 103

        Chapter 1

        Introduction

        Dark matter (DM) was first postulated over 80 years ago when Swiss astronomer FritzZwicky observed a discrepancy between the amount of light emitted by a cluster of galaxiesand the total mass contained within the cluster inferred from the relative motion of thosegalaxies by a simple application of the theory of Newtonian gravitation The surprising resultof this observation was that the vast majority of the mass in the cluster did not emit lightwhich was contrary to the expectation that most of the mass would be carried by the starsSince that time further observations over a wide range of scales and experimental techniqueshave continued to point to the same result and refine it Some of these observations and otherevidence are discussed in section 22 We now know with certainty that in the entire Universeall of the matter we know about - stars planets gases and other cosmic objects such as blackholes can only account for less than 5 of the mass that we calculate to be there

        A recent phenomenon that has received much attention is the significant deviation frombackground expectations of the Fermi Large Area Telescope(Fermi-LAT) gamma ray flux atthe galactic centre [1] A number of astrophysical explanations have been proposed includingmillisecond pulsars of supernova remnants [2] or burst-like continuous events at the galacticcentre but these are unresolved However it has also been noted that the observed Fermi-LATexcess is consistent with the annihilation of dark matter particles which would naturally beconcentrated at the Galactic centre in a manner consistent with the Navarro-Frenk-Whitedistribution of dark matter [3]

        There are a number of other purely theoretical (particle physics) reasons to postulatethe existence of weakly interacting matter particles that could supply the missing mass andyet remain unobservable Weakly interacting massive particle (WIMPS) have been a majorfocus of Run I and ongoing Run II searches of the Large Hadron Collider (LHC) In spite

        2

        of substantial theoretical and experimental effort the microscopic properties of dark matterparticles are still unknown

        One of the main problems with extracting these properties is the plethora of competingtheoretical models that provide dark matter candidates Another is the very weak interactionof these particles and the lack of experimental observables allowing one to discriminatebetween the competing models

        A particularly fruitful approach is the simplified model approach which makes minimalassumptions about the DM candidates as extensions to the Standard Model interacting weaklywith ordinary matter through a mediator These can be coded as Lagrangian interaction termsusing tools such as FeynRules [4] and Calchep [5] and used to simulate Dark Matter relicdensities indirect detection expectations as well as direct detection expectations throughMicrOmegas [6] This in turn can be run under a Bayesian inference tool (see below)

        Using a Bayesian inference tool such as MultiNest [7][8][9] one can scan the parameterspace of a particular model to find regions of highest probability for that model based onobservations of a range of phenomena including direct detection experiments such as ZEPLIN[10] XENON [11] DEAP [12] ARDM [13] DarkSide [14] PandaX [15] and LUX [16]indirect detection - eg Fermi-LAT gamma ray excess [17] Pamela-positron and antiprotonflux [18] and production at colliders MultiNest calculates the evidence value which is theaverage likelihood of the model weighted by prior probability and a posterior distributionwhich is the distribution of the parameter(s) after taking into account the observed databy scanning over a range of parameters based on prior probabilities This can be used tocompare models on a consistent basis The scans also provide information about the best fitparameters for the overall fit as well as for the seperate experimentsobservations

        The simplified model approach has been taken in a number of papers to study the Fermi-LAT gamma ray excess (see literature review in section 12) and one in particular comparesthese models using a Bayesian inference tool which found that a Majorana fermion DMcandidate interacting via a scalar mediator with arbitrary couplings to third generationfermions showed the best fit [19] A subsequent paper [20] examined the most likelyparameter combinations that simultaneously fit the Planck relic density Fermi-LAT galacticcentre gamma ray AMS-02 positron flux and Pamela antiproton flux data

        There has not as yet been a thorough study of the collider constraints on DM simplifiedmodels that fit the Fermi-LAT galactic centre excess In this thesis we reproduce the

        11 Motivation 3

        previous results of [20] using similar astrophysical constraints and then implement detailedsimulations of collider constraints on DM and mediator production

        In Section 11 we review the motivation for this study and in Section 12 we considerrecent astronomical data which is consistent with the annihilation of DM particles andreview the literature and searches which attempt to explain it with a range of extensions tothe standard model

        In Chapter 2 we review the Standard Model of particle physics (SM) the evidence fordark matter and the ATLAS and CMS experiments at the LHC

        In Chapter 3 we describe the models considered in this paper the observables used inthe Bayesian scans and the collider experiments which provide limits against which theMagraph simulations of the three models can be compared

        In Chapter 4 we review the calculation tools that have been used in this paper

        In Chapters 5 6 and 7 we give the results of the Bayesian scans and collider cutsanalyses for each of the models and in Chapter 8 we summarise our conclusions

        11 Motivation

        The Fermi gamma ray space telescope has over the past few years detected a gamma raysignal from the inner few degrees around the galactic centre corresponding to a regionseveral hundred parsecs in radius The spectrum and angular distribution are compatible withthat from annihilating dark matter particles

        A number of papers have recently attempted to explain the Gamma Ray excess in termsof extensions to the Standard Model taking into account constraints from ATLAS and CMScollider experiments direct detection of dark matter colliding off nucleii in well shieldeddetectors (eg LUX) as well as other indirect measurements (astrophysical observations)These papers take the approach of simplified models which describe the contact interactionsof a mediator beween DM and ordinary matter using an effective field theory approach(EFT) Until recently it has been difficult to discriminate between the results of these paperswhich look at different models and produce different statistical results A recent paper[19] Simplified Dark Matter Models Confront the Gamma Ray Excess used a Bayesianinference tool to compare a variety of simplified models using the Bayesian evidence values

        12 Literature review 4

        calculated for each model to find the most preferred explanation The motivation for thechosen models is discussed in section 121

        A subsequent paper [20] Interpreting the Fermi-LAT gamma ray excess in the simplifiedframework studied the most likely model again using Bayesian inference to find the mostlikely parameters for the (most likely) model The data inputs (observables) to the scanwere the Planck measurement of the dark matter relic density [21] Fermi-LAT gamma rayflux data from the galactic center [17] cosmic positron flux data [22] cosmic anti-protonflux data [18] cosmic microwave background data and direct detection data from the LUXexperiment [16] The result of the scan was a posterior probability density function in theparameter space of the model

        This paper extends the findings in [19] by adding collider constraints to the likelihoodestimations for the three models It is not a simple matter to add collider constraints becauseeach point in parameter space demands a significant amount of computing to calculate theexpected excess amount of observable particles corresponding to the particular experimentgiven the proposed simplified model A sampling method has been employed to calculatethe collider constraints using the probability of the parameters based on the astrophysicalobservables

        12 Literature review

        121 Simplified Models

        A recent paper [23] summarised the requirements and properties that simplified models forLHC dark matter searches should have

        The general principles are

        bull Besides the Standard Model (SM) the Lagrangian should contain a DM candidate thatis absolutely stable or lives long enough to escape the LHC detector and a mediatorthat couples the two sectors

        bull The Lagrangian should contain all terms that are renormalizable consistent withLorentz invariance SM symmetries and DM stability

        bull Additional interactions should not violate exact and approximate global symmetries ofthe SM

        12 Literature review 5

        The examples of models that satisfy these requirements are

        1 Scalar (or pseudo-scalar) s-channel mediator coupled to fermionic dark matter (Diracor Majorana)

        2 Higgs portal DM where the DM is a scalar singlet or fermion singlet under the gaugesymmetries of the SM coupling to a scalar boson which mixes with the Higgs (Thefermion singlet is a specific realisation of 1 above)

        3 DM is a mixture of an electroweak singlet and doublet as in the Minimal Supersym-metric SM (MSSM)

        4 Vector (or Axial vector) s-channel mediator obtained by extending the SM gaugesymmetry by a new U(1)rsquo which is spontaneously broken such that the mediatoracquires mass

        5 t-channel flavoured mediator (if the DM particle is a fermion χ the mediator can bea coloured scalar or vector particle φ -eg assuming a scalar mediator a couplingof the form φ χq requires either χ or φ to carry a flavor index to be consistent withminimal flavour violation (MFV) which postulates the flavour changing neutral currentstructure in the SM is preserved

        Another recent paper [24] summarised the requirements and properties that simplifiedmodels for LHC dark matter searches should have The models studied in this paper arepossible candidates competing with Dirac fermion and complex scalar dark matter andmodels with vector and pseudo-scalar mediators Other models would have an elasticscattering cross section that will remain beyond the reach of direct detection experiments dueto the neutrino floor (the irreducible background produced by neutrino scattering) or arealready close to sensitivity to existing LUX and XENONIT experiments

        A recent paper [25] Scalar Simplified Models for Dark Matter looks at Dirac fermiondark matter mediated by either a new scalar or pseudo-scalar to SM fermions This paperplaces bounds on the coupling combination gχgφ (coupling of DM to mediator times couplingof mediator to SM fermions) calculated from direct detection indirect detection and relicdensity by the mass of DM and mass of mediator Bounds are also calculated for colliderconstraints (mono-jet searches heavy-flavour searches and collider bounds on the 125 GeVHiggs are used to place limits on the monojet and heavy flavor channels which can betranslated into limits on the Higgs coupling to dark matter and experimental measurements

        12 Literature review 6

        of the Higgs width are used to constrain the addition of new channels to Higgs decay) Thepaper models the top loop in a full theory using the Monte Carlo generator MCFM [26]and extends the process to accommodate off-shell mediator production and decay to a darkmatter pair While this paper does not consider the Majorana fermion dark matter model theconstraints on the coupling product may have some relevence to the present study if only asan order of magnitude indicator

        Constraining dark sectors with monojets and dijets [27] studies a simplified model withDirac fermion dark matter and an axial-vector mediator They use two classes of constraint-searches for DM production in events with large amounts of missing transverse energy inassociation with SM particles (eg monojet events) and direct constraints on the mediatorfrom monojet searches to show that for example where the choice of couplings of themediator to quarks and DM gA

        q =gAχ=1 all mediator masses in the range 130 GeV ltMR lt3

        TeV are excluded

        The paper Constraining Dark Sectors at Colliders Beyond the Effective Theory Ap-proach [28] considers Higgs portal models with scalar pseudo-scalar vector and axial-vectormediators between the SM and the dark sector The paper takes account of loop processesand gluon fusion as well as quark-mediator interactions using MCFM Apart from mediatortype the models are characterised by free parameters of mediator width and mass darkmatter mass and the effective coupling combination gχgφ (coupling of DM to mediator timescoupling of mediator to SM fermions) The conclusion is that LHC searches can proberegions of the parameter space that are not able to be explored in direct detection experiments(since the latter measurements suffer from a velocity suppression of the WIMP-nucleoncross-section)

        The paper Constraining the Fermi-LAT excess with multi-jet plus EmissT collider searches

        [29] shows that models with a pseudo-scalar or scalar mediator are best constrained by multi-jet final states with Emiss

        T The analysis was done using POWHEG-BOX-V2 at LO and usesM2

        T variables together with multi-jet binning to exploit differences in signal and backgroundin the different regions of phase space to obtain a more stringent set of constraints on DMproduction This paper largely motivated the current work in that it showed that collidersearches that were not traditionally thought of as dark matter searches were neverthelessoften sensitive to dark matter produced in association with heavy quarks The combinationof searches used in this paper reflects this fact

        12 Literature review 7

        122 Collider Constraints

        In order to constrain this model with collider constraints (direct detection limits) we considerthe following papers by the CMS and ATLAS collaborations which describe searches at theLHC

        ATLAS Experiments

        bull Search for dark matter in events with heavy quarks and missing transverse momentumin p p collisions with the ATLAS detector [30]

        bull Search for direct third-generation squark pair production in final states with missingtransverse momentum and two b-jets in

        radics= 8 TeV pp collisions with the ATLAS

        detector[31]

        bull Search for direct pair production of the top squark in all-hadronic final states inprotonndashproton collisions at

        radics=8 TeV with the ATLAS detector [32]

        bull Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

        radic(s)=8TeV pp collisions using 21 f bminus1 of

        ATLAS data [33]

        bull Search for direct top squark pair production in final states with two leptons inradic

        s = 8TeV pp collisions using 20 f bminus1 of ATLAS data [34]

        bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

        radics=8 TeV [35]

        CMS Experiments

        bull Searches for anomalous tt production in p p collisions atradic

        s=8 TeV [36]

        bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

        radics=8 TeV [37]

        bull Search for the production of dark matter in association with top-quark pairs in thesingle-lepton final state in proton-proton collisions at

        radics = 8 TeV [38]

        bull Search for new physics in monojet events in p p collisions atradic

        s = 8 TeV(CMS) [39]

        bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

        s = 8 TeV [40]

        12 Literature review 8

        bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theSingle-lepton Final State in pp collisions at

        radics=8 TeV [41]

        bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

        s=8 TeV [42]

        bull Search for neutral MSSM Higgs bosons decaying into a pair of bottom quarks [43]

        bull Search for neutral MSSM Higgs bosons decaying into a pair of τ leptons in p p collision[44]

        bull Search for dark matter extra dimensions and unparticles in monojet events in proton-proton collisions at

        radics=8 TeV [45]

        In particular we have implemented Madgraph simulations of certain data analysesapplying to collider experiments described in papers [32][33][34][31][30][41] Theremaining models were not implemented in Madgraph because of inability to validate themodels against published results in the time available We developed C++ analysis code toimplement the collider cuts after showering with Pythia and basic analysis with Delpheswith output to root files The C++ code allowed multiple experimental cuts to be applied tothe same simulation output The C++ code was first checked against the results of the variousexperiments above and then the simulations were rerun using the three modelsrsquo Lagrangians(implemented in FeynRules) described in this paper

        Chapter 2

        Review of Physics

        21 Standard Model

        211 Introduction

        The Standard Model of particle physics (SM) has sometimes been described as the theoryof Almost Everything The SM is a field theory which emerged in the second half of thetwentieth century from quantum mechanics and special relativity which themselves wereonly conceived after 1905 by Einsten Pauli Bohr Dirac and others In the first decades ofthe twenty first century the discovery of the Higgs boson [46][47] completed the pictureand confirmed the last previously unverified part of the SM The SM is a self consistenttheory that describes and classifies all known particles and their interactions (other thangravity) to an astonishing level of accuracy and detail However the theory is incomplete-it does not describe the complete theory of gravitation or account for DM (see below) ordark energy (the accelerating expansion of the universe) Related to these shortcomings aretheoretical deficiencies such as the asymmetry of matter and anti-matter the fact that the SMis inconsistent with general relativity (to the extent that one or both theories break down atspacetime singularities like the Big Bang and black hole event horizons)

        212 Quantum Mechanics

        Quantum Mechanics (QM) describes particles as probability waves which can be thoughtof as packets of energy which can be localised or spread through all space depending on how

        21 Standard Model 10

        accurately one measures the particlersquos momentum However QM is not itself consistent withspecial relativity It was realised too that because of relativity particles could be created anddestroyed and because of the uncertainty principle the theory should describe the probabilityof particles being created and destroyed on arbitrarily short timescales A quantum fieldtheory (QFT) is a way to describe this probability at every point of space and time

        213 Field Theory

        A field theory is described by a mathematical function of space and time called a LagrangianAs in classical mechanics where Lagrangians were introduced by Louis Lagrange at thebeginning of the nineteenth century to summarise dynamical systems the Lagrangian sum-marises the dynamics of space which is permeated by force fields The equations of motionarising from a Lagrangian are determined by finding the contours of the Lagrangian describedby stationary points (ie where the infinitesimal change in the Lagrangian is zero for a smallchange in field) Particles are generally described by singularities in the Lagrangian (wherethe values tend to infinity also called resonances)

        214 Spin and Statistics

        It was also realised that different types of particle needed to be represented by different fieldsin the Lagrangian Diracrsquos discovery that particles with spin 12 quantum numbers couldbe represented by a field obtained by effectively taking the square root of the relativisticequation (Klein Gordon equation) was probably the most significant advance in quantumfield theory It required the introduction of a matrix operators called the gamma matricesor Dirac matrices which helped explain the already known fact that spin half particles orfermions obey Fermi-Dirac statistics ie when two spin half particles are exchanged thesign of the wavefunction must change and hence they cannot occupy the same quantum stateunlike bosons with integer spin that can occupy the same quantum state

        Other particles need to be represented by fields called vector pseudo-vector scalar (as inthe Klein Gordon equation) pseudo-scalar and complex fields A vector field is really fourfields one for each dimension of space plus time and is parameterised by the four gammamatrices γmicro which transform among themselves in a relativistically invariant way (Lorentzinvariance) A pseudo-vector field is a vector field which transforms in the same way but

        21 Standard Model 11

        with additional sign change under parity transformations (a simultaneous reversal of the threespatial directions) and is represented by γmicroγ5 A pseudo-scalar is a scalar which changes signunder parity transformations and a complex scalar field is represented by complex numbers

        215 Feynman Diagrams

        QFT took another giant leap forward when Richard Feynman realised that the propagationof particles described by the field could be described by expanding the so called S-matrixor scattering matrix derived from calculating the probability of initial states transitioning tofinal states The expansion is similar to a Taylor expansion in mathematics allowing for thespecial commuting properties of elements of the field The Feynman diagrams representingindividual terms in the expansion can be interpreted as interactions between the particlecomponents The S-Matrix (or more correctly the non trivial part called the T-Matrix) isderived using the interaction Hamiltonian of the field The Hamiltonian is an operator whoseeigenvalues or characteristic values give the energy quanta of the field The T-matrix canalso be described by a path integral which is an integral over all of quantum space weightedby the exponential of the Lagrangian This is completely equivalent to the Hamiltonianformulation and shows how the Lagrangian arises naturally out of the Hamiltonian (in factthe Lagrangian is the Legendre transformation of the Hamiltonian)

        Figure 21 Feynman Diagram of electron interacting with a muon

        γ

        eminus

        e+

        micro+

        microminus

        The diagram Fig21 is a tree diagram (diagrams with closed loops indicate self in-teractions and are higher order terms in the expansion) and shows an electron (eminus) and ananti-muon (micro+) interacting by exchanging a photon

        21 Standard Model 12

        216 Gauge Symmetries and Quantum Electrodynamics (QED)

        The Dirac equation with its gamma matrices ushered in the use of group theory to describethe transformation of fields (because the multiplication table of the gamma matrices isclosed and described by group theory) It also ushered in the idea of symmetries within theLagrangian- for example consider a Lagrangian

        ψ(ipart minusm)ψ (21)

        The Lagrangian is invariant under a global U(1) transformation which acts on the fieldsand hence the derivatives as follows

        ψ rarr eiqαψ ψ rarr ψeminusiqα partmicroψ rarr eiqα

        partmicroψ (22)

        where qα is a global phase and α is a continuous parameter

        A symmetry leads by Noetherrsquos theorem to a conserved current jmicro = qψγmicroψ and is thebasis of charge conservation since partmicro jmicro = 0 and q =

        intd3x j0(x)

        By promoting this transformation to a local one (ie α rarr α(x) where x = xmicro are thespace time co-ordinates) the transformations become

        ψ rarr eiqα(x)ψ ψ rarr ψeminusiqα(x)partmicroψ rarr eiqα(x)(partmicroψ + iq(partmicroα(x))ψ) (23)

        The derivative transformation introduces an extra term which would spoil the gaugeinvariance This can be solved by introducing a new field the photon field Amicro(x) whichinteracts with the fermion field ψ and transforms under a U(1) transformation

        Amicro rarr Amicro minuspartmicroα(x) (24)

        If we replace normal derivatives partmicro with covariant derivatives Dmicro = (partmicro + iqAmicro(x)) in theLagrangian then the extra term is cancelled and we can now see that the covariant derivativetransforms like ψ - under a local U(1) transform ie

        Dmicro rarr Dmicroψprime = eiqα(x)Dmicroψ (25)

        The additional term iqAmicro(x) can be interpreted in a geometric sense as a connectionwhich facilitates the parallel transport of vectors along a curve in a parallel consistent manner

        21 Standard Model 13

        We must add a gauge-invariant kinetic term to the Lagrangian to describe the dynamicsof the photon field This is given by the electromagnetic field strength tensor

        Fmicroν = partmicroAν minuspartνAmicro (26)

        The full Quantum Electrodynamics (QED) Lagrangian is then for electrons with mass mand charge -e

        LQED = ψ(i Dminusm)ψ minus 14

        Fmicroν(X)Fmicroν(x) (27)

        This Lagrangian is Lorentz and U(1) guage invariant The interaction term is

        Lint =+eψ Aψ = eψγmicro

        ψAmicro = jmicro

        EMAmicro (28)

        where jmicro

        EM is the electromagnetic four current

        217 The Standard Electroweak Model

        The Standard Electroweak (EW) Model was created by Glashow-Weinberg-Salam (also calledthe GWS model) and unifies the electomagnetic and weak interactions of the fundamentalparticles

        The idea of gauge symmetry was extended to the idea of a non-abelian gauge symmetry(essentially the operators defining the symmetry do not commute and so are represented bymatrices- in this case the Pauli matrices for the SU(2) group) - the gauge group is the directproduct G = SU(2)

        otimesU(1) It was known that weak interactions were mediated by Wplusmn

        and Z0 particles where the superscripts are the signs of the electromagnetic charge Thesewere fairly massive but it was also known that through spontaneous symmetry breaking ofa continuous global non-abelian symmetry there would be one massive propagation modefor each spontaneously broken-symmetry generator and massless modes for the unbrokengenerators (the so-called Goldstone bosons)

        This required an extra field to represent the gauge bosons called Bmicro (a vector fieldhaving the 4 spacetime degrees of freedom) With this identification the covariant derivativebecomes

        Dmicro = partmicro minus igAmicro τ

        2minus i

        gprime

        2Y Bmicro (29)

        21 Standard Model 14

        Here Y is the so called weak hypercharge of the field Bmicro which is a quantum numberdepending on particle (lepton or quark) g and gprime are two independent parameters in theGWS theory and the expression Amicro τ implies a sum over 3 components Aa

        micro a=123 and thePauli matrices τa

        This isnrsquot the full story because it had been known since 1956 when Chin Shuing Wuand her collaborators confirmed in an experiment prompted by Lee and Yang that nature isnot ambivalent to so-called parity It had been assumed that the laws of nature would notchange if all spatial coordinates were reversed (a parity transformation) The experimentinvolved the beta decay of cooled 60C atoms (more electrons were emitted in the backwardhemisphere with respect to spin direction of the atoms) This emission involved the weakinteraction and required that the Lagrangian for weak interactions contain terms which aremaximally parity violating- which turn out to be currents of the form

        ψ(1minus γ5)γmicro

        ψ (210)

        The term

        12(1minus γ

        5) (211)

        projects out a left-handed spinor and can be absorbed into the spinor to describe left-handedleptons (and as later discovered quarks) The right handed leptons are blind to the weakinteraction

        The processes describing left-handed current interactions are shown in Fig 22

        Analogously to isospin where the proton and neutron are considered two eigenstates ofthe nucleon we postulate that Bmicro is the gauge boson of an SU(2)transformation coupled toweak SU(2) doublets (

        νe

        eminus

        )

        (ud

        ) (212)

        We may now write the weak SU(2) currents as eg

        jimicro = (ν e)Lγmicro

        τ i

        2

        e

        )L (213)

        21 Standard Model 15

        Figure 22 Weak Interaction Vertices [48]

        where τ i are the Pauli-spin matrices and the subscript L denotes the left-handed componentof the spinor projected out by Eqn211 The third current corresponding to i=3 is called theneutral current as it doesnrsquot change the charge of the particle involved in the interaction

        We could postulate that the electromagnetic current is a linear combination of the leftand right elecromagnetic fields eL = 1

        2(1minus γ5)e and eR = 12(1+ γ5)e

        jemmicro = eLγmicroQeL + eRγmicroQeR (214)

        where Q is the electromagnetic operator and eR is a singlet However this is not invariantunder SU(2)L To construct an invariant current we need to use the SU(2)L doublet 212The current which is an SU(2)L invariant U(1) current is

        jYmicro = (ν e)LγmicroYL

        e

        )L+ eRγmicroYReR (215)

        where hypercharges YL and YR are associated with the U(1)Y symmetry jYmicro is a linearcombination of the electromagnetic current and the weak neutral current thus explaining whythe hypercharge differs for the left and right-handed components since the electromagneticcurrent has the same charge for both components whilst the weak current exists only forleft-handed particles j3

        micro and the third component of weak isospin T 3 allows us to calculate

        21 Standard Model 16

        the relationship Y = 2(QminusT 3) which can be seen by writing jYmicro as a linear combination ofj3micro (the weak neutral current that doesnrsquot change the charge of the particle involved in the

        interaction) and jemmicro (using the convention that the jYmicro is multiplied by 1

        2 to match the samefactor implicit in j3

        micro ) Substituting

        τ3 =

        (1 00 minus1

        )(216)

        into equation 213 for i=3 and subtracting equation 215 and equating to 12 times equation

        214

        we get

        eLγmicroQeL + eRγmicroQeR minus (νLγmicro

        12

        νL minus eLγmicro

        12

        eL) =12

        eRγmicroYReR +12(ν e)LγmicroYL

        e

        )L (217)

        from which we can read out

        YR = 2QYL = 2Q+1 (218)

        and T3(eR) = 0 T3(νL) =12 and T3(eL) =

        12 The latter three identities are implied by

        the fraction 12 inserted into the definition of equation 213

        The Lagrangian kinetic terms of the fermions can then be written

        L =minus14

        FmicroνFmicroν minus 14

        GmicroνGmicroν

        + sumgenerations

        LL(i D)LL + lR(i D)lR + νR(i D)νR

        + sumgenerations

        QL(i D)QL +UR(i D)UR + DR(i D)DR

        (219)

        LL are the left-handed lepton doublets QL are the left-handed quark doublets lRνR theright handed lepton singlets URDR the right handed quark singlets

        The field strength tensors are given by

        Fmicroν = partmicroAν minuspartνAmicro minus ig[Amicro Aν ] (220)

        21 Standard Model 17

        andGmicroν = partmicroBν minuspartνBmicro (221)

        Bmicro is an abelian U(1) gauge field while Amicro is non-abelian and Fmicroν contains the com-mutator of the field through the action of the covariant derivative [Dmicro Dν ] This equationcontains no mass terms terms for the field Amicro for good reason- this would violate gaugeinvariance We know that the gauge bosons of the weak force have substantial mass and theway that this problem has been solved is through the Higgs mechanism

        218 Higgs Mechanism

        To solve the problem of getting gauge invariant mass terms three independent groupspublished papers in 1964 The mechanism was named after Peter Higgs To introduce theconcept consider a complex scalar field φ that permeates all space Consider a potential forthe Higgs field of the form

        minusmicro2φ

        daggerφ +λ (φ dagger

        φ2) (222)

        which has the form of a mexican hat Fig 23 This potential is still U(1) gauge invariant andcan be added to our Lagrangian together with kinetic+field strength terms to give

        L = (Dmicroφ)dagger(Dmicroφ)minusmicro2φ

        daggerφ +λ (φ dagger

        φ2)minus 1

        4FmicroνFmicroν (223)

        It is easily seen that this is invariant to the transformations

        Amicro rarr Amicro minuspartmicroη(x) (224)

        φ(x)rarr eieη(x)φ(x) (225)

        The minimum of this field is not at 0 but in a circular ring around 0 with a vacuum

        expectation value(vev)radic

        micro2

        2λequiv vradic

        2

        We can parameterise φ as v+h(x)radic2

        ei π

        Fπ where h and π are referred to as the Higgs Bosonand Goldstone Boson respectively and are real scalar fields with no vevs Fπ is a constant

        21 Standard Model 18

        Figure 23 Higgs Potential [49]

        Substituting this back into the Lagrangian 223 we get

        minus14

        FmicroνFmicroν minusevAmicropartmicro

        π+e2v2

        2AmicroAmicro +

        12(partmicrohpart

        microhminus2micro2h2)+

        12

        partmicroπpartmicro

        π+(hπinteractions)(226)

        This Lagrangian now seems to describe a theory with a photon Amicro of mass = ev a Higgsboson h with mass =

        radic2micro and a massless Goldstone π

        However there is a term mixing Amicro and part microπ in this expression and this shows Amicro and π

        are not independent normal coordinates and we can not conclude that the third term is a massterm This π minusAmicro mixing can be removed by making a gauge transformation in equations224 and 225 making

        φrarrv+h(x)radic2

        ei π

        Fπminusieη(x) (227)

        and setting πrarr π

        Fπminus eη(x) = 0 This choosing of a point on the circle for the vev is called

        spontaneous symmetry breaking much like a vertical pencil falling and pointing horizontallyin the plane This choice of gauge is called unitary gauge The Goldstone π will thencompletely disappear from the theory and one says that the Goldstone has been eaten to givethe photon mass

        21 Standard Model 19

        This can be confirmed by counting degrees of freedom before and after spontaneoussymmetry breaking The original Lagrangian had four degrees of freedom- two from thereal massless vector field Amicro (massless photons have only two independent polarisationstates) and two from the complex scalar field (one each from the h and π fields) The newLagrangian has one degree of freedom from the h field and 3 degrees of freedom from themassive real vector field Amicro and the π field has disappeared

        The actual Higgs field is postulated to be a complex scalar doublet with four degrees offreedom

        Φ =

        (φ+

        φ0

        )(228)

        which transforms under SU(2) in the fundamental representation and also transforms underU(1) hypercharge with quantum numbers

        Table 21 Quantum numbers of the Higgs field

        T 3 Q Yφ+

        12 1 1

        φ0 minus12 1 0

        We can parameterise the Higgs field in terms of deviations from the vacuum

        Φ(x) =(

        η1(x)+ iη2(x)v+σ(x)+ iη3(x)

        ) (229)

        It can be shown that the three η fields can be transformed away by working in unitarygauge By analogy with the scalar Higgs field example above the classical action will beminimised for a constant value of the Higgs field Φ = Φ0 for which Φ

        dagger0Φ0 = v2 This again

        defines a range of minima and we can pick one arbitarily (this is equivalent to applying anSU(2)timesU(1) transform to the Higgs doublet transforming the top component to 0 and thebottom component to Φ0 = v) which corresponds to a vacuum of zero charge density andlt φ0 gt=+v

        In this gauge we can write the Higgs doublet as

        Φ =

        (φ+

        φ0

        )rarr M

        (0

        v+ H(x)radic2

        ) (230)

        where H(x) is a real Higgs field v is the vacuum expectation and M is an SU(2)timesU(1)matrix The matrix Mdagger converts the isospinor into a down form (an SU(2) rotation) followedby a U(1) rotation to a real number

        21 Standard Model 20

        If we consider the Higgs part of the Lagrangian

        minus14(Fmicroν)

        2 minus 14(Bmicroν)

        2 +(DmicroΦ)dagger(DmicroΦ)minusλ (Φdagger

        Φminus v2)2 (231)

        Substituting from equation 230 into this and noting that

        DmicroΦ = partmicroΦminus igW amicro τ

        aΦminus 1

        2ig

        primeBmicroΦ (232)

        We can express as

        DmicroΦ = (partmicro minus i2

        (gA3

        micro +gprimeBmicro g(A1micro minusA2

        micro)

        g(A1micro +A2

        micro) minusgA3micro +gprimeBmicro

        ))Φ equiv (partmicro minus i

        2Amicro)Φ (233)

        After some calculation the kinetic term is

        (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

        14(v+

        Hradic2)2[A 2]22 (234)

        where the 22 subscript is the index in the matrix

        If we defineWplusmn

        micro =1radic2(A1

        micro∓iA2micro) (235)

        then [A 2]22 is given by

        [A 2]22 =

        (gprimeBmicro +gA3

        micro

        radic2gW+

        microradic2gWminus

        micro gprimeBmicro minusgA3micro

        ) (236)

        We can now substitute this expression for [A 2]22 into equation 234 and get

        (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

        14(v+

        Hradic2)2(2g2Wminus

        micro W+micro +(gprimeBmicro minusgA3micro)

        2) (237)

        This expression contains mass terms for three massive gauge bosons Wplusmn and the combi-nation of vector fields gprimeBmicro minusgA3

        micro where note

        21 Standard Model 21

        Table 22 Weak Quantum numbers of Lepton and Quarks

        T 3 Q YνL

        12 0 -1

        lminusL minus12 -1 -1

        νR 0 0 0lminusR 0 -1 -2UL

        12

        23

        13

        DL minus12 minus1

        313

        UR 0 23

        43

        DR 0 minus13 minus2

        3

        Wminusmicro = (W+

        micro )dagger equivW 1micro minus iW 2

        micro (238)

        Then the mass terms can be written

        12

        v2g2|Wmicro |2 +14

        v2(gprimeBmicro minusgA3micro)

        2 (239)

        W+micro and W microminus can be associated with the W boson and its anti-particle and (minusgprimeBmicro +

        gA3micro) with the Z Boson (after normalisation by

        radicg2 +(gprime

        )2) The combination gprimeA3micro +gBmicro

        is orthogonal to Z and remains massless This can be associated with the Photon (afternormalisation) These predict mass terms for the W and Z bosons mW = vgradic

        2and mZ =

        vradic2

        radicg2 +(gprime

        )2It is again instructive to count the degrees of freedom before and after the Higgs mech-

        anism Before we had a complex doublet Φ with four degrees of freedom one masslessB with two degrees of freedom and three massless W a fields each with two each givinga total of twelve degrees of freedom After spontaneous symmetry breaking we have onereal scalar Higgs field with one degree of freedom three massive weak bosons with nineand one massless photon with two again giving twelve degrees of freedom One says thescalar degrees of freedom have been eaten to give the Wplusmn and Z bosons their longitudinalcomponents (ie mass)

        Table 22 shows T 3 the weak isospin quantum number (which describes how a particletransforms under SU(2)) Y the weak hypercharge (which describes how the particle trans-

        21 Standard Model 22

        forms under U(1)) and Q the electric charge of leptons and quarks The former two enterdirectly into the Lagrangian while the latter can be derived as a function of the former two

        Strictly speaking the right handed neutrino which has no charge may not exist and is notpart of the standard model although some theories postulate a mass for this (hypothetical)particle

        Writing gauge invariant couplings between ELeRφ with the vacuum expectation ofφ = v (Yukawa matrices λ

        i ju and λ

        i jd respectively for the up and down quarks) we get mass

        terms for the quarks (and similarly for the leptons)

        Mass terms for quarks minussumi j[(λi jd Qi

        Lφd jR)+λ

        i ju εab(Qi

        L)aφlowastb u j

        R +hc]

        Mass terms for leptonsminussumi j[(λi jl Li

        Lφ l jR)+λ

        i jν εab(Li

        L)aφlowastb ν

        jR +hc]

        Here hc is the hermitian conjugate of the first terms L and Q are lepton and quarkdoublets l and ν are the lepton singlet and corresponding neutrino d and u are singlet downand up quarks εab is the Levi-Civita (anti-symmetric) tensor

        If we write the quark generations in a basis that diagonalises the Higgs couplings weget a unitary transformations Uu and Ud for the up and down quarks respectively which iscompounded in the boson current (Udagger

        u Ud) called the CKM matrix where off-diagonal termsallow mixing between quark generations (this is also called the mass basis as opposed tothe flavour basis) A similar argument can give masses to right handed sterile neutrinos butthese can never be measured However a lepton analogue of the CKM matrix the PNMSmatrix allows left-handed neutrinos to mix between various generations and oscillate betweenflavours as they propagate through space allowing a calculation of mass differences

        219 Quantum Chromodynamics

        The final plank in SM theory tackling the strong force Quantum Chromodynamics wasproposed by Gell-Mann and Zweig in 1963 Three generations of quarks were proposedwith charges of 23 for uc and t and -13 for ds and b Initially SU(2) isospin was consideredtransforming u and d states but later irreducible representations of SU(3) were found to fit theexperimental data The need for an additional quantum number was motivated by the fact thatthe lightest excited state of the nucleon consisted of sss (the ∆++) To reconcile the baryon

        21 Standard Model 23

        spectrum with spin statistics colour was proposed Quarks transform under the fundamentalor 3 representation of colour symmetry and anti-quarks under 3 The inner product of 3and 3 is an invariant under SU(3) as is a totally anti-symmetric combination ε i jkqiq jqk (andthe anti-quark equivalent) It was discovered though deep inelastic scattering that quarksshowed asymptotic freedom (the binding force grew as the seperation grew) Non-abeliangauge theories showed this same property and colour was identified with the gauge quantumnumbers of the quarks The quanta of the gauge field are called gluons Because the forcesshow asymptotic freedom - ie the interaction becomes weaker at high energies and shorterdistances perturbation theory can be used (ie Feynman diagrams) at these energies but forlonger distances and lower energies lattice simulations must be done

        2110 Full SM Lagrangian

        The full SM can be written

        L =minus14

        BmicroνBmicroν minus 18

        tr(FmicroνFmicroν)minus 12

        tr(GmicroνGmicroν)

        + sumgenerations

        (ν eL)σmicro iDmicro

        (νL

        eL

        )+ eRσ

        micro iDmicroeR + νRσmicro iDmicroνR +hc

        + sumgenerations

        (u dL)σmicro iDmicro

        (uL

        dL

        )+ uRσ

        micro iDmicrouR + dRσmicro iDmicrodR +hc

        minussumi j[(λ

        i jl Li

        Lφ l jR)+λ

        i jν ε

        ab(LiL)aφ

        lowastb ν

        jR +hc]

        minussumi j[(λ

        i jd Qi

        Lφd jR)+λ

        i ju ε

        ab(QiL)aφ

        lowastb u j

        R +hc]

        + (Dmicroφ)dagger(Dmicroφ)minusmh[φφ minus v22]22v2

        (240)

        where σ micro are the extended Pauli matrices

        (1 00 1

        )

        (0 11 0

        )

        (0 minusii 0

        )

        (1 00 minus1

        )

        The first line gives the gauge field strength terms the second the lepton dynamical termsthe third the quark dynamical terms the fourth the lepton (including neutrino) mass termsthe fifth the quark mass terms and the sixth the Higgs terms

        The SU(3) gauge field is given by Gmicroν and which provides the colour quantum numbers

        21 Standard Model 24

        Figure 24 Standard Model Particles and Forces [50]

        Note The covariant derivatives operating on the left-handed leptons contain only theU(1) and SU(2) gauge fields Bmicro Fmicro whereas those operating on left-handed quarks containU(1)SU(2) and SU(3) gauge fields Bmicro Fmicro Gmicro The covariant derivatives operating onright handed electrons and quarks are as above but do not contain the SU(2) gauge fieldsRight handed neutrinos have no gauge fields

        The sums over i j above are over the different generations of leptons and quarks

        The particles and forces that emerge from the SM are shown in Fig 24

        22 Dark Matter 25

        22 Dark Matter

        221 Evidence for the existence of dark matter

        2211 Bullet Cluster of galaxies

        Most of the mass of the cluster as measured by a gravitational lensing map is concentratedaway from the centre of the object The visible matter (as measured by X-ray emission) ismostly present in the centre of the object The interpretation is that the two galaxy clustershave collided perpendicular to the line of sight with the mostly collisionless dark matterpassing straight through and the visible matter interacting and being trapped in the middleIt was estimated that the ratio of actual to visible matter was about 16 [51]

        Figure 25 Bullet Cluster [52]

        2212 Coma Cluster

        The first evidence for dark matter came in the 1930s when Fritz Zwicky observed the Comacluster of galaxies over 1000 galaxies gravitationally bound to one another He observed thevelocities of individual galaxies in the cluster and estimated the amount of matter based onthe average galaxy He calculated that many of the galaxies should be able to escape fromthe cluster unless the mass was about 10 times higher than estimated

        22 Dark Matter 26

        2213 Rotation Curves [53]

        Since those early observations many spiral galaxies have been observed with the rotationcurves calculated by doppler shift with the result that in all cases the outermost stars arerotating at rates fast enough to escape their host galaxies Indeed the rotation curves arealmost flat indicating large amounts of dark matter rotating with the galaxies and holdingthe galaxies together

        Figure 26 Galaxy Rotation Curves [54]

        2214 WIMPS MACHOS

        The question is if the particles that make up most of the mass of the universe are not baryonswhat are they It is clear that the matter does not interact significantly with radiation so mustbe electrically neutral It has also not lost much of its kinetic energy sufficiently to relax intothe disks of galaxies like ordinary matter Massive particles may survive to the present ifthey carry a conserved additive or multiplicative quantum number This is so even if equalnumbers of particles and anti-particles can annihilate if the number densities become too lowto reduce the density further These particles have been dubbed Weakly interacting MassiveParticles (WIMPS) as they interact with ordinary matter only through a coupling strengthtypically encountered in the weak interaction They are predicted to have frozen out ofthe hot plasma and would give the correct relic density based solely on the weak interactioncross section (sometimes called the WIMP Miracle) Another possibility called MassiveCompact Halo Objects (MACHOS)- are brown dwarf stars Jupiter mass objects or remnantsfrom an earlier generation of star formation

        22 Dark Matter 27

        2215 MACHO Collaboration [55]

        In order to estimate the mass of MACHOS a number of experiments have been conductedusing microlensing which makes use of the fact that if a massive object lies in the line ofsight of a much more distant star the light from the more distant star will be lensed and forma ring around the star It is extemely unlikely that the MACHO will pass directly through theline of sight but if there is a near miss the two images will be seperated by a small angleThe signature for microlensing is a time-symmetric brightening of a star as the MACHOpasses near the line of sight called a light curve One can obtain statistical information aboutthe MACHO masses from the shape and duration of the light curves and a large number ofstars (many million) must be monitored The MACHO Collaboration [55] have monitoredover 11 million stars and less than 20 events were detected The conclusion was that thiscould account for a small addition (perhaps up to 20 ) to the visible mass but not enoughto account for Dark Matter

        2216 Big Bang Nucleosynthesis (BBN) [56]

        Another source of evidence for DM comes from the theory of Nucleosynthesis in the fewseconds after the Big Bang In the first few minutes after the big bang the temperatures wereso high that all matter was dissociated but after about 3 minutes the temperature rapidlycooled to about 109 Kelvin Nucleosynthesis could begin when protons and neutrons couldstick together to form Deuterium (Heavy Hydrogen) and isotopes of Helium The ratio ofDeuterium to Hydrogen and Helium is very much a function of the baryon density and alsothe rate of expansion of the universe at the time of Nucleosynthesis There is a small windowwhen the density of Baryons is sufficient to account for the ratios of the various elementsThe rate of expansion of the universe is also a function of the total matter density The ratiosof these lighter elements have been calculated by spectral measurements with Helium some25 of the mass of the universe Deuterium 001 The theory of BBN accounts for theratios of all of the lighter elements if the ratio of Baryon mass to total mass is a few percentsuggesting that the missing matter is some exotic non-baryonic form

        22 Dark Matter 28

        2217 Cosmic Microwave Background [57]

        The discovery of a uniform Microwave Background radiation by Penzias and Wilson in1965 was confirmation of the theory first proposed by Gamow and collaborators that theuniverse should be filled with black body radiation an afterglow of the big bang TheCOBE satelite launched in 1989 confirmed this black body radiation with temperature of2725+-002K with 95 confidence While extremely uniform it found tiny irregularitieswhich could be accounted for by irregularities in the density of the plasma at last scattering(gravitational redshifts or blueshifts called the Sachs Wolfe effect) and temperature andvelocity fluctuations in the plasma The anisotropy of the background has been furtheranalysed by WMAP (the Wilkinson Microwave Anisotropy Probe) launched in 2001 Thefluctuations in the density of the plasma are what account for the seeds of galaxy formationHowever detailed calculations have found that the fluctuations were too small to account forthe observed galaxy distribution without an electrically neutral gravitating matter in additionto ordinary matter that only became electrically neutral at the time of recombination [58]

        In May 2009 the European Space Agency (ESA) launched the Planck sattelite a spacebased observatory to observe the universe at wavelengths between 03mm and 111 mmbroadly covering the far-infrared microwave and high frequency domains The missionrsquosmain goal was to study the cosmic microwave background across the whole sky at greatersensitivity and resolution than ever before Using the information gathered and the standardmodel of cosmology ESA was able to calculate cosmological parameters including the relicdensity of dark matter remaining after the Big Bang to a high degree of precision withinthe framework of the standard model of cosmology The standard model of cosmology canbe described by a relatively small number of parameters including the density of ordinarymatter dark matter and dark energy the speed of cosmic expansion at the present epoch (alsoknown as the Hubble constant) the geometry of the Universe and the relative amount ofthe primordial fluctuations embedded during inflation on different scales and their amplitude[59]

        2218 LUX Experiment - Large Underground Xenon experiment [16]

        The LUX experiment is conducted at the Sanford Underground research facility one mileunderground to minimise the background of cosmic rays The detector is a time projectionchamber allowing a 3D positioning of interactions occurring within its active volume LUX

        22 Dark Matter 29

        Figure 27 WMAP Cosmic Microwave Background Fluctuations [58]

        Figure 28 Dark Matter Interactions [60]

        uses as its target 368 kiliograms of liquefied ultra pure xenon which is a scintillator Interac-tions inside the xenon will create an amount of light proportional to the energy depositedThat light can be collected on arrays of light detectors sensitive to a single photon

        Xenon is very pure and the amount of radiation originating within the target is limitedAlso being three times denser than water it can stop radiation originating from outside thedetector The 3D detector focuses on interactions near the center which is a quiet regionwhere rare dark matter interactions might be detected [16]

        22 Dark Matter 30

        222 Searches for dark matter

        2221 Dark Matter Detection

        Fig 28 illustrates the reactions studied under the different forms of experiment designed todetect DM

        Direct detection experiments are mainly underground experiments with huge detectorsattempting to detect the recoil of dark matter off ordinary matter such as LUX [16] CDMS-II [61] XENON100 [11] COUPP [62] PICASSO [63] ZEPLIN [10] These involveinteractions between DM and SM particles with final states involving DM and SM particles

        Indirect detection involves the search for the annihilation of DM particles in distantastrophysical sources where the DM content is expected to be large Example facilitiesinclude space based experiments (EGRET [64] PAMELA [18] FERMI-LAT [65]) balloonflights (ATIC [66] PEBS [67] PPB-BETS [68]) or atmospheric Cherenkov telescopes(HESS [69]MAGIC [70]) and neutrino telescopes (SuperKamiokande [71] IceCube [72]ANTARES [73] KM3NET [74]) Indirect Detection involves DM particles in the initial stateand SM particles in the final state

        Collider detection involves SM particles in the initial state creating DM particles in thefinal state which can be detected by missing transverse energy and explained by variousmodels extending the SM

        223 Possible signals of dark matter

        224 Gamma Ray Excess at the Centre of the Galaxy [65]

        The center of the galaxy is the most dense and active region and it is no surprise that itproduces intense gamma ray sources When astronomers subtract all known gamma raysources there remains an excess which extends out some 5000 light years from the centerHooper et al find that the annihilation of dark matter particles with a mass between 31 and40 Gev fits the excess to a remarkable degree based on Gamma Ray spectrum distributionand brightness

        23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 31

        Figure 29 Gamma Ray Excess from the Milky Way Center [75]

        23 Background on ATLAS and CMS Experiments at theLarge Hadron collider (LHC)

        The collider constraints in this paper are based on two of the experiments that are part of theLHC in CERN

        Figure 210 ATLAS Experiment

        The LHC accelerates proton beams in opposite directions around a 27 km ring of super-conducting magnets until they are made to collide at an interaction point inside one of the 4experiments placed in the ring An upgrade to the LHC has just been performed providing65 TeV per beam

        23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 32

        231 ATLAS Experiment

        The ATLAS detector consists of concentric cylinders around the interaction point whichmeasure different aspects of the particles that are flung out of the collision point There are 4major parts the Inner Detector the Calorimeters the Muon spectrometer and the magnetsystems

        2311 Inner Detector

        The inner detector extends from a few centimeters to 12 meters from the interaction point andmeasures details of the charged particles and their momentum interacting with the materialIf particles seem to originate from a point other than the interaction point this may indicatethat the particle comes from the decay of a hadron with a bottom quark This is used to b-tagcertain particles

        The inner detector consists of a pixel detector with three concentric layers and threeendcaps containing detecting silicon modules 250 microm thick The pixel detector has over 80million readout channels about 50 percent of the total channels for the experiment All thecomponents are radiation hardened to cope with the extremely high radiation in the proximityof the interaction point

        The next layer is the semi-conductor layer (SCT) similar in concept to the pixel detectorbut with long narrow strips instead of pixels to cover a larger area It has 4 double layers ofsilicon strips with 63 million readout channels

        The outermost component of the inner detector is the Transition Radiation Tracker It is acombination of a straw tracker and a transition radiation detector The detecting elementsare drift tubes (straws) each 4mm in diameter and up to 144 cm long Each straw is filledwith gas which becomes ionised when the radiation passes through The straws are held at-1500V with a thin wire down the center This drives negatively ionised atoms to the centercausing a pulse of electricity in the wire This causes a pattern of hit straws allowing the pathof the particles to be determined Between the straws materials with widely varying indicesof refraction cause transition radiation when particles pass through leaving much strongersignals in some straws Transition radiation is strongest for highly relativistic particles (thelightest particles- electrons and positrons) allowing strong signals to be associated with theseparticles

        23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 33

        2312 Calorimeters

        The inner detector is surrounded by a solenoidal magnet of 2 Tesla (there is also an outertoroidal magnet) but surrounding the inner magnet are the calorimeters to measure theenergy of particles by absorbing them There is an inner electromagnetic calorimeter whichmeasures the angle between the detectorrsquos beam axis and the particle (or more preciselypseudo-rapidity) and its angle in the perpendicular plane to a high degree of accuracy Theabsorbing material is lead and stainless steel and the sampling material liquid argon Thehadron calorimeter measures the energy of particles that interact via the strong force Theenergy absorbing material is scintillating steel

        2313 Muon Specrometer

        The muon spectrometer surrounds the calorimeters and has a set of 1200 chambers tomeasure the tracks of outgoing muons with high precision using a magnetic field andtriggering chambers Very few particles of other types pass through the calorimeters and aresubsequently measured

        2314 Magnets

        The magnets are in two layers the inner solenoidal and outer toroidal which together causecharged particles to curve in the magetic field allowing their momentum to be determinedThe outer toroidal magnet consists of 8 very large air core superconducting magnets whichsupply the field in the muon spectrometer

        232 CMS Experiment

        The CMS Detector is similar in design to the ATLAS Detector with a single solenoidalmagnet which is the largest of its type ever built (length 13m and diameter 7m) and allowsthe tracker and calorimeter detectors to be placed inside the coil in a compact way Thestrong magnetic field gives a high resolution for measuring muons electrons and photons ahigh quality central tracking system and a hermetic hadron calorimeter designed to preventhadrons from escaping

        23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 34

        Figure 211 CMS Experiment

        Chapter 3

        Fitting Models to the Observables

        31 Simplified Models Considered

        In order to extend the findings of previous studies [19] to include collider constraints wefirst performed Bayesian scans [7] over three simplified models real scalar dark matter φ Majorana fermion χ and real vector Xmicro The purpose was to obtain the most probable regionsof each model then to use the resulting posterior samples to investigate whether the LHCexperiments have any sensitivity to the preferred regions

        The three models couple to the mediator with interactions shown in the following table

        Table 31 Simplified Models

        Hypothesis real scalar DM Majorana fermion DM real vector DM

        DM mediator int Lφ sup microφ mφ φ 2S Lχ sup iλχ

        2 χγ5χS LX sup microX mX2 X microXmicroS

        The interactions between the mediator and the standard fermions is assumed to be

        LS sup f f S (31)

        and in line with minimal flavour violation we only consider third generation fermions- ief = b tτ

        For the purposes of these scans we consider the following observables

        32 Observables 36

        32 Observables

        321 Dark Matter Abundance

        We have assumed a single dark matter candidate which freezes out in the early universe witha central value as determined by Planck [21] of

        ΩDMh2 = 01199plusmn 0031 (32)

        h is the reduced hubble constant

        The experimental uncertainty of the background relic density [59] is dominated by atheoretical uncertainty of 50 of the value of the scanned Ωh2 [76] We have added thetheoretical and the experimental uncertainties in quadrature

        SD =radic(05Ωh2)2 + 00312 (33)

        This gives a log likelihood of

        minus05lowast (Ωh2 minus 1199)2

        SD2 minus log(radic

        2πSD) (34)

        322 Gamma Rays from the Galactic Center

        Assuming that the gamma ray flux excess from the galactic centre observed by Fermi-LAT iscaused by self annihilation of dark matter particles the differential flux

        d2Φ

        dEdΩ=

        lt σv gt8πmχ

        2 J(ψ)sumf

        B fdN f

        γ

        dE(35)

        has been calculated using both Micromegas v425 with a Navarro-Frenk-White (NFW) darkmatter galactic distribution function

        ρ(r) = ρ0(rrs)

        minusγ

        (1+ rrs)3minusγ (36)

        with γ = 126 and an angle of 5 to the galactic centre [19]

        32 Observables 37

        Here Φ is the flux dE is the differential energy lt σv gt the velocity averaged annihi-lation cross section of the dark matter across the galactic centre mχ the mass of the darkmatter particle B f =lt σv gt f σv is the annihilation fraction into the f f final state anddN f

        γ dE is the energy distribution of photons produced in the annihilation channel with thefinal state f f The J factor is described below

        The angle ψ to the galactic centre determines the normalisation of the gamma ray fluxthrough the factor

        J(ψ) =int

        losρ(l2 + r⊙2 minus2lr⊙cosψ)dl (37)

        where ρ(r) is the dark matter density at distance r from the centre of the galaxy r⊙=85 kpcis the solar distance from the Galactic center ρ0 is set to reproduce local dark matter densityof 03GeVcm3

        The central values for the gamma ray spectrum were taken from Fig 5 of [77] and thestandard deviations of individual flux points from the error bars on the graph In commonwith [78] we ignore possible segment to segment correlations

        For the Fermi-LAT log likelihood we assumed each point was distributed with indepen-dent Gaussian likelihood with the experimental error for each data point given by the errorbar presented in Fig5 of [76] for each point We also assumed an equal theoretical errorfor each point which when added to the experimental error in quadrature gave an effectivelog likelihood for each point of minus05lowast (giminusdi)

        2

        2lowastσ2i

        where gi are the calculated values and di theexperimental values and σi the experimental errors

        323 Direct Detection - LUX

        The LUX experiment [16] is the most constraining direct search experiment to date for awide range of WIMP models We have used LUXCalc [79] (modified to handle momentumdependent nucleon cross sections) to calculate the likelihood of the expected number ofsignal events for the Majorana fermion model

        The likelihood function is taken as the Poisson distribution

        L(mχ σ | N) = P(N | mχ σ) =(b+micro)Neminus(b+micro)

        N (38)

        32 Observables 38

        where b= expected background of 064 and micro = micro(mχσ ) is the expected number of signalevents calculated for a given model

        micro = MTint

        infin

        0dEφ(E)

        dRdE

        (E) (39)

        where MT is the detector mass times time exposure and φ(E) is a global efficiency factorincorporating trigger efficiencies energy resolution and analysis cuts

        The differential recoil rate of dark matter on nucleii as a function of recoil energy E

        dRdE

        =ρX

        mχmA

        intdvv f (v)

        dσASI

        dER (310)

        where mA is the nucleon mass f (v) is the dark matter velocity distribution and

        dσSIA

        dER= Gχ(q2)

        4micro2A

        Emaxπ[Z f χ

        p +(AminusZ) f χn ]

        2F2A (q) (311)

        where Emax = 2micro2Av2mA Gχ(q2) = q2

        4m2χ

        [24] is an adjustment for a momentum dependantcross section which applies only to the Majorana model

        f χ

        N =λχ

        2m2SgSNN assuming that the relic density is the central value of 1199 We have

        implemented a scaling of the direct search cross section in our code by multiplying gSNN by( Ω

        Ω0)05 as a convenient method of allowing for the fact that the actual relic density enters

        into the calculation of the cross section as a square

        FA(q) is the nucleus form factor and

        microA =mχmA

        (mχ +mA)(312)

        is the reduced WIMP-nucleon mass

        The strength of the dark matter nucleon interaction between dark matter particles andthird generation quarks is

        gSNN =2

        27mN fT G sum

        f=bt

        λ f

        m f (313)

        where fT G = 1minus f NTuminus f N

        Tdminus fTs and f N

        Tu= 02 f N

        Td= 026 fTs = 043 [20]

        33 Calculations 39

        For the real scalar and real vector models where the spin independent cross sections arenot momentum dependent we based the likelihood on the LUX 95 CL from [16] using acomplementary error function Li(d|mχ I) = 05lowastCEr f (dminusx(mχ )

        σ) where x is the LUX limit

        and d the calculated SI cross section We assumed a LUX error σ of 01lowastd lowastradic

        2

        33 Calculations

        We coded the Lagrangians for the three models in Feynrules [4] Calculation of the expectedobservables (dark matter relic density Fermi-LAT gamma ray spectrum and LUX nucleonscattering amplitudes) were performed using Micromegas v425

        331 Mediator Decay

        A number of searches have previously been performed at the LHC which show sensitivity tothe decay of scalar mediators in the form of Higgs bosons (in beyond the standard modeltheories) to bottom quarks top quarks and τ leptons (all third generation fermions) Two inparticular [43 44] provide limits on the cross-sections for the production of the scalar timesthe branching ratios to brsquos and τrsquos In order to check these limits against each of the modelswe ran parameter scans in Madgraph at leading order over the top 900 points from the nestedsampling output from MultiNest [19]

        The two processes were

        1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to apair of τ leptons in pp collisions [44]

        bull generate p p gt b b S where S is the scalar mediator

        The process measures the decay of a scalar in the presence of b-jets that decay to τ+τminus

        leptons Fig31 shows the main diagrams contributing to the process The leftmost diagramis a loop process and the centre and right are tree level however we only got Madgraph togenerated the middle diagram since we are interested in τ leptons and b quarks in the finalstate

        The limit given in [44] only applies in the case of narrow resonances where narrow istaken to mean that the width of the resonance is less than 10 of its mass To test the validity

        33 Calculations 40

        Figure 31 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof τ leptons

        of applying these limits to our models we show in Figures 32 to 34 the percentage widthof the resonance vs mS[GeV ] (pole mass of the scalar mediator) λb and λt (the couplingconstants to the b and t quarks) The probabilities are the posterior probabilities obtainedfrom MultiNest scans of the Majorana model varying the coupling constants and scalarmediator and calculating the decay width of the mediator The results are represented as 2dposterior probability heat maps in the central panels and 1d marginalised distributions on thex and y axes of each graph

        16 18 20 22 24 26 28 30

        log10(mS[GeV])

        001

        002

        003

        004

        005

        Widthm

        S

        00

        04

        08

        12

        0 100 200

        Posterior Probability

        Figure 32 WidthmS vs mS

        The WidthmS was lt 01 at almost all points except where the mass of the scalar wasgreater than 1000 GeV λb was greater than 1 or λτ was greater than 1

        This can be seen from the graphs in Figs 323334

        33 Calculations 41

        4 3 2 1 0

        λb

        001

        002

        003

        004

        005

        WidthmS

        000

        015

        030

        045

        0 100 200

        Posterior Probability

        Figure 33 WidthmS vs λb

        5 4 3 2 1 0

        λτ

        001

        002

        003

        004

        005

        WidthmS

        000

        015

        030

        045

        0 100 200

        Posterior Probability

        Figure 34 WidthmS vs λτ

        The decay widths are calculated in Madgraph at leading order and branching fractionscan be calculated to b t τ and χ (DM) The formulae for the calculations can be found indecayspy which is a subroutine of Madgraph which we have documented in Appendix BB1 Using these we calculated the branching ratio to τ+τminus so that the cross section timesbranching ratio can be compared to the limits in [80] shown in Chapters 56 and 7 in Figs5564 and 74 for each model

        2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

        This scan calculates the contributions of b-quark associated Higgs boson production inthe center and right diagrams of Fig 35 reproduced from [43] The scans did not includePythia and Delphes as only production cross sections were required and the producedparticles were in the final state and required no matching or merging

        The Madgraph processes were

        bull generate p p gt b S where S is the scalar mediator

        bull add process p p gt b S j

        bull add process p p gt b S

        33 Calculations 42

        Figure 35 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof b quarks in the presence of at least one b quark

        bull add process p p gt b S j

        The cross section times branching ratios to bb for mediator production in association withat least one b quark is shown for each model in Chapters 56 and 7 in Figs 5765 and 75

        332 Collider Cuts Analyses

        We generated two parameter scans with parameter combinations from the Multinest outputwhich were the basis of the plots of Fig 6 in [19] The plots in Fig 6 were marginalisedto two parameters each but the raw Multinest output contained 5 parameters plus the priorprobabilities of each combination normalised to total 10

        The first scan was for χ produced with b quarks In order to force the mediator to decayto χ only we had χ in the final state and defined jets to include b quarks in Madgraph Thenwe generated the processes

        bull generate p p gt χ χ j

        bull add process p p gt χ χ j j

        Jet matching was on

        The second scan was for t quarks produced in the final state

        bull generate p p gt χ χ tt

        33 Calculations 43

        No jet matching was required because we did not mix final state jet multiplicities at thematrix element level

        The outputs from these two processes were normalised to 21 f bminus1 and combined

        The results of these scans prior to collider cuts calculations are shown in Figs 5161and 71 These show the posterior probabilities of the various parameter combinations withdifferent constraints applied seperately and then together A more detailed commentary isgiven in each of the sections

        We then sampled 900 points based on posterior probability from these scans and ran thesethrough Madgraph v425 then applied collider cuts with the C++ code using the limits fromthe following six ATLAS and CMS searches shown in Table32 and described below to findany exclusions

        333 Description of Collider Cuts Analyses

        In the following all masses and energies are in GeV and angles in radians unless specificallystated

        3331 Lepstop0

        Search for direct pair production of the top squark in all-hadronic final states in pro-tonndashproton collisions at

        radics=8 TeV with the ATLAS detector[32]

        This analysis searched for direct pair production of the supersymmetric partner to thetop quark using integrated luminosity of 201 f bminus1 The top squark was assumed to de-cay via t rarr t χ0

        1 or t rarr bχ01 or t rarr bχ

        plusmn1 rarr bW (lowast)χ1

        0 where χ01 (χ

        plusmn1 ) denotes the lightest

        neutralino(chargino) in supersymmetric models The search targets a fully hadronic finalstate with four or more jets and large missing momentum The production of a mediator inassociation with top quarks followed by an invisible mediator decay to dark matter particleshas a similar final state to that considered in the ATLAS stop searches Thus we expect theATLAS reach to have sensitivity to the three dark matter models considered in this thesis

        The collider cuts program mimicked the object reconstruction through jetlepton overlapremoval and selected signal electrons muons and τ particles with pT gt 10 GeV and

        33 Calculations 44

        Table 32 95 CL by Signal Region

        Experiment Region Number

        Lepstop0

        SRA1 66SRA2 57SRA3 67SRA4 65SRC1 157SRC2 124SRC3 80

        Lepstop1

        SRtN1-1 857SRtN1-2 498SRtN1-3 389SRtN2 107SRtN3 85SRbC1 832SRbC2 195SRbC3 76

        Lepstop2

        L90 740L100 56L110 90L120 170

        2bstop

        SRA 857SRB 270SRA150 380SRA200 260SRA250 90SRA300 75SRA350 52

        CMSTopDM1L SRA 1385

        ATLASMonobjetSR1 1240SR2 790

        33 Calculations 45

        |η | lt 24724 and 247 respectively (for eminusmicrominusτ) Jets were selected with pT gt 20 GeVand |η |lt 45 b-jets were selected if they were tagged by random selection lt 70 and had|η |lt 25 and pT gt 20 GeV

        These jets and leptons were further managed for overlap removal within ∆R lt 04 withnon b-jets removed if ∆R lt 02

        The b-jets and non b-jets were further selected as signal jets if pT gt 35 GeV and |η |lt 28

        These signal jets electrons and muons were finally binned according to the signal regionsin Tables 333435 below

        Table 33 Selection criteria common to all signal regions

        Trigger EmissT

        Nlep 0b-tagged jets ⩾ 2

        EmissT 150 GeV

        |∆φ( jet pmissT )| gtπ5

        mbminT gt175 GeV

        Table 34 Selection criteria for signal regions A

        SRA1 SRA2 SRA3 SRA4anti-kt R=04 jets ⩾ 6 pT gt 808035353535 GeV

        m0b j j lt 225 GeV [50250] GeV

        m1b j j lt 225 GeV [50400] GeV

        min( jet i pmissT ) - gt50 GeV

        τ veto yesEmiss

        T gt150 GeV gt250 GeV gt300 GeV gt 350 GeV

        Table 35 Selection criteria for signal regions C

        SRC1 SRC2 SRC3anti-kt R=04 jets ⩾ 5 pT gt 8080353535 GeV

        |∆φ(bb)| gt02 π

        mbminT gt185 GeV gt200 GeV gt200 GeV

        mbmaxT gt205 GeV gt290 GeV gt325 GeV

        τ veto yesEmiss

        T gt160 GeV gt160 GeV gt215 GeV

        wherembmin

        T =radic

        2pbt Emiss

        T [1minus cos∆φ(pbT pmiss

        T )]gt 175 (314)

        33 Calculations 46

        and pbT is calculated for the b-tagged jet with the smallest ∆φ to the pmiss

        T direction andmbmax

        T is calculated with the pbT for the b-tagged jet with the largest ∆φ to the pmiss

        T direction

        m0b j j is obtained by first selecting the two jets with the highest b-tagging weight- from

        the remaining jets the two jets closest in the η minusφ plane are combined to form a W bosoncandidate this candidate is then combined with the closest of the 2 b-tagged jets in the η minusφ

        plane to form the first top candidate with mass m0b j j A second W boson candidate is formed

        by repeating the procedure with the remaining jets this candidate is combined with thesecond of the selected b-tagged jets to form the second top candidate with mass m1

        b j j Thecuts shown in Appendix1 Table A1 show the cumulative effect of the individual cuts appliedto the data the signal region cuts and the cumulative effects of individual cuts in SignalRegion C

        3332 Lepstop1

        Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

        radics=8 TeV pp collisions using 21 f bminus1 of

        ATLAS data[33]

        The stop decay modes considered were those to a top quark and the lightest supersym-metric particle (LSP) as well as a bottom quark and the lightest chargino where the charginodecays to the LSP by emitting a W boson We should expect the three dark matter models inthis paper with couplings to t and b quarks to show some visibility in this search

        The collider cuts program mimicked the object construction by selecting electrons andmuons with pT gt 10 GeV and |η |lt 247 and 24 respectively and jets with pT gt 20 GeVand |η |lt 10 b-jets with |η |lt 25 and pT gt 25 GeV were selected with random efficiency75 and mistag rate 2 (where non b-jets were mis-classified) We then removed any jetwithin ∆R = 02 of an electron and non overlapping jets with |η |lt 28 were used to removeany electrons and muons within ∆R = 04 of these while non-overlapping jets with |η |lt 25and pT gt 25 GeV were classified as signal jets Non overlapping electrons and muons withpT gt 25 GeV were classified as signal electrons and muons

        Signal selection defined six regions to optimize sensitivity for different stop and LSPmasses as well as sensitivity to two considered stop decay scenarios Three signal regionslabelled SRbC 1-3 where bC is a moniker for b+Chargino and 3 regions labelled SRtN 1-3where tN is a moniker for t+Neutralino were optimised for the scenarios where t1 decaysto the respective particles indicated by the labels Increasing label numbers correspond toincreasingly strict selection criteria

        33 Calculations 47

        The dominant background in all signal regions arises from dileptonic tt events All threeSRtN selectons impose a requirement on the 3-jet mass m j j j of the hadronically decayingtop quark to specifically reject the tt background where both W bosons from the top quarkdecay leponically

        For increasing stop mass and increasing difference between the stop and the LSP therequirents are tightened on Emiss

        T on the ratio EmissT

        radicHT where HT is the scalar sum of the

        momenta of the four selected jets and also tightened on mT

        To further reduce the dileptonic tt background for an event characterised by two one stepdecay chains a and b each producing a missing particle C use is made of the variable

        mT 2 =min

        pCTa + pC

        T b = pmissT

        [max(mTamtb)] (315)

        where mTi is the transverse mass 1 of branch i for a given hypothetical allocation (pCTa pC

        T b)

        of the missing particle momenta Eqn 315 is extended for more general decay chains byamT 2 [81] which is an asymmetric form of mT 2 in which the missing particle is the W bosonin the branch for the lost lepton and the neutrino is the missing particle for the branch withthe observed charged lepton

        ∆φ( jet12 pmissT ) the azimuthal angle between leading or subleading jets and missing mo-

        mentum is used to suppress the background from mostly multijet events with mis-measuredEmiss

        T

        Events with four or more jets are selected where the jets must satisfy |η | lt 25 and pT gt80604025 GeV respectively

        These signal particles were then classified according to the cutflows analysis in Fig36below and compared with the paper in Appendix1 Table A2

        3333 Lepstop2

        Search for direct top squark pair production in final states with two leptons in p pcollisions at

        radics=8TeV with the ATLAS detector[34]

        Three different analysis strategies are used to search for t1 pair production Two targett1 rarr b+ χ

        plusmn1 decay and the three body t1 rarr bW χ0

        1 decay via an off-shell top quark whilst

        1The transverse mass is defined as m2T = 2plep

        T EmissT (1minus cos(∆φ)) of branch i where ∆φ is the azimuthal

        angle between the lepton and the missing transverse momentum

        33 Calculations 48

        Table 36 Signal Regions - Lepstop1

        Requirement SRtN2 SRtN3 SRbC1 SRbC2 SRbC3∆φ( jet1 pmiss

        t )gt - 08 08 08 08∆φ( jet2 pmiss

        T )gt 08 08 08 08 08Emiss

        T [GeV ]gt 200 275 150 160 160Emiss

        T radic

        HT [GeV12 ]gt 13 11 7 8 8

        mT [GeV ]gt 140 200 120 120 120me f f [GeV ]gt - - - 550 700amT 2[GeV ]gt 170 175 - 175 200mT

        T 2[GeV ]gt - 80 - - -m j j j Yes Yes - - -

        one targets t1 rarr t + χ01 to an on shell quark decay mode The analysis is arranged to look

        at complementary mass splittings between χplusmn1 and χ0

        1 smaller and larger than the W bosonmass The smaller splitting provides sensitivity to t1 three body decay

        Object reconstruction was carried out by selecting baseline electronsmuons and τrsquos withpT gt 10 GeV and |η |lt 24724 and 247 respectively Baseline jets and b-jets were selectedwith pT gt 20 GeV and |η |lt 25 and b-jets with 70 efficiency Baseline jets were removedif within ∆R lt 02 of an electron and electrons and muons were removed if within ∆R lt 04of surviving jets Signal electrons required pT gt 10 GeV and signal jets pT gt 20 GeV and|η |lt 25

        The analysis made use of the variable mT 2 [82] the stransverse mass used to mea-sure the masses of pair-produced semi-invisibly decaying heavy particles It is definedas mT 2(pT1 pT2qT ) =

        minqT1+qT2=qT

        max[mT (pT1qT1)mT (pT2qT2)] where mT indicatesthe transverse mass pT1 and pT2 are vectors and qT = qT1 + qT2 The minimisation isperformed over all possible decompositions of qT

        Events were selected if they passed either a single-electron a single-muon a double-electron a double-muon or an electron-muon trigger Events are required to have exactlytwo opposite sign leptons and at least one of the leptons or muons must have a momentumgreater than 25 GeV and the invariant mass (rest mass) of the two leptons is required to begreater than 20 GeV In order to reduce the number of background events containing twoleptons produced by the on-shell decay of the Z boson the invariant mas must be outside711-111 GeV range

        Two additional selections are applied to reduce the number of background events withhigh mT 2 arising from events with large Emiss

        T due to mismeasured jets ∆φb lt 15 and∆φ gt 10 The quantity ∆φb is the azimuthal angle between pll

        T b = pmissT + pl1

        T +Pl2T The

        33 Calculations 49

        vector pllT b is the opposite of the vector sum of all the hadronic activity in the event For WW

        and tt backgrounds it measures the transverse boost of the WW system and for the signalthe transverse boost of the chargino-chargino system The ∆φ variable is the azimuthal anglebetween the pmiss

        T vector and the direction of the closest jet

        By requiring mT 2 gt 40 GeV the tt background is the dominant background in both thesame-flavoured (SF) and different-flavoured (DF) events

        Four signal regions are then defined as shown in Table 37 SR M90 has the loosestselection requiring MT 2gt 90 GeV and no additional requirement on jets and providessensitivity to scenarios with a small difference between the masses of the top-squark andchargino SR M110 and M120 have a loose selection on jets requiring two jets with pT

        gt 20 GeV and mT 2 to be greater than 110 GeV and 120 GeV respectively They providesensitivity to scenarios with small to moderate difference between the invariant masses of thestop quark and the charginos SR M100 has a tighter jet slection and provides sensitivity toscenarios with a large mass difference between the both stop quark and the chargino and thestop quark and the neutralino

        The analysis cut regions are summarised in Fig37 below

        Table 37 Signal Regions Lepstop2

        SR M90 M100 M110 M120pT leading lepton gt 25 GeV

        ∆φ(pmissT closest jet) gt10

        ∆φ(pmissT pll

        T b) lt15mT 2 gt90 GeV gt100 GeV gt110 GeV gt120 GeV

        pT leading jet no selection gt100 GeV gt20 GeV gt20 GeVpT second jet no selection gt50 GeV gt20 GeV gt20 GeV

        To validate the code we ran the cuts shown in Appendix 1 Table A3 with the givenMadgraph simulation and compared to [83] It was not clear what baseline cuts had beenemployed in the ATLAS paper so our results were normalised to the 2 SF electrons beforecomparison Some of the ATLAS cut numbers were less than 3 and we did not produce anyevents passing these cuts but clearly the margin of error at this level was greater than thenumber of ATLAS events

        33 Calculations 50

        3334 2bstop

        Search for direct third-generation squark pair production in final states with miss-ing transverse momentum and two b-jets in

        radics= 8 TeV pp collisions with the ATLAS

        detector[31]

        Two sets of scenarios are considered In the first the coloured sbottom is the onlyproduction process decaying via b1 rarr bχ0

        1 In the second the stop is the only productionprocess decaying via t1 rarr t χplusmn

        1 and the lightest chargino (χplusmn1 ) decays via a virtual W boson

        into the final state χ01 f f prime Both scenarios are characterised by the presence of two jets

        resulting from the hadronisation of b quarks and large missing transverse momentum Theproduction of a mediator in association with top quarks followed by an invisible mediatordecay to dark matter particles has a similar final state as that considered in the ATLAS stopsearches Thus we expect the ATLAS reach to have sensitivity to the three dark mattermodels considered in this thesis

        The collider cuts program mimicked object reconstruction by selecting baseline electronswith pT gt 7 GeV and |η |lt 247 and muons with pT gt 6 GeV and |η |lt 24 Baseline jetswere selected with pT gt 20 GeV and |η |lt 28 We then removed any jets within ∆R lt 02of an electron and any electron within ∆R lt 04 of surviving jets Signal electrons andmuons were required to have pT gt 25 GeV b-jets were selected with 60 tagging efficiencyand |η |lt 25 and pT gt 20 GeV

        Two sets of signal regions are defined to provide sensitivity to the kinematic topologiesassociated with the different mass splittings between the sbottom or the stop and the neu-tralino In all cases the presence of at least one primary vertex (with at least five tracks withpT gt 04 GeV) is required Events are selected with Emiss

        T gt 150 GeV and no electrons ormuons identified in the final state For signal region selections jets with |η | lt 28 are orderedaccording to their pT and 2 jets are required to be b-tagged

        The variables are defined as follows

        bull ∆φmin is the minimal azimuthal angle ∆φ between any of the three leading jets andpmiss

        T

        bull me f f (k) = sumki=1(p jet

        T )i +EmissT where the index refers to the pT ordered list of jets

        bull HT3 is defined as the scalar sum of the pT of the n leading pT ordered jets withoutincluding the 3 leading jets HT3 = sum

        ni=4(p jet

        T )i

        bull mbb is the invariant mass of the two b-tagged jets in the event

        33 Calculations 51

        bull mCT is the contransverse mass [82] and is a kinematical variable that can be usedto measure the masses of pair produced semi-invisibly decaying particles For twoidentical decays of heavy particles into two invisible particles v1 and v2 and twoinvisible particles mCT is defined as m2

        CT (v1v2) = [ET (v1)+ET (v2)]2 minus [pT (v1)minus

        pT (v2)]2 where ET =

        radicp2

        T +m2 The the contransverse mass is invariant under equaland opposite boosts of the parent particles in the transverse plane and is bounded fromabove by an analytical expression of the particle masses For production of sbottompairs the bound is given by mmax

        CT =m2(b)minusm2(χ0

        1 )

        m(b) and for tt events the bound is 135

        GeV A similar equation can be written for stop pairs in term of the stop and charginomasses

        A definition of the signal regions is given in the Table38

        Table 38 Signal Regions 2bstop

        Description SRA SRBEvent cleaning All signal regions

        Lepton veto No emicro after overlap removal with pT gt7(6) GeV for e(micro)Emiss

        T gt 150 GeV gt 250 GeVLeading jet pT ( j1) gt 130 GeV gt 150 GeVSecond jet pT ( j2) gt 50 GeV gt 30 GeVThird jet pT ( j3) veto if gt 50 GeV gt 30 GeV

        ∆φ(pmissT j1) - gt 25

        b-tagging leading 2 jets(pT gt 50 GeV |η | lt 25)

        2nd- and 3rd- leading jets(pT gt 30 GeV|η | lt 25)

        ∆φmin gt 04 gt 04Emiss

        T me f f (k) EmissT me f f (2) gt 025 Emiss

        T me f f (3) gt 04mCT gt 150200250300350 GeV -HT3 - lt 50 GeVmbb gt200 GeV -

        The analysis cuts are summarised in Table A4 of Appendix 1

        3335 ATLASMonobjet

        Search for dark matter in events with heavy quarks and missing transverse momentumin pp collisions with the ATLAS detector[30]

        Events were selected with high missing transverse momentum when produced in associa-tion with one or more jets identified as containing b quarks Final states with top quarks wereselected by requiring a high jet multiplicity and in some cases a single lepton The research

        33 Calculations 52

        studied effective contact operators in the form of a scalar operator Oscalar = sumqmqMN

        lowastqqχχ

        where MNlowast is the effective mass of the coupling and the quark and DM fields are denoted by

        q and χ respectively These are the equivalent of the simplified models in this paper with themediator integrated out

        Object reconstruction selected baseline electrons muons jets and τrsquos with pT gt 20 GeVand |η |lt 2525 and 45 respectively Signal jets were selected by removing any jets within∆R lt 02 of an electron and electrons and muons were selected by removing them if theywere within ∆R lt 04 of surviving jets b-jets were tagged with an efficiency of 60

        Only signal regions SR1 and SR2 were analysed

        The analysis split events into signal regions SR1 and SR2 according to the followingCriteria

        Table 39 Signal Region ATLASmonobjet

        Cut SR1 SR2Jet multiplicity n j 1minus2 3minus4

        bminusjet multiplicity nb gt 0 (60 eff) gt 0 (60 eff)Lepton multiplicity nl 0 0

        EmissT gt300 GeV gt200 GeV

        Jet kinematics pb1T gt100 GeV pb1

        T gt100 GeV p j2T gt100 (60) GeV

        ∆φ( j1EmissT ) gt 10 i = 12 gt 10 i = 14

        Where p jiT (pbi

        T ) represent the transverse momentum of the i-th jet b-tagged jet respec-tively If there is a second b-tagged jet the pT must be greater than 60 GeV

        3336 CMSTop1L

        Search for top-squark pair production in the single-lepton final state in pp collisionsat

        radics=8 TeV[41]

        This search was for pair production of top squarks in events with a single isolatedelectron or muon jets large missing transverse momentum and large transverse mass

        (MT =radic

        2EmissT pl

        T (1minus cos(∆φ))) where plT is the leptonrsquos transverse momentum and ∆(φ)

        is the difference between the azimuthal angles of the lepton and EmissT The 3 models

        considered should have some visibility at certain parameter combinations in this regionThe search focuses on two decay modes of the top squark leading to the two processespp rarr t tlowast rarr tt χ0

        1 χ01 rarr bbW+Wminusχ0

        1 χ01 and pp rarr t tlowast rarr bbχ

        +1 χ

        minus1 rarr bbW+Wminusχ0

        1 χ01 The

        33 Calculations 53

        lightest neutralino χ01 is considered the lightest supersymmetric particle and escapes without

        detection The analysis is based on events where one of the W bosons decays leptonicallyand the other hadronically resulting in one isolated lepton and 4 jets two of which originatefrom b quarks The two neutralinos and the neutrino from the W decay can result in largemissing transverse momentom The kinematical end point of MT is lt the mass of the Wboson However for signal events the presence of LSPs in the final state allows MT to exceedthe mass of the W boson The search is therefore for events with large MT The dominantbackground with large MT arises from tt decays where W bosons decay leptonically withone of the leptons not being identified

        To reduce the dominant tt background use was made of the MWT 2 variable defined as

        the minimum mother particle mass compatible with the transverse momenta and mass shellconstraints [84]

        Object reconstruction required electrons and muons to have pT gt 30 and 25 GeV respand |η |lt 25 and 21 respectively Baseline jets were required to have pT gt 30 and |η |lt 4and b-jets were tagged with an efficiency of 75 for |η |lt 24

        Events were chosen if they contained one isolated electron or muon at least 4 jets and atleast one b-tagged jet and Emiss

        T gt 100 GeV MT gt 160 GeV MWT 2 was required to be greater

        than 200 GeV and the smallest azimuthal angle difference between baseline jets and EmissT

        gt12

        Chapter 4

        Calculation Tools

        41 Summary

        Figure 41 Calculation Tools

        The simplified models were implemented in Feynrules [4] and the model files importedinto MicrOmegas v425 which was used to calculate the log-likelihoods of the astrophysicalobservables MicrOmegas was modified to allow only s-channel decays in both relic densityand gamma ray calculations Direct detection constraints were easy to implement for thescalar and vector models by coding the analytical expression for the spin independent WIMP-nucleon scattering cross section but due to the momentum dependence of the WIMP-nucleon

        42 FeynRules 55

        scattering cross section of the Majorana fermion model LUXCalc v101 [79] was modifiedfor this model to implement momentum dependent cross sections [85] We used MicrOmegasv425 to calculate relic density values and to produce gamma ray spectra for comparisonwith the Fermi-LAT results Parameter scans were carried out using MultiNest v310 withthe parameter ranges given in Table 51 and using astrophysical constraints only Oncethese scans had converged a random sample of 900 points was chosen (weighted by theposterior probability) and then used to calculate collider constraints using the MadgraphMonte Carlo generator This included comparison with mediator production and decay limitsfrom the LHC experiments and the reproduction of searches for dark matter plus other finalstate objects which required dedicated C++ code that reproduces the cut and count analysesperformed by CMS and ATLAS

        42 FeynRules

        FeynRules is a Mathematicareg package that allows the calculation of Feynman rules inmomentum space for any QFT physics model The user needs to provide FeynRules withminimal information to describe the new model contained in the model-file This informationis then used to calculate a set of Fenman rules associated with the Lagrangian The Feynmanrules calculated by the code can then be used to implement new physics models into otherexisting tools such as Monte Carlo (MC) generators via a set of interfaces developed andmaintained by the MC authors

        Feynrules can be run with a base model = SM (or MSSM) and user defined extensions tothe base model

        43 LUXCalc

        LUXCalc is a utility program for calculating and deriving WIMP-nucleon coupling limitsfrom the recent results of the LUX direct search dark matter experiment The program hasbeen used to derive limits on both spin-dependent and spin-independent WIMP-nucleoncouplings and it was found under standard astrophysical assumptions and common otherassumptions that LUX excludes the entire spin-dependent parameter space consistent witha dark matter interpretation of DAMArsquos anomalous signal and also found good agreementwith published LUX results [79]

        We modified the program for the Majorana fermion model to implement momentumdependent cross -sections [85]

        44 Multinest 56

        44 Multinest

        Multinest is an efficient Bayesian Inference Tool which uses a simultaneous ellipsoidalnested sampling method a Monte Carlo technique aimed at an efficient Bayesian Evidencecalculation with posterior inferences as a by product

        Bayes theorem states that

        Pr(θ |DH) =Pr(D|θ H)Pr(θ |H)

        Pr(D|H) (41)

        Here D is the data and H is the Hypothesis Pr(θ |DH) is the posterior probability ofthe parameters Pr(D|θ H) = L(θ) is the likelihood Pr(θ |H) is the prior and Pr(D|H) isthe evidence

        The evidence Pr(D|H) =int

        Pr(θ |DH)Pr(θ |H)d(θ) =int

        L(θ)Pr(θ |H)d(θ) Thus ifwe define the prior volume dX = Pr(θ |H)d(θ) and

        X(λ ) =int

        L(θ)gtλ

        Pr(θ |H)d(θ) (42)

        where the integral extends over the region of parameter space within the iso likelihoodcontour L(θ) = λ then the evidence integral can be written

        int 10 L (X)dX where L (X) the

        inverse of equation 42 and is a monotonically decreasing function of X (the prior volumewhich is 1 for the total sample space and 0 for a single point)

        Thus if one can evaluate likelihoods L (Xi) where 0 lt Xn lt lt X2 lt X1 lt X0 = 1the evidence can be estimated as sumLiwi where wi = 5(Ximinus1 minusXi+1) Ellipsoidal samplingapproximates the iso likelihood contour by a D dimensional covariance matrix of the currentset of active points New points are selected from the prior within this ellipsoidal bounduntil one is obtained that has a likelihood greater than the likelihood of the removed lowestlikelihood point This method is extended for multimodal sampling where there may bemore than one local maximum by clustering points that are well separated and constructingindividual ellipsoidal bounds for each cluster For more details consult [7]

        The results of the analysis are expressed as a data file containing the sample probability-2loglikelihood followed by the parameters of the model for the particular sampling Theprobabilities are normalised to total 10

        45 Madgraph 57

        45 Madgraph

        Madgraph is a general purpose matrix element calculator and Monte Carlo generator Basedon user-supplied Feynman rules (eg the output of Feynrules) the package calculates the tree-level matrix elements for any processes that the user specifies and generates samples of LHCevents including an accurate simulation of the proton collisions themselves (which requires aknowledge of the parton distribution functions that determine the relative proportion of eachparton inside the nucleus) Hadronisation of final state quarks is not possible in Madgraphand hence the code must be used in conjunction with Pythia which performs a parton shower

        The Madgraph output was automatically showered by Pythia (Pythia describes partonradiation as successive parton emissions using Markov chain techniques based on Sudakovform factors and this description is only formally correct for the soft and collinear emissions)A Madgraph simulation is at the level of the fundamental particles (quarks and leptons)Madgraph generates a limited set of matrix element processes at LO representing hard (highenergy) emissions and passes these to Pythia for showering The final state partons in theMadgraph simulation must be matched to the Pythia final state partons to avoid doublecounting

        The output from Pythia is automatically fed through Delphes where basic collider cutscan be performed (calculation of the numbers of particles and jets that would be observed atdifferent energies and locations within the collider) The output of the Delphes simulation is afile containing reconstructed events that can be analysed using the standard ROOT framework(the most popular data analysis package in particle physics) [86]

        The parameters that gave the best fits to the direct [16] and indirect DM experiments [87]and relic density [59] are ranked by posterior probability from the Multinest [7] output

        The parameters in the Madgraph simulations are stored in files in the cards sub directoryand in particular the run card param card and Delphes card record items relevent to thesimulation

        In the run card the number of events the Beam 1 and 2 total energy in GeV (4000 each)and the type of jet matching were modified (to MLM type matching where the final statepartons in Madgraph are clustered according to the KT jet algorithm to find the equivalentparton shower history of the event there is a minimum cutoff scale for the momentum pT

        given by the xqcut parameter) The maxjetflavour (=5 to include b quarks) and xqcut = 30GeV parameters were also modified

        46 Collider Cuts C++ Code 58

        The param card was modified to specify the masses of the new particles and the autodecay modes The Delphes card is overwritten with either the ATLAS or CMS defaultcards which record tracking efficiencies and isolation momentum and energy resolutionscalorimeter details jet finder and b and τ -tagging parameters However the isolation andtagging in this study were done by the collider cuts C++ code since multiple sets of cutswere applied to the same output from Pythia

        When performing simulations of the three dark matter models it was necessary to definethe proton and jets to contain b quarks as this is not the default in Madgraph The processesgenerated included 1 and 2 jets

        46 Collider Cuts C++ Code

        Because of the volume of simulations required for the collider constraints we used C++code (originally developed by Martin White and Sky French and modified by me) that readsthe ROOT files and outputs the numbers of events that would have been observed under thechosen collider experiments had the model been true for the given parameters Thus the sameoutput files can be used for all of the collider cuts and hence no cuts were performed throughthe Delphes cards

        In order to validate the C++ code we replicated the observed events with the papersdescribing the various collider searches (see Appendix)

        Chapter 5

        Majorana Model Results

        51 Bayesian Scans

        To build some intuition we first examined the constraining effect of each observable on itsown We ran separate scans for each constraint turned on and then for all the constraintstogether In Fig 51 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together The paramaters were scanned over the range in Table 51 below

        Table 51 Scanned Ranges

        Parameter mχ [GeV ] mS[GeV ] λt λb λτ

        Range 0minus103 0minus103 10minus5 minus10 10minus5 minus10 10minus5 minus10

        In Fig51 we can see that the preferred DM mass is centered around 35 GeV primarily dueto the influence of the Fermi-LAT gamma ray constraint whilst the relic density constraintallows a wide range of possible mediator and DM masses Direct detection (LUX) constraintssplit the mass regions into a top left and bottom right region and together these split theoverall mass parameter regions into two regions by mediator mass but with DM mass around35 GeV Later studies using a lower excess gamma ray spectrum tend to increase the preferredDM mass up to 50 GeV [76] and [88]

        The marginalised posterior probabilities for all combinations of parameters taken twoat a time are shown in Fig52 These plots have all constraints (Gamma LUX and RelicDensity) applied together The idea of these graphs is to study the posterior probabilities ofcombinations of two paramaters taken at a time (the off-diagonal graphs) The diagonal plotsshow marginalised histograms of a single parameter at a time- this gives an indication of thespread of each parameter The plots show that mS and λb show a high degree of correlationfor mS gt 100 GeV but there is also a range of values of mS around 35 GeV which are

        51 Bayesian Scans 60

        1 0 1 2 3 4log10(mχ)[GeV]

        1

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        s)[GeV

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        1 0 1 2 3 4log10(mχ)[GeV]

        1

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        s)[GeV

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        1 0 1 2 3 4log10(mχ)[GeV]

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        s)[GeV

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        05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

        05

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        (d) All Constraints

        Figure 51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether

        51 Bayesian Scans 61

        00

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        χ)[GeV

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        b)

        00 05 10 15 20 25 30

        log10(mχ)[GeV]

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        τ)

        00 05 10 15 20 25 30

        ms[Gev]5 4 3 2 1 0 1

        log10(λt)5 4 3 2 1 0 1

        log10(λb)5 4 3 2 1 0 1

        log10(λτ)

        Figure 52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter

        52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 62

        possible for λb around 001 λt and λτ show a wide range of possible values for two possibleranges of mS lt 35 GeV and greater than 100 GeV mχ shows a narrow range of possiblevalues around 35 GeV for all values of the other parameters

        52 Best fit Gamma Ray Spectrum for the Majorana FermionDM model

        We can see from the Fig51(a) scan that the maximum log-likelihood Fermi-LAT gammaray spectrum on its own shows values that centre around 35 GeV This is not changed byadding a relic density constraint or a direct detection (LUX) constraint

        The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraintsis shown in Fig53 The parameters for this point were

        Table 52 Best Fit Parameters

        Parameter mχ [GeV ] mS[GeV ] λt λb λτ

        Value 3332 49266 0322371 409990 0008106

        10-1 100 101 102

        E(GeV)

        10

        05

        00

        05

        10

        15

        20

        25

        30

        35

        E2dφd

        E(G

        eVc

        m2ss

        r)

        1e 6

        Best fitData

        Figure 53 Gamma Ray Spectrum

        The relic density (Ωh2) at this point has a value of 0135 but the point is separated onlyslightly in posterior probability from points with a value of 01199 with mS lt 10 GeV (iethe posterior probability is almost the same differing by less than 1)

        To better understand the interplay of the different constraints the three graphs in Fig54show the log likelihood surfaces and individual best fit points (best log likelihoods) for the γ

        and Ω scans seperately and then combined The diagrams each choose a pair of couplings(fixing the remaining parameters) and display the 2d surfaces of minuslog(χ2) (proportional to

        52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 63

        the log likelihoods of the coupling combinations) obtained by varying the pair of couplingsThese graphs do not show marginalised posterior probabilities but rather are based on the-2log likelihood calculations directly from MicrOmegas for the given coupling combinationsWe have termed this value as χ2 for convenience and used the log of this value to give abetter range for the colours These log likelihood values are what drive the MultiNest searchof parameter space for the best parameter combinations

        The graphs show a complex interplay of the different constraints resulting often in acompromise parameter combination The arrow on each graph points to the maximum point We have chosen a single representative set of couplings near the best overall fit point for allthe graphs

        52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 64

        00 05 10 15 20 25 30

        log10(mχ)

        00

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        10(m

        S)

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        minuslog10(χ2(Γ)) λt = 487 λτ = 024 λb = 0344

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        minuslog10(χ2(Omega)) λt = 487 λτ = 024 λb = 0344

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        minuslog10(χ2(Both)) λt = 487 λτ = 024 λb = 0344

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        Both Maximum at mχ=8317 GeV mS=2884GeV The best fit point for these couplings maybe above below or between the individual bestfit points but will on average be between thepoints

        Figure 54 Plots of log likelihoods by individual and combined constraints Masses in GeV

        53 Collider Constraints 65

        53 Collider Constraints

        531 Mediator Decay

        1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof τ leptons in pp collisions [44] Refer also to Chapter 3 for details

        We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

        The results of the calculations are shown in the graph Fig55 and can be compared withthe limits in the table from the paper [44] in Fig 56 The black line is the observed ATLASlimit

        Figure 55 σ lowastBr(σ rarr ττ) versus Mass of Scalar

        0 200 400 600 800

        mS[GeV]

        10

        5

        0

        log 1

        0(σ

        (bbS

        )lowastB

        (Sgtττ

        ))[pb]

        Observed LimitLikely PointsExcluded Points

        0

        20

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        60

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        120

        0 5 10 15 20 25 30 35 40 45

        We can see that only one point here exceeded the limit and the points showed a widespread over σ(bbS)lowastBr(σ rarr ττ) by mS space which is consistent with the spread of pointson the MultiNest scan on the mS by λτ plane causing a spread of branching ratios to the τ

        quark It may seem odd that only one point was excluded here but this point was sampledwith a very low probability having high coupling constants of 597 to the τ quark and 278 tothe t quark and only just exceeds the limit As such it can be explained as a random outlier

        53 Collider Constraints 66

        Figure 56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar

        2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

        This scan calculates the contributions of b-quark associated Higgs boson productionreproduced from [43] Refer to Chapter 3 for details

        We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

        The cross section times branching ratios to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig57 The black line is the CMSLimit

        Figure 57 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar

        0 200 400 600 800

        mS[GeV]

        15

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        10(σ

        (bS

        +X

        )lowastB

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        bb))

        [pb]

        Observed LimitLikely PointsExcluded Points

        0

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        53 Collider Constraints 67

        The results of this scan were compared to the limits in [89] with the plot shown inFig58

        Figure 58 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

        We can see that there are a number of exclusions across the range of mS which occuredfor high values of the coupling constants gt 1 for λbλt λtau - these can be seen also on Fig59The excluded parameter combinations represent less than 2 of the total sampled number

        532 Collider Cuts Analyses

        We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude and the results are shown against theposterior probabilities of the parameter combinations in Fig 59

        The circles in Figure 59 show which points were excluded by the collider cuts as well asthe limits on cross sections times branching ratios to each of τ+τminus and bb

        All the excluded points under the heading Collider Cuts showed exclusion by the Lepstop0experiment (Search for direct pair production of the top quark in all hadronic final states

        53 Collider Constraints 68

        0 1 2 3

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        σ lowastBr(σgt bS+X)

        σ lowastBr(σgt ττ)

        (a) mχ by mS

        6 5 4 3 2 1 0 1 2

        log10(λt)

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        Figure 59 Excluded points from Collider Cuts and σBranching Ratio

        53 Collider Constraints 69

        [32]) Some of these same points were also excluded by Lepstop2 2bstop CMSTop andATLASMonobjet experiments There were expected events at other points but none weresignificant at the 95 exclusion level These significant exclusions occurred mainly wherethe mass of the mediator was over 100 GeV and higher values of the couplings to b and tquarks but there were a few points with mediator mass around 25 GeV The exclusions withmediator mass around 25 GeV were all accompanied by couplings to t quarks of order 10which is probably non-perturbative

        Looking at the exclusions in Fig 59 we can see that these occur mainly in the regionof higher mediator mass gt 100 GeV and lt 400 GeV These exclusions come from theastrophysical constraints (Gamma) which show a strong correlation between λb and mS andhigh values of λt gt 1 However these mass regions do not overlap at the same couplingcombinations as the high likelihood region The green and white circles correspond to theexcluded points in Figs 57 and 58 respectively

        The remaining regions of highest likelihood can be seen to be where 0001 λb lt 01accompanied by λt lt 0001 and mass of the mediator mS lt 10 GeV or 400 GeV lt mS lt1000GeV with mχ sim 35 GeV

        The plot of λt by λb shows the significant regions do not overlap with the collider cutsand cross section constraints since the latter occur at high values of either coupling while thelikely region has low values of these couplings (also accompanied by low couplings to λτ )

        Chapter 6

        Real Scalar Model Results

        61 Bayesian Scans

        To build some intuition we first examined the constraining effect of each observable on itsown We ran seperate scans for each constraint turned on and then for all the constraintstogether In Figure 61 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together

        In Figure 61 we can see that there are is a narrow region which dominates when allconstraints are taken into account This centers on mχ=9 GeV and a range of values for mS

        from 35 GeV to 400 GeV The region is not the highest likelihood region for the Fermi-LATgamma ray spectrum on its own (as can be seen from the individual plot - there is a narrowband to the left of the highest likelihood band) nor for the LUX likelihood or relic densitybut represents a compromise between the three constraints

        The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig62 The diagonal plots show marginalised histograms of a singleparameter at a time

        61 Bayesian Scans 71

        05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

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        Figure 61 Real Scalar Dark Matter - By Individual Constraint and All Together

        61 Bayesian Scans 72

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        Figure 62 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter

        62 Best fit Gamma Ray Spectrum for the Real Scalar DM model 73

        62 Best fit Gamma Ray Spectrum for the Real Scalar DMmodel

        We can see from Fig61 that the maximum log-likelihood Fermi-LAT Gamma ray spectrumon its own shows values of mχ that centre around 35 GeV However the LUX constraintin particular moves the best fit to a point around 9 GeV- the tight LUX constraint onlydisappears for low masses and at that point the proportion of τ+τminus annihilation needs toincrease to fit the constraints

        The gamma ray spectrum at the points of best fit for Fermi-LAT including all constraintsis shown in Fig63 The parameters for these points were respectively and involve decays tomainly τ and b quarks respectively

        Table 61 Best Fit Parameters

        Parameter mχ [GeV ] mS[GeV ] λt λb λτ

        Value 932 3526 000049 0002561 000781

        10-1 100 101 102

        E(GeV)

        10

        05

        00

        05

        10

        15

        20

        25

        30

        35

        E2dφdE

        (GeVc

        m2ss

        r)

        1e 6

        Best fitData

        Figure 63 Gamma Ray Spectrum

        This is not a particularly good fit and it occcurs at a relic density Ωh2 of 335 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

        63 Collider Constraints 74

        63 Collider Constraints

        631 Mediator Decay

        1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44]

        We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

        The results of the scan are shown in the graph Fig64 and can be compared with thelimits in the table from the paper [44] in Fig 56The black line is the observed ATLAS limit

        Figure 64 σ lowastBr(σ rarr ττ) versus Mass of Scalar

        0 200 400 600 800

        mS[GeV]

        8

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        log 1

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        (Sgtττ

        ))[pb]

        Observed LimitLikely PointsExcluded Points

        050

        100150200250300350

        0 10 20 30 40 50 60

        We can see that there is a narrow band in the region of interest in the σ(bbS)lowastB(S gt ττ)

        by mS plane due to the high degree of correlation between mS and λτ seen in the likelihoodscan Values for mS gt 300Gev are excluded

        2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

        We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinations

        63 Collider Constraints 75

        randomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

        The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig65 The black line is the CMSlimit

        Figure 65 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

        0 200 400 600 800

        mS[GeV]

        8

        6

        4

        2

        0

        2

        4

        log

        10(σ

        (bS

        +X

        )lowastB

        (Sgt

        bb))

        [pb]

        Observed LimitLikely PointsExcluded Points

        050

        100150200250300350

        0 10 20 30 40 50 60

        The results of this scan were compared to the limits in [89] with the plot shown inFig58

        We can see a similar shape for the region of interest to that in Fig64 due to the highcorrelation between mS and λb over the region preferred by astrophysical constraints resultingin branching ratios highly correlated to ms Only one point is excluded with mS gt 500GeV

        632 Collider Cuts Analyses

        We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expected

        63 Collider Constraints 76

        with 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 66

        We can see that collider cuts show some exclusions only for masses mχ below the highlikelihood region but these exclusions do not occur at the same λb and λt combinations asseen from the plot of these two couplings except for a small region at the highest mS andcouplings

        All but one of the collider cut exclusions were excluded by the Lepstop0 experiment (thiswas excluded by 2bstop with high coupling to b quarks of 73) A number of the excludedpoints were also excluded by Lepstop2 2bstop CMSTopDM1L and ATLASMonobJet-(around 30 of the total for each)

        63 Collider Constraints 77

        0 1 2 3

        log10(mχ)[GeV]

        0

        1

        2

        3

        log 1

        0(m

        s)[GeV

        ]Collider Cuts

        σ lowastBr(σgt bS+X)

        σ lowastBr(σgt ττ)

        (a) mχ by mS

        5 4 3 2 1 0 1

        log10(λt)

        0

        1

        2

        3

        log 1

        0(m

        s)[GeV

        ](b) λt by mS

        5 4 3 2 1 0 1

        log10(λb)

        5

        4

        3

        2

        1

        0

        1

        log 1

        0(λ

        t)

        (c) λb by λt

        5 4 3 2 1 0 1

        log10(λb)

        6

        5

        4

        3

        2

        1

        0

        1

        2

        log 1

        0(λ

        τ)

        (d) λb by λτ

        5 4 3 2 1 0 1

        log10(λt)

        6

        5

        4

        3

        2

        1

        0

        1

        2

        log 1

        0(λ

        τ)

        (e) λt by λτ

        5 4 3 2 1 0 1

        log10(λb)

        0

        1

        2

        3

        log 1

        0(m

        s)[GeV

        ]

        (f) λb by mS

        Figure 66 Excluded points from Collider Cuts and σBranching Ratio

        Chapter 7

        Real Vector Dark Matter Results

        71 Bayesian Scans

        In the Figure 71 we can see that the preferred DM mass is centered around 35 GeV primarilydue to the influence of the Fermi-LAT gamma ray constraint with relic density having a widerange of possible mediator and DM masses The result of combining all the constraints is todefine a small region around mχ = 8 GeV and mS = 20 GeV which dominates the posteriorprobability

        The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig72 The diagonal plots show marginalised histograms of a singleparameter at a time

        71 Bayesian Scans 79

        05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

        1

        0

        1

        2

        3

        4

        log 1

        0(m

        s)[GeV

        ]

        (a) Gamma Only

        05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

        05

        00

        05

        10

        15

        20

        25

        30

        35

        log 1

        0(m

        s)[GeV

        ]

        (b) Relic Density

        1 0 1 2 3 4log10(mχ)[GeV]

        05

        00

        05

        10

        15

        20

        25

        30

        35

        log 1

        0(m

        s)[GeV

        ]

        (c) LUX

        05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

        05

        00

        05

        10

        15

        20

        25

        30

        35

        log 1

        0(m

        s)[GeV

        ]

        (d) All Constraints

        Figure 71 Real Vector Dark Matter - By Individual Constraint and All Together

        71 Bayesian Scans 80

        00

        05

        10

        15

        20

        25

        30

        log 1

        0(m

        χ)[GeV

        ]

        00

        05

        10

        15

        20

        25

        30

        ms[Gev

        ]

        5

        4

        3

        2

        1

        0

        1

        log 1

        0(λ

        t)

        5

        4

        3

        2

        1

        0

        1

        log 1

        0(λ

        b)

        00 05 10 15 20 25 30

        log10(mχ)[GeV]

        5

        4

        3

        2

        1

        0

        1

        log 1

        0(λ

        τ)

        00 05 10 15 20 25 30

        ms[Gev]5 4 3 2 1 0 1

        log10(λt)5 4 3 2 1 0 1

        log10(λb)5 4 3 2 1 0 1

        log10(λτ)

        Figure 72 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter

        72 Best fit Gamma Ray Spectrum for the Real Vector DM model 81

        72 Best fit Gamma Ray Spectrum for the Real Vector DMmodel

        The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraints isshown in Fig73

        Table 71 Best Fit Parameters

        Parameter mχ [GeV ] mS[GeV ] λt λb λτ

        Value 8447 20685 0000022 0000746 0002439

        10-1 100 101 102

        E(GeV)

        10

        05

        00

        05

        10

        15

        20

        25

        30

        35

        E2dφdE

        (GeVc

        m2s

        sr)

        1e 6

        Best fitData

        Figure 73 Gamma Ray Spectrum

        This is not a particularly good fit and it occcurs at a relic density Ωh2 of 025 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

        73 Collider Constraints

        731 Mediator Decay

        1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44] (See Chapter 3 for more details)

        We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

        73 Collider Constraints 82

        The results of the scan are shown in the graph Fig74 and can be compared with thelimits in the table from the paper [44] in Fig 56

        Figure 74 σ lowastBr(σ rarr ττ) versus Mass of Scalar

        0 200 400 600 800

        mS[GeV]

        8

        6

        4

        2

        0

        2

        log 1

        0(σ

        (bbS

        )lowastB

        (Sgtττ

        ))[pb]

        Observed LimitLikely PointsExcluded Points

        0100200300400500600700800

        0 20 40 60 80 100120140

        We can see that the region of highest likelihood is limited to mS lt 100 GeV and a narrowrange of couplings to λτ giving a narrow region of interest which falls well below the limit

        2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

        We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

        The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig75 The black line is the CMSlimit

        The results of this scan were compared to the limits in [89] with the plot shown in Fig58

        We can see that again the region of interest for mS is below 100 GeV and the points fallwell below the limit

        73 Collider Constraints 83

        Figure 75 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

        0 200 400 600 800

        mS[GeV]

        8

        6

        4

        2

        0

        2

        4

        log

        10(σ

        (bS

        +X

        )lowastB

        (Sgt

        bb))

        [pb]

        Observed LimitLikely PointsExcluded Points

        0100200300400500600700800

        0 20 40 60 80 100120140

        732 Collider Cuts Analyses

        We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 76

        We can see from Figure 76 that like the real scalar model the region of highest likelihoodhas λb and λt lt 001 while the excluded points from collider cuts and cross sections areabove 01 Most of the excluded points have mχ gt than the values in the region of highestlikelihood and all points are widely separated in parameter space from this region

        Once again 89 of excluded points were excluded by Lepstop0 with many also beingexcluded by the other experiments (50 by Lepstop2 and 56 by CMSTopDM1L 35 byLepstop2 and 20 by ATLASMonobjet)

        73 Collider Constraints 84

        0 1 2 3

        log10(mχ)[GeV]

        0

        1

        2

        3

        log 1

        0(m

        s)[GeV

        ]Collider Cuts

        σ lowastBr(σgt bS+X)

        σ lowastBr(σgt ττ)

        (a) mχ by mS

        5 4 3 2 1 0 1

        log10(λt)

        0

        1

        2

        3

        log 1

        0(m

        s)[GeV

        ](b) λt by mS

        5 4 3 2 1 0 1

        log10(λb)

        5

        4

        3

        2

        1

        0

        1

        log 1

        0(λ

        t)

        (c) λb by λt

        5 4 3 2 1 0 1

        log10(λb)

        5

        4

        3

        2

        1

        0

        1

        log 1

        0(λ

        τ)

        (d) λb by λτ

        5 4 3 2 1 0 1

        log10(λt)

        5

        4

        3

        2

        1

        0

        1

        log 1

        0(λ

        τ)

        (e) λt by λτ

        5 4 3 2 1 0 1

        log10(λb)

        0

        1

        2

        3

        log 1

        0(m

        s)[GeV

        ]

        (f) λb by mS

        Figure 76 Excluded points from Collider Cuts and σBranching Ratio

        Chapter 8

        Conclusion

        We first conducted Bayesian scans over mass and coupling parameters for three simplifiedmodels - Majorana fermion real scalar and real vector using direct detecion (LUX) indirectdetection (GCE γ rays) and relic density constraints to find the regions of highest likelihoodWe ran these scans one constraint at a time and then together to better understand the interplayof the different constraints on best fit and likely parameter regions

        We also developed and tested C++ collider cut software specifically designed to mimickthe cuts applied in a number of ATLAS and CMS experiments that searched for Emiss

        T inconnection with specific beyond the standard model (BSM) theories of Dark Matter

        We then selected parameter combinations by randomly sampling the scanned resultsbased on the posterior probability distribution of the points and applied collider cuts andcross section times branching ratio calculations to these points to see if there would havebeen any sensitivity from the ATLAS and CMS experiments

        We found that the Majorana fermion model showed the best overall fit to all the constraintswith relic density close to the experimental value of 01199 and a low χ2 error on the gammaspectrum fit The preferred mass mχ was around 35 GeV and mS ranging from lt10 GeV to500 GeV The smaller masses give a better fit for relic density and also are well separatedfrom current collider constraints mainly due the the fact that the exclusions occur mainlywith higher couplings to t quarks (greater than 1) which is consistent with the fact that manyof the experiments were targeting dark matter pair production associated with t quark decaysNevertheless there were some exclusions that did fall in the region sampled by randomselection from the posterior probability distribution

        The preferred parameter regions were also consistent with the observation in [20] thatthe Majorana model showed the best overall fit with similar mass constraints We find thatthe couplings to the SM fermions should all be less that 001

        86

        The scalar and vector DM models give slightly less good fits especially to the currentrelic density estimate of 01199 but again the current collider constraints are well separatedfrom the best fit region and in fact none of the points randomly sampled from the posteriordistribution showed any exclusion The best fit parameter combination favours couplings toSM fermions of less than 001

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        Appendix A

        Validation of Calculation Tools

        Lepstop0Search for direct pair production of the top squarkin all hadronic final states in proton=proton collisions atradic

        s=8 TeV with the ATLAS detector [32]

        Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1) mt1=600 GeV mn1=1 GeVWhere t1 is the top squark and n1 is the neutralino of the MSSM

        94

        Table A1 0 Leptons in the final state

        Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 10000 460 460 100Emiss

        T gt 130GeV 8574 395 451 88Leptonveto 6621 307 285 108Emiss

        T gt 150GeV 6316 293 294 100Jet multiplicityand pT 1099 48 51 94|∆φ( jet pmiss

        T )|gt π5 902 40 42 96gt= 2b jets 396 19 18 106τ veto 396 19 18 106mbmin

        T gt 175GeV 270 13 13 105SRA1 79 4 4 104SRA2 70 3 3 93SRA3 67 3 3 97SRA4 52 2 2 91SRC exactly5 jets 1297 50 60 83SRC |∆φ( jet pmiss

        T )|gt π5 1082 50 50 99SRC gt= 2b jets 430 20 20 97SRC τ veto 430 20 20 97SRC |∆φ(bb)|gt 02π 367 17 17 99SRC1 246 12 11 105SRC2 177 9 8 116SRC3 152 8 7 116

        95

        Lepstop1Search for direct top squark pair production infinal states with one isolated lepton jets and missing trans-verse momentum in

        radics = 8 TeV pp collisions using 21 f bminus1

        of ATLAS data[33]

        Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1)(t1 gt b x+1 )(t1 gt b xminus1 )mt1=650 GeV mn1=1 GeV Where t1 is the top squark n1 is the neutralino xplusmn1 are thecharginos

        96

        Table A2 1 Lepton in the Final state

        Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 100000 99990 99990 100Electron(= 1signal) 11558 11557 11785 984 jets(80604025) 5187 5186 6271 83gt= 1btag 4998 4997 5389 93Emiss

        T gt 100GeV [allSRs] 4131 4131 4393 94Emiss

        T radic

        HT gt 5 [allSRs] 4026 4026 4286 94∆φ( jet2 pmiss

        T )gt 08 [allSRs] 3565 3565 3817 93∆φ( jet1Pmiss

        T )gt 08 [not SRtN2] 3446 3446 3723 93Emiss

        T gt 200GeV (SRtN2) 2027 2027 2186 93Emiss

        T radic

        HT gt 13(SRtN2) 1156 1156 1172 99mT gt 140GeV (SRtN2) 839 839 974 86Emiss

        T gt 275GeV (SRtN3) 948 948 965 98Emiss

        T radic

        HT gt 11(SRtN3) 932 932 937 99mT gt 200GeV (SRtN3) 456 456 566 81Emiss

        T gt 150GeV (SRbC1minusSRbC3) 2803 2803 3080 91Emiss

        T radic

        HT gt 7(SRbC1minusSRbC3) 2742 2742 3017 91mT gt 12GeV (SRbC1minusSRbC3) 1917 1917 2347 82Emiss

        T gt 160GeV (SRbC2SRbC3) 1836 1836 2257 81Emiss

        T radic

        HT gt 8(SRbC2SRbC3) 1788 1788 2175 82me f f gt 550GeV (SRbC2) 1741 1741 2042 85me f f gt 700GeV (SRbC3) 1075 1075 1398 77Muon(= 1signal) 11379 11378 11975 954 jets(80604025) 5264 5263 6046 87gt= 1btag 5101 5100 5711 89Emiss

        T gt 100GeV [allSRs] 4199 4199 4770 88Emiss

        T radic

        HT gt 5 [all SRs] 4115 4115 4653 88∆φ( jet2 pmiss

        T )gt 08 [all SRs] 3622 3622 4077 89∆φ( jet1 pmiss

        T )gt 08 [not SRtN2] 3515 3515 3970 89Emiss

        T gt 200GeV (SRtN2) 2136 2136 2534 84Emiss

        T radic

        HT gt 13(SRtN2) 1261 1261 1441 87mT gt 140GeV (SRtN2) 928 928 1033 90Emiss

        T gt 275GeV (SRtN3) 1036 1036 1274 81Emiss

        T radic

        HT gt 11(SRtN3) 1018 1018 1235 82mT gt 200GeV (SRtN3) 497 497 612 81Emiss

        T gt 150GeV (SRbC1minusSRbC3) 2891 2891 3380 86Emiss

        T radic

        HT gt 7(SRbC1minusSRbC3) 2821 2821 3290 86mT gt 120GeV (SRbC1minusSRbC3) 1977 1977 2290 86Emiss

        T gt 160GeV (SRbC2SRbC3) 1913 1913 2220 86Emiss

        T radic

        HT gt 8(SRbC2SRbC3) 1863 1863 2152 87me f f gt 550GeV (SRbC2) 1826 1826 2116 86me f f gt 700GeV (SRbC3) 1131 1131 1452 78

        97

        Lepstop2Search for direct top squark pair production infinal states with two leptons in

        radics =8 TeV pp collisions using

        20 f bminus1 of ATLAS data[83][34]

        Simulated in Madgraph with p p gt t1 t1(t1 gt b w+ n1)(t1 gt b wminusn1) mt1=180 GeV mn1=60 GeV Where t1 is the top squark n1 is the neutralino b1 is the bottom squark

        Table A3 2 Leptons in the final state

        Cut RawMe Scaled ATLAS MeATLAS PercentNoCuts 100000 324308 199000 1632BaselineCuts 3593 11652 7203 1622SF Signal Leptons 1699 5510 5510 1002OSSF Signal Leptons 1632 5293 5233 101mll gt 20Gev 1553 5037 5028 100leadinglepton pT 1479 4797 5048 95|mll minusmZ|gt 20Gev 1015 3292 3484 94∆φmin gt 10 850 2757 2160 128∆φb lt 15 680 2205 1757 126SRM90 [SF ] 13 42 60 70SRM120 [SF ] 0 0 19 0SRM100 +2 jets [SF ] 0 0 17 0SRM110 +2 jets [SF ] 0 0 1 02DF Signal Leptons 1737 5633 7164 792OSDF Signal Leptons 1670 5416 5396 100mll20Gev 1587 5147 5136 100leadinglepton pT 1501 4868 4931 99∆φmin gt 10 1247 4044 3122 130∆φb lt 15 1010 3276 2558 128SRM90 [SF ] 21 68 95 72SRM120 [SF ] 0 0 23 0SRM100 +2 jets [SF ] 0 0 19 0SRM110 +2 jets [SF ] 0 0 14 0

        98

        2bstopSearch for direct third-generation squark pair pro-duction in final states with missing transverse momentumand two b-jets[31]

        Simulated in Madgraph with p p gt b1 b1(b1 gt b n1)(b1 gt b n1)

        SRAmb1=500 GeV mn1=1 GeV SRBmb1=350 GeV mn1=320 GeVWhere b1 is the bottom squark and n1 is the neutralino

        Table A4 2b jets in the final state

        Cut RawMe Scaled ATLAS MeATLAS PercentSRA Nocuts 100000 173800 1738 100SRA MET gt 80 93407 162341 1606 101SRA Leptonveto 93302 162159 1505 108SRA MET gt 150 79449 138082 1323 104SRA Jet selection 10213 17750 119 149SRA mbb gt 200 8344 14502 96 151SRA mCT gt 150 6535 11358 82 139SRA mCT gt 200 5175 8994 67 134SRA mCT gt 250 3793 6592 51 129SRA mCT gt 300 2487 4322 35 123SRB NoCuts 100000 1624100 16241 100SRB leptonveto 11489 186593 4069 46SRB MET gt 250 2530 41090 757 54SRB Jet selection 63 1023 79 130SRB HT 3 lt 50 51 828 52 159

        99

        CMSTop1LSearch for the Production of Dark Matter in As-sociation with Top Quark Pairs in the Single-lepton FinalState in pp collisions[41]

        Simulated in Madgraph with p p gt t t p1 p1

        Where p1= Majorana fermion Table 3 shows Signal efficiencies and cross sections TheM scale was read from Figure 7 for 90 CL

        Table A5 Signal Efficiencies 90 CL on σ limexp[ f b] on pp gt tt +χχ

        Mχ[GeV ] My Signal Efficiency Quoted Efficiency My σ [ f b] Quoted σ [ f b]1 102 101 63 47

        10 132 101 62 4650 143 120 50 38100 170 146 56 32200 185 173 69 27600 265 240 32 19

        1000 320 276 41 17

        Appendix B

        Branching ratio calculations for narrowwidth approximation

        B1 Code obtained from decayspy in Madgraph

        Br(S rarr bb) = (minus24λ2b m2

        b +6λ2b m2

        s

        radicminus4m2

        bm2S +m4

        S)16πm3S

        Br(S rarr tt) = (6λ2t m2

        S minus24λ2t m2

        t

        radicm4

        S minus4ms2m2t )16πm3

        S

        Br(S rarr τ+

        τminus) = (2λ

        2τ m2

        S minus8λ2τ m2

        τ

        radicm4

        S minus4m2Sm2

        τ)16πm3S

        Br(S rarr χχ) = (2λ2χm2

        S

        radicm4

        S minus4m2Sm2

        χ)32πm3S

        (B1)

        Where

        mS is the mass of the scalar mediator

        mχ is the mass of the Dark Matter particle

        mb is the mass of the b quark

        mt is the mass of the t quark

        mτ is the mass of the τ lepton

        The coupling constants λ follow the same pattern

        • TITLE Collider Constraints applied to Simplified Models of Dark Matter fitted to the Fermi-LAT gamma ray excess using Bayesian Techniques
          • Dedication
          • Declaration
          • Acknowledgements
          • Contents
          • List of Figures
          • List of Tables
            • Chapter 1 Introduction
            • Chapter 2 Review of Physics
            • Chapter 3 Fitting Models to the Observables
            • Chapter 4 Calculation Tools
            • Chapter 5 Majorana Model Results
            • Chapter 6 Real Scalar Model Results
            • Chapter 7 Real Vector Dark Matter Results
            • Chapter 8 Conclusion
            • Bibliography
            • Appendix A Validation of Calculation Tools
            • Appendix B Branching ratio calculations for narrow width approximation

          Contents

          List of Figures xiii

          List of Tables xv

          1 Introduction 111 Motivation 312 Literature review 4

          121 Simplified Models 4122 Collider Constraints 7

          2 Review of Physics 921 Standard Model 9

          211 Introduction 9212 Quantum Mechanics 9213 Field Theory 10214 Spin and Statistics 10215 Feynman Diagrams 11216 Gauge Symmetries and Quantum Electrodynamics (QED) 12217 The Standard Electroweak Model 13218 Higgs Mechanism 17219 Quantum Chromodynamics 222110 Full SM Lagrangian 23

          22 Dark Matter 25221 Evidence for the existence of dark matter 25222 Searches for dark matter 30223 Possible signals of dark matter 30224 Gamma Ray Excess at the Centre of the Galaxy [65] 30

          Contents vi

          23 Background on ATLAS and CMS Experiments at the Large Hadron collider(LHC) 31231 ATLAS Experiment 32232 CMS Experiment 33

          3 Fitting Models to the Observables 3531 Simplified Models Considered 3532 Observables 36

          321 Dark Matter Abundance 36322 Gamma Rays from the Galactic Center 36323 Direct Detection - LUX 37

          33 Calculations 39331 Mediator Decay 39332 Collider Cuts Analyses 42333 Description of Collider Cuts Analyses 43

          4 Calculation Tools 5541 Summary 5542 FeynRules 5643 LUXCalc 5644 Multinest 5745 Madgraph 5846 Collider Cuts C++ Code 59

          5 Majorana Model Results 6151 Bayesian Scans 6152 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 6453 Collider Constraints 67

          531 Mediator Decay 67532 Collider Cuts Analyses 69

          6 Real Scalar Model Results 7361 Bayesian Scans 7362 Best fit Gamma Ray Spectrum for the Real Scalar DM model 7663 Collider Constraints 77

          631 Mediator Decay 77632 Collider Cuts Analyses 78

          Contents vii

          7 Real Vector Dark Matter Results 8171 Bayesian Scans 8172 Best fit Gamma Ray Spectrum for the Real Vector DM model 8473 Collider Constraints 84

          731 Mediator Decay 84732 Collider Cuts Analyses 86

          8 Conclusion 89

          Bibliography 91

          Appendix A Validation of Calculation Tools 97

          Appendix B Branching ratio calculations for narrow width approximation 105B1 Code obtained from decayspy in Madgraph 105

          List of Figures

          21 Feynman Diagram of electron interacting with a muon 1122 Weak Interaction Vertices [48] 1523 Higgs Potential [49] 1824 Standard Model Particles and Forces [50] 2425 Bullet Cluster [52] 2526 Galaxy Rotation Curves [54] 2627 WMAP Cosmic Microwave Background Fluctuations [58] 2928 Dark Matter Interactions [60] 2929 Gamma Ray Excess from the Milky Way Center [75] 31210 ATLAS Experiment 31211 CMS Experiment 34

          31 Main Feyman diagrams leading to the cross section for scalar decaying to apair of τ leptons 40

          32 WidthmS vs mS 4033 WidthmS vs λb 4134 WidthmS vs λτ 4135 Main Feyman diagrams leading to the cross section for scalar decaying to a

          pair of b quarks in the presence of at least one b quark 42

          41 Calculation Tools 55

          51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether 62

          52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter 6353 Gamma Ray Spectrum 6454 Plots of log likelihoods by individual and combined constraints Masses in

          GeV 6655 σ lowastBr(σ rarr ττ) versus Mass of Scalar 67

          List of Figures ix

          56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar 6857 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar 6858 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 6959 Excluded points from Collider Cuts and σBranching Ratio 70

          61 Real Scalar Dark Matter - By Individual Constraint and All Together 7462 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter 7563 Gamma Ray Spectrum 7664 σ lowastBr(σ rarr ττ) versus Mass of Scalar 7765 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 7866 Excluded points from Collider Cuts and σBranching Ratio 80

          71 Real Vector Dark Matter - By Individual Constraint and All Together 8272 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter 8373 Gamma Ray Spectrum 8474 σ lowastBr(σ rarr ττ) versus Mass of Scalar 8575 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 8676 Excluded points from Collider Cuts and σBranching Ratio 87

          List of Tables

          21 Quantum numbers of the Higgs field 1922 Weak Quantum numbers of Lepton and Quarks 21

          31 Simplified Models 3532 95 CL by Signal Region 4433 Selection criteria common to all signal regions 4534 Selection criteria for signal regions A 4535 Selection criteria for signal regions C 4536 Signal Regions - Lepstop1 4837 Signal Regions Lepstop2 4938 Signal Regions 2bstop 5139 Signal Region ATLASmonobjet 52

          51 Scanned Ranges 6152 Best Fit Parameters 64

          61 Best Fit Parameters 76

          71 Best Fit Parameters 84

          A1 0 Leptons in the final state 98A2 1 Lepton in the Final state 100A3 2 Leptons in the final state 101A4 2b jets in the final state 102A5 Signal Efficiencies 90 CL on σ lim

          exp[ f b] on pp gt tt +χχ 103

          Chapter 1

          Introduction

          Dark matter (DM) was first postulated over 80 years ago when Swiss astronomer FritzZwicky observed a discrepancy between the amount of light emitted by a cluster of galaxiesand the total mass contained within the cluster inferred from the relative motion of thosegalaxies by a simple application of the theory of Newtonian gravitation The surprising resultof this observation was that the vast majority of the mass in the cluster did not emit lightwhich was contrary to the expectation that most of the mass would be carried by the starsSince that time further observations over a wide range of scales and experimental techniqueshave continued to point to the same result and refine it Some of these observations and otherevidence are discussed in section 22 We now know with certainty that in the entire Universeall of the matter we know about - stars planets gases and other cosmic objects such as blackholes can only account for less than 5 of the mass that we calculate to be there

          A recent phenomenon that has received much attention is the significant deviation frombackground expectations of the Fermi Large Area Telescope(Fermi-LAT) gamma ray flux atthe galactic centre [1] A number of astrophysical explanations have been proposed includingmillisecond pulsars of supernova remnants [2] or burst-like continuous events at the galacticcentre but these are unresolved However it has also been noted that the observed Fermi-LATexcess is consistent with the annihilation of dark matter particles which would naturally beconcentrated at the Galactic centre in a manner consistent with the Navarro-Frenk-Whitedistribution of dark matter [3]

          There are a number of other purely theoretical (particle physics) reasons to postulatethe existence of weakly interacting matter particles that could supply the missing mass andyet remain unobservable Weakly interacting massive particle (WIMPS) have been a majorfocus of Run I and ongoing Run II searches of the Large Hadron Collider (LHC) In spite

          2

          of substantial theoretical and experimental effort the microscopic properties of dark matterparticles are still unknown

          One of the main problems with extracting these properties is the plethora of competingtheoretical models that provide dark matter candidates Another is the very weak interactionof these particles and the lack of experimental observables allowing one to discriminatebetween the competing models

          A particularly fruitful approach is the simplified model approach which makes minimalassumptions about the DM candidates as extensions to the Standard Model interacting weaklywith ordinary matter through a mediator These can be coded as Lagrangian interaction termsusing tools such as FeynRules [4] and Calchep [5] and used to simulate Dark Matter relicdensities indirect detection expectations as well as direct detection expectations throughMicrOmegas [6] This in turn can be run under a Bayesian inference tool (see below)

          Using a Bayesian inference tool such as MultiNest [7][8][9] one can scan the parameterspace of a particular model to find regions of highest probability for that model based onobservations of a range of phenomena including direct detection experiments such as ZEPLIN[10] XENON [11] DEAP [12] ARDM [13] DarkSide [14] PandaX [15] and LUX [16]indirect detection - eg Fermi-LAT gamma ray excess [17] Pamela-positron and antiprotonflux [18] and production at colliders MultiNest calculates the evidence value which is theaverage likelihood of the model weighted by prior probability and a posterior distributionwhich is the distribution of the parameter(s) after taking into account the observed databy scanning over a range of parameters based on prior probabilities This can be used tocompare models on a consistent basis The scans also provide information about the best fitparameters for the overall fit as well as for the seperate experimentsobservations

          The simplified model approach has been taken in a number of papers to study the Fermi-LAT gamma ray excess (see literature review in section 12) and one in particular comparesthese models using a Bayesian inference tool which found that a Majorana fermion DMcandidate interacting via a scalar mediator with arbitrary couplings to third generationfermions showed the best fit [19] A subsequent paper [20] examined the most likelyparameter combinations that simultaneously fit the Planck relic density Fermi-LAT galacticcentre gamma ray AMS-02 positron flux and Pamela antiproton flux data

          There has not as yet been a thorough study of the collider constraints on DM simplifiedmodels that fit the Fermi-LAT galactic centre excess In this thesis we reproduce the

          11 Motivation 3

          previous results of [20] using similar astrophysical constraints and then implement detailedsimulations of collider constraints on DM and mediator production

          In Section 11 we review the motivation for this study and in Section 12 we considerrecent astronomical data which is consistent with the annihilation of DM particles andreview the literature and searches which attempt to explain it with a range of extensions tothe standard model

          In Chapter 2 we review the Standard Model of particle physics (SM) the evidence fordark matter and the ATLAS and CMS experiments at the LHC

          In Chapter 3 we describe the models considered in this paper the observables used inthe Bayesian scans and the collider experiments which provide limits against which theMagraph simulations of the three models can be compared

          In Chapter 4 we review the calculation tools that have been used in this paper

          In Chapters 5 6 and 7 we give the results of the Bayesian scans and collider cutsanalyses for each of the models and in Chapter 8 we summarise our conclusions

          11 Motivation

          The Fermi gamma ray space telescope has over the past few years detected a gamma raysignal from the inner few degrees around the galactic centre corresponding to a regionseveral hundred parsecs in radius The spectrum and angular distribution are compatible withthat from annihilating dark matter particles

          A number of papers have recently attempted to explain the Gamma Ray excess in termsof extensions to the Standard Model taking into account constraints from ATLAS and CMScollider experiments direct detection of dark matter colliding off nucleii in well shieldeddetectors (eg LUX) as well as other indirect measurements (astrophysical observations)These papers take the approach of simplified models which describe the contact interactionsof a mediator beween DM and ordinary matter using an effective field theory approach(EFT) Until recently it has been difficult to discriminate between the results of these paperswhich look at different models and produce different statistical results A recent paper[19] Simplified Dark Matter Models Confront the Gamma Ray Excess used a Bayesianinference tool to compare a variety of simplified models using the Bayesian evidence values

          12 Literature review 4

          calculated for each model to find the most preferred explanation The motivation for thechosen models is discussed in section 121

          A subsequent paper [20] Interpreting the Fermi-LAT gamma ray excess in the simplifiedframework studied the most likely model again using Bayesian inference to find the mostlikely parameters for the (most likely) model The data inputs (observables) to the scanwere the Planck measurement of the dark matter relic density [21] Fermi-LAT gamma rayflux data from the galactic center [17] cosmic positron flux data [22] cosmic anti-protonflux data [18] cosmic microwave background data and direct detection data from the LUXexperiment [16] The result of the scan was a posterior probability density function in theparameter space of the model

          This paper extends the findings in [19] by adding collider constraints to the likelihoodestimations for the three models It is not a simple matter to add collider constraints becauseeach point in parameter space demands a significant amount of computing to calculate theexpected excess amount of observable particles corresponding to the particular experimentgiven the proposed simplified model A sampling method has been employed to calculatethe collider constraints using the probability of the parameters based on the astrophysicalobservables

          12 Literature review

          121 Simplified Models

          A recent paper [23] summarised the requirements and properties that simplified models forLHC dark matter searches should have

          The general principles are

          bull Besides the Standard Model (SM) the Lagrangian should contain a DM candidate thatis absolutely stable or lives long enough to escape the LHC detector and a mediatorthat couples the two sectors

          bull The Lagrangian should contain all terms that are renormalizable consistent withLorentz invariance SM symmetries and DM stability

          bull Additional interactions should not violate exact and approximate global symmetries ofthe SM

          12 Literature review 5

          The examples of models that satisfy these requirements are

          1 Scalar (or pseudo-scalar) s-channel mediator coupled to fermionic dark matter (Diracor Majorana)

          2 Higgs portal DM where the DM is a scalar singlet or fermion singlet under the gaugesymmetries of the SM coupling to a scalar boson which mixes with the Higgs (Thefermion singlet is a specific realisation of 1 above)

          3 DM is a mixture of an electroweak singlet and doublet as in the Minimal Supersym-metric SM (MSSM)

          4 Vector (or Axial vector) s-channel mediator obtained by extending the SM gaugesymmetry by a new U(1)rsquo which is spontaneously broken such that the mediatoracquires mass

          5 t-channel flavoured mediator (if the DM particle is a fermion χ the mediator can bea coloured scalar or vector particle φ -eg assuming a scalar mediator a couplingof the form φ χq requires either χ or φ to carry a flavor index to be consistent withminimal flavour violation (MFV) which postulates the flavour changing neutral currentstructure in the SM is preserved

          Another recent paper [24] summarised the requirements and properties that simplifiedmodels for LHC dark matter searches should have The models studied in this paper arepossible candidates competing with Dirac fermion and complex scalar dark matter andmodels with vector and pseudo-scalar mediators Other models would have an elasticscattering cross section that will remain beyond the reach of direct detection experiments dueto the neutrino floor (the irreducible background produced by neutrino scattering) or arealready close to sensitivity to existing LUX and XENONIT experiments

          A recent paper [25] Scalar Simplified Models for Dark Matter looks at Dirac fermiondark matter mediated by either a new scalar or pseudo-scalar to SM fermions This paperplaces bounds on the coupling combination gχgφ (coupling of DM to mediator times couplingof mediator to SM fermions) calculated from direct detection indirect detection and relicdensity by the mass of DM and mass of mediator Bounds are also calculated for colliderconstraints (mono-jet searches heavy-flavour searches and collider bounds on the 125 GeVHiggs are used to place limits on the monojet and heavy flavor channels which can betranslated into limits on the Higgs coupling to dark matter and experimental measurements

          12 Literature review 6

          of the Higgs width are used to constrain the addition of new channels to Higgs decay) Thepaper models the top loop in a full theory using the Monte Carlo generator MCFM [26]and extends the process to accommodate off-shell mediator production and decay to a darkmatter pair While this paper does not consider the Majorana fermion dark matter model theconstraints on the coupling product may have some relevence to the present study if only asan order of magnitude indicator

          Constraining dark sectors with monojets and dijets [27] studies a simplified model withDirac fermion dark matter and an axial-vector mediator They use two classes of constraint-searches for DM production in events with large amounts of missing transverse energy inassociation with SM particles (eg monojet events) and direct constraints on the mediatorfrom monojet searches to show that for example where the choice of couplings of themediator to quarks and DM gA

          q =gAχ=1 all mediator masses in the range 130 GeV ltMR lt3

          TeV are excluded

          The paper Constraining Dark Sectors at Colliders Beyond the Effective Theory Ap-proach [28] considers Higgs portal models with scalar pseudo-scalar vector and axial-vectormediators between the SM and the dark sector The paper takes account of loop processesand gluon fusion as well as quark-mediator interactions using MCFM Apart from mediatortype the models are characterised by free parameters of mediator width and mass darkmatter mass and the effective coupling combination gχgφ (coupling of DM to mediator timescoupling of mediator to SM fermions) The conclusion is that LHC searches can proberegions of the parameter space that are not able to be explored in direct detection experiments(since the latter measurements suffer from a velocity suppression of the WIMP-nucleoncross-section)

          The paper Constraining the Fermi-LAT excess with multi-jet plus EmissT collider searches

          [29] shows that models with a pseudo-scalar or scalar mediator are best constrained by multi-jet final states with Emiss

          T The analysis was done using POWHEG-BOX-V2 at LO and usesM2

          T variables together with multi-jet binning to exploit differences in signal and backgroundin the different regions of phase space to obtain a more stringent set of constraints on DMproduction This paper largely motivated the current work in that it showed that collidersearches that were not traditionally thought of as dark matter searches were neverthelessoften sensitive to dark matter produced in association with heavy quarks The combinationof searches used in this paper reflects this fact

          12 Literature review 7

          122 Collider Constraints

          In order to constrain this model with collider constraints (direct detection limits) we considerthe following papers by the CMS and ATLAS collaborations which describe searches at theLHC

          ATLAS Experiments

          bull Search for dark matter in events with heavy quarks and missing transverse momentumin p p collisions with the ATLAS detector [30]

          bull Search for direct third-generation squark pair production in final states with missingtransverse momentum and two b-jets in

          radics= 8 TeV pp collisions with the ATLAS

          detector[31]

          bull Search for direct pair production of the top squark in all-hadronic final states inprotonndashproton collisions at

          radics=8 TeV with the ATLAS detector [32]

          bull Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

          radic(s)=8TeV pp collisions using 21 f bminus1 of

          ATLAS data [33]

          bull Search for direct top squark pair production in final states with two leptons inradic

          s = 8TeV pp collisions using 20 f bminus1 of ATLAS data [34]

          bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

          radics=8 TeV [35]

          CMS Experiments

          bull Searches for anomalous tt production in p p collisions atradic

          s=8 TeV [36]

          bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

          radics=8 TeV [37]

          bull Search for the production of dark matter in association with top-quark pairs in thesingle-lepton final state in proton-proton collisions at

          radics = 8 TeV [38]

          bull Search for new physics in monojet events in p p collisions atradic

          s = 8 TeV(CMS) [39]

          bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

          s = 8 TeV [40]

          12 Literature review 8

          bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theSingle-lepton Final State in pp collisions at

          radics=8 TeV [41]

          bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

          s=8 TeV [42]

          bull Search for neutral MSSM Higgs bosons decaying into a pair of bottom quarks [43]

          bull Search for neutral MSSM Higgs bosons decaying into a pair of τ leptons in p p collision[44]

          bull Search for dark matter extra dimensions and unparticles in monojet events in proton-proton collisions at

          radics=8 TeV [45]

          In particular we have implemented Madgraph simulations of certain data analysesapplying to collider experiments described in papers [32][33][34][31][30][41] Theremaining models were not implemented in Madgraph because of inability to validate themodels against published results in the time available We developed C++ analysis code toimplement the collider cuts after showering with Pythia and basic analysis with Delpheswith output to root files The C++ code allowed multiple experimental cuts to be applied tothe same simulation output The C++ code was first checked against the results of the variousexperiments above and then the simulations were rerun using the three modelsrsquo Lagrangians(implemented in FeynRules) described in this paper

          Chapter 2

          Review of Physics

          21 Standard Model

          211 Introduction

          The Standard Model of particle physics (SM) has sometimes been described as the theoryof Almost Everything The SM is a field theory which emerged in the second half of thetwentieth century from quantum mechanics and special relativity which themselves wereonly conceived after 1905 by Einsten Pauli Bohr Dirac and others In the first decades ofthe twenty first century the discovery of the Higgs boson [46][47] completed the pictureand confirmed the last previously unverified part of the SM The SM is a self consistenttheory that describes and classifies all known particles and their interactions (other thangravity) to an astonishing level of accuracy and detail However the theory is incomplete-it does not describe the complete theory of gravitation or account for DM (see below) ordark energy (the accelerating expansion of the universe) Related to these shortcomings aretheoretical deficiencies such as the asymmetry of matter and anti-matter the fact that the SMis inconsistent with general relativity (to the extent that one or both theories break down atspacetime singularities like the Big Bang and black hole event horizons)

          212 Quantum Mechanics

          Quantum Mechanics (QM) describes particles as probability waves which can be thoughtof as packets of energy which can be localised or spread through all space depending on how

          21 Standard Model 10

          accurately one measures the particlersquos momentum However QM is not itself consistent withspecial relativity It was realised too that because of relativity particles could be created anddestroyed and because of the uncertainty principle the theory should describe the probabilityof particles being created and destroyed on arbitrarily short timescales A quantum fieldtheory (QFT) is a way to describe this probability at every point of space and time

          213 Field Theory

          A field theory is described by a mathematical function of space and time called a LagrangianAs in classical mechanics where Lagrangians were introduced by Louis Lagrange at thebeginning of the nineteenth century to summarise dynamical systems the Lagrangian sum-marises the dynamics of space which is permeated by force fields The equations of motionarising from a Lagrangian are determined by finding the contours of the Lagrangian describedby stationary points (ie where the infinitesimal change in the Lagrangian is zero for a smallchange in field) Particles are generally described by singularities in the Lagrangian (wherethe values tend to infinity also called resonances)

          214 Spin and Statistics

          It was also realised that different types of particle needed to be represented by different fieldsin the Lagrangian Diracrsquos discovery that particles with spin 12 quantum numbers couldbe represented by a field obtained by effectively taking the square root of the relativisticequation (Klein Gordon equation) was probably the most significant advance in quantumfield theory It required the introduction of a matrix operators called the gamma matricesor Dirac matrices which helped explain the already known fact that spin half particles orfermions obey Fermi-Dirac statistics ie when two spin half particles are exchanged thesign of the wavefunction must change and hence they cannot occupy the same quantum stateunlike bosons with integer spin that can occupy the same quantum state

          Other particles need to be represented by fields called vector pseudo-vector scalar (as inthe Klein Gordon equation) pseudo-scalar and complex fields A vector field is really fourfields one for each dimension of space plus time and is parameterised by the four gammamatrices γmicro which transform among themselves in a relativistically invariant way (Lorentzinvariance) A pseudo-vector field is a vector field which transforms in the same way but

          21 Standard Model 11

          with additional sign change under parity transformations (a simultaneous reversal of the threespatial directions) and is represented by γmicroγ5 A pseudo-scalar is a scalar which changes signunder parity transformations and a complex scalar field is represented by complex numbers

          215 Feynman Diagrams

          QFT took another giant leap forward when Richard Feynman realised that the propagationof particles described by the field could be described by expanding the so called S-matrixor scattering matrix derived from calculating the probability of initial states transitioning tofinal states The expansion is similar to a Taylor expansion in mathematics allowing for thespecial commuting properties of elements of the field The Feynman diagrams representingindividual terms in the expansion can be interpreted as interactions between the particlecomponents The S-Matrix (or more correctly the non trivial part called the T-Matrix) isderived using the interaction Hamiltonian of the field The Hamiltonian is an operator whoseeigenvalues or characteristic values give the energy quanta of the field The T-matrix canalso be described by a path integral which is an integral over all of quantum space weightedby the exponential of the Lagrangian This is completely equivalent to the Hamiltonianformulation and shows how the Lagrangian arises naturally out of the Hamiltonian (in factthe Lagrangian is the Legendre transformation of the Hamiltonian)

          Figure 21 Feynman Diagram of electron interacting with a muon

          γ

          eminus

          e+

          micro+

          microminus

          The diagram Fig21 is a tree diagram (diagrams with closed loops indicate self in-teractions and are higher order terms in the expansion) and shows an electron (eminus) and ananti-muon (micro+) interacting by exchanging a photon

          21 Standard Model 12

          216 Gauge Symmetries and Quantum Electrodynamics (QED)

          The Dirac equation with its gamma matrices ushered in the use of group theory to describethe transformation of fields (because the multiplication table of the gamma matrices isclosed and described by group theory) It also ushered in the idea of symmetries within theLagrangian- for example consider a Lagrangian

          ψ(ipart minusm)ψ (21)

          The Lagrangian is invariant under a global U(1) transformation which acts on the fieldsand hence the derivatives as follows

          ψ rarr eiqαψ ψ rarr ψeminusiqα partmicroψ rarr eiqα

          partmicroψ (22)

          where qα is a global phase and α is a continuous parameter

          A symmetry leads by Noetherrsquos theorem to a conserved current jmicro = qψγmicroψ and is thebasis of charge conservation since partmicro jmicro = 0 and q =

          intd3x j0(x)

          By promoting this transformation to a local one (ie α rarr α(x) where x = xmicro are thespace time co-ordinates) the transformations become

          ψ rarr eiqα(x)ψ ψ rarr ψeminusiqα(x)partmicroψ rarr eiqα(x)(partmicroψ + iq(partmicroα(x))ψ) (23)

          The derivative transformation introduces an extra term which would spoil the gaugeinvariance This can be solved by introducing a new field the photon field Amicro(x) whichinteracts with the fermion field ψ and transforms under a U(1) transformation

          Amicro rarr Amicro minuspartmicroα(x) (24)

          If we replace normal derivatives partmicro with covariant derivatives Dmicro = (partmicro + iqAmicro(x)) in theLagrangian then the extra term is cancelled and we can now see that the covariant derivativetransforms like ψ - under a local U(1) transform ie

          Dmicro rarr Dmicroψprime = eiqα(x)Dmicroψ (25)

          The additional term iqAmicro(x) can be interpreted in a geometric sense as a connectionwhich facilitates the parallel transport of vectors along a curve in a parallel consistent manner

          21 Standard Model 13

          We must add a gauge-invariant kinetic term to the Lagrangian to describe the dynamicsof the photon field This is given by the electromagnetic field strength tensor

          Fmicroν = partmicroAν minuspartνAmicro (26)

          The full Quantum Electrodynamics (QED) Lagrangian is then for electrons with mass mand charge -e

          LQED = ψ(i Dminusm)ψ minus 14

          Fmicroν(X)Fmicroν(x) (27)

          This Lagrangian is Lorentz and U(1) guage invariant The interaction term is

          Lint =+eψ Aψ = eψγmicro

          ψAmicro = jmicro

          EMAmicro (28)

          where jmicro

          EM is the electromagnetic four current

          217 The Standard Electroweak Model

          The Standard Electroweak (EW) Model was created by Glashow-Weinberg-Salam (also calledthe GWS model) and unifies the electomagnetic and weak interactions of the fundamentalparticles

          The idea of gauge symmetry was extended to the idea of a non-abelian gauge symmetry(essentially the operators defining the symmetry do not commute and so are represented bymatrices- in this case the Pauli matrices for the SU(2) group) - the gauge group is the directproduct G = SU(2)

          otimesU(1) It was known that weak interactions were mediated by Wplusmn

          and Z0 particles where the superscripts are the signs of the electromagnetic charge Thesewere fairly massive but it was also known that through spontaneous symmetry breaking ofa continuous global non-abelian symmetry there would be one massive propagation modefor each spontaneously broken-symmetry generator and massless modes for the unbrokengenerators (the so-called Goldstone bosons)

          This required an extra field to represent the gauge bosons called Bmicro (a vector fieldhaving the 4 spacetime degrees of freedom) With this identification the covariant derivativebecomes

          Dmicro = partmicro minus igAmicro τ

          2minus i

          gprime

          2Y Bmicro (29)

          21 Standard Model 14

          Here Y is the so called weak hypercharge of the field Bmicro which is a quantum numberdepending on particle (lepton or quark) g and gprime are two independent parameters in theGWS theory and the expression Amicro τ implies a sum over 3 components Aa

          micro a=123 and thePauli matrices τa

          This isnrsquot the full story because it had been known since 1956 when Chin Shuing Wuand her collaborators confirmed in an experiment prompted by Lee and Yang that nature isnot ambivalent to so-called parity It had been assumed that the laws of nature would notchange if all spatial coordinates were reversed (a parity transformation) The experimentinvolved the beta decay of cooled 60C atoms (more electrons were emitted in the backwardhemisphere with respect to spin direction of the atoms) This emission involved the weakinteraction and required that the Lagrangian for weak interactions contain terms which aremaximally parity violating- which turn out to be currents of the form

          ψ(1minus γ5)γmicro

          ψ (210)

          The term

          12(1minus γ

          5) (211)

          projects out a left-handed spinor and can be absorbed into the spinor to describe left-handedleptons (and as later discovered quarks) The right handed leptons are blind to the weakinteraction

          The processes describing left-handed current interactions are shown in Fig 22

          Analogously to isospin where the proton and neutron are considered two eigenstates ofthe nucleon we postulate that Bmicro is the gauge boson of an SU(2)transformation coupled toweak SU(2) doublets (

          νe

          eminus

          )

          (ud

          ) (212)

          We may now write the weak SU(2) currents as eg

          jimicro = (ν e)Lγmicro

          τ i

          2

          e

          )L (213)

          21 Standard Model 15

          Figure 22 Weak Interaction Vertices [48]

          where τ i are the Pauli-spin matrices and the subscript L denotes the left-handed componentof the spinor projected out by Eqn211 The third current corresponding to i=3 is called theneutral current as it doesnrsquot change the charge of the particle involved in the interaction

          We could postulate that the electromagnetic current is a linear combination of the leftand right elecromagnetic fields eL = 1

          2(1minus γ5)e and eR = 12(1+ γ5)e

          jemmicro = eLγmicroQeL + eRγmicroQeR (214)

          where Q is the electromagnetic operator and eR is a singlet However this is not invariantunder SU(2)L To construct an invariant current we need to use the SU(2)L doublet 212The current which is an SU(2)L invariant U(1) current is

          jYmicro = (ν e)LγmicroYL

          e

          )L+ eRγmicroYReR (215)

          where hypercharges YL and YR are associated with the U(1)Y symmetry jYmicro is a linearcombination of the electromagnetic current and the weak neutral current thus explaining whythe hypercharge differs for the left and right-handed components since the electromagneticcurrent has the same charge for both components whilst the weak current exists only forleft-handed particles j3

          micro and the third component of weak isospin T 3 allows us to calculate

          21 Standard Model 16

          the relationship Y = 2(QminusT 3) which can be seen by writing jYmicro as a linear combination ofj3micro (the weak neutral current that doesnrsquot change the charge of the particle involved in the

          interaction) and jemmicro (using the convention that the jYmicro is multiplied by 1

          2 to match the samefactor implicit in j3

          micro ) Substituting

          τ3 =

          (1 00 minus1

          )(216)

          into equation 213 for i=3 and subtracting equation 215 and equating to 12 times equation

          214

          we get

          eLγmicroQeL + eRγmicroQeR minus (νLγmicro

          12

          νL minus eLγmicro

          12

          eL) =12

          eRγmicroYReR +12(ν e)LγmicroYL

          e

          )L (217)

          from which we can read out

          YR = 2QYL = 2Q+1 (218)

          and T3(eR) = 0 T3(νL) =12 and T3(eL) =

          12 The latter three identities are implied by

          the fraction 12 inserted into the definition of equation 213

          The Lagrangian kinetic terms of the fermions can then be written

          L =minus14

          FmicroνFmicroν minus 14

          GmicroνGmicroν

          + sumgenerations

          LL(i D)LL + lR(i D)lR + νR(i D)νR

          + sumgenerations

          QL(i D)QL +UR(i D)UR + DR(i D)DR

          (219)

          LL are the left-handed lepton doublets QL are the left-handed quark doublets lRνR theright handed lepton singlets URDR the right handed quark singlets

          The field strength tensors are given by

          Fmicroν = partmicroAν minuspartνAmicro minus ig[Amicro Aν ] (220)

          21 Standard Model 17

          andGmicroν = partmicroBν minuspartνBmicro (221)

          Bmicro is an abelian U(1) gauge field while Amicro is non-abelian and Fmicroν contains the com-mutator of the field through the action of the covariant derivative [Dmicro Dν ] This equationcontains no mass terms terms for the field Amicro for good reason- this would violate gaugeinvariance We know that the gauge bosons of the weak force have substantial mass and theway that this problem has been solved is through the Higgs mechanism

          218 Higgs Mechanism

          To solve the problem of getting gauge invariant mass terms three independent groupspublished papers in 1964 The mechanism was named after Peter Higgs To introduce theconcept consider a complex scalar field φ that permeates all space Consider a potential forthe Higgs field of the form

          minusmicro2φ

          daggerφ +λ (φ dagger

          φ2) (222)

          which has the form of a mexican hat Fig 23 This potential is still U(1) gauge invariant andcan be added to our Lagrangian together with kinetic+field strength terms to give

          L = (Dmicroφ)dagger(Dmicroφ)minusmicro2φ

          daggerφ +λ (φ dagger

          φ2)minus 1

          4FmicroνFmicroν (223)

          It is easily seen that this is invariant to the transformations

          Amicro rarr Amicro minuspartmicroη(x) (224)

          φ(x)rarr eieη(x)φ(x) (225)

          The minimum of this field is not at 0 but in a circular ring around 0 with a vacuum

          expectation value(vev)radic

          micro2

          2λequiv vradic

          2

          We can parameterise φ as v+h(x)radic2

          ei π

          Fπ where h and π are referred to as the Higgs Bosonand Goldstone Boson respectively and are real scalar fields with no vevs Fπ is a constant

          21 Standard Model 18

          Figure 23 Higgs Potential [49]

          Substituting this back into the Lagrangian 223 we get

          minus14

          FmicroνFmicroν minusevAmicropartmicro

          π+e2v2

          2AmicroAmicro +

          12(partmicrohpart

          microhminus2micro2h2)+

          12

          partmicroπpartmicro

          π+(hπinteractions)(226)

          This Lagrangian now seems to describe a theory with a photon Amicro of mass = ev a Higgsboson h with mass =

          radic2micro and a massless Goldstone π

          However there is a term mixing Amicro and part microπ in this expression and this shows Amicro and π

          are not independent normal coordinates and we can not conclude that the third term is a massterm This π minusAmicro mixing can be removed by making a gauge transformation in equations224 and 225 making

          φrarrv+h(x)radic2

          ei π

          Fπminusieη(x) (227)

          and setting πrarr π

          Fπminus eη(x) = 0 This choosing of a point on the circle for the vev is called

          spontaneous symmetry breaking much like a vertical pencil falling and pointing horizontallyin the plane This choice of gauge is called unitary gauge The Goldstone π will thencompletely disappear from the theory and one says that the Goldstone has been eaten to givethe photon mass

          21 Standard Model 19

          This can be confirmed by counting degrees of freedom before and after spontaneoussymmetry breaking The original Lagrangian had four degrees of freedom- two from thereal massless vector field Amicro (massless photons have only two independent polarisationstates) and two from the complex scalar field (one each from the h and π fields) The newLagrangian has one degree of freedom from the h field and 3 degrees of freedom from themassive real vector field Amicro and the π field has disappeared

          The actual Higgs field is postulated to be a complex scalar doublet with four degrees offreedom

          Φ =

          (φ+

          φ0

          )(228)

          which transforms under SU(2) in the fundamental representation and also transforms underU(1) hypercharge with quantum numbers

          Table 21 Quantum numbers of the Higgs field

          T 3 Q Yφ+

          12 1 1

          φ0 minus12 1 0

          We can parameterise the Higgs field in terms of deviations from the vacuum

          Φ(x) =(

          η1(x)+ iη2(x)v+σ(x)+ iη3(x)

          ) (229)

          It can be shown that the three η fields can be transformed away by working in unitarygauge By analogy with the scalar Higgs field example above the classical action will beminimised for a constant value of the Higgs field Φ = Φ0 for which Φ

          dagger0Φ0 = v2 This again

          defines a range of minima and we can pick one arbitarily (this is equivalent to applying anSU(2)timesU(1) transform to the Higgs doublet transforming the top component to 0 and thebottom component to Φ0 = v) which corresponds to a vacuum of zero charge density andlt φ0 gt=+v

          In this gauge we can write the Higgs doublet as

          Φ =

          (φ+

          φ0

          )rarr M

          (0

          v+ H(x)radic2

          ) (230)

          where H(x) is a real Higgs field v is the vacuum expectation and M is an SU(2)timesU(1)matrix The matrix Mdagger converts the isospinor into a down form (an SU(2) rotation) followedby a U(1) rotation to a real number

          21 Standard Model 20

          If we consider the Higgs part of the Lagrangian

          minus14(Fmicroν)

          2 minus 14(Bmicroν)

          2 +(DmicroΦ)dagger(DmicroΦ)minusλ (Φdagger

          Φminus v2)2 (231)

          Substituting from equation 230 into this and noting that

          DmicroΦ = partmicroΦminus igW amicro τ

          aΦminus 1

          2ig

          primeBmicroΦ (232)

          We can express as

          DmicroΦ = (partmicro minus i2

          (gA3

          micro +gprimeBmicro g(A1micro minusA2

          micro)

          g(A1micro +A2

          micro) minusgA3micro +gprimeBmicro

          ))Φ equiv (partmicro minus i

          2Amicro)Φ (233)

          After some calculation the kinetic term is

          (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

          14(v+

          Hradic2)2[A 2]22 (234)

          where the 22 subscript is the index in the matrix

          If we defineWplusmn

          micro =1radic2(A1

          micro∓iA2micro) (235)

          then [A 2]22 is given by

          [A 2]22 =

          (gprimeBmicro +gA3

          micro

          radic2gW+

          microradic2gWminus

          micro gprimeBmicro minusgA3micro

          ) (236)

          We can now substitute this expression for [A 2]22 into equation 234 and get

          (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

          14(v+

          Hradic2)2(2g2Wminus

          micro W+micro +(gprimeBmicro minusgA3micro)

          2) (237)

          This expression contains mass terms for three massive gauge bosons Wplusmn and the combi-nation of vector fields gprimeBmicro minusgA3

          micro where note

          21 Standard Model 21

          Table 22 Weak Quantum numbers of Lepton and Quarks

          T 3 Q YνL

          12 0 -1

          lminusL minus12 -1 -1

          νR 0 0 0lminusR 0 -1 -2UL

          12

          23

          13

          DL minus12 minus1

          313

          UR 0 23

          43

          DR 0 minus13 minus2

          3

          Wminusmicro = (W+

          micro )dagger equivW 1micro minus iW 2

          micro (238)

          Then the mass terms can be written

          12

          v2g2|Wmicro |2 +14

          v2(gprimeBmicro minusgA3micro)

          2 (239)

          W+micro and W microminus can be associated with the W boson and its anti-particle and (minusgprimeBmicro +

          gA3micro) with the Z Boson (after normalisation by

          radicg2 +(gprime

          )2) The combination gprimeA3micro +gBmicro

          is orthogonal to Z and remains massless This can be associated with the Photon (afternormalisation) These predict mass terms for the W and Z bosons mW = vgradic

          2and mZ =

          vradic2

          radicg2 +(gprime

          )2It is again instructive to count the degrees of freedom before and after the Higgs mech-

          anism Before we had a complex doublet Φ with four degrees of freedom one masslessB with two degrees of freedom and three massless W a fields each with two each givinga total of twelve degrees of freedom After spontaneous symmetry breaking we have onereal scalar Higgs field with one degree of freedom three massive weak bosons with nineand one massless photon with two again giving twelve degrees of freedom One says thescalar degrees of freedom have been eaten to give the Wplusmn and Z bosons their longitudinalcomponents (ie mass)

          Table 22 shows T 3 the weak isospin quantum number (which describes how a particletransforms under SU(2)) Y the weak hypercharge (which describes how the particle trans-

          21 Standard Model 22

          forms under U(1)) and Q the electric charge of leptons and quarks The former two enterdirectly into the Lagrangian while the latter can be derived as a function of the former two

          Strictly speaking the right handed neutrino which has no charge may not exist and is notpart of the standard model although some theories postulate a mass for this (hypothetical)particle

          Writing gauge invariant couplings between ELeRφ with the vacuum expectation ofφ = v (Yukawa matrices λ

          i ju and λ

          i jd respectively for the up and down quarks) we get mass

          terms for the quarks (and similarly for the leptons)

          Mass terms for quarks minussumi j[(λi jd Qi

          Lφd jR)+λ

          i ju εab(Qi

          L)aφlowastb u j

          R +hc]

          Mass terms for leptonsminussumi j[(λi jl Li

          Lφ l jR)+λ

          i jν εab(Li

          L)aφlowastb ν

          jR +hc]

          Here hc is the hermitian conjugate of the first terms L and Q are lepton and quarkdoublets l and ν are the lepton singlet and corresponding neutrino d and u are singlet downand up quarks εab is the Levi-Civita (anti-symmetric) tensor

          If we write the quark generations in a basis that diagonalises the Higgs couplings weget a unitary transformations Uu and Ud for the up and down quarks respectively which iscompounded in the boson current (Udagger

          u Ud) called the CKM matrix where off-diagonal termsallow mixing between quark generations (this is also called the mass basis as opposed tothe flavour basis) A similar argument can give masses to right handed sterile neutrinos butthese can never be measured However a lepton analogue of the CKM matrix the PNMSmatrix allows left-handed neutrinos to mix between various generations and oscillate betweenflavours as they propagate through space allowing a calculation of mass differences

          219 Quantum Chromodynamics

          The final plank in SM theory tackling the strong force Quantum Chromodynamics wasproposed by Gell-Mann and Zweig in 1963 Three generations of quarks were proposedwith charges of 23 for uc and t and -13 for ds and b Initially SU(2) isospin was consideredtransforming u and d states but later irreducible representations of SU(3) were found to fit theexperimental data The need for an additional quantum number was motivated by the fact thatthe lightest excited state of the nucleon consisted of sss (the ∆++) To reconcile the baryon

          21 Standard Model 23

          spectrum with spin statistics colour was proposed Quarks transform under the fundamentalor 3 representation of colour symmetry and anti-quarks under 3 The inner product of 3and 3 is an invariant under SU(3) as is a totally anti-symmetric combination ε i jkqiq jqk (andthe anti-quark equivalent) It was discovered though deep inelastic scattering that quarksshowed asymptotic freedom (the binding force grew as the seperation grew) Non-abeliangauge theories showed this same property and colour was identified with the gauge quantumnumbers of the quarks The quanta of the gauge field are called gluons Because the forcesshow asymptotic freedom - ie the interaction becomes weaker at high energies and shorterdistances perturbation theory can be used (ie Feynman diagrams) at these energies but forlonger distances and lower energies lattice simulations must be done

          2110 Full SM Lagrangian

          The full SM can be written

          L =minus14

          BmicroνBmicroν minus 18

          tr(FmicroνFmicroν)minus 12

          tr(GmicroνGmicroν)

          + sumgenerations

          (ν eL)σmicro iDmicro

          (νL

          eL

          )+ eRσ

          micro iDmicroeR + νRσmicro iDmicroνR +hc

          + sumgenerations

          (u dL)σmicro iDmicro

          (uL

          dL

          )+ uRσ

          micro iDmicrouR + dRσmicro iDmicrodR +hc

          minussumi j[(λ

          i jl Li

          Lφ l jR)+λ

          i jν ε

          ab(LiL)aφ

          lowastb ν

          jR +hc]

          minussumi j[(λ

          i jd Qi

          Lφd jR)+λ

          i ju ε

          ab(QiL)aφ

          lowastb u j

          R +hc]

          + (Dmicroφ)dagger(Dmicroφ)minusmh[φφ minus v22]22v2

          (240)

          where σ micro are the extended Pauli matrices

          (1 00 1

          )

          (0 11 0

          )

          (0 minusii 0

          )

          (1 00 minus1

          )

          The first line gives the gauge field strength terms the second the lepton dynamical termsthe third the quark dynamical terms the fourth the lepton (including neutrino) mass termsthe fifth the quark mass terms and the sixth the Higgs terms

          The SU(3) gauge field is given by Gmicroν and which provides the colour quantum numbers

          21 Standard Model 24

          Figure 24 Standard Model Particles and Forces [50]

          Note The covariant derivatives operating on the left-handed leptons contain only theU(1) and SU(2) gauge fields Bmicro Fmicro whereas those operating on left-handed quarks containU(1)SU(2) and SU(3) gauge fields Bmicro Fmicro Gmicro The covariant derivatives operating onright handed electrons and quarks are as above but do not contain the SU(2) gauge fieldsRight handed neutrinos have no gauge fields

          The sums over i j above are over the different generations of leptons and quarks

          The particles and forces that emerge from the SM are shown in Fig 24

          22 Dark Matter 25

          22 Dark Matter

          221 Evidence for the existence of dark matter

          2211 Bullet Cluster of galaxies

          Most of the mass of the cluster as measured by a gravitational lensing map is concentratedaway from the centre of the object The visible matter (as measured by X-ray emission) ismostly present in the centre of the object The interpretation is that the two galaxy clustershave collided perpendicular to the line of sight with the mostly collisionless dark matterpassing straight through and the visible matter interacting and being trapped in the middleIt was estimated that the ratio of actual to visible matter was about 16 [51]

          Figure 25 Bullet Cluster [52]

          2212 Coma Cluster

          The first evidence for dark matter came in the 1930s when Fritz Zwicky observed the Comacluster of galaxies over 1000 galaxies gravitationally bound to one another He observed thevelocities of individual galaxies in the cluster and estimated the amount of matter based onthe average galaxy He calculated that many of the galaxies should be able to escape fromthe cluster unless the mass was about 10 times higher than estimated

          22 Dark Matter 26

          2213 Rotation Curves [53]

          Since those early observations many spiral galaxies have been observed with the rotationcurves calculated by doppler shift with the result that in all cases the outermost stars arerotating at rates fast enough to escape their host galaxies Indeed the rotation curves arealmost flat indicating large amounts of dark matter rotating with the galaxies and holdingthe galaxies together

          Figure 26 Galaxy Rotation Curves [54]

          2214 WIMPS MACHOS

          The question is if the particles that make up most of the mass of the universe are not baryonswhat are they It is clear that the matter does not interact significantly with radiation so mustbe electrically neutral It has also not lost much of its kinetic energy sufficiently to relax intothe disks of galaxies like ordinary matter Massive particles may survive to the present ifthey carry a conserved additive or multiplicative quantum number This is so even if equalnumbers of particles and anti-particles can annihilate if the number densities become too lowto reduce the density further These particles have been dubbed Weakly interacting MassiveParticles (WIMPS) as they interact with ordinary matter only through a coupling strengthtypically encountered in the weak interaction They are predicted to have frozen out ofthe hot plasma and would give the correct relic density based solely on the weak interactioncross section (sometimes called the WIMP Miracle) Another possibility called MassiveCompact Halo Objects (MACHOS)- are brown dwarf stars Jupiter mass objects or remnantsfrom an earlier generation of star formation

          22 Dark Matter 27

          2215 MACHO Collaboration [55]

          In order to estimate the mass of MACHOS a number of experiments have been conductedusing microlensing which makes use of the fact that if a massive object lies in the line ofsight of a much more distant star the light from the more distant star will be lensed and forma ring around the star It is extemely unlikely that the MACHO will pass directly through theline of sight but if there is a near miss the two images will be seperated by a small angleThe signature for microlensing is a time-symmetric brightening of a star as the MACHOpasses near the line of sight called a light curve One can obtain statistical information aboutthe MACHO masses from the shape and duration of the light curves and a large number ofstars (many million) must be monitored The MACHO Collaboration [55] have monitoredover 11 million stars and less than 20 events were detected The conclusion was that thiscould account for a small addition (perhaps up to 20 ) to the visible mass but not enoughto account for Dark Matter

          2216 Big Bang Nucleosynthesis (BBN) [56]

          Another source of evidence for DM comes from the theory of Nucleosynthesis in the fewseconds after the Big Bang In the first few minutes after the big bang the temperatures wereso high that all matter was dissociated but after about 3 minutes the temperature rapidlycooled to about 109 Kelvin Nucleosynthesis could begin when protons and neutrons couldstick together to form Deuterium (Heavy Hydrogen) and isotopes of Helium The ratio ofDeuterium to Hydrogen and Helium is very much a function of the baryon density and alsothe rate of expansion of the universe at the time of Nucleosynthesis There is a small windowwhen the density of Baryons is sufficient to account for the ratios of the various elementsThe rate of expansion of the universe is also a function of the total matter density The ratiosof these lighter elements have been calculated by spectral measurements with Helium some25 of the mass of the universe Deuterium 001 The theory of BBN accounts for theratios of all of the lighter elements if the ratio of Baryon mass to total mass is a few percentsuggesting that the missing matter is some exotic non-baryonic form

          22 Dark Matter 28

          2217 Cosmic Microwave Background [57]

          The discovery of a uniform Microwave Background radiation by Penzias and Wilson in1965 was confirmation of the theory first proposed by Gamow and collaborators that theuniverse should be filled with black body radiation an afterglow of the big bang TheCOBE satelite launched in 1989 confirmed this black body radiation with temperature of2725+-002K with 95 confidence While extremely uniform it found tiny irregularitieswhich could be accounted for by irregularities in the density of the plasma at last scattering(gravitational redshifts or blueshifts called the Sachs Wolfe effect) and temperature andvelocity fluctuations in the plasma The anisotropy of the background has been furtheranalysed by WMAP (the Wilkinson Microwave Anisotropy Probe) launched in 2001 Thefluctuations in the density of the plasma are what account for the seeds of galaxy formationHowever detailed calculations have found that the fluctuations were too small to account forthe observed galaxy distribution without an electrically neutral gravitating matter in additionto ordinary matter that only became electrically neutral at the time of recombination [58]

          In May 2009 the European Space Agency (ESA) launched the Planck sattelite a spacebased observatory to observe the universe at wavelengths between 03mm and 111 mmbroadly covering the far-infrared microwave and high frequency domains The missionrsquosmain goal was to study the cosmic microwave background across the whole sky at greatersensitivity and resolution than ever before Using the information gathered and the standardmodel of cosmology ESA was able to calculate cosmological parameters including the relicdensity of dark matter remaining after the Big Bang to a high degree of precision withinthe framework of the standard model of cosmology The standard model of cosmology canbe described by a relatively small number of parameters including the density of ordinarymatter dark matter and dark energy the speed of cosmic expansion at the present epoch (alsoknown as the Hubble constant) the geometry of the Universe and the relative amount ofthe primordial fluctuations embedded during inflation on different scales and their amplitude[59]

          2218 LUX Experiment - Large Underground Xenon experiment [16]

          The LUX experiment is conducted at the Sanford Underground research facility one mileunderground to minimise the background of cosmic rays The detector is a time projectionchamber allowing a 3D positioning of interactions occurring within its active volume LUX

          22 Dark Matter 29

          Figure 27 WMAP Cosmic Microwave Background Fluctuations [58]

          Figure 28 Dark Matter Interactions [60]

          uses as its target 368 kiliograms of liquefied ultra pure xenon which is a scintillator Interac-tions inside the xenon will create an amount of light proportional to the energy depositedThat light can be collected on arrays of light detectors sensitive to a single photon

          Xenon is very pure and the amount of radiation originating within the target is limitedAlso being three times denser than water it can stop radiation originating from outside thedetector The 3D detector focuses on interactions near the center which is a quiet regionwhere rare dark matter interactions might be detected [16]

          22 Dark Matter 30

          222 Searches for dark matter

          2221 Dark Matter Detection

          Fig 28 illustrates the reactions studied under the different forms of experiment designed todetect DM

          Direct detection experiments are mainly underground experiments with huge detectorsattempting to detect the recoil of dark matter off ordinary matter such as LUX [16] CDMS-II [61] XENON100 [11] COUPP [62] PICASSO [63] ZEPLIN [10] These involveinteractions between DM and SM particles with final states involving DM and SM particles

          Indirect detection involves the search for the annihilation of DM particles in distantastrophysical sources where the DM content is expected to be large Example facilitiesinclude space based experiments (EGRET [64] PAMELA [18] FERMI-LAT [65]) balloonflights (ATIC [66] PEBS [67] PPB-BETS [68]) or atmospheric Cherenkov telescopes(HESS [69]MAGIC [70]) and neutrino telescopes (SuperKamiokande [71] IceCube [72]ANTARES [73] KM3NET [74]) Indirect Detection involves DM particles in the initial stateand SM particles in the final state

          Collider detection involves SM particles in the initial state creating DM particles in thefinal state which can be detected by missing transverse energy and explained by variousmodels extending the SM

          223 Possible signals of dark matter

          224 Gamma Ray Excess at the Centre of the Galaxy [65]

          The center of the galaxy is the most dense and active region and it is no surprise that itproduces intense gamma ray sources When astronomers subtract all known gamma raysources there remains an excess which extends out some 5000 light years from the centerHooper et al find that the annihilation of dark matter particles with a mass between 31 and40 Gev fits the excess to a remarkable degree based on Gamma Ray spectrum distributionand brightness

          23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 31

          Figure 29 Gamma Ray Excess from the Milky Way Center [75]

          23 Background on ATLAS and CMS Experiments at theLarge Hadron collider (LHC)

          The collider constraints in this paper are based on two of the experiments that are part of theLHC in CERN

          Figure 210 ATLAS Experiment

          The LHC accelerates proton beams in opposite directions around a 27 km ring of super-conducting magnets until they are made to collide at an interaction point inside one of the 4experiments placed in the ring An upgrade to the LHC has just been performed providing65 TeV per beam

          23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 32

          231 ATLAS Experiment

          The ATLAS detector consists of concentric cylinders around the interaction point whichmeasure different aspects of the particles that are flung out of the collision point There are 4major parts the Inner Detector the Calorimeters the Muon spectrometer and the magnetsystems

          2311 Inner Detector

          The inner detector extends from a few centimeters to 12 meters from the interaction point andmeasures details of the charged particles and their momentum interacting with the materialIf particles seem to originate from a point other than the interaction point this may indicatethat the particle comes from the decay of a hadron with a bottom quark This is used to b-tagcertain particles

          The inner detector consists of a pixel detector with three concentric layers and threeendcaps containing detecting silicon modules 250 microm thick The pixel detector has over 80million readout channels about 50 percent of the total channels for the experiment All thecomponents are radiation hardened to cope with the extremely high radiation in the proximityof the interaction point

          The next layer is the semi-conductor layer (SCT) similar in concept to the pixel detectorbut with long narrow strips instead of pixels to cover a larger area It has 4 double layers ofsilicon strips with 63 million readout channels

          The outermost component of the inner detector is the Transition Radiation Tracker It is acombination of a straw tracker and a transition radiation detector The detecting elementsare drift tubes (straws) each 4mm in diameter and up to 144 cm long Each straw is filledwith gas which becomes ionised when the radiation passes through The straws are held at-1500V with a thin wire down the center This drives negatively ionised atoms to the centercausing a pulse of electricity in the wire This causes a pattern of hit straws allowing the pathof the particles to be determined Between the straws materials with widely varying indicesof refraction cause transition radiation when particles pass through leaving much strongersignals in some straws Transition radiation is strongest for highly relativistic particles (thelightest particles- electrons and positrons) allowing strong signals to be associated with theseparticles

          23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 33

          2312 Calorimeters

          The inner detector is surrounded by a solenoidal magnet of 2 Tesla (there is also an outertoroidal magnet) but surrounding the inner magnet are the calorimeters to measure theenergy of particles by absorbing them There is an inner electromagnetic calorimeter whichmeasures the angle between the detectorrsquos beam axis and the particle (or more preciselypseudo-rapidity) and its angle in the perpendicular plane to a high degree of accuracy Theabsorbing material is lead and stainless steel and the sampling material liquid argon Thehadron calorimeter measures the energy of particles that interact via the strong force Theenergy absorbing material is scintillating steel

          2313 Muon Specrometer

          The muon spectrometer surrounds the calorimeters and has a set of 1200 chambers tomeasure the tracks of outgoing muons with high precision using a magnetic field andtriggering chambers Very few particles of other types pass through the calorimeters and aresubsequently measured

          2314 Magnets

          The magnets are in two layers the inner solenoidal and outer toroidal which together causecharged particles to curve in the magetic field allowing their momentum to be determinedThe outer toroidal magnet consists of 8 very large air core superconducting magnets whichsupply the field in the muon spectrometer

          232 CMS Experiment

          The CMS Detector is similar in design to the ATLAS Detector with a single solenoidalmagnet which is the largest of its type ever built (length 13m and diameter 7m) and allowsthe tracker and calorimeter detectors to be placed inside the coil in a compact way Thestrong magnetic field gives a high resolution for measuring muons electrons and photons ahigh quality central tracking system and a hermetic hadron calorimeter designed to preventhadrons from escaping

          23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 34

          Figure 211 CMS Experiment

          Chapter 3

          Fitting Models to the Observables

          31 Simplified Models Considered

          In order to extend the findings of previous studies [19] to include collider constraints wefirst performed Bayesian scans [7] over three simplified models real scalar dark matter φ Majorana fermion χ and real vector Xmicro The purpose was to obtain the most probable regionsof each model then to use the resulting posterior samples to investigate whether the LHCexperiments have any sensitivity to the preferred regions

          The three models couple to the mediator with interactions shown in the following table

          Table 31 Simplified Models

          Hypothesis real scalar DM Majorana fermion DM real vector DM

          DM mediator int Lφ sup microφ mφ φ 2S Lχ sup iλχ

          2 χγ5χS LX sup microX mX2 X microXmicroS

          The interactions between the mediator and the standard fermions is assumed to be

          LS sup f f S (31)

          and in line with minimal flavour violation we only consider third generation fermions- ief = b tτ

          For the purposes of these scans we consider the following observables

          32 Observables 36

          32 Observables

          321 Dark Matter Abundance

          We have assumed a single dark matter candidate which freezes out in the early universe witha central value as determined by Planck [21] of

          ΩDMh2 = 01199plusmn 0031 (32)

          h is the reduced hubble constant

          The experimental uncertainty of the background relic density [59] is dominated by atheoretical uncertainty of 50 of the value of the scanned Ωh2 [76] We have added thetheoretical and the experimental uncertainties in quadrature

          SD =radic(05Ωh2)2 + 00312 (33)

          This gives a log likelihood of

          minus05lowast (Ωh2 minus 1199)2

          SD2 minus log(radic

          2πSD) (34)

          322 Gamma Rays from the Galactic Center

          Assuming that the gamma ray flux excess from the galactic centre observed by Fermi-LAT iscaused by self annihilation of dark matter particles the differential flux

          d2Φ

          dEdΩ=

          lt σv gt8πmχ

          2 J(ψ)sumf

          B fdN f

          γ

          dE(35)

          has been calculated using both Micromegas v425 with a Navarro-Frenk-White (NFW) darkmatter galactic distribution function

          ρ(r) = ρ0(rrs)

          minusγ

          (1+ rrs)3minusγ (36)

          with γ = 126 and an angle of 5 to the galactic centre [19]

          32 Observables 37

          Here Φ is the flux dE is the differential energy lt σv gt the velocity averaged annihi-lation cross section of the dark matter across the galactic centre mχ the mass of the darkmatter particle B f =lt σv gt f σv is the annihilation fraction into the f f final state anddN f

          γ dE is the energy distribution of photons produced in the annihilation channel with thefinal state f f The J factor is described below

          The angle ψ to the galactic centre determines the normalisation of the gamma ray fluxthrough the factor

          J(ψ) =int

          losρ(l2 + r⊙2 minus2lr⊙cosψ)dl (37)

          where ρ(r) is the dark matter density at distance r from the centre of the galaxy r⊙=85 kpcis the solar distance from the Galactic center ρ0 is set to reproduce local dark matter densityof 03GeVcm3

          The central values for the gamma ray spectrum were taken from Fig 5 of [77] and thestandard deviations of individual flux points from the error bars on the graph In commonwith [78] we ignore possible segment to segment correlations

          For the Fermi-LAT log likelihood we assumed each point was distributed with indepen-dent Gaussian likelihood with the experimental error for each data point given by the errorbar presented in Fig5 of [76] for each point We also assumed an equal theoretical errorfor each point which when added to the experimental error in quadrature gave an effectivelog likelihood for each point of minus05lowast (giminusdi)

          2

          2lowastσ2i

          where gi are the calculated values and di theexperimental values and σi the experimental errors

          323 Direct Detection - LUX

          The LUX experiment [16] is the most constraining direct search experiment to date for awide range of WIMP models We have used LUXCalc [79] (modified to handle momentumdependent nucleon cross sections) to calculate the likelihood of the expected number ofsignal events for the Majorana fermion model

          The likelihood function is taken as the Poisson distribution

          L(mχ σ | N) = P(N | mχ σ) =(b+micro)Neminus(b+micro)

          N (38)

          32 Observables 38

          where b= expected background of 064 and micro = micro(mχσ ) is the expected number of signalevents calculated for a given model

          micro = MTint

          infin

          0dEφ(E)

          dRdE

          (E) (39)

          where MT is the detector mass times time exposure and φ(E) is a global efficiency factorincorporating trigger efficiencies energy resolution and analysis cuts

          The differential recoil rate of dark matter on nucleii as a function of recoil energy E

          dRdE

          =ρX

          mχmA

          intdvv f (v)

          dσASI

          dER (310)

          where mA is the nucleon mass f (v) is the dark matter velocity distribution and

          dσSIA

          dER= Gχ(q2)

          4micro2A

          Emaxπ[Z f χ

          p +(AminusZ) f χn ]

          2F2A (q) (311)

          where Emax = 2micro2Av2mA Gχ(q2) = q2

          4m2χ

          [24] is an adjustment for a momentum dependantcross section which applies only to the Majorana model

          f χ

          N =λχ

          2m2SgSNN assuming that the relic density is the central value of 1199 We have

          implemented a scaling of the direct search cross section in our code by multiplying gSNN by( Ω

          Ω0)05 as a convenient method of allowing for the fact that the actual relic density enters

          into the calculation of the cross section as a square

          FA(q) is the nucleus form factor and

          microA =mχmA

          (mχ +mA)(312)

          is the reduced WIMP-nucleon mass

          The strength of the dark matter nucleon interaction between dark matter particles andthird generation quarks is

          gSNN =2

          27mN fT G sum

          f=bt

          λ f

          m f (313)

          where fT G = 1minus f NTuminus f N

          Tdminus fTs and f N

          Tu= 02 f N

          Td= 026 fTs = 043 [20]

          33 Calculations 39

          For the real scalar and real vector models where the spin independent cross sections arenot momentum dependent we based the likelihood on the LUX 95 CL from [16] using acomplementary error function Li(d|mχ I) = 05lowastCEr f (dminusx(mχ )

          σ) where x is the LUX limit

          and d the calculated SI cross section We assumed a LUX error σ of 01lowastd lowastradic

          2

          33 Calculations

          We coded the Lagrangians for the three models in Feynrules [4] Calculation of the expectedobservables (dark matter relic density Fermi-LAT gamma ray spectrum and LUX nucleonscattering amplitudes) were performed using Micromegas v425

          331 Mediator Decay

          A number of searches have previously been performed at the LHC which show sensitivity tothe decay of scalar mediators in the form of Higgs bosons (in beyond the standard modeltheories) to bottom quarks top quarks and τ leptons (all third generation fermions) Two inparticular [43 44] provide limits on the cross-sections for the production of the scalar timesthe branching ratios to brsquos and τrsquos In order to check these limits against each of the modelswe ran parameter scans in Madgraph at leading order over the top 900 points from the nestedsampling output from MultiNest [19]

          The two processes were

          1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to apair of τ leptons in pp collisions [44]

          bull generate p p gt b b S where S is the scalar mediator

          The process measures the decay of a scalar in the presence of b-jets that decay to τ+τminus

          leptons Fig31 shows the main diagrams contributing to the process The leftmost diagramis a loop process and the centre and right are tree level however we only got Madgraph togenerated the middle diagram since we are interested in τ leptons and b quarks in the finalstate

          The limit given in [44] only applies in the case of narrow resonances where narrow istaken to mean that the width of the resonance is less than 10 of its mass To test the validity

          33 Calculations 40

          Figure 31 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof τ leptons

          of applying these limits to our models we show in Figures 32 to 34 the percentage widthof the resonance vs mS[GeV ] (pole mass of the scalar mediator) λb and λt (the couplingconstants to the b and t quarks) The probabilities are the posterior probabilities obtainedfrom MultiNest scans of the Majorana model varying the coupling constants and scalarmediator and calculating the decay width of the mediator The results are represented as 2dposterior probability heat maps in the central panels and 1d marginalised distributions on thex and y axes of each graph

          16 18 20 22 24 26 28 30

          log10(mS[GeV])

          001

          002

          003

          004

          005

          Widthm

          S

          00

          04

          08

          12

          0 100 200

          Posterior Probability

          Figure 32 WidthmS vs mS

          The WidthmS was lt 01 at almost all points except where the mass of the scalar wasgreater than 1000 GeV λb was greater than 1 or λτ was greater than 1

          This can be seen from the graphs in Figs 323334

          33 Calculations 41

          4 3 2 1 0

          λb

          001

          002

          003

          004

          005

          WidthmS

          000

          015

          030

          045

          0 100 200

          Posterior Probability

          Figure 33 WidthmS vs λb

          5 4 3 2 1 0

          λτ

          001

          002

          003

          004

          005

          WidthmS

          000

          015

          030

          045

          0 100 200

          Posterior Probability

          Figure 34 WidthmS vs λτ

          The decay widths are calculated in Madgraph at leading order and branching fractionscan be calculated to b t τ and χ (DM) The formulae for the calculations can be found indecayspy which is a subroutine of Madgraph which we have documented in Appendix BB1 Using these we calculated the branching ratio to τ+τminus so that the cross section timesbranching ratio can be compared to the limits in [80] shown in Chapters 56 and 7 in Figs5564 and 74 for each model

          2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

          This scan calculates the contributions of b-quark associated Higgs boson production inthe center and right diagrams of Fig 35 reproduced from [43] The scans did not includePythia and Delphes as only production cross sections were required and the producedparticles were in the final state and required no matching or merging

          The Madgraph processes were

          bull generate p p gt b S where S is the scalar mediator

          bull add process p p gt b S j

          bull add process p p gt b S

          33 Calculations 42

          Figure 35 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof b quarks in the presence of at least one b quark

          bull add process p p gt b S j

          The cross section times branching ratios to bb for mediator production in association withat least one b quark is shown for each model in Chapters 56 and 7 in Figs 5765 and 75

          332 Collider Cuts Analyses

          We generated two parameter scans with parameter combinations from the Multinest outputwhich were the basis of the plots of Fig 6 in [19] The plots in Fig 6 were marginalisedto two parameters each but the raw Multinest output contained 5 parameters plus the priorprobabilities of each combination normalised to total 10

          The first scan was for χ produced with b quarks In order to force the mediator to decayto χ only we had χ in the final state and defined jets to include b quarks in Madgraph Thenwe generated the processes

          bull generate p p gt χ χ j

          bull add process p p gt χ χ j j

          Jet matching was on

          The second scan was for t quarks produced in the final state

          bull generate p p gt χ χ tt

          33 Calculations 43

          No jet matching was required because we did not mix final state jet multiplicities at thematrix element level

          The outputs from these two processes were normalised to 21 f bminus1 and combined

          The results of these scans prior to collider cuts calculations are shown in Figs 5161and 71 These show the posterior probabilities of the various parameter combinations withdifferent constraints applied seperately and then together A more detailed commentary isgiven in each of the sections

          We then sampled 900 points based on posterior probability from these scans and ran thesethrough Madgraph v425 then applied collider cuts with the C++ code using the limits fromthe following six ATLAS and CMS searches shown in Table32 and described below to findany exclusions

          333 Description of Collider Cuts Analyses

          In the following all masses and energies are in GeV and angles in radians unless specificallystated

          3331 Lepstop0

          Search for direct pair production of the top squark in all-hadronic final states in pro-tonndashproton collisions at

          radics=8 TeV with the ATLAS detector[32]

          This analysis searched for direct pair production of the supersymmetric partner to thetop quark using integrated luminosity of 201 f bminus1 The top squark was assumed to de-cay via t rarr t χ0

          1 or t rarr bχ01 or t rarr bχ

          plusmn1 rarr bW (lowast)χ1

          0 where χ01 (χ

          plusmn1 ) denotes the lightest

          neutralino(chargino) in supersymmetric models The search targets a fully hadronic finalstate with four or more jets and large missing momentum The production of a mediator inassociation with top quarks followed by an invisible mediator decay to dark matter particleshas a similar final state to that considered in the ATLAS stop searches Thus we expect theATLAS reach to have sensitivity to the three dark matter models considered in this thesis

          The collider cuts program mimicked the object reconstruction through jetlepton overlapremoval and selected signal electrons muons and τ particles with pT gt 10 GeV and

          33 Calculations 44

          Table 32 95 CL by Signal Region

          Experiment Region Number

          Lepstop0

          SRA1 66SRA2 57SRA3 67SRA4 65SRC1 157SRC2 124SRC3 80

          Lepstop1

          SRtN1-1 857SRtN1-2 498SRtN1-3 389SRtN2 107SRtN3 85SRbC1 832SRbC2 195SRbC3 76

          Lepstop2

          L90 740L100 56L110 90L120 170

          2bstop

          SRA 857SRB 270SRA150 380SRA200 260SRA250 90SRA300 75SRA350 52

          CMSTopDM1L SRA 1385

          ATLASMonobjetSR1 1240SR2 790

          33 Calculations 45

          |η | lt 24724 and 247 respectively (for eminusmicrominusτ) Jets were selected with pT gt 20 GeVand |η |lt 45 b-jets were selected if they were tagged by random selection lt 70 and had|η |lt 25 and pT gt 20 GeV

          These jets and leptons were further managed for overlap removal within ∆R lt 04 withnon b-jets removed if ∆R lt 02

          The b-jets and non b-jets were further selected as signal jets if pT gt 35 GeV and |η |lt 28

          These signal jets electrons and muons were finally binned according to the signal regionsin Tables 333435 below

          Table 33 Selection criteria common to all signal regions

          Trigger EmissT

          Nlep 0b-tagged jets ⩾ 2

          EmissT 150 GeV

          |∆φ( jet pmissT )| gtπ5

          mbminT gt175 GeV

          Table 34 Selection criteria for signal regions A

          SRA1 SRA2 SRA3 SRA4anti-kt R=04 jets ⩾ 6 pT gt 808035353535 GeV

          m0b j j lt 225 GeV [50250] GeV

          m1b j j lt 225 GeV [50400] GeV

          min( jet i pmissT ) - gt50 GeV

          τ veto yesEmiss

          T gt150 GeV gt250 GeV gt300 GeV gt 350 GeV

          Table 35 Selection criteria for signal regions C

          SRC1 SRC2 SRC3anti-kt R=04 jets ⩾ 5 pT gt 8080353535 GeV

          |∆φ(bb)| gt02 π

          mbminT gt185 GeV gt200 GeV gt200 GeV

          mbmaxT gt205 GeV gt290 GeV gt325 GeV

          τ veto yesEmiss

          T gt160 GeV gt160 GeV gt215 GeV

          wherembmin

          T =radic

          2pbt Emiss

          T [1minus cos∆φ(pbT pmiss

          T )]gt 175 (314)

          33 Calculations 46

          and pbT is calculated for the b-tagged jet with the smallest ∆φ to the pmiss

          T direction andmbmax

          T is calculated with the pbT for the b-tagged jet with the largest ∆φ to the pmiss

          T direction

          m0b j j is obtained by first selecting the two jets with the highest b-tagging weight- from

          the remaining jets the two jets closest in the η minusφ plane are combined to form a W bosoncandidate this candidate is then combined with the closest of the 2 b-tagged jets in the η minusφ

          plane to form the first top candidate with mass m0b j j A second W boson candidate is formed

          by repeating the procedure with the remaining jets this candidate is combined with thesecond of the selected b-tagged jets to form the second top candidate with mass m1

          b j j Thecuts shown in Appendix1 Table A1 show the cumulative effect of the individual cuts appliedto the data the signal region cuts and the cumulative effects of individual cuts in SignalRegion C

          3332 Lepstop1

          Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

          radics=8 TeV pp collisions using 21 f bminus1 of

          ATLAS data[33]

          The stop decay modes considered were those to a top quark and the lightest supersym-metric particle (LSP) as well as a bottom quark and the lightest chargino where the charginodecays to the LSP by emitting a W boson We should expect the three dark matter models inthis paper with couplings to t and b quarks to show some visibility in this search

          The collider cuts program mimicked the object construction by selecting electrons andmuons with pT gt 10 GeV and |η |lt 247 and 24 respectively and jets with pT gt 20 GeVand |η |lt 10 b-jets with |η |lt 25 and pT gt 25 GeV were selected with random efficiency75 and mistag rate 2 (where non b-jets were mis-classified) We then removed any jetwithin ∆R = 02 of an electron and non overlapping jets with |η |lt 28 were used to removeany electrons and muons within ∆R = 04 of these while non-overlapping jets with |η |lt 25and pT gt 25 GeV were classified as signal jets Non overlapping electrons and muons withpT gt 25 GeV were classified as signal electrons and muons

          Signal selection defined six regions to optimize sensitivity for different stop and LSPmasses as well as sensitivity to two considered stop decay scenarios Three signal regionslabelled SRbC 1-3 where bC is a moniker for b+Chargino and 3 regions labelled SRtN 1-3where tN is a moniker for t+Neutralino were optimised for the scenarios where t1 decaysto the respective particles indicated by the labels Increasing label numbers correspond toincreasingly strict selection criteria

          33 Calculations 47

          The dominant background in all signal regions arises from dileptonic tt events All threeSRtN selectons impose a requirement on the 3-jet mass m j j j of the hadronically decayingtop quark to specifically reject the tt background where both W bosons from the top quarkdecay leponically

          For increasing stop mass and increasing difference between the stop and the LSP therequirents are tightened on Emiss

          T on the ratio EmissT

          radicHT where HT is the scalar sum of the

          momenta of the four selected jets and also tightened on mT

          To further reduce the dileptonic tt background for an event characterised by two one stepdecay chains a and b each producing a missing particle C use is made of the variable

          mT 2 =min

          pCTa + pC

          T b = pmissT

          [max(mTamtb)] (315)

          where mTi is the transverse mass 1 of branch i for a given hypothetical allocation (pCTa pC

          T b)

          of the missing particle momenta Eqn 315 is extended for more general decay chains byamT 2 [81] which is an asymmetric form of mT 2 in which the missing particle is the W bosonin the branch for the lost lepton and the neutrino is the missing particle for the branch withthe observed charged lepton

          ∆φ( jet12 pmissT ) the azimuthal angle between leading or subleading jets and missing mo-

          mentum is used to suppress the background from mostly multijet events with mis-measuredEmiss

          T

          Events with four or more jets are selected where the jets must satisfy |η | lt 25 and pT gt80604025 GeV respectively

          These signal particles were then classified according to the cutflows analysis in Fig36below and compared with the paper in Appendix1 Table A2

          3333 Lepstop2

          Search for direct top squark pair production in final states with two leptons in p pcollisions at

          radics=8TeV with the ATLAS detector[34]

          Three different analysis strategies are used to search for t1 pair production Two targett1 rarr b+ χ

          plusmn1 decay and the three body t1 rarr bW χ0

          1 decay via an off-shell top quark whilst

          1The transverse mass is defined as m2T = 2plep

          T EmissT (1minus cos(∆φ)) of branch i where ∆φ is the azimuthal

          angle between the lepton and the missing transverse momentum

          33 Calculations 48

          Table 36 Signal Regions - Lepstop1

          Requirement SRtN2 SRtN3 SRbC1 SRbC2 SRbC3∆φ( jet1 pmiss

          t )gt - 08 08 08 08∆φ( jet2 pmiss

          T )gt 08 08 08 08 08Emiss

          T [GeV ]gt 200 275 150 160 160Emiss

          T radic

          HT [GeV12 ]gt 13 11 7 8 8

          mT [GeV ]gt 140 200 120 120 120me f f [GeV ]gt - - - 550 700amT 2[GeV ]gt 170 175 - 175 200mT

          T 2[GeV ]gt - 80 - - -m j j j Yes Yes - - -

          one targets t1 rarr t + χ01 to an on shell quark decay mode The analysis is arranged to look

          at complementary mass splittings between χplusmn1 and χ0

          1 smaller and larger than the W bosonmass The smaller splitting provides sensitivity to t1 three body decay

          Object reconstruction was carried out by selecting baseline electronsmuons and τrsquos withpT gt 10 GeV and |η |lt 24724 and 247 respectively Baseline jets and b-jets were selectedwith pT gt 20 GeV and |η |lt 25 and b-jets with 70 efficiency Baseline jets were removedif within ∆R lt 02 of an electron and electrons and muons were removed if within ∆R lt 04of surviving jets Signal electrons required pT gt 10 GeV and signal jets pT gt 20 GeV and|η |lt 25

          The analysis made use of the variable mT 2 [82] the stransverse mass used to mea-sure the masses of pair-produced semi-invisibly decaying heavy particles It is definedas mT 2(pT1 pT2qT ) =

          minqT1+qT2=qT

          max[mT (pT1qT1)mT (pT2qT2)] where mT indicatesthe transverse mass pT1 and pT2 are vectors and qT = qT1 + qT2 The minimisation isperformed over all possible decompositions of qT

          Events were selected if they passed either a single-electron a single-muon a double-electron a double-muon or an electron-muon trigger Events are required to have exactlytwo opposite sign leptons and at least one of the leptons or muons must have a momentumgreater than 25 GeV and the invariant mass (rest mass) of the two leptons is required to begreater than 20 GeV In order to reduce the number of background events containing twoleptons produced by the on-shell decay of the Z boson the invariant mas must be outside711-111 GeV range

          Two additional selections are applied to reduce the number of background events withhigh mT 2 arising from events with large Emiss

          T due to mismeasured jets ∆φb lt 15 and∆φ gt 10 The quantity ∆φb is the azimuthal angle between pll

          T b = pmissT + pl1

          T +Pl2T The

          33 Calculations 49

          vector pllT b is the opposite of the vector sum of all the hadronic activity in the event For WW

          and tt backgrounds it measures the transverse boost of the WW system and for the signalthe transverse boost of the chargino-chargino system The ∆φ variable is the azimuthal anglebetween the pmiss

          T vector and the direction of the closest jet

          By requiring mT 2 gt 40 GeV the tt background is the dominant background in both thesame-flavoured (SF) and different-flavoured (DF) events

          Four signal regions are then defined as shown in Table 37 SR M90 has the loosestselection requiring MT 2gt 90 GeV and no additional requirement on jets and providessensitivity to scenarios with a small difference between the masses of the top-squark andchargino SR M110 and M120 have a loose selection on jets requiring two jets with pT

          gt 20 GeV and mT 2 to be greater than 110 GeV and 120 GeV respectively They providesensitivity to scenarios with small to moderate difference between the invariant masses of thestop quark and the charginos SR M100 has a tighter jet slection and provides sensitivity toscenarios with a large mass difference between the both stop quark and the chargino and thestop quark and the neutralino

          The analysis cut regions are summarised in Fig37 below

          Table 37 Signal Regions Lepstop2

          SR M90 M100 M110 M120pT leading lepton gt 25 GeV

          ∆φ(pmissT closest jet) gt10

          ∆φ(pmissT pll

          T b) lt15mT 2 gt90 GeV gt100 GeV gt110 GeV gt120 GeV

          pT leading jet no selection gt100 GeV gt20 GeV gt20 GeVpT second jet no selection gt50 GeV gt20 GeV gt20 GeV

          To validate the code we ran the cuts shown in Appendix 1 Table A3 with the givenMadgraph simulation and compared to [83] It was not clear what baseline cuts had beenemployed in the ATLAS paper so our results were normalised to the 2 SF electrons beforecomparison Some of the ATLAS cut numbers were less than 3 and we did not produce anyevents passing these cuts but clearly the margin of error at this level was greater than thenumber of ATLAS events

          33 Calculations 50

          3334 2bstop

          Search for direct third-generation squark pair production in final states with miss-ing transverse momentum and two b-jets in

          radics= 8 TeV pp collisions with the ATLAS

          detector[31]

          Two sets of scenarios are considered In the first the coloured sbottom is the onlyproduction process decaying via b1 rarr bχ0

          1 In the second the stop is the only productionprocess decaying via t1 rarr t χplusmn

          1 and the lightest chargino (χplusmn1 ) decays via a virtual W boson

          into the final state χ01 f f prime Both scenarios are characterised by the presence of two jets

          resulting from the hadronisation of b quarks and large missing transverse momentum Theproduction of a mediator in association with top quarks followed by an invisible mediatordecay to dark matter particles has a similar final state as that considered in the ATLAS stopsearches Thus we expect the ATLAS reach to have sensitivity to the three dark mattermodels considered in this thesis

          The collider cuts program mimicked object reconstruction by selecting baseline electronswith pT gt 7 GeV and |η |lt 247 and muons with pT gt 6 GeV and |η |lt 24 Baseline jetswere selected with pT gt 20 GeV and |η |lt 28 We then removed any jets within ∆R lt 02of an electron and any electron within ∆R lt 04 of surviving jets Signal electrons andmuons were required to have pT gt 25 GeV b-jets were selected with 60 tagging efficiencyand |η |lt 25 and pT gt 20 GeV

          Two sets of signal regions are defined to provide sensitivity to the kinematic topologiesassociated with the different mass splittings between the sbottom or the stop and the neu-tralino In all cases the presence of at least one primary vertex (with at least five tracks withpT gt 04 GeV) is required Events are selected with Emiss

          T gt 150 GeV and no electrons ormuons identified in the final state For signal region selections jets with |η | lt 28 are orderedaccording to their pT and 2 jets are required to be b-tagged

          The variables are defined as follows

          bull ∆φmin is the minimal azimuthal angle ∆φ between any of the three leading jets andpmiss

          T

          bull me f f (k) = sumki=1(p jet

          T )i +EmissT where the index refers to the pT ordered list of jets

          bull HT3 is defined as the scalar sum of the pT of the n leading pT ordered jets withoutincluding the 3 leading jets HT3 = sum

          ni=4(p jet

          T )i

          bull mbb is the invariant mass of the two b-tagged jets in the event

          33 Calculations 51

          bull mCT is the contransverse mass [82] and is a kinematical variable that can be usedto measure the masses of pair produced semi-invisibly decaying particles For twoidentical decays of heavy particles into two invisible particles v1 and v2 and twoinvisible particles mCT is defined as m2

          CT (v1v2) = [ET (v1)+ET (v2)]2 minus [pT (v1)minus

          pT (v2)]2 where ET =

          radicp2

          T +m2 The the contransverse mass is invariant under equaland opposite boosts of the parent particles in the transverse plane and is bounded fromabove by an analytical expression of the particle masses For production of sbottompairs the bound is given by mmax

          CT =m2(b)minusm2(χ0

          1 )

          m(b) and for tt events the bound is 135

          GeV A similar equation can be written for stop pairs in term of the stop and charginomasses

          A definition of the signal regions is given in the Table38

          Table 38 Signal Regions 2bstop

          Description SRA SRBEvent cleaning All signal regions

          Lepton veto No emicro after overlap removal with pT gt7(6) GeV for e(micro)Emiss

          T gt 150 GeV gt 250 GeVLeading jet pT ( j1) gt 130 GeV gt 150 GeVSecond jet pT ( j2) gt 50 GeV gt 30 GeVThird jet pT ( j3) veto if gt 50 GeV gt 30 GeV

          ∆φ(pmissT j1) - gt 25

          b-tagging leading 2 jets(pT gt 50 GeV |η | lt 25)

          2nd- and 3rd- leading jets(pT gt 30 GeV|η | lt 25)

          ∆φmin gt 04 gt 04Emiss

          T me f f (k) EmissT me f f (2) gt 025 Emiss

          T me f f (3) gt 04mCT gt 150200250300350 GeV -HT3 - lt 50 GeVmbb gt200 GeV -

          The analysis cuts are summarised in Table A4 of Appendix 1

          3335 ATLASMonobjet

          Search for dark matter in events with heavy quarks and missing transverse momentumin pp collisions with the ATLAS detector[30]

          Events were selected with high missing transverse momentum when produced in associa-tion with one or more jets identified as containing b quarks Final states with top quarks wereselected by requiring a high jet multiplicity and in some cases a single lepton The research

          33 Calculations 52

          studied effective contact operators in the form of a scalar operator Oscalar = sumqmqMN

          lowastqqχχ

          where MNlowast is the effective mass of the coupling and the quark and DM fields are denoted by

          q and χ respectively These are the equivalent of the simplified models in this paper with themediator integrated out

          Object reconstruction selected baseline electrons muons jets and τrsquos with pT gt 20 GeVand |η |lt 2525 and 45 respectively Signal jets were selected by removing any jets within∆R lt 02 of an electron and electrons and muons were selected by removing them if theywere within ∆R lt 04 of surviving jets b-jets were tagged with an efficiency of 60

          Only signal regions SR1 and SR2 were analysed

          The analysis split events into signal regions SR1 and SR2 according to the followingCriteria

          Table 39 Signal Region ATLASmonobjet

          Cut SR1 SR2Jet multiplicity n j 1minus2 3minus4

          bminusjet multiplicity nb gt 0 (60 eff) gt 0 (60 eff)Lepton multiplicity nl 0 0

          EmissT gt300 GeV gt200 GeV

          Jet kinematics pb1T gt100 GeV pb1

          T gt100 GeV p j2T gt100 (60) GeV

          ∆φ( j1EmissT ) gt 10 i = 12 gt 10 i = 14

          Where p jiT (pbi

          T ) represent the transverse momentum of the i-th jet b-tagged jet respec-tively If there is a second b-tagged jet the pT must be greater than 60 GeV

          3336 CMSTop1L

          Search for top-squark pair production in the single-lepton final state in pp collisionsat

          radics=8 TeV[41]

          This search was for pair production of top squarks in events with a single isolatedelectron or muon jets large missing transverse momentum and large transverse mass

          (MT =radic

          2EmissT pl

          T (1minus cos(∆φ))) where plT is the leptonrsquos transverse momentum and ∆(φ)

          is the difference between the azimuthal angles of the lepton and EmissT The 3 models

          considered should have some visibility at certain parameter combinations in this regionThe search focuses on two decay modes of the top squark leading to the two processespp rarr t tlowast rarr tt χ0

          1 χ01 rarr bbW+Wminusχ0

          1 χ01 and pp rarr t tlowast rarr bbχ

          +1 χ

          minus1 rarr bbW+Wminusχ0

          1 χ01 The

          33 Calculations 53

          lightest neutralino χ01 is considered the lightest supersymmetric particle and escapes without

          detection The analysis is based on events where one of the W bosons decays leptonicallyand the other hadronically resulting in one isolated lepton and 4 jets two of which originatefrom b quarks The two neutralinos and the neutrino from the W decay can result in largemissing transverse momentom The kinematical end point of MT is lt the mass of the Wboson However for signal events the presence of LSPs in the final state allows MT to exceedthe mass of the W boson The search is therefore for events with large MT The dominantbackground with large MT arises from tt decays where W bosons decay leptonically withone of the leptons not being identified

          To reduce the dominant tt background use was made of the MWT 2 variable defined as

          the minimum mother particle mass compatible with the transverse momenta and mass shellconstraints [84]

          Object reconstruction required electrons and muons to have pT gt 30 and 25 GeV respand |η |lt 25 and 21 respectively Baseline jets were required to have pT gt 30 and |η |lt 4and b-jets were tagged with an efficiency of 75 for |η |lt 24

          Events were chosen if they contained one isolated electron or muon at least 4 jets and atleast one b-tagged jet and Emiss

          T gt 100 GeV MT gt 160 GeV MWT 2 was required to be greater

          than 200 GeV and the smallest azimuthal angle difference between baseline jets and EmissT

          gt12

          Chapter 4

          Calculation Tools

          41 Summary

          Figure 41 Calculation Tools

          The simplified models were implemented in Feynrules [4] and the model files importedinto MicrOmegas v425 which was used to calculate the log-likelihoods of the astrophysicalobservables MicrOmegas was modified to allow only s-channel decays in both relic densityand gamma ray calculations Direct detection constraints were easy to implement for thescalar and vector models by coding the analytical expression for the spin independent WIMP-nucleon scattering cross section but due to the momentum dependence of the WIMP-nucleon

          42 FeynRules 55

          scattering cross section of the Majorana fermion model LUXCalc v101 [79] was modifiedfor this model to implement momentum dependent cross sections [85] We used MicrOmegasv425 to calculate relic density values and to produce gamma ray spectra for comparisonwith the Fermi-LAT results Parameter scans were carried out using MultiNest v310 withthe parameter ranges given in Table 51 and using astrophysical constraints only Oncethese scans had converged a random sample of 900 points was chosen (weighted by theposterior probability) and then used to calculate collider constraints using the MadgraphMonte Carlo generator This included comparison with mediator production and decay limitsfrom the LHC experiments and the reproduction of searches for dark matter plus other finalstate objects which required dedicated C++ code that reproduces the cut and count analysesperformed by CMS and ATLAS

          42 FeynRules

          FeynRules is a Mathematicareg package that allows the calculation of Feynman rules inmomentum space for any QFT physics model The user needs to provide FeynRules withminimal information to describe the new model contained in the model-file This informationis then used to calculate a set of Fenman rules associated with the Lagrangian The Feynmanrules calculated by the code can then be used to implement new physics models into otherexisting tools such as Monte Carlo (MC) generators via a set of interfaces developed andmaintained by the MC authors

          Feynrules can be run with a base model = SM (or MSSM) and user defined extensions tothe base model

          43 LUXCalc

          LUXCalc is a utility program for calculating and deriving WIMP-nucleon coupling limitsfrom the recent results of the LUX direct search dark matter experiment The program hasbeen used to derive limits on both spin-dependent and spin-independent WIMP-nucleoncouplings and it was found under standard astrophysical assumptions and common otherassumptions that LUX excludes the entire spin-dependent parameter space consistent witha dark matter interpretation of DAMArsquos anomalous signal and also found good agreementwith published LUX results [79]

          We modified the program for the Majorana fermion model to implement momentumdependent cross -sections [85]

          44 Multinest 56

          44 Multinest

          Multinest is an efficient Bayesian Inference Tool which uses a simultaneous ellipsoidalnested sampling method a Monte Carlo technique aimed at an efficient Bayesian Evidencecalculation with posterior inferences as a by product

          Bayes theorem states that

          Pr(θ |DH) =Pr(D|θ H)Pr(θ |H)

          Pr(D|H) (41)

          Here D is the data and H is the Hypothesis Pr(θ |DH) is the posterior probability ofthe parameters Pr(D|θ H) = L(θ) is the likelihood Pr(θ |H) is the prior and Pr(D|H) isthe evidence

          The evidence Pr(D|H) =int

          Pr(θ |DH)Pr(θ |H)d(θ) =int

          L(θ)Pr(θ |H)d(θ) Thus ifwe define the prior volume dX = Pr(θ |H)d(θ) and

          X(λ ) =int

          L(θ)gtλ

          Pr(θ |H)d(θ) (42)

          where the integral extends over the region of parameter space within the iso likelihoodcontour L(θ) = λ then the evidence integral can be written

          int 10 L (X)dX where L (X) the

          inverse of equation 42 and is a monotonically decreasing function of X (the prior volumewhich is 1 for the total sample space and 0 for a single point)

          Thus if one can evaluate likelihoods L (Xi) where 0 lt Xn lt lt X2 lt X1 lt X0 = 1the evidence can be estimated as sumLiwi where wi = 5(Ximinus1 minusXi+1) Ellipsoidal samplingapproximates the iso likelihood contour by a D dimensional covariance matrix of the currentset of active points New points are selected from the prior within this ellipsoidal bounduntil one is obtained that has a likelihood greater than the likelihood of the removed lowestlikelihood point This method is extended for multimodal sampling where there may bemore than one local maximum by clustering points that are well separated and constructingindividual ellipsoidal bounds for each cluster For more details consult [7]

          The results of the analysis are expressed as a data file containing the sample probability-2loglikelihood followed by the parameters of the model for the particular sampling Theprobabilities are normalised to total 10

          45 Madgraph 57

          45 Madgraph

          Madgraph is a general purpose matrix element calculator and Monte Carlo generator Basedon user-supplied Feynman rules (eg the output of Feynrules) the package calculates the tree-level matrix elements for any processes that the user specifies and generates samples of LHCevents including an accurate simulation of the proton collisions themselves (which requires aknowledge of the parton distribution functions that determine the relative proportion of eachparton inside the nucleus) Hadronisation of final state quarks is not possible in Madgraphand hence the code must be used in conjunction with Pythia which performs a parton shower

          The Madgraph output was automatically showered by Pythia (Pythia describes partonradiation as successive parton emissions using Markov chain techniques based on Sudakovform factors and this description is only formally correct for the soft and collinear emissions)A Madgraph simulation is at the level of the fundamental particles (quarks and leptons)Madgraph generates a limited set of matrix element processes at LO representing hard (highenergy) emissions and passes these to Pythia for showering The final state partons in theMadgraph simulation must be matched to the Pythia final state partons to avoid doublecounting

          The output from Pythia is automatically fed through Delphes where basic collider cutscan be performed (calculation of the numbers of particles and jets that would be observed atdifferent energies and locations within the collider) The output of the Delphes simulation is afile containing reconstructed events that can be analysed using the standard ROOT framework(the most popular data analysis package in particle physics) [86]

          The parameters that gave the best fits to the direct [16] and indirect DM experiments [87]and relic density [59] are ranked by posterior probability from the Multinest [7] output

          The parameters in the Madgraph simulations are stored in files in the cards sub directoryand in particular the run card param card and Delphes card record items relevent to thesimulation

          In the run card the number of events the Beam 1 and 2 total energy in GeV (4000 each)and the type of jet matching were modified (to MLM type matching where the final statepartons in Madgraph are clustered according to the KT jet algorithm to find the equivalentparton shower history of the event there is a minimum cutoff scale for the momentum pT

          given by the xqcut parameter) The maxjetflavour (=5 to include b quarks) and xqcut = 30GeV parameters were also modified

          46 Collider Cuts C++ Code 58

          The param card was modified to specify the masses of the new particles and the autodecay modes The Delphes card is overwritten with either the ATLAS or CMS defaultcards which record tracking efficiencies and isolation momentum and energy resolutionscalorimeter details jet finder and b and τ -tagging parameters However the isolation andtagging in this study were done by the collider cuts C++ code since multiple sets of cutswere applied to the same output from Pythia

          When performing simulations of the three dark matter models it was necessary to definethe proton and jets to contain b quarks as this is not the default in Madgraph The processesgenerated included 1 and 2 jets

          46 Collider Cuts C++ Code

          Because of the volume of simulations required for the collider constraints we used C++code (originally developed by Martin White and Sky French and modified by me) that readsthe ROOT files and outputs the numbers of events that would have been observed under thechosen collider experiments had the model been true for the given parameters Thus the sameoutput files can be used for all of the collider cuts and hence no cuts were performed throughthe Delphes cards

          In order to validate the C++ code we replicated the observed events with the papersdescribing the various collider searches (see Appendix)

          Chapter 5

          Majorana Model Results

          51 Bayesian Scans

          To build some intuition we first examined the constraining effect of each observable on itsown We ran separate scans for each constraint turned on and then for all the constraintstogether In Fig 51 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together The paramaters were scanned over the range in Table 51 below

          Table 51 Scanned Ranges

          Parameter mχ [GeV ] mS[GeV ] λt λb λτ

          Range 0minus103 0minus103 10minus5 minus10 10minus5 minus10 10minus5 minus10

          In Fig51 we can see that the preferred DM mass is centered around 35 GeV primarily dueto the influence of the Fermi-LAT gamma ray constraint whilst the relic density constraintallows a wide range of possible mediator and DM masses Direct detection (LUX) constraintssplit the mass regions into a top left and bottom right region and together these split theoverall mass parameter regions into two regions by mediator mass but with DM mass around35 GeV Later studies using a lower excess gamma ray spectrum tend to increase the preferredDM mass up to 50 GeV [76] and [88]

          The marginalised posterior probabilities for all combinations of parameters taken twoat a time are shown in Fig52 These plots have all constraints (Gamma LUX and RelicDensity) applied together The idea of these graphs is to study the posterior probabilities ofcombinations of two paramaters taken at a time (the off-diagonal graphs) The diagonal plotsshow marginalised histograms of a single parameter at a time- this gives an indication of thespread of each parameter The plots show that mS and λb show a high degree of correlationfor mS gt 100 GeV but there is also a range of values of mS around 35 GeV which are

          51 Bayesian Scans 60

          1 0 1 2 3 4log10(mχ)[GeV]

          1

          0

          1

          2

          3

          4

          log 1

          0(m

          s)[GeV

          ]

          (a) Gamma Only

          1 0 1 2 3 4log10(mχ)[GeV]

          1

          0

          1

          2

          3

          4

          log 1

          0(m

          s)[GeV

          ]

          (b) Relic Density

          1 0 1 2 3 4log10(mχ)[GeV]

          1

          0

          1

          2

          3

          4

          log 1

          0(m

          s)[GeV

          ]

          (c) LUX

          05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

          05

          00

          05

          10

          15

          20

          25

          30

          35

          log 1

          0(m

          s)[GeV

          ]

          (d) All Constraints

          Figure 51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether

          51 Bayesian Scans 61

          00

          05

          10

          15

          20

          25

          30

          log 1

          0(m

          χ)[GeV

          ]

          00

          05

          10

          15

          20

          25

          30

          ms[Gev

          ]

          5

          4

          3

          2

          1

          0

          1

          log 1

          0(λ

          t)

          5

          4

          3

          2

          1

          0

          1

          log 1

          0(λ

          b)

          00 05 10 15 20 25 30

          log10(mχ)[GeV]

          5

          4

          3

          2

          1

          0

          1

          log 1

          0(λ

          τ)

          00 05 10 15 20 25 30

          ms[Gev]5 4 3 2 1 0 1

          log10(λt)5 4 3 2 1 0 1

          log10(λb)5 4 3 2 1 0 1

          log10(λτ)

          Figure 52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter

          52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 62

          possible for λb around 001 λt and λτ show a wide range of possible values for two possibleranges of mS lt 35 GeV and greater than 100 GeV mχ shows a narrow range of possiblevalues around 35 GeV for all values of the other parameters

          52 Best fit Gamma Ray Spectrum for the Majorana FermionDM model

          We can see from the Fig51(a) scan that the maximum log-likelihood Fermi-LAT gammaray spectrum on its own shows values that centre around 35 GeV This is not changed byadding a relic density constraint or a direct detection (LUX) constraint

          The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraintsis shown in Fig53 The parameters for this point were

          Table 52 Best Fit Parameters

          Parameter mχ [GeV ] mS[GeV ] λt λb λτ

          Value 3332 49266 0322371 409990 0008106

          10-1 100 101 102

          E(GeV)

          10

          05

          00

          05

          10

          15

          20

          25

          30

          35

          E2dφd

          E(G

          eVc

          m2ss

          r)

          1e 6

          Best fitData

          Figure 53 Gamma Ray Spectrum

          The relic density (Ωh2) at this point has a value of 0135 but the point is separated onlyslightly in posterior probability from points with a value of 01199 with mS lt 10 GeV (iethe posterior probability is almost the same differing by less than 1)

          To better understand the interplay of the different constraints the three graphs in Fig54show the log likelihood surfaces and individual best fit points (best log likelihoods) for the γ

          and Ω scans seperately and then combined The diagrams each choose a pair of couplings(fixing the remaining parameters) and display the 2d surfaces of minuslog(χ2) (proportional to

          52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 63

          the log likelihoods of the coupling combinations) obtained by varying the pair of couplingsThese graphs do not show marginalised posterior probabilities but rather are based on the-2log likelihood calculations directly from MicrOmegas for the given coupling combinationsWe have termed this value as χ2 for convenience and used the log of this value to give abetter range for the colours These log likelihood values are what drive the MultiNest searchof parameter space for the best parameter combinations

          The graphs show a complex interplay of the different constraints resulting often in acompromise parameter combination The arrow on each graph points to the maximum point We have chosen a single representative set of couplings near the best overall fit point for allthe graphs

          52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 64

          00 05 10 15 20 25 30

          log10(mχ)

          00

          05

          10

          15

          20

          25

          30

          log

          10(m

          S)

          Max

          minuslog10(χ2(Γ)) λt = 487 λτ = 024 λb = 0344

          16

          14

          12

          10

          8

          6

          4

          2

          0

          γ Maximum at mχ=416 GeV mS=2188 GeV

          00 05 10 15 20 25 30

          log10(mχ)

          00

          05

          10

          15

          20

          25

          30

          log

          10(m

          S)

          Max

          minuslog10(χ2(Omega)) λt = 487 λτ = 024 λb = 0344

          28

          24

          20

          16

          12

          08

          04

          00

          04

          Ω Maximum at mχ=363 GeV mS=1659 GeV

          00 05 10 15 20 25 30

          log10(mχ)

          00

          05

          10

          15

          20

          25

          30

          log

          10(m

          S)

          Max

          minuslog10(χ2(Both)) λt = 487 λτ = 024 λb = 0344

          16

          14

          12

          10

          8

          6

          4

          2

          0

          Both Maximum at mχ=8317 GeV mS=2884GeV The best fit point for these couplings maybe above below or between the individual bestfit points but will on average be between thepoints

          Figure 54 Plots of log likelihoods by individual and combined constraints Masses in GeV

          53 Collider Constraints 65

          53 Collider Constraints

          531 Mediator Decay

          1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof τ leptons in pp collisions [44] Refer also to Chapter 3 for details

          We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

          The results of the calculations are shown in the graph Fig55 and can be compared withthe limits in the table from the paper [44] in Fig 56 The black line is the observed ATLASlimit

          Figure 55 σ lowastBr(σ rarr ττ) versus Mass of Scalar

          0 200 400 600 800

          mS[GeV]

          10

          5

          0

          log 1

          0(σ

          (bbS

          )lowastB

          (Sgtττ

          ))[pb]

          Observed LimitLikely PointsExcluded Points

          0

          20

          40

          60

          80

          100

          120

          0 5 10 15 20 25 30 35 40 45

          We can see that only one point here exceeded the limit and the points showed a widespread over σ(bbS)lowastBr(σ rarr ττ) by mS space which is consistent with the spread of pointson the MultiNest scan on the mS by λτ plane causing a spread of branching ratios to the τ

          quark It may seem odd that only one point was excluded here but this point was sampledwith a very low probability having high coupling constants of 597 to the τ quark and 278 tothe t quark and only just exceeds the limit As such it can be explained as a random outlier

          53 Collider Constraints 66

          Figure 56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar

          2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

          This scan calculates the contributions of b-quark associated Higgs boson productionreproduced from [43] Refer to Chapter 3 for details

          We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

          The cross section times branching ratios to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig57 The black line is the CMSLimit

          Figure 57 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar

          0 200 400 600 800

          mS[GeV]

          15

          10

          5

          0

          5

          log

          10(σ

          (bS

          +X

          )lowastB

          (Sgt

          bb))

          [pb]

          Observed LimitLikely PointsExcluded Points

          0

          20

          40

          60

          80

          100

          120

          0 50 100 150 200 250

          53 Collider Constraints 67

          The results of this scan were compared to the limits in [89] with the plot shown inFig58

          Figure 58 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

          We can see that there are a number of exclusions across the range of mS which occuredfor high values of the coupling constants gt 1 for λbλt λtau - these can be seen also on Fig59The excluded parameter combinations represent less than 2 of the total sampled number

          532 Collider Cuts Analyses

          We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude and the results are shown against theposterior probabilities of the parameter combinations in Fig 59

          The circles in Figure 59 show which points were excluded by the collider cuts as well asthe limits on cross sections times branching ratios to each of τ+τminus and bb

          All the excluded points under the heading Collider Cuts showed exclusion by the Lepstop0experiment (Search for direct pair production of the top quark in all hadronic final states

          53 Collider Constraints 68

          0 1 2 3

          log10(mχ)[GeV]

          0

          1

          2

          3

          log 1

          0(m

          s)[GeV

          ]Collider Cuts

          σ lowastBr(σgt bS+X)

          σ lowastBr(σgt ττ)

          (a) mχ by mS

          6 5 4 3 2 1 0 1 2

          log10(λt)

          0

          1

          2

          3

          log 1

          0(m

          s)[GeV

          ](b) λt by mS

          5 4 3 2 1 0 1

          log10(λb)

          6

          5

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          2

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          0(λ

          t)

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          5 4 3 2 1 0 1

          log10(λb)

          6

          5

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          2

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          0(λ

          τ)

          (d) λb by λτ

          6 5 4 3 2 1 0 1 2

          log10(λt)

          6

          5

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          3

          2

          1

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          log 1

          0(λ

          τ)

          (e) λt by λτ

          5 4 3 2 1 0 1

          log10(λb)

          0

          1

          2

          3

          log 1

          0(m

          s)[GeV

          ]

          (f) λb by mS

          Figure 59 Excluded points from Collider Cuts and σBranching Ratio

          53 Collider Constraints 69

          [32]) Some of these same points were also excluded by Lepstop2 2bstop CMSTop andATLASMonobjet experiments There were expected events at other points but none weresignificant at the 95 exclusion level These significant exclusions occurred mainly wherethe mass of the mediator was over 100 GeV and higher values of the couplings to b and tquarks but there were a few points with mediator mass around 25 GeV The exclusions withmediator mass around 25 GeV were all accompanied by couplings to t quarks of order 10which is probably non-perturbative

          Looking at the exclusions in Fig 59 we can see that these occur mainly in the regionof higher mediator mass gt 100 GeV and lt 400 GeV These exclusions come from theastrophysical constraints (Gamma) which show a strong correlation between λb and mS andhigh values of λt gt 1 However these mass regions do not overlap at the same couplingcombinations as the high likelihood region The green and white circles correspond to theexcluded points in Figs 57 and 58 respectively

          The remaining regions of highest likelihood can be seen to be where 0001 λb lt 01accompanied by λt lt 0001 and mass of the mediator mS lt 10 GeV or 400 GeV lt mS lt1000GeV with mχ sim 35 GeV

          The plot of λt by λb shows the significant regions do not overlap with the collider cutsand cross section constraints since the latter occur at high values of either coupling while thelikely region has low values of these couplings (also accompanied by low couplings to λτ )

          Chapter 6

          Real Scalar Model Results

          61 Bayesian Scans

          To build some intuition we first examined the constraining effect of each observable on itsown We ran seperate scans for each constraint turned on and then for all the constraintstogether In Figure 61 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together

          In Figure 61 we can see that there are is a narrow region which dominates when allconstraints are taken into account This centers on mχ=9 GeV and a range of values for mS

          from 35 GeV to 400 GeV The region is not the highest likelihood region for the Fermi-LATgamma ray spectrum on its own (as can be seen from the individual plot - there is a narrowband to the left of the highest likelihood band) nor for the LUX likelihood or relic densitybut represents a compromise between the three constraints

          The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig62 The diagonal plots show marginalised histograms of a singleparameter at a time

          61 Bayesian Scans 71

          05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

          05

          00

          05

          10

          15

          20

          25

          30

          35

          log 1

          0(m

          s)[GeV

          ]

          (a) Gamma Only

          05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

          05

          00

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          25

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          35

          log 1

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          s)[GeV

          ]

          (b) Relic Density

          05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

          05

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          log 1

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          s)[GeV

          ]

          (c) LUX

          05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

          05

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          log 1

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          s)[GeV

          ]

          (d) All Constraints

          Figure 61 Real Scalar Dark Matter - By Individual Constraint and All Together

          61 Bayesian Scans 72

          00

          05

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          20

          25

          30

          log 1

          0(m

          χ)[GeV

          ]

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          ms[Gev

          ]

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          t)

          5

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          b)

          00 05 10 15 20 25 30

          log10(mχ)[GeV]

          5

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          1

          log 1

          0(λ

          τ)

          00 05 10 15 20 25 30

          ms[Gev]5 4 3 2 1 0 1

          log10(λt)5 4 3 2 1 0 1

          log10(λb)5 4 3 2 1 0 1

          log10(λτ)

          Figure 62 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter

          62 Best fit Gamma Ray Spectrum for the Real Scalar DM model 73

          62 Best fit Gamma Ray Spectrum for the Real Scalar DMmodel

          We can see from Fig61 that the maximum log-likelihood Fermi-LAT Gamma ray spectrumon its own shows values of mχ that centre around 35 GeV However the LUX constraintin particular moves the best fit to a point around 9 GeV- the tight LUX constraint onlydisappears for low masses and at that point the proportion of τ+τminus annihilation needs toincrease to fit the constraints

          The gamma ray spectrum at the points of best fit for Fermi-LAT including all constraintsis shown in Fig63 The parameters for these points were respectively and involve decays tomainly τ and b quarks respectively

          Table 61 Best Fit Parameters

          Parameter mχ [GeV ] mS[GeV ] λt λb λτ

          Value 932 3526 000049 0002561 000781

          10-1 100 101 102

          E(GeV)

          10

          05

          00

          05

          10

          15

          20

          25

          30

          35

          E2dφdE

          (GeVc

          m2ss

          r)

          1e 6

          Best fitData

          Figure 63 Gamma Ray Spectrum

          This is not a particularly good fit and it occcurs at a relic density Ωh2 of 335 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

          63 Collider Constraints 74

          63 Collider Constraints

          631 Mediator Decay

          1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44]

          We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

          The results of the scan are shown in the graph Fig64 and can be compared with thelimits in the table from the paper [44] in Fig 56The black line is the observed ATLAS limit

          Figure 64 σ lowastBr(σ rarr ττ) versus Mass of Scalar

          0 200 400 600 800

          mS[GeV]

          8

          6

          4

          2

          0

          2

          4

          log 1

          0(σ

          (bbS

          )lowastB

          (Sgtττ

          ))[pb]

          Observed LimitLikely PointsExcluded Points

          050

          100150200250300350

          0 10 20 30 40 50 60

          We can see that there is a narrow band in the region of interest in the σ(bbS)lowastB(S gt ττ)

          by mS plane due to the high degree of correlation between mS and λτ seen in the likelihoodscan Values for mS gt 300Gev are excluded

          2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

          We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinations

          63 Collider Constraints 75

          randomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

          The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig65 The black line is the CMSlimit

          Figure 65 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

          0 200 400 600 800

          mS[GeV]

          8

          6

          4

          2

          0

          2

          4

          log

          10(σ

          (bS

          +X

          )lowastB

          (Sgt

          bb))

          [pb]

          Observed LimitLikely PointsExcluded Points

          050

          100150200250300350

          0 10 20 30 40 50 60

          The results of this scan were compared to the limits in [89] with the plot shown inFig58

          We can see a similar shape for the region of interest to that in Fig64 due to the highcorrelation between mS and λb over the region preferred by astrophysical constraints resultingin branching ratios highly correlated to ms Only one point is excluded with mS gt 500GeV

          632 Collider Cuts Analyses

          We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expected

          63 Collider Constraints 76

          with 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 66

          We can see that collider cuts show some exclusions only for masses mχ below the highlikelihood region but these exclusions do not occur at the same λb and λt combinations asseen from the plot of these two couplings except for a small region at the highest mS andcouplings

          All but one of the collider cut exclusions were excluded by the Lepstop0 experiment (thiswas excluded by 2bstop with high coupling to b quarks of 73) A number of the excludedpoints were also excluded by Lepstop2 2bstop CMSTopDM1L and ATLASMonobJet-(around 30 of the total for each)

          63 Collider Constraints 77

          0 1 2 3

          log10(mχ)[GeV]

          0

          1

          2

          3

          log 1

          0(m

          s)[GeV

          ]Collider Cuts

          σ lowastBr(σgt bS+X)

          σ lowastBr(σgt ττ)

          (a) mχ by mS

          5 4 3 2 1 0 1

          log10(λt)

          0

          1

          2

          3

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          0(m

          s)[GeV

          ](b) λt by mS

          5 4 3 2 1 0 1

          log10(λb)

          5

          4

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          2

          1

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          1

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          0(λ

          t)

          (c) λb by λt

          5 4 3 2 1 0 1

          log10(λb)

          6

          5

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          2

          1

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          0(λ

          τ)

          (d) λb by λτ

          5 4 3 2 1 0 1

          log10(λt)

          6

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          1

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          0(λ

          τ)

          (e) λt by λτ

          5 4 3 2 1 0 1

          log10(λb)

          0

          1

          2

          3

          log 1

          0(m

          s)[GeV

          ]

          (f) λb by mS

          Figure 66 Excluded points from Collider Cuts and σBranching Ratio

          Chapter 7

          Real Vector Dark Matter Results

          71 Bayesian Scans

          In the Figure 71 we can see that the preferred DM mass is centered around 35 GeV primarilydue to the influence of the Fermi-LAT gamma ray constraint with relic density having a widerange of possible mediator and DM masses The result of combining all the constraints is todefine a small region around mχ = 8 GeV and mS = 20 GeV which dominates the posteriorprobability

          The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig72 The diagonal plots show marginalised histograms of a singleparameter at a time

          71 Bayesian Scans 79

          05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

          1

          0

          1

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          0(m

          s)[GeV

          ]

          (a) Gamma Only

          05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

          05

          00

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          s)[GeV

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          1 0 1 2 3 4log10(mχ)[GeV]

          05

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          s)[GeV

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          (c) LUX

          05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

          05

          00

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          log 1

          0(m

          s)[GeV

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          (d) All Constraints

          Figure 71 Real Vector Dark Matter - By Individual Constraint and All Together

          71 Bayesian Scans 80

          00

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          χ)[GeV

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          ]

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          t)

          5

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          b)

          00 05 10 15 20 25 30

          log10(mχ)[GeV]

          5

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          1

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          log 1

          0(λ

          τ)

          00 05 10 15 20 25 30

          ms[Gev]5 4 3 2 1 0 1

          log10(λt)5 4 3 2 1 0 1

          log10(λb)5 4 3 2 1 0 1

          log10(λτ)

          Figure 72 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter

          72 Best fit Gamma Ray Spectrum for the Real Vector DM model 81

          72 Best fit Gamma Ray Spectrum for the Real Vector DMmodel

          The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraints isshown in Fig73

          Table 71 Best Fit Parameters

          Parameter mχ [GeV ] mS[GeV ] λt λb λτ

          Value 8447 20685 0000022 0000746 0002439

          10-1 100 101 102

          E(GeV)

          10

          05

          00

          05

          10

          15

          20

          25

          30

          35

          E2dφdE

          (GeVc

          m2s

          sr)

          1e 6

          Best fitData

          Figure 73 Gamma Ray Spectrum

          This is not a particularly good fit and it occcurs at a relic density Ωh2 of 025 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

          73 Collider Constraints

          731 Mediator Decay

          1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44] (See Chapter 3 for more details)

          We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

          73 Collider Constraints 82

          The results of the scan are shown in the graph Fig74 and can be compared with thelimits in the table from the paper [44] in Fig 56

          Figure 74 σ lowastBr(σ rarr ττ) versus Mass of Scalar

          0 200 400 600 800

          mS[GeV]

          8

          6

          4

          2

          0

          2

          log 1

          0(σ

          (bbS

          )lowastB

          (Sgtττ

          ))[pb]

          Observed LimitLikely PointsExcluded Points

          0100200300400500600700800

          0 20 40 60 80 100120140

          We can see that the region of highest likelihood is limited to mS lt 100 GeV and a narrowrange of couplings to λτ giving a narrow region of interest which falls well below the limit

          2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

          We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

          The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig75 The black line is the CMSlimit

          The results of this scan were compared to the limits in [89] with the plot shown in Fig58

          We can see that again the region of interest for mS is below 100 GeV and the points fallwell below the limit

          73 Collider Constraints 83

          Figure 75 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

          0 200 400 600 800

          mS[GeV]

          8

          6

          4

          2

          0

          2

          4

          log

          10(σ

          (bS

          +X

          )lowastB

          (Sgt

          bb))

          [pb]

          Observed LimitLikely PointsExcluded Points

          0100200300400500600700800

          0 20 40 60 80 100120140

          732 Collider Cuts Analyses

          We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 76

          We can see from Figure 76 that like the real scalar model the region of highest likelihoodhas λb and λt lt 001 while the excluded points from collider cuts and cross sections areabove 01 Most of the excluded points have mχ gt than the values in the region of highestlikelihood and all points are widely separated in parameter space from this region

          Once again 89 of excluded points were excluded by Lepstop0 with many also beingexcluded by the other experiments (50 by Lepstop2 and 56 by CMSTopDM1L 35 byLepstop2 and 20 by ATLASMonobjet)

          73 Collider Constraints 84

          0 1 2 3

          log10(mχ)[GeV]

          0

          1

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          0(m

          s)[GeV

          ]Collider Cuts

          σ lowastBr(σgt bS+X)

          σ lowastBr(σgt ττ)

          (a) mχ by mS

          5 4 3 2 1 0 1

          log10(λt)

          0

          1

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          0(m

          s)[GeV

          ](b) λt by mS

          5 4 3 2 1 0 1

          log10(λb)

          5

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          1

          0

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          0(λ

          t)

          (c) λb by λt

          5 4 3 2 1 0 1

          log10(λb)

          5

          4

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          τ)

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          5 4 3 2 1 0 1

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          5

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          τ)

          (e) λt by λτ

          5 4 3 2 1 0 1

          log10(λb)

          0

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          3

          log 1

          0(m

          s)[GeV

          ]

          (f) λb by mS

          Figure 76 Excluded points from Collider Cuts and σBranching Ratio

          Chapter 8

          Conclusion

          We first conducted Bayesian scans over mass and coupling parameters for three simplifiedmodels - Majorana fermion real scalar and real vector using direct detecion (LUX) indirectdetection (GCE γ rays) and relic density constraints to find the regions of highest likelihoodWe ran these scans one constraint at a time and then together to better understand the interplayof the different constraints on best fit and likely parameter regions

          We also developed and tested C++ collider cut software specifically designed to mimickthe cuts applied in a number of ATLAS and CMS experiments that searched for Emiss

          T inconnection with specific beyond the standard model (BSM) theories of Dark Matter

          We then selected parameter combinations by randomly sampling the scanned resultsbased on the posterior probability distribution of the points and applied collider cuts andcross section times branching ratio calculations to these points to see if there would havebeen any sensitivity from the ATLAS and CMS experiments

          We found that the Majorana fermion model showed the best overall fit to all the constraintswith relic density close to the experimental value of 01199 and a low χ2 error on the gammaspectrum fit The preferred mass mχ was around 35 GeV and mS ranging from lt10 GeV to500 GeV The smaller masses give a better fit for relic density and also are well separatedfrom current collider constraints mainly due the the fact that the exclusions occur mainlywith higher couplings to t quarks (greater than 1) which is consistent with the fact that manyof the experiments were targeting dark matter pair production associated with t quark decaysNevertheless there were some exclusions that did fall in the region sampled by randomselection from the posterior probability distribution

          The preferred parameter regions were also consistent with the observation in [20] thatthe Majorana model showed the best overall fit with similar mass constraints We find thatthe couplings to the SM fermions should all be less that 001

          86

          The scalar and vector DM models give slightly less good fits especially to the currentrelic density estimate of 01199 but again the current collider constraints are well separatedfrom the best fit region and in fact none of the points randomly sampled from the posteriordistribution showed any exclusion The best fit parameter combination favours couplings toSM fermions of less than 001

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          Appendix A

          Validation of Calculation Tools

          Lepstop0Search for direct pair production of the top squarkin all hadronic final states in proton=proton collisions atradic

          s=8 TeV with the ATLAS detector [32]

          Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1) mt1=600 GeV mn1=1 GeVWhere t1 is the top squark and n1 is the neutralino of the MSSM

          94

          Table A1 0 Leptons in the final state

          Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 10000 460 460 100Emiss

          T gt 130GeV 8574 395 451 88Leptonveto 6621 307 285 108Emiss

          T gt 150GeV 6316 293 294 100Jet multiplicityand pT 1099 48 51 94|∆φ( jet pmiss

          T )|gt π5 902 40 42 96gt= 2b jets 396 19 18 106τ veto 396 19 18 106mbmin

          T gt 175GeV 270 13 13 105SRA1 79 4 4 104SRA2 70 3 3 93SRA3 67 3 3 97SRA4 52 2 2 91SRC exactly5 jets 1297 50 60 83SRC |∆φ( jet pmiss

          T )|gt π5 1082 50 50 99SRC gt= 2b jets 430 20 20 97SRC τ veto 430 20 20 97SRC |∆φ(bb)|gt 02π 367 17 17 99SRC1 246 12 11 105SRC2 177 9 8 116SRC3 152 8 7 116

          95

          Lepstop1Search for direct top squark pair production infinal states with one isolated lepton jets and missing trans-verse momentum in

          radics = 8 TeV pp collisions using 21 f bminus1

          of ATLAS data[33]

          Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1)(t1 gt b x+1 )(t1 gt b xminus1 )mt1=650 GeV mn1=1 GeV Where t1 is the top squark n1 is the neutralino xplusmn1 are thecharginos

          96

          Table A2 1 Lepton in the Final state

          Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 100000 99990 99990 100Electron(= 1signal) 11558 11557 11785 984 jets(80604025) 5187 5186 6271 83gt= 1btag 4998 4997 5389 93Emiss

          T gt 100GeV [allSRs] 4131 4131 4393 94Emiss

          T radic

          HT gt 5 [allSRs] 4026 4026 4286 94∆φ( jet2 pmiss

          T )gt 08 [allSRs] 3565 3565 3817 93∆φ( jet1Pmiss

          T )gt 08 [not SRtN2] 3446 3446 3723 93Emiss

          T gt 200GeV (SRtN2) 2027 2027 2186 93Emiss

          T radic

          HT gt 13(SRtN2) 1156 1156 1172 99mT gt 140GeV (SRtN2) 839 839 974 86Emiss

          T gt 275GeV (SRtN3) 948 948 965 98Emiss

          T radic

          HT gt 11(SRtN3) 932 932 937 99mT gt 200GeV (SRtN3) 456 456 566 81Emiss

          T gt 150GeV (SRbC1minusSRbC3) 2803 2803 3080 91Emiss

          T radic

          HT gt 7(SRbC1minusSRbC3) 2742 2742 3017 91mT gt 12GeV (SRbC1minusSRbC3) 1917 1917 2347 82Emiss

          T gt 160GeV (SRbC2SRbC3) 1836 1836 2257 81Emiss

          T radic

          HT gt 8(SRbC2SRbC3) 1788 1788 2175 82me f f gt 550GeV (SRbC2) 1741 1741 2042 85me f f gt 700GeV (SRbC3) 1075 1075 1398 77Muon(= 1signal) 11379 11378 11975 954 jets(80604025) 5264 5263 6046 87gt= 1btag 5101 5100 5711 89Emiss

          T gt 100GeV [allSRs] 4199 4199 4770 88Emiss

          T radic

          HT gt 5 [all SRs] 4115 4115 4653 88∆φ( jet2 pmiss

          T )gt 08 [all SRs] 3622 3622 4077 89∆φ( jet1 pmiss

          T )gt 08 [not SRtN2] 3515 3515 3970 89Emiss

          T gt 200GeV (SRtN2) 2136 2136 2534 84Emiss

          T radic

          HT gt 13(SRtN2) 1261 1261 1441 87mT gt 140GeV (SRtN2) 928 928 1033 90Emiss

          T gt 275GeV (SRtN3) 1036 1036 1274 81Emiss

          T radic

          HT gt 11(SRtN3) 1018 1018 1235 82mT gt 200GeV (SRtN3) 497 497 612 81Emiss

          T gt 150GeV (SRbC1minusSRbC3) 2891 2891 3380 86Emiss

          T radic

          HT gt 7(SRbC1minusSRbC3) 2821 2821 3290 86mT gt 120GeV (SRbC1minusSRbC3) 1977 1977 2290 86Emiss

          T gt 160GeV (SRbC2SRbC3) 1913 1913 2220 86Emiss

          T radic

          HT gt 8(SRbC2SRbC3) 1863 1863 2152 87me f f gt 550GeV (SRbC2) 1826 1826 2116 86me f f gt 700GeV (SRbC3) 1131 1131 1452 78

          97

          Lepstop2Search for direct top squark pair production infinal states with two leptons in

          radics =8 TeV pp collisions using

          20 f bminus1 of ATLAS data[83][34]

          Simulated in Madgraph with p p gt t1 t1(t1 gt b w+ n1)(t1 gt b wminusn1) mt1=180 GeV mn1=60 GeV Where t1 is the top squark n1 is the neutralino b1 is the bottom squark

          Table A3 2 Leptons in the final state

          Cut RawMe Scaled ATLAS MeATLAS PercentNoCuts 100000 324308 199000 1632BaselineCuts 3593 11652 7203 1622SF Signal Leptons 1699 5510 5510 1002OSSF Signal Leptons 1632 5293 5233 101mll gt 20Gev 1553 5037 5028 100leadinglepton pT 1479 4797 5048 95|mll minusmZ|gt 20Gev 1015 3292 3484 94∆φmin gt 10 850 2757 2160 128∆φb lt 15 680 2205 1757 126SRM90 [SF ] 13 42 60 70SRM120 [SF ] 0 0 19 0SRM100 +2 jets [SF ] 0 0 17 0SRM110 +2 jets [SF ] 0 0 1 02DF Signal Leptons 1737 5633 7164 792OSDF Signal Leptons 1670 5416 5396 100mll20Gev 1587 5147 5136 100leadinglepton pT 1501 4868 4931 99∆φmin gt 10 1247 4044 3122 130∆φb lt 15 1010 3276 2558 128SRM90 [SF ] 21 68 95 72SRM120 [SF ] 0 0 23 0SRM100 +2 jets [SF ] 0 0 19 0SRM110 +2 jets [SF ] 0 0 14 0

          98

          2bstopSearch for direct third-generation squark pair pro-duction in final states with missing transverse momentumand two b-jets[31]

          Simulated in Madgraph with p p gt b1 b1(b1 gt b n1)(b1 gt b n1)

          SRAmb1=500 GeV mn1=1 GeV SRBmb1=350 GeV mn1=320 GeVWhere b1 is the bottom squark and n1 is the neutralino

          Table A4 2b jets in the final state

          Cut RawMe Scaled ATLAS MeATLAS PercentSRA Nocuts 100000 173800 1738 100SRA MET gt 80 93407 162341 1606 101SRA Leptonveto 93302 162159 1505 108SRA MET gt 150 79449 138082 1323 104SRA Jet selection 10213 17750 119 149SRA mbb gt 200 8344 14502 96 151SRA mCT gt 150 6535 11358 82 139SRA mCT gt 200 5175 8994 67 134SRA mCT gt 250 3793 6592 51 129SRA mCT gt 300 2487 4322 35 123SRB NoCuts 100000 1624100 16241 100SRB leptonveto 11489 186593 4069 46SRB MET gt 250 2530 41090 757 54SRB Jet selection 63 1023 79 130SRB HT 3 lt 50 51 828 52 159

          99

          CMSTop1LSearch for the Production of Dark Matter in As-sociation with Top Quark Pairs in the Single-lepton FinalState in pp collisions[41]

          Simulated in Madgraph with p p gt t t p1 p1

          Where p1= Majorana fermion Table 3 shows Signal efficiencies and cross sections TheM scale was read from Figure 7 for 90 CL

          Table A5 Signal Efficiencies 90 CL on σ limexp[ f b] on pp gt tt +χχ

          Mχ[GeV ] My Signal Efficiency Quoted Efficiency My σ [ f b] Quoted σ [ f b]1 102 101 63 47

          10 132 101 62 4650 143 120 50 38100 170 146 56 32200 185 173 69 27600 265 240 32 19

          1000 320 276 41 17

          Appendix B

          Branching ratio calculations for narrowwidth approximation

          B1 Code obtained from decayspy in Madgraph

          Br(S rarr bb) = (minus24λ2b m2

          b +6λ2b m2

          s

          radicminus4m2

          bm2S +m4

          S)16πm3S

          Br(S rarr tt) = (6λ2t m2

          S minus24λ2t m2

          t

          radicm4

          S minus4ms2m2t )16πm3

          S

          Br(S rarr τ+

          τminus) = (2λ

          2τ m2

          S minus8λ2τ m2

          τ

          radicm4

          S minus4m2Sm2

          τ)16πm3S

          Br(S rarr χχ) = (2λ2χm2

          S

          radicm4

          S minus4m2Sm2

          χ)32πm3S

          (B1)

          Where

          mS is the mass of the scalar mediator

          mχ is the mass of the Dark Matter particle

          mb is the mass of the b quark

          mt is the mass of the t quark

          mτ is the mass of the τ lepton

          The coupling constants λ follow the same pattern

          • TITLE Collider Constraints applied to Simplified Models of Dark Matter fitted to the Fermi-LAT gamma ray excess using Bayesian Techniques
            • Dedication
            • Declaration
            • Acknowledgements
            • Contents
            • List of Figures
            • List of Tables
              • Chapter 1 Introduction
              • Chapter 2 Review of Physics
              • Chapter 3 Fitting Models to the Observables
              • Chapter 4 Calculation Tools
              • Chapter 5 Majorana Model Results
              • Chapter 6 Real Scalar Model Results
              • Chapter 7 Real Vector Dark Matter Results
              • Chapter 8 Conclusion
              • Bibliography
              • Appendix A Validation of Calculation Tools
              • Appendix B Branching ratio calculations for narrow width approximation

            Contents vi

            23 Background on ATLAS and CMS Experiments at the Large Hadron collider(LHC) 31231 ATLAS Experiment 32232 CMS Experiment 33

            3 Fitting Models to the Observables 3531 Simplified Models Considered 3532 Observables 36

            321 Dark Matter Abundance 36322 Gamma Rays from the Galactic Center 36323 Direct Detection - LUX 37

            33 Calculations 39331 Mediator Decay 39332 Collider Cuts Analyses 42333 Description of Collider Cuts Analyses 43

            4 Calculation Tools 5541 Summary 5542 FeynRules 5643 LUXCalc 5644 Multinest 5745 Madgraph 5846 Collider Cuts C++ Code 59

            5 Majorana Model Results 6151 Bayesian Scans 6152 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 6453 Collider Constraints 67

            531 Mediator Decay 67532 Collider Cuts Analyses 69

            6 Real Scalar Model Results 7361 Bayesian Scans 7362 Best fit Gamma Ray Spectrum for the Real Scalar DM model 7663 Collider Constraints 77

            631 Mediator Decay 77632 Collider Cuts Analyses 78

            Contents vii

            7 Real Vector Dark Matter Results 8171 Bayesian Scans 8172 Best fit Gamma Ray Spectrum for the Real Vector DM model 8473 Collider Constraints 84

            731 Mediator Decay 84732 Collider Cuts Analyses 86

            8 Conclusion 89

            Bibliography 91

            Appendix A Validation of Calculation Tools 97

            Appendix B Branching ratio calculations for narrow width approximation 105B1 Code obtained from decayspy in Madgraph 105

            List of Figures

            21 Feynman Diagram of electron interacting with a muon 1122 Weak Interaction Vertices [48] 1523 Higgs Potential [49] 1824 Standard Model Particles and Forces [50] 2425 Bullet Cluster [52] 2526 Galaxy Rotation Curves [54] 2627 WMAP Cosmic Microwave Background Fluctuations [58] 2928 Dark Matter Interactions [60] 2929 Gamma Ray Excess from the Milky Way Center [75] 31210 ATLAS Experiment 31211 CMS Experiment 34

            31 Main Feyman diagrams leading to the cross section for scalar decaying to apair of τ leptons 40

            32 WidthmS vs mS 4033 WidthmS vs λb 4134 WidthmS vs λτ 4135 Main Feyman diagrams leading to the cross section for scalar decaying to a

            pair of b quarks in the presence of at least one b quark 42

            41 Calculation Tools 55

            51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether 62

            52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter 6353 Gamma Ray Spectrum 6454 Plots of log likelihoods by individual and combined constraints Masses in

            GeV 6655 σ lowastBr(σ rarr ττ) versus Mass of Scalar 67

            List of Figures ix

            56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar 6857 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar 6858 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 6959 Excluded points from Collider Cuts and σBranching Ratio 70

            61 Real Scalar Dark Matter - By Individual Constraint and All Together 7462 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter 7563 Gamma Ray Spectrum 7664 σ lowastBr(σ rarr ττ) versus Mass of Scalar 7765 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 7866 Excluded points from Collider Cuts and σBranching Ratio 80

            71 Real Vector Dark Matter - By Individual Constraint and All Together 8272 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter 8373 Gamma Ray Spectrum 8474 σ lowastBr(σ rarr ττ) versus Mass of Scalar 8575 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 8676 Excluded points from Collider Cuts and σBranching Ratio 87

            List of Tables

            21 Quantum numbers of the Higgs field 1922 Weak Quantum numbers of Lepton and Quarks 21

            31 Simplified Models 3532 95 CL by Signal Region 4433 Selection criteria common to all signal regions 4534 Selection criteria for signal regions A 4535 Selection criteria for signal regions C 4536 Signal Regions - Lepstop1 4837 Signal Regions Lepstop2 4938 Signal Regions 2bstop 5139 Signal Region ATLASmonobjet 52

            51 Scanned Ranges 6152 Best Fit Parameters 64

            61 Best Fit Parameters 76

            71 Best Fit Parameters 84

            A1 0 Leptons in the final state 98A2 1 Lepton in the Final state 100A3 2 Leptons in the final state 101A4 2b jets in the final state 102A5 Signal Efficiencies 90 CL on σ lim

            exp[ f b] on pp gt tt +χχ 103

            Chapter 1

            Introduction

            Dark matter (DM) was first postulated over 80 years ago when Swiss astronomer FritzZwicky observed a discrepancy between the amount of light emitted by a cluster of galaxiesand the total mass contained within the cluster inferred from the relative motion of thosegalaxies by a simple application of the theory of Newtonian gravitation The surprising resultof this observation was that the vast majority of the mass in the cluster did not emit lightwhich was contrary to the expectation that most of the mass would be carried by the starsSince that time further observations over a wide range of scales and experimental techniqueshave continued to point to the same result and refine it Some of these observations and otherevidence are discussed in section 22 We now know with certainty that in the entire Universeall of the matter we know about - stars planets gases and other cosmic objects such as blackholes can only account for less than 5 of the mass that we calculate to be there

            A recent phenomenon that has received much attention is the significant deviation frombackground expectations of the Fermi Large Area Telescope(Fermi-LAT) gamma ray flux atthe galactic centre [1] A number of astrophysical explanations have been proposed includingmillisecond pulsars of supernova remnants [2] or burst-like continuous events at the galacticcentre but these are unresolved However it has also been noted that the observed Fermi-LATexcess is consistent with the annihilation of dark matter particles which would naturally beconcentrated at the Galactic centre in a manner consistent with the Navarro-Frenk-Whitedistribution of dark matter [3]

            There are a number of other purely theoretical (particle physics) reasons to postulatethe existence of weakly interacting matter particles that could supply the missing mass andyet remain unobservable Weakly interacting massive particle (WIMPS) have been a majorfocus of Run I and ongoing Run II searches of the Large Hadron Collider (LHC) In spite

            2

            of substantial theoretical and experimental effort the microscopic properties of dark matterparticles are still unknown

            One of the main problems with extracting these properties is the plethora of competingtheoretical models that provide dark matter candidates Another is the very weak interactionof these particles and the lack of experimental observables allowing one to discriminatebetween the competing models

            A particularly fruitful approach is the simplified model approach which makes minimalassumptions about the DM candidates as extensions to the Standard Model interacting weaklywith ordinary matter through a mediator These can be coded as Lagrangian interaction termsusing tools such as FeynRules [4] and Calchep [5] and used to simulate Dark Matter relicdensities indirect detection expectations as well as direct detection expectations throughMicrOmegas [6] This in turn can be run under a Bayesian inference tool (see below)

            Using a Bayesian inference tool such as MultiNest [7][8][9] one can scan the parameterspace of a particular model to find regions of highest probability for that model based onobservations of a range of phenomena including direct detection experiments such as ZEPLIN[10] XENON [11] DEAP [12] ARDM [13] DarkSide [14] PandaX [15] and LUX [16]indirect detection - eg Fermi-LAT gamma ray excess [17] Pamela-positron and antiprotonflux [18] and production at colliders MultiNest calculates the evidence value which is theaverage likelihood of the model weighted by prior probability and a posterior distributionwhich is the distribution of the parameter(s) after taking into account the observed databy scanning over a range of parameters based on prior probabilities This can be used tocompare models on a consistent basis The scans also provide information about the best fitparameters for the overall fit as well as for the seperate experimentsobservations

            The simplified model approach has been taken in a number of papers to study the Fermi-LAT gamma ray excess (see literature review in section 12) and one in particular comparesthese models using a Bayesian inference tool which found that a Majorana fermion DMcandidate interacting via a scalar mediator with arbitrary couplings to third generationfermions showed the best fit [19] A subsequent paper [20] examined the most likelyparameter combinations that simultaneously fit the Planck relic density Fermi-LAT galacticcentre gamma ray AMS-02 positron flux and Pamela antiproton flux data

            There has not as yet been a thorough study of the collider constraints on DM simplifiedmodels that fit the Fermi-LAT galactic centre excess In this thesis we reproduce the

            11 Motivation 3

            previous results of [20] using similar astrophysical constraints and then implement detailedsimulations of collider constraints on DM and mediator production

            In Section 11 we review the motivation for this study and in Section 12 we considerrecent astronomical data which is consistent with the annihilation of DM particles andreview the literature and searches which attempt to explain it with a range of extensions tothe standard model

            In Chapter 2 we review the Standard Model of particle physics (SM) the evidence fordark matter and the ATLAS and CMS experiments at the LHC

            In Chapter 3 we describe the models considered in this paper the observables used inthe Bayesian scans and the collider experiments which provide limits against which theMagraph simulations of the three models can be compared

            In Chapter 4 we review the calculation tools that have been used in this paper

            In Chapters 5 6 and 7 we give the results of the Bayesian scans and collider cutsanalyses for each of the models and in Chapter 8 we summarise our conclusions

            11 Motivation

            The Fermi gamma ray space telescope has over the past few years detected a gamma raysignal from the inner few degrees around the galactic centre corresponding to a regionseveral hundred parsecs in radius The spectrum and angular distribution are compatible withthat from annihilating dark matter particles

            A number of papers have recently attempted to explain the Gamma Ray excess in termsof extensions to the Standard Model taking into account constraints from ATLAS and CMScollider experiments direct detection of dark matter colliding off nucleii in well shieldeddetectors (eg LUX) as well as other indirect measurements (astrophysical observations)These papers take the approach of simplified models which describe the contact interactionsof a mediator beween DM and ordinary matter using an effective field theory approach(EFT) Until recently it has been difficult to discriminate between the results of these paperswhich look at different models and produce different statistical results A recent paper[19] Simplified Dark Matter Models Confront the Gamma Ray Excess used a Bayesianinference tool to compare a variety of simplified models using the Bayesian evidence values

            12 Literature review 4

            calculated for each model to find the most preferred explanation The motivation for thechosen models is discussed in section 121

            A subsequent paper [20] Interpreting the Fermi-LAT gamma ray excess in the simplifiedframework studied the most likely model again using Bayesian inference to find the mostlikely parameters for the (most likely) model The data inputs (observables) to the scanwere the Planck measurement of the dark matter relic density [21] Fermi-LAT gamma rayflux data from the galactic center [17] cosmic positron flux data [22] cosmic anti-protonflux data [18] cosmic microwave background data and direct detection data from the LUXexperiment [16] The result of the scan was a posterior probability density function in theparameter space of the model

            This paper extends the findings in [19] by adding collider constraints to the likelihoodestimations for the three models It is not a simple matter to add collider constraints becauseeach point in parameter space demands a significant amount of computing to calculate theexpected excess amount of observable particles corresponding to the particular experimentgiven the proposed simplified model A sampling method has been employed to calculatethe collider constraints using the probability of the parameters based on the astrophysicalobservables

            12 Literature review

            121 Simplified Models

            A recent paper [23] summarised the requirements and properties that simplified models forLHC dark matter searches should have

            The general principles are

            bull Besides the Standard Model (SM) the Lagrangian should contain a DM candidate thatis absolutely stable or lives long enough to escape the LHC detector and a mediatorthat couples the two sectors

            bull The Lagrangian should contain all terms that are renormalizable consistent withLorentz invariance SM symmetries and DM stability

            bull Additional interactions should not violate exact and approximate global symmetries ofthe SM

            12 Literature review 5

            The examples of models that satisfy these requirements are

            1 Scalar (or pseudo-scalar) s-channel mediator coupled to fermionic dark matter (Diracor Majorana)

            2 Higgs portal DM where the DM is a scalar singlet or fermion singlet under the gaugesymmetries of the SM coupling to a scalar boson which mixes with the Higgs (Thefermion singlet is a specific realisation of 1 above)

            3 DM is a mixture of an electroweak singlet and doublet as in the Minimal Supersym-metric SM (MSSM)

            4 Vector (or Axial vector) s-channel mediator obtained by extending the SM gaugesymmetry by a new U(1)rsquo which is spontaneously broken such that the mediatoracquires mass

            5 t-channel flavoured mediator (if the DM particle is a fermion χ the mediator can bea coloured scalar or vector particle φ -eg assuming a scalar mediator a couplingof the form φ χq requires either χ or φ to carry a flavor index to be consistent withminimal flavour violation (MFV) which postulates the flavour changing neutral currentstructure in the SM is preserved

            Another recent paper [24] summarised the requirements and properties that simplifiedmodels for LHC dark matter searches should have The models studied in this paper arepossible candidates competing with Dirac fermion and complex scalar dark matter andmodels with vector and pseudo-scalar mediators Other models would have an elasticscattering cross section that will remain beyond the reach of direct detection experiments dueto the neutrino floor (the irreducible background produced by neutrino scattering) or arealready close to sensitivity to existing LUX and XENONIT experiments

            A recent paper [25] Scalar Simplified Models for Dark Matter looks at Dirac fermiondark matter mediated by either a new scalar or pseudo-scalar to SM fermions This paperplaces bounds on the coupling combination gχgφ (coupling of DM to mediator times couplingof mediator to SM fermions) calculated from direct detection indirect detection and relicdensity by the mass of DM and mass of mediator Bounds are also calculated for colliderconstraints (mono-jet searches heavy-flavour searches and collider bounds on the 125 GeVHiggs are used to place limits on the monojet and heavy flavor channels which can betranslated into limits on the Higgs coupling to dark matter and experimental measurements

            12 Literature review 6

            of the Higgs width are used to constrain the addition of new channels to Higgs decay) Thepaper models the top loop in a full theory using the Monte Carlo generator MCFM [26]and extends the process to accommodate off-shell mediator production and decay to a darkmatter pair While this paper does not consider the Majorana fermion dark matter model theconstraints on the coupling product may have some relevence to the present study if only asan order of magnitude indicator

            Constraining dark sectors with monojets and dijets [27] studies a simplified model withDirac fermion dark matter and an axial-vector mediator They use two classes of constraint-searches for DM production in events with large amounts of missing transverse energy inassociation with SM particles (eg monojet events) and direct constraints on the mediatorfrom monojet searches to show that for example where the choice of couplings of themediator to quarks and DM gA

            q =gAχ=1 all mediator masses in the range 130 GeV ltMR lt3

            TeV are excluded

            The paper Constraining Dark Sectors at Colliders Beyond the Effective Theory Ap-proach [28] considers Higgs portal models with scalar pseudo-scalar vector and axial-vectormediators between the SM and the dark sector The paper takes account of loop processesand gluon fusion as well as quark-mediator interactions using MCFM Apart from mediatortype the models are characterised by free parameters of mediator width and mass darkmatter mass and the effective coupling combination gχgφ (coupling of DM to mediator timescoupling of mediator to SM fermions) The conclusion is that LHC searches can proberegions of the parameter space that are not able to be explored in direct detection experiments(since the latter measurements suffer from a velocity suppression of the WIMP-nucleoncross-section)

            The paper Constraining the Fermi-LAT excess with multi-jet plus EmissT collider searches

            [29] shows that models with a pseudo-scalar or scalar mediator are best constrained by multi-jet final states with Emiss

            T The analysis was done using POWHEG-BOX-V2 at LO and usesM2

            T variables together with multi-jet binning to exploit differences in signal and backgroundin the different regions of phase space to obtain a more stringent set of constraints on DMproduction This paper largely motivated the current work in that it showed that collidersearches that were not traditionally thought of as dark matter searches were neverthelessoften sensitive to dark matter produced in association with heavy quarks The combinationof searches used in this paper reflects this fact

            12 Literature review 7

            122 Collider Constraints

            In order to constrain this model with collider constraints (direct detection limits) we considerthe following papers by the CMS and ATLAS collaborations which describe searches at theLHC

            ATLAS Experiments

            bull Search for dark matter in events with heavy quarks and missing transverse momentumin p p collisions with the ATLAS detector [30]

            bull Search for direct third-generation squark pair production in final states with missingtransverse momentum and two b-jets in

            radics= 8 TeV pp collisions with the ATLAS

            detector[31]

            bull Search for direct pair production of the top squark in all-hadronic final states inprotonndashproton collisions at

            radics=8 TeV with the ATLAS detector [32]

            bull Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

            radic(s)=8TeV pp collisions using 21 f bminus1 of

            ATLAS data [33]

            bull Search for direct top squark pair production in final states with two leptons inradic

            s = 8TeV pp collisions using 20 f bminus1 of ATLAS data [34]

            bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

            radics=8 TeV [35]

            CMS Experiments

            bull Searches for anomalous tt production in p p collisions atradic

            s=8 TeV [36]

            bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

            radics=8 TeV [37]

            bull Search for the production of dark matter in association with top-quark pairs in thesingle-lepton final state in proton-proton collisions at

            radics = 8 TeV [38]

            bull Search for new physics in monojet events in p p collisions atradic

            s = 8 TeV(CMS) [39]

            bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

            s = 8 TeV [40]

            12 Literature review 8

            bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theSingle-lepton Final State in pp collisions at

            radics=8 TeV [41]

            bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

            s=8 TeV [42]

            bull Search for neutral MSSM Higgs bosons decaying into a pair of bottom quarks [43]

            bull Search for neutral MSSM Higgs bosons decaying into a pair of τ leptons in p p collision[44]

            bull Search for dark matter extra dimensions and unparticles in monojet events in proton-proton collisions at

            radics=8 TeV [45]

            In particular we have implemented Madgraph simulations of certain data analysesapplying to collider experiments described in papers [32][33][34][31][30][41] Theremaining models were not implemented in Madgraph because of inability to validate themodels against published results in the time available We developed C++ analysis code toimplement the collider cuts after showering with Pythia and basic analysis with Delpheswith output to root files The C++ code allowed multiple experimental cuts to be applied tothe same simulation output The C++ code was first checked against the results of the variousexperiments above and then the simulations were rerun using the three modelsrsquo Lagrangians(implemented in FeynRules) described in this paper

            Chapter 2

            Review of Physics

            21 Standard Model

            211 Introduction

            The Standard Model of particle physics (SM) has sometimes been described as the theoryof Almost Everything The SM is a field theory which emerged in the second half of thetwentieth century from quantum mechanics and special relativity which themselves wereonly conceived after 1905 by Einsten Pauli Bohr Dirac and others In the first decades ofthe twenty first century the discovery of the Higgs boson [46][47] completed the pictureand confirmed the last previously unverified part of the SM The SM is a self consistenttheory that describes and classifies all known particles and their interactions (other thangravity) to an astonishing level of accuracy and detail However the theory is incomplete-it does not describe the complete theory of gravitation or account for DM (see below) ordark energy (the accelerating expansion of the universe) Related to these shortcomings aretheoretical deficiencies such as the asymmetry of matter and anti-matter the fact that the SMis inconsistent with general relativity (to the extent that one or both theories break down atspacetime singularities like the Big Bang and black hole event horizons)

            212 Quantum Mechanics

            Quantum Mechanics (QM) describes particles as probability waves which can be thoughtof as packets of energy which can be localised or spread through all space depending on how

            21 Standard Model 10

            accurately one measures the particlersquos momentum However QM is not itself consistent withspecial relativity It was realised too that because of relativity particles could be created anddestroyed and because of the uncertainty principle the theory should describe the probabilityof particles being created and destroyed on arbitrarily short timescales A quantum fieldtheory (QFT) is a way to describe this probability at every point of space and time

            213 Field Theory

            A field theory is described by a mathematical function of space and time called a LagrangianAs in classical mechanics where Lagrangians were introduced by Louis Lagrange at thebeginning of the nineteenth century to summarise dynamical systems the Lagrangian sum-marises the dynamics of space which is permeated by force fields The equations of motionarising from a Lagrangian are determined by finding the contours of the Lagrangian describedby stationary points (ie where the infinitesimal change in the Lagrangian is zero for a smallchange in field) Particles are generally described by singularities in the Lagrangian (wherethe values tend to infinity also called resonances)

            214 Spin and Statistics

            It was also realised that different types of particle needed to be represented by different fieldsin the Lagrangian Diracrsquos discovery that particles with spin 12 quantum numbers couldbe represented by a field obtained by effectively taking the square root of the relativisticequation (Klein Gordon equation) was probably the most significant advance in quantumfield theory It required the introduction of a matrix operators called the gamma matricesor Dirac matrices which helped explain the already known fact that spin half particles orfermions obey Fermi-Dirac statistics ie when two spin half particles are exchanged thesign of the wavefunction must change and hence they cannot occupy the same quantum stateunlike bosons with integer spin that can occupy the same quantum state

            Other particles need to be represented by fields called vector pseudo-vector scalar (as inthe Klein Gordon equation) pseudo-scalar and complex fields A vector field is really fourfields one for each dimension of space plus time and is parameterised by the four gammamatrices γmicro which transform among themselves in a relativistically invariant way (Lorentzinvariance) A pseudo-vector field is a vector field which transforms in the same way but

            21 Standard Model 11

            with additional sign change under parity transformations (a simultaneous reversal of the threespatial directions) and is represented by γmicroγ5 A pseudo-scalar is a scalar which changes signunder parity transformations and a complex scalar field is represented by complex numbers

            215 Feynman Diagrams

            QFT took another giant leap forward when Richard Feynman realised that the propagationof particles described by the field could be described by expanding the so called S-matrixor scattering matrix derived from calculating the probability of initial states transitioning tofinal states The expansion is similar to a Taylor expansion in mathematics allowing for thespecial commuting properties of elements of the field The Feynman diagrams representingindividual terms in the expansion can be interpreted as interactions between the particlecomponents The S-Matrix (or more correctly the non trivial part called the T-Matrix) isderived using the interaction Hamiltonian of the field The Hamiltonian is an operator whoseeigenvalues or characteristic values give the energy quanta of the field The T-matrix canalso be described by a path integral which is an integral over all of quantum space weightedby the exponential of the Lagrangian This is completely equivalent to the Hamiltonianformulation and shows how the Lagrangian arises naturally out of the Hamiltonian (in factthe Lagrangian is the Legendre transformation of the Hamiltonian)

            Figure 21 Feynman Diagram of electron interacting with a muon

            γ

            eminus

            e+

            micro+

            microminus

            The diagram Fig21 is a tree diagram (diagrams with closed loops indicate self in-teractions and are higher order terms in the expansion) and shows an electron (eminus) and ananti-muon (micro+) interacting by exchanging a photon

            21 Standard Model 12

            216 Gauge Symmetries and Quantum Electrodynamics (QED)

            The Dirac equation with its gamma matrices ushered in the use of group theory to describethe transformation of fields (because the multiplication table of the gamma matrices isclosed and described by group theory) It also ushered in the idea of symmetries within theLagrangian- for example consider a Lagrangian

            ψ(ipart minusm)ψ (21)

            The Lagrangian is invariant under a global U(1) transformation which acts on the fieldsand hence the derivatives as follows

            ψ rarr eiqαψ ψ rarr ψeminusiqα partmicroψ rarr eiqα

            partmicroψ (22)

            where qα is a global phase and α is a continuous parameter

            A symmetry leads by Noetherrsquos theorem to a conserved current jmicro = qψγmicroψ and is thebasis of charge conservation since partmicro jmicro = 0 and q =

            intd3x j0(x)

            By promoting this transformation to a local one (ie α rarr α(x) where x = xmicro are thespace time co-ordinates) the transformations become

            ψ rarr eiqα(x)ψ ψ rarr ψeminusiqα(x)partmicroψ rarr eiqα(x)(partmicroψ + iq(partmicroα(x))ψ) (23)

            The derivative transformation introduces an extra term which would spoil the gaugeinvariance This can be solved by introducing a new field the photon field Amicro(x) whichinteracts with the fermion field ψ and transforms under a U(1) transformation

            Amicro rarr Amicro minuspartmicroα(x) (24)

            If we replace normal derivatives partmicro with covariant derivatives Dmicro = (partmicro + iqAmicro(x)) in theLagrangian then the extra term is cancelled and we can now see that the covariant derivativetransforms like ψ - under a local U(1) transform ie

            Dmicro rarr Dmicroψprime = eiqα(x)Dmicroψ (25)

            The additional term iqAmicro(x) can be interpreted in a geometric sense as a connectionwhich facilitates the parallel transport of vectors along a curve in a parallel consistent manner

            21 Standard Model 13

            We must add a gauge-invariant kinetic term to the Lagrangian to describe the dynamicsof the photon field This is given by the electromagnetic field strength tensor

            Fmicroν = partmicroAν minuspartνAmicro (26)

            The full Quantum Electrodynamics (QED) Lagrangian is then for electrons with mass mand charge -e

            LQED = ψ(i Dminusm)ψ minus 14

            Fmicroν(X)Fmicroν(x) (27)

            This Lagrangian is Lorentz and U(1) guage invariant The interaction term is

            Lint =+eψ Aψ = eψγmicro

            ψAmicro = jmicro

            EMAmicro (28)

            where jmicro

            EM is the electromagnetic four current

            217 The Standard Electroweak Model

            The Standard Electroweak (EW) Model was created by Glashow-Weinberg-Salam (also calledthe GWS model) and unifies the electomagnetic and weak interactions of the fundamentalparticles

            The idea of gauge symmetry was extended to the idea of a non-abelian gauge symmetry(essentially the operators defining the symmetry do not commute and so are represented bymatrices- in this case the Pauli matrices for the SU(2) group) - the gauge group is the directproduct G = SU(2)

            otimesU(1) It was known that weak interactions were mediated by Wplusmn

            and Z0 particles where the superscripts are the signs of the electromagnetic charge Thesewere fairly massive but it was also known that through spontaneous symmetry breaking ofa continuous global non-abelian symmetry there would be one massive propagation modefor each spontaneously broken-symmetry generator and massless modes for the unbrokengenerators (the so-called Goldstone bosons)

            This required an extra field to represent the gauge bosons called Bmicro (a vector fieldhaving the 4 spacetime degrees of freedom) With this identification the covariant derivativebecomes

            Dmicro = partmicro minus igAmicro τ

            2minus i

            gprime

            2Y Bmicro (29)

            21 Standard Model 14

            Here Y is the so called weak hypercharge of the field Bmicro which is a quantum numberdepending on particle (lepton or quark) g and gprime are two independent parameters in theGWS theory and the expression Amicro τ implies a sum over 3 components Aa

            micro a=123 and thePauli matrices τa

            This isnrsquot the full story because it had been known since 1956 when Chin Shuing Wuand her collaborators confirmed in an experiment prompted by Lee and Yang that nature isnot ambivalent to so-called parity It had been assumed that the laws of nature would notchange if all spatial coordinates were reversed (a parity transformation) The experimentinvolved the beta decay of cooled 60C atoms (more electrons were emitted in the backwardhemisphere with respect to spin direction of the atoms) This emission involved the weakinteraction and required that the Lagrangian for weak interactions contain terms which aremaximally parity violating- which turn out to be currents of the form

            ψ(1minus γ5)γmicro

            ψ (210)

            The term

            12(1minus γ

            5) (211)

            projects out a left-handed spinor and can be absorbed into the spinor to describe left-handedleptons (and as later discovered quarks) The right handed leptons are blind to the weakinteraction

            The processes describing left-handed current interactions are shown in Fig 22

            Analogously to isospin where the proton and neutron are considered two eigenstates ofthe nucleon we postulate that Bmicro is the gauge boson of an SU(2)transformation coupled toweak SU(2) doublets (

            νe

            eminus

            )

            (ud

            ) (212)

            We may now write the weak SU(2) currents as eg

            jimicro = (ν e)Lγmicro

            τ i

            2

            e

            )L (213)

            21 Standard Model 15

            Figure 22 Weak Interaction Vertices [48]

            where τ i are the Pauli-spin matrices and the subscript L denotes the left-handed componentof the spinor projected out by Eqn211 The third current corresponding to i=3 is called theneutral current as it doesnrsquot change the charge of the particle involved in the interaction

            We could postulate that the electromagnetic current is a linear combination of the leftand right elecromagnetic fields eL = 1

            2(1minus γ5)e and eR = 12(1+ γ5)e

            jemmicro = eLγmicroQeL + eRγmicroQeR (214)

            where Q is the electromagnetic operator and eR is a singlet However this is not invariantunder SU(2)L To construct an invariant current we need to use the SU(2)L doublet 212The current which is an SU(2)L invariant U(1) current is

            jYmicro = (ν e)LγmicroYL

            e

            )L+ eRγmicroYReR (215)

            where hypercharges YL and YR are associated with the U(1)Y symmetry jYmicro is a linearcombination of the electromagnetic current and the weak neutral current thus explaining whythe hypercharge differs for the left and right-handed components since the electromagneticcurrent has the same charge for both components whilst the weak current exists only forleft-handed particles j3

            micro and the third component of weak isospin T 3 allows us to calculate

            21 Standard Model 16

            the relationship Y = 2(QminusT 3) which can be seen by writing jYmicro as a linear combination ofj3micro (the weak neutral current that doesnrsquot change the charge of the particle involved in the

            interaction) and jemmicro (using the convention that the jYmicro is multiplied by 1

            2 to match the samefactor implicit in j3

            micro ) Substituting

            τ3 =

            (1 00 minus1

            )(216)

            into equation 213 for i=3 and subtracting equation 215 and equating to 12 times equation

            214

            we get

            eLγmicroQeL + eRγmicroQeR minus (νLγmicro

            12

            νL minus eLγmicro

            12

            eL) =12

            eRγmicroYReR +12(ν e)LγmicroYL

            e

            )L (217)

            from which we can read out

            YR = 2QYL = 2Q+1 (218)

            and T3(eR) = 0 T3(νL) =12 and T3(eL) =

            12 The latter three identities are implied by

            the fraction 12 inserted into the definition of equation 213

            The Lagrangian kinetic terms of the fermions can then be written

            L =minus14

            FmicroνFmicroν minus 14

            GmicroνGmicroν

            + sumgenerations

            LL(i D)LL + lR(i D)lR + νR(i D)νR

            + sumgenerations

            QL(i D)QL +UR(i D)UR + DR(i D)DR

            (219)

            LL are the left-handed lepton doublets QL are the left-handed quark doublets lRνR theright handed lepton singlets URDR the right handed quark singlets

            The field strength tensors are given by

            Fmicroν = partmicroAν minuspartνAmicro minus ig[Amicro Aν ] (220)

            21 Standard Model 17

            andGmicroν = partmicroBν minuspartνBmicro (221)

            Bmicro is an abelian U(1) gauge field while Amicro is non-abelian and Fmicroν contains the com-mutator of the field through the action of the covariant derivative [Dmicro Dν ] This equationcontains no mass terms terms for the field Amicro for good reason- this would violate gaugeinvariance We know that the gauge bosons of the weak force have substantial mass and theway that this problem has been solved is through the Higgs mechanism

            218 Higgs Mechanism

            To solve the problem of getting gauge invariant mass terms three independent groupspublished papers in 1964 The mechanism was named after Peter Higgs To introduce theconcept consider a complex scalar field φ that permeates all space Consider a potential forthe Higgs field of the form

            minusmicro2φ

            daggerφ +λ (φ dagger

            φ2) (222)

            which has the form of a mexican hat Fig 23 This potential is still U(1) gauge invariant andcan be added to our Lagrangian together with kinetic+field strength terms to give

            L = (Dmicroφ)dagger(Dmicroφ)minusmicro2φ

            daggerφ +λ (φ dagger

            φ2)minus 1

            4FmicroνFmicroν (223)

            It is easily seen that this is invariant to the transformations

            Amicro rarr Amicro minuspartmicroη(x) (224)

            φ(x)rarr eieη(x)φ(x) (225)

            The minimum of this field is not at 0 but in a circular ring around 0 with a vacuum

            expectation value(vev)radic

            micro2

            2λequiv vradic

            2

            We can parameterise φ as v+h(x)radic2

            ei π

            Fπ where h and π are referred to as the Higgs Bosonand Goldstone Boson respectively and are real scalar fields with no vevs Fπ is a constant

            21 Standard Model 18

            Figure 23 Higgs Potential [49]

            Substituting this back into the Lagrangian 223 we get

            minus14

            FmicroνFmicroν minusevAmicropartmicro

            π+e2v2

            2AmicroAmicro +

            12(partmicrohpart

            microhminus2micro2h2)+

            12

            partmicroπpartmicro

            π+(hπinteractions)(226)

            This Lagrangian now seems to describe a theory with a photon Amicro of mass = ev a Higgsboson h with mass =

            radic2micro and a massless Goldstone π

            However there is a term mixing Amicro and part microπ in this expression and this shows Amicro and π

            are not independent normal coordinates and we can not conclude that the third term is a massterm This π minusAmicro mixing can be removed by making a gauge transformation in equations224 and 225 making

            φrarrv+h(x)radic2

            ei π

            Fπminusieη(x) (227)

            and setting πrarr π

            Fπminus eη(x) = 0 This choosing of a point on the circle for the vev is called

            spontaneous symmetry breaking much like a vertical pencil falling and pointing horizontallyin the plane This choice of gauge is called unitary gauge The Goldstone π will thencompletely disappear from the theory and one says that the Goldstone has been eaten to givethe photon mass

            21 Standard Model 19

            This can be confirmed by counting degrees of freedom before and after spontaneoussymmetry breaking The original Lagrangian had four degrees of freedom- two from thereal massless vector field Amicro (massless photons have only two independent polarisationstates) and two from the complex scalar field (one each from the h and π fields) The newLagrangian has one degree of freedom from the h field and 3 degrees of freedom from themassive real vector field Amicro and the π field has disappeared

            The actual Higgs field is postulated to be a complex scalar doublet with four degrees offreedom

            Φ =

            (φ+

            φ0

            )(228)

            which transforms under SU(2) in the fundamental representation and also transforms underU(1) hypercharge with quantum numbers

            Table 21 Quantum numbers of the Higgs field

            T 3 Q Yφ+

            12 1 1

            φ0 minus12 1 0

            We can parameterise the Higgs field in terms of deviations from the vacuum

            Φ(x) =(

            η1(x)+ iη2(x)v+σ(x)+ iη3(x)

            ) (229)

            It can be shown that the three η fields can be transformed away by working in unitarygauge By analogy with the scalar Higgs field example above the classical action will beminimised for a constant value of the Higgs field Φ = Φ0 for which Φ

            dagger0Φ0 = v2 This again

            defines a range of minima and we can pick one arbitarily (this is equivalent to applying anSU(2)timesU(1) transform to the Higgs doublet transforming the top component to 0 and thebottom component to Φ0 = v) which corresponds to a vacuum of zero charge density andlt φ0 gt=+v

            In this gauge we can write the Higgs doublet as

            Φ =

            (φ+

            φ0

            )rarr M

            (0

            v+ H(x)radic2

            ) (230)

            where H(x) is a real Higgs field v is the vacuum expectation and M is an SU(2)timesU(1)matrix The matrix Mdagger converts the isospinor into a down form (an SU(2) rotation) followedby a U(1) rotation to a real number

            21 Standard Model 20

            If we consider the Higgs part of the Lagrangian

            minus14(Fmicroν)

            2 minus 14(Bmicroν)

            2 +(DmicroΦ)dagger(DmicroΦ)minusλ (Φdagger

            Φminus v2)2 (231)

            Substituting from equation 230 into this and noting that

            DmicroΦ = partmicroΦminus igW amicro τ

            aΦminus 1

            2ig

            primeBmicroΦ (232)

            We can express as

            DmicroΦ = (partmicro minus i2

            (gA3

            micro +gprimeBmicro g(A1micro minusA2

            micro)

            g(A1micro +A2

            micro) minusgA3micro +gprimeBmicro

            ))Φ equiv (partmicro minus i

            2Amicro)Φ (233)

            After some calculation the kinetic term is

            (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

            14(v+

            Hradic2)2[A 2]22 (234)

            where the 22 subscript is the index in the matrix

            If we defineWplusmn

            micro =1radic2(A1

            micro∓iA2micro) (235)

            then [A 2]22 is given by

            [A 2]22 =

            (gprimeBmicro +gA3

            micro

            radic2gW+

            microradic2gWminus

            micro gprimeBmicro minusgA3micro

            ) (236)

            We can now substitute this expression for [A 2]22 into equation 234 and get

            (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

            14(v+

            Hradic2)2(2g2Wminus

            micro W+micro +(gprimeBmicro minusgA3micro)

            2) (237)

            This expression contains mass terms for three massive gauge bosons Wplusmn and the combi-nation of vector fields gprimeBmicro minusgA3

            micro where note

            21 Standard Model 21

            Table 22 Weak Quantum numbers of Lepton and Quarks

            T 3 Q YνL

            12 0 -1

            lminusL minus12 -1 -1

            νR 0 0 0lminusR 0 -1 -2UL

            12

            23

            13

            DL minus12 minus1

            313

            UR 0 23

            43

            DR 0 minus13 minus2

            3

            Wminusmicro = (W+

            micro )dagger equivW 1micro minus iW 2

            micro (238)

            Then the mass terms can be written

            12

            v2g2|Wmicro |2 +14

            v2(gprimeBmicro minusgA3micro)

            2 (239)

            W+micro and W microminus can be associated with the W boson and its anti-particle and (minusgprimeBmicro +

            gA3micro) with the Z Boson (after normalisation by

            radicg2 +(gprime

            )2) The combination gprimeA3micro +gBmicro

            is orthogonal to Z and remains massless This can be associated with the Photon (afternormalisation) These predict mass terms for the W and Z bosons mW = vgradic

            2and mZ =

            vradic2

            radicg2 +(gprime

            )2It is again instructive to count the degrees of freedom before and after the Higgs mech-

            anism Before we had a complex doublet Φ with four degrees of freedom one masslessB with two degrees of freedom and three massless W a fields each with two each givinga total of twelve degrees of freedom After spontaneous symmetry breaking we have onereal scalar Higgs field with one degree of freedom three massive weak bosons with nineand one massless photon with two again giving twelve degrees of freedom One says thescalar degrees of freedom have been eaten to give the Wplusmn and Z bosons their longitudinalcomponents (ie mass)

            Table 22 shows T 3 the weak isospin quantum number (which describes how a particletransforms under SU(2)) Y the weak hypercharge (which describes how the particle trans-

            21 Standard Model 22

            forms under U(1)) and Q the electric charge of leptons and quarks The former two enterdirectly into the Lagrangian while the latter can be derived as a function of the former two

            Strictly speaking the right handed neutrino which has no charge may not exist and is notpart of the standard model although some theories postulate a mass for this (hypothetical)particle

            Writing gauge invariant couplings between ELeRφ with the vacuum expectation ofφ = v (Yukawa matrices λ

            i ju and λ

            i jd respectively for the up and down quarks) we get mass

            terms for the quarks (and similarly for the leptons)

            Mass terms for quarks minussumi j[(λi jd Qi

            Lφd jR)+λ

            i ju εab(Qi

            L)aφlowastb u j

            R +hc]

            Mass terms for leptonsminussumi j[(λi jl Li

            Lφ l jR)+λ

            i jν εab(Li

            L)aφlowastb ν

            jR +hc]

            Here hc is the hermitian conjugate of the first terms L and Q are lepton and quarkdoublets l and ν are the lepton singlet and corresponding neutrino d and u are singlet downand up quarks εab is the Levi-Civita (anti-symmetric) tensor

            If we write the quark generations in a basis that diagonalises the Higgs couplings weget a unitary transformations Uu and Ud for the up and down quarks respectively which iscompounded in the boson current (Udagger

            u Ud) called the CKM matrix where off-diagonal termsallow mixing between quark generations (this is also called the mass basis as opposed tothe flavour basis) A similar argument can give masses to right handed sterile neutrinos butthese can never be measured However a lepton analogue of the CKM matrix the PNMSmatrix allows left-handed neutrinos to mix between various generations and oscillate betweenflavours as they propagate through space allowing a calculation of mass differences

            219 Quantum Chromodynamics

            The final plank in SM theory tackling the strong force Quantum Chromodynamics wasproposed by Gell-Mann and Zweig in 1963 Three generations of quarks were proposedwith charges of 23 for uc and t and -13 for ds and b Initially SU(2) isospin was consideredtransforming u and d states but later irreducible representations of SU(3) were found to fit theexperimental data The need for an additional quantum number was motivated by the fact thatthe lightest excited state of the nucleon consisted of sss (the ∆++) To reconcile the baryon

            21 Standard Model 23

            spectrum with spin statistics colour was proposed Quarks transform under the fundamentalor 3 representation of colour symmetry and anti-quarks under 3 The inner product of 3and 3 is an invariant under SU(3) as is a totally anti-symmetric combination ε i jkqiq jqk (andthe anti-quark equivalent) It was discovered though deep inelastic scattering that quarksshowed asymptotic freedom (the binding force grew as the seperation grew) Non-abeliangauge theories showed this same property and colour was identified with the gauge quantumnumbers of the quarks The quanta of the gauge field are called gluons Because the forcesshow asymptotic freedom - ie the interaction becomes weaker at high energies and shorterdistances perturbation theory can be used (ie Feynman diagrams) at these energies but forlonger distances and lower energies lattice simulations must be done

            2110 Full SM Lagrangian

            The full SM can be written

            L =minus14

            BmicroνBmicroν minus 18

            tr(FmicroνFmicroν)minus 12

            tr(GmicroνGmicroν)

            + sumgenerations

            (ν eL)σmicro iDmicro

            (νL

            eL

            )+ eRσ

            micro iDmicroeR + νRσmicro iDmicroνR +hc

            + sumgenerations

            (u dL)σmicro iDmicro

            (uL

            dL

            )+ uRσ

            micro iDmicrouR + dRσmicro iDmicrodR +hc

            minussumi j[(λ

            i jl Li

            Lφ l jR)+λ

            i jν ε

            ab(LiL)aφ

            lowastb ν

            jR +hc]

            minussumi j[(λ

            i jd Qi

            Lφd jR)+λ

            i ju ε

            ab(QiL)aφ

            lowastb u j

            R +hc]

            + (Dmicroφ)dagger(Dmicroφ)minusmh[φφ minus v22]22v2

            (240)

            where σ micro are the extended Pauli matrices

            (1 00 1

            )

            (0 11 0

            )

            (0 minusii 0

            )

            (1 00 minus1

            )

            The first line gives the gauge field strength terms the second the lepton dynamical termsthe third the quark dynamical terms the fourth the lepton (including neutrino) mass termsthe fifth the quark mass terms and the sixth the Higgs terms

            The SU(3) gauge field is given by Gmicroν and which provides the colour quantum numbers

            21 Standard Model 24

            Figure 24 Standard Model Particles and Forces [50]

            Note The covariant derivatives operating on the left-handed leptons contain only theU(1) and SU(2) gauge fields Bmicro Fmicro whereas those operating on left-handed quarks containU(1)SU(2) and SU(3) gauge fields Bmicro Fmicro Gmicro The covariant derivatives operating onright handed electrons and quarks are as above but do not contain the SU(2) gauge fieldsRight handed neutrinos have no gauge fields

            The sums over i j above are over the different generations of leptons and quarks

            The particles and forces that emerge from the SM are shown in Fig 24

            22 Dark Matter 25

            22 Dark Matter

            221 Evidence for the existence of dark matter

            2211 Bullet Cluster of galaxies

            Most of the mass of the cluster as measured by a gravitational lensing map is concentratedaway from the centre of the object The visible matter (as measured by X-ray emission) ismostly present in the centre of the object The interpretation is that the two galaxy clustershave collided perpendicular to the line of sight with the mostly collisionless dark matterpassing straight through and the visible matter interacting and being trapped in the middleIt was estimated that the ratio of actual to visible matter was about 16 [51]

            Figure 25 Bullet Cluster [52]

            2212 Coma Cluster

            The first evidence for dark matter came in the 1930s when Fritz Zwicky observed the Comacluster of galaxies over 1000 galaxies gravitationally bound to one another He observed thevelocities of individual galaxies in the cluster and estimated the amount of matter based onthe average galaxy He calculated that many of the galaxies should be able to escape fromthe cluster unless the mass was about 10 times higher than estimated

            22 Dark Matter 26

            2213 Rotation Curves [53]

            Since those early observations many spiral galaxies have been observed with the rotationcurves calculated by doppler shift with the result that in all cases the outermost stars arerotating at rates fast enough to escape their host galaxies Indeed the rotation curves arealmost flat indicating large amounts of dark matter rotating with the galaxies and holdingthe galaxies together

            Figure 26 Galaxy Rotation Curves [54]

            2214 WIMPS MACHOS

            The question is if the particles that make up most of the mass of the universe are not baryonswhat are they It is clear that the matter does not interact significantly with radiation so mustbe electrically neutral It has also not lost much of its kinetic energy sufficiently to relax intothe disks of galaxies like ordinary matter Massive particles may survive to the present ifthey carry a conserved additive or multiplicative quantum number This is so even if equalnumbers of particles and anti-particles can annihilate if the number densities become too lowto reduce the density further These particles have been dubbed Weakly interacting MassiveParticles (WIMPS) as they interact with ordinary matter only through a coupling strengthtypically encountered in the weak interaction They are predicted to have frozen out ofthe hot plasma and would give the correct relic density based solely on the weak interactioncross section (sometimes called the WIMP Miracle) Another possibility called MassiveCompact Halo Objects (MACHOS)- are brown dwarf stars Jupiter mass objects or remnantsfrom an earlier generation of star formation

            22 Dark Matter 27

            2215 MACHO Collaboration [55]

            In order to estimate the mass of MACHOS a number of experiments have been conductedusing microlensing which makes use of the fact that if a massive object lies in the line ofsight of a much more distant star the light from the more distant star will be lensed and forma ring around the star It is extemely unlikely that the MACHO will pass directly through theline of sight but if there is a near miss the two images will be seperated by a small angleThe signature for microlensing is a time-symmetric brightening of a star as the MACHOpasses near the line of sight called a light curve One can obtain statistical information aboutthe MACHO masses from the shape and duration of the light curves and a large number ofstars (many million) must be monitored The MACHO Collaboration [55] have monitoredover 11 million stars and less than 20 events were detected The conclusion was that thiscould account for a small addition (perhaps up to 20 ) to the visible mass but not enoughto account for Dark Matter

            2216 Big Bang Nucleosynthesis (BBN) [56]

            Another source of evidence for DM comes from the theory of Nucleosynthesis in the fewseconds after the Big Bang In the first few minutes after the big bang the temperatures wereso high that all matter was dissociated but after about 3 minutes the temperature rapidlycooled to about 109 Kelvin Nucleosynthesis could begin when protons and neutrons couldstick together to form Deuterium (Heavy Hydrogen) and isotopes of Helium The ratio ofDeuterium to Hydrogen and Helium is very much a function of the baryon density and alsothe rate of expansion of the universe at the time of Nucleosynthesis There is a small windowwhen the density of Baryons is sufficient to account for the ratios of the various elementsThe rate of expansion of the universe is also a function of the total matter density The ratiosof these lighter elements have been calculated by spectral measurements with Helium some25 of the mass of the universe Deuterium 001 The theory of BBN accounts for theratios of all of the lighter elements if the ratio of Baryon mass to total mass is a few percentsuggesting that the missing matter is some exotic non-baryonic form

            22 Dark Matter 28

            2217 Cosmic Microwave Background [57]

            The discovery of a uniform Microwave Background radiation by Penzias and Wilson in1965 was confirmation of the theory first proposed by Gamow and collaborators that theuniverse should be filled with black body radiation an afterglow of the big bang TheCOBE satelite launched in 1989 confirmed this black body radiation with temperature of2725+-002K with 95 confidence While extremely uniform it found tiny irregularitieswhich could be accounted for by irregularities in the density of the plasma at last scattering(gravitational redshifts or blueshifts called the Sachs Wolfe effect) and temperature andvelocity fluctuations in the plasma The anisotropy of the background has been furtheranalysed by WMAP (the Wilkinson Microwave Anisotropy Probe) launched in 2001 Thefluctuations in the density of the plasma are what account for the seeds of galaxy formationHowever detailed calculations have found that the fluctuations were too small to account forthe observed galaxy distribution without an electrically neutral gravitating matter in additionto ordinary matter that only became electrically neutral at the time of recombination [58]

            In May 2009 the European Space Agency (ESA) launched the Planck sattelite a spacebased observatory to observe the universe at wavelengths between 03mm and 111 mmbroadly covering the far-infrared microwave and high frequency domains The missionrsquosmain goal was to study the cosmic microwave background across the whole sky at greatersensitivity and resolution than ever before Using the information gathered and the standardmodel of cosmology ESA was able to calculate cosmological parameters including the relicdensity of dark matter remaining after the Big Bang to a high degree of precision withinthe framework of the standard model of cosmology The standard model of cosmology canbe described by a relatively small number of parameters including the density of ordinarymatter dark matter and dark energy the speed of cosmic expansion at the present epoch (alsoknown as the Hubble constant) the geometry of the Universe and the relative amount ofthe primordial fluctuations embedded during inflation on different scales and their amplitude[59]

            2218 LUX Experiment - Large Underground Xenon experiment [16]

            The LUX experiment is conducted at the Sanford Underground research facility one mileunderground to minimise the background of cosmic rays The detector is a time projectionchamber allowing a 3D positioning of interactions occurring within its active volume LUX

            22 Dark Matter 29

            Figure 27 WMAP Cosmic Microwave Background Fluctuations [58]

            Figure 28 Dark Matter Interactions [60]

            uses as its target 368 kiliograms of liquefied ultra pure xenon which is a scintillator Interac-tions inside the xenon will create an amount of light proportional to the energy depositedThat light can be collected on arrays of light detectors sensitive to a single photon

            Xenon is very pure and the amount of radiation originating within the target is limitedAlso being three times denser than water it can stop radiation originating from outside thedetector The 3D detector focuses on interactions near the center which is a quiet regionwhere rare dark matter interactions might be detected [16]

            22 Dark Matter 30

            222 Searches for dark matter

            2221 Dark Matter Detection

            Fig 28 illustrates the reactions studied under the different forms of experiment designed todetect DM

            Direct detection experiments are mainly underground experiments with huge detectorsattempting to detect the recoil of dark matter off ordinary matter such as LUX [16] CDMS-II [61] XENON100 [11] COUPP [62] PICASSO [63] ZEPLIN [10] These involveinteractions between DM and SM particles with final states involving DM and SM particles

            Indirect detection involves the search for the annihilation of DM particles in distantastrophysical sources where the DM content is expected to be large Example facilitiesinclude space based experiments (EGRET [64] PAMELA [18] FERMI-LAT [65]) balloonflights (ATIC [66] PEBS [67] PPB-BETS [68]) or atmospheric Cherenkov telescopes(HESS [69]MAGIC [70]) and neutrino telescopes (SuperKamiokande [71] IceCube [72]ANTARES [73] KM3NET [74]) Indirect Detection involves DM particles in the initial stateand SM particles in the final state

            Collider detection involves SM particles in the initial state creating DM particles in thefinal state which can be detected by missing transverse energy and explained by variousmodels extending the SM

            223 Possible signals of dark matter

            224 Gamma Ray Excess at the Centre of the Galaxy [65]

            The center of the galaxy is the most dense and active region and it is no surprise that itproduces intense gamma ray sources When astronomers subtract all known gamma raysources there remains an excess which extends out some 5000 light years from the centerHooper et al find that the annihilation of dark matter particles with a mass between 31 and40 Gev fits the excess to a remarkable degree based on Gamma Ray spectrum distributionand brightness

            23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 31

            Figure 29 Gamma Ray Excess from the Milky Way Center [75]

            23 Background on ATLAS and CMS Experiments at theLarge Hadron collider (LHC)

            The collider constraints in this paper are based on two of the experiments that are part of theLHC in CERN

            Figure 210 ATLAS Experiment

            The LHC accelerates proton beams in opposite directions around a 27 km ring of super-conducting magnets until they are made to collide at an interaction point inside one of the 4experiments placed in the ring An upgrade to the LHC has just been performed providing65 TeV per beam

            23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 32

            231 ATLAS Experiment

            The ATLAS detector consists of concentric cylinders around the interaction point whichmeasure different aspects of the particles that are flung out of the collision point There are 4major parts the Inner Detector the Calorimeters the Muon spectrometer and the magnetsystems

            2311 Inner Detector

            The inner detector extends from a few centimeters to 12 meters from the interaction point andmeasures details of the charged particles and their momentum interacting with the materialIf particles seem to originate from a point other than the interaction point this may indicatethat the particle comes from the decay of a hadron with a bottom quark This is used to b-tagcertain particles

            The inner detector consists of a pixel detector with three concentric layers and threeendcaps containing detecting silicon modules 250 microm thick The pixel detector has over 80million readout channels about 50 percent of the total channels for the experiment All thecomponents are radiation hardened to cope with the extremely high radiation in the proximityof the interaction point

            The next layer is the semi-conductor layer (SCT) similar in concept to the pixel detectorbut with long narrow strips instead of pixels to cover a larger area It has 4 double layers ofsilicon strips with 63 million readout channels

            The outermost component of the inner detector is the Transition Radiation Tracker It is acombination of a straw tracker and a transition radiation detector The detecting elementsare drift tubes (straws) each 4mm in diameter and up to 144 cm long Each straw is filledwith gas which becomes ionised when the radiation passes through The straws are held at-1500V with a thin wire down the center This drives negatively ionised atoms to the centercausing a pulse of electricity in the wire This causes a pattern of hit straws allowing the pathof the particles to be determined Between the straws materials with widely varying indicesof refraction cause transition radiation when particles pass through leaving much strongersignals in some straws Transition radiation is strongest for highly relativistic particles (thelightest particles- electrons and positrons) allowing strong signals to be associated with theseparticles

            23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 33

            2312 Calorimeters

            The inner detector is surrounded by a solenoidal magnet of 2 Tesla (there is also an outertoroidal magnet) but surrounding the inner magnet are the calorimeters to measure theenergy of particles by absorbing them There is an inner electromagnetic calorimeter whichmeasures the angle between the detectorrsquos beam axis and the particle (or more preciselypseudo-rapidity) and its angle in the perpendicular plane to a high degree of accuracy Theabsorbing material is lead and stainless steel and the sampling material liquid argon Thehadron calorimeter measures the energy of particles that interact via the strong force Theenergy absorbing material is scintillating steel

            2313 Muon Specrometer

            The muon spectrometer surrounds the calorimeters and has a set of 1200 chambers tomeasure the tracks of outgoing muons with high precision using a magnetic field andtriggering chambers Very few particles of other types pass through the calorimeters and aresubsequently measured

            2314 Magnets

            The magnets are in two layers the inner solenoidal and outer toroidal which together causecharged particles to curve in the magetic field allowing their momentum to be determinedThe outer toroidal magnet consists of 8 very large air core superconducting magnets whichsupply the field in the muon spectrometer

            232 CMS Experiment

            The CMS Detector is similar in design to the ATLAS Detector with a single solenoidalmagnet which is the largest of its type ever built (length 13m and diameter 7m) and allowsthe tracker and calorimeter detectors to be placed inside the coil in a compact way Thestrong magnetic field gives a high resolution for measuring muons electrons and photons ahigh quality central tracking system and a hermetic hadron calorimeter designed to preventhadrons from escaping

            23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 34

            Figure 211 CMS Experiment

            Chapter 3

            Fitting Models to the Observables

            31 Simplified Models Considered

            In order to extend the findings of previous studies [19] to include collider constraints wefirst performed Bayesian scans [7] over three simplified models real scalar dark matter φ Majorana fermion χ and real vector Xmicro The purpose was to obtain the most probable regionsof each model then to use the resulting posterior samples to investigate whether the LHCexperiments have any sensitivity to the preferred regions

            The three models couple to the mediator with interactions shown in the following table

            Table 31 Simplified Models

            Hypothesis real scalar DM Majorana fermion DM real vector DM

            DM mediator int Lφ sup microφ mφ φ 2S Lχ sup iλχ

            2 χγ5χS LX sup microX mX2 X microXmicroS

            The interactions between the mediator and the standard fermions is assumed to be

            LS sup f f S (31)

            and in line with minimal flavour violation we only consider third generation fermions- ief = b tτ

            For the purposes of these scans we consider the following observables

            32 Observables 36

            32 Observables

            321 Dark Matter Abundance

            We have assumed a single dark matter candidate which freezes out in the early universe witha central value as determined by Planck [21] of

            ΩDMh2 = 01199plusmn 0031 (32)

            h is the reduced hubble constant

            The experimental uncertainty of the background relic density [59] is dominated by atheoretical uncertainty of 50 of the value of the scanned Ωh2 [76] We have added thetheoretical and the experimental uncertainties in quadrature

            SD =radic(05Ωh2)2 + 00312 (33)

            This gives a log likelihood of

            minus05lowast (Ωh2 minus 1199)2

            SD2 minus log(radic

            2πSD) (34)

            322 Gamma Rays from the Galactic Center

            Assuming that the gamma ray flux excess from the galactic centre observed by Fermi-LAT iscaused by self annihilation of dark matter particles the differential flux

            d2Φ

            dEdΩ=

            lt σv gt8πmχ

            2 J(ψ)sumf

            B fdN f

            γ

            dE(35)

            has been calculated using both Micromegas v425 with a Navarro-Frenk-White (NFW) darkmatter galactic distribution function

            ρ(r) = ρ0(rrs)

            minusγ

            (1+ rrs)3minusγ (36)

            with γ = 126 and an angle of 5 to the galactic centre [19]

            32 Observables 37

            Here Φ is the flux dE is the differential energy lt σv gt the velocity averaged annihi-lation cross section of the dark matter across the galactic centre mχ the mass of the darkmatter particle B f =lt σv gt f σv is the annihilation fraction into the f f final state anddN f

            γ dE is the energy distribution of photons produced in the annihilation channel with thefinal state f f The J factor is described below

            The angle ψ to the galactic centre determines the normalisation of the gamma ray fluxthrough the factor

            J(ψ) =int

            losρ(l2 + r⊙2 minus2lr⊙cosψ)dl (37)

            where ρ(r) is the dark matter density at distance r from the centre of the galaxy r⊙=85 kpcis the solar distance from the Galactic center ρ0 is set to reproduce local dark matter densityof 03GeVcm3

            The central values for the gamma ray spectrum were taken from Fig 5 of [77] and thestandard deviations of individual flux points from the error bars on the graph In commonwith [78] we ignore possible segment to segment correlations

            For the Fermi-LAT log likelihood we assumed each point was distributed with indepen-dent Gaussian likelihood with the experimental error for each data point given by the errorbar presented in Fig5 of [76] for each point We also assumed an equal theoretical errorfor each point which when added to the experimental error in quadrature gave an effectivelog likelihood for each point of minus05lowast (giminusdi)

            2

            2lowastσ2i

            where gi are the calculated values and di theexperimental values and σi the experimental errors

            323 Direct Detection - LUX

            The LUX experiment [16] is the most constraining direct search experiment to date for awide range of WIMP models We have used LUXCalc [79] (modified to handle momentumdependent nucleon cross sections) to calculate the likelihood of the expected number ofsignal events for the Majorana fermion model

            The likelihood function is taken as the Poisson distribution

            L(mχ σ | N) = P(N | mχ σ) =(b+micro)Neminus(b+micro)

            N (38)

            32 Observables 38

            where b= expected background of 064 and micro = micro(mχσ ) is the expected number of signalevents calculated for a given model

            micro = MTint

            infin

            0dEφ(E)

            dRdE

            (E) (39)

            where MT is the detector mass times time exposure and φ(E) is a global efficiency factorincorporating trigger efficiencies energy resolution and analysis cuts

            The differential recoil rate of dark matter on nucleii as a function of recoil energy E

            dRdE

            =ρX

            mχmA

            intdvv f (v)

            dσASI

            dER (310)

            where mA is the nucleon mass f (v) is the dark matter velocity distribution and

            dσSIA

            dER= Gχ(q2)

            4micro2A

            Emaxπ[Z f χ

            p +(AminusZ) f χn ]

            2F2A (q) (311)

            where Emax = 2micro2Av2mA Gχ(q2) = q2

            4m2χ

            [24] is an adjustment for a momentum dependantcross section which applies only to the Majorana model

            f χ

            N =λχ

            2m2SgSNN assuming that the relic density is the central value of 1199 We have

            implemented a scaling of the direct search cross section in our code by multiplying gSNN by( Ω

            Ω0)05 as a convenient method of allowing for the fact that the actual relic density enters

            into the calculation of the cross section as a square

            FA(q) is the nucleus form factor and

            microA =mχmA

            (mχ +mA)(312)

            is the reduced WIMP-nucleon mass

            The strength of the dark matter nucleon interaction between dark matter particles andthird generation quarks is

            gSNN =2

            27mN fT G sum

            f=bt

            λ f

            m f (313)

            where fT G = 1minus f NTuminus f N

            Tdminus fTs and f N

            Tu= 02 f N

            Td= 026 fTs = 043 [20]

            33 Calculations 39

            For the real scalar and real vector models where the spin independent cross sections arenot momentum dependent we based the likelihood on the LUX 95 CL from [16] using acomplementary error function Li(d|mχ I) = 05lowastCEr f (dminusx(mχ )

            σ) where x is the LUX limit

            and d the calculated SI cross section We assumed a LUX error σ of 01lowastd lowastradic

            2

            33 Calculations

            We coded the Lagrangians for the three models in Feynrules [4] Calculation of the expectedobservables (dark matter relic density Fermi-LAT gamma ray spectrum and LUX nucleonscattering amplitudes) were performed using Micromegas v425

            331 Mediator Decay

            A number of searches have previously been performed at the LHC which show sensitivity tothe decay of scalar mediators in the form of Higgs bosons (in beyond the standard modeltheories) to bottom quarks top quarks and τ leptons (all third generation fermions) Two inparticular [43 44] provide limits on the cross-sections for the production of the scalar timesthe branching ratios to brsquos and τrsquos In order to check these limits against each of the modelswe ran parameter scans in Madgraph at leading order over the top 900 points from the nestedsampling output from MultiNest [19]

            The two processes were

            1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to apair of τ leptons in pp collisions [44]

            bull generate p p gt b b S where S is the scalar mediator

            The process measures the decay of a scalar in the presence of b-jets that decay to τ+τminus

            leptons Fig31 shows the main diagrams contributing to the process The leftmost diagramis a loop process and the centre and right are tree level however we only got Madgraph togenerated the middle diagram since we are interested in τ leptons and b quarks in the finalstate

            The limit given in [44] only applies in the case of narrow resonances where narrow istaken to mean that the width of the resonance is less than 10 of its mass To test the validity

            33 Calculations 40

            Figure 31 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof τ leptons

            of applying these limits to our models we show in Figures 32 to 34 the percentage widthof the resonance vs mS[GeV ] (pole mass of the scalar mediator) λb and λt (the couplingconstants to the b and t quarks) The probabilities are the posterior probabilities obtainedfrom MultiNest scans of the Majorana model varying the coupling constants and scalarmediator and calculating the decay width of the mediator The results are represented as 2dposterior probability heat maps in the central panels and 1d marginalised distributions on thex and y axes of each graph

            16 18 20 22 24 26 28 30

            log10(mS[GeV])

            001

            002

            003

            004

            005

            Widthm

            S

            00

            04

            08

            12

            0 100 200

            Posterior Probability

            Figure 32 WidthmS vs mS

            The WidthmS was lt 01 at almost all points except where the mass of the scalar wasgreater than 1000 GeV λb was greater than 1 or λτ was greater than 1

            This can be seen from the graphs in Figs 323334

            33 Calculations 41

            4 3 2 1 0

            λb

            001

            002

            003

            004

            005

            WidthmS

            000

            015

            030

            045

            0 100 200

            Posterior Probability

            Figure 33 WidthmS vs λb

            5 4 3 2 1 0

            λτ

            001

            002

            003

            004

            005

            WidthmS

            000

            015

            030

            045

            0 100 200

            Posterior Probability

            Figure 34 WidthmS vs λτ

            The decay widths are calculated in Madgraph at leading order and branching fractionscan be calculated to b t τ and χ (DM) The formulae for the calculations can be found indecayspy which is a subroutine of Madgraph which we have documented in Appendix BB1 Using these we calculated the branching ratio to τ+τminus so that the cross section timesbranching ratio can be compared to the limits in [80] shown in Chapters 56 and 7 in Figs5564 and 74 for each model

            2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

            This scan calculates the contributions of b-quark associated Higgs boson production inthe center and right diagrams of Fig 35 reproduced from [43] The scans did not includePythia and Delphes as only production cross sections were required and the producedparticles were in the final state and required no matching or merging

            The Madgraph processes were

            bull generate p p gt b S where S is the scalar mediator

            bull add process p p gt b S j

            bull add process p p gt b S

            33 Calculations 42

            Figure 35 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof b quarks in the presence of at least one b quark

            bull add process p p gt b S j

            The cross section times branching ratios to bb for mediator production in association withat least one b quark is shown for each model in Chapters 56 and 7 in Figs 5765 and 75

            332 Collider Cuts Analyses

            We generated two parameter scans with parameter combinations from the Multinest outputwhich were the basis of the plots of Fig 6 in [19] The plots in Fig 6 were marginalisedto two parameters each but the raw Multinest output contained 5 parameters plus the priorprobabilities of each combination normalised to total 10

            The first scan was for χ produced with b quarks In order to force the mediator to decayto χ only we had χ in the final state and defined jets to include b quarks in Madgraph Thenwe generated the processes

            bull generate p p gt χ χ j

            bull add process p p gt χ χ j j

            Jet matching was on

            The second scan was for t quarks produced in the final state

            bull generate p p gt χ χ tt

            33 Calculations 43

            No jet matching was required because we did not mix final state jet multiplicities at thematrix element level

            The outputs from these two processes were normalised to 21 f bminus1 and combined

            The results of these scans prior to collider cuts calculations are shown in Figs 5161and 71 These show the posterior probabilities of the various parameter combinations withdifferent constraints applied seperately and then together A more detailed commentary isgiven in each of the sections

            We then sampled 900 points based on posterior probability from these scans and ran thesethrough Madgraph v425 then applied collider cuts with the C++ code using the limits fromthe following six ATLAS and CMS searches shown in Table32 and described below to findany exclusions

            333 Description of Collider Cuts Analyses

            In the following all masses and energies are in GeV and angles in radians unless specificallystated

            3331 Lepstop0

            Search for direct pair production of the top squark in all-hadronic final states in pro-tonndashproton collisions at

            radics=8 TeV with the ATLAS detector[32]

            This analysis searched for direct pair production of the supersymmetric partner to thetop quark using integrated luminosity of 201 f bminus1 The top squark was assumed to de-cay via t rarr t χ0

            1 or t rarr bχ01 or t rarr bχ

            plusmn1 rarr bW (lowast)χ1

            0 where χ01 (χ

            plusmn1 ) denotes the lightest

            neutralino(chargino) in supersymmetric models The search targets a fully hadronic finalstate with four or more jets and large missing momentum The production of a mediator inassociation with top quarks followed by an invisible mediator decay to dark matter particleshas a similar final state to that considered in the ATLAS stop searches Thus we expect theATLAS reach to have sensitivity to the three dark matter models considered in this thesis

            The collider cuts program mimicked the object reconstruction through jetlepton overlapremoval and selected signal electrons muons and τ particles with pT gt 10 GeV and

            33 Calculations 44

            Table 32 95 CL by Signal Region

            Experiment Region Number

            Lepstop0

            SRA1 66SRA2 57SRA3 67SRA4 65SRC1 157SRC2 124SRC3 80

            Lepstop1

            SRtN1-1 857SRtN1-2 498SRtN1-3 389SRtN2 107SRtN3 85SRbC1 832SRbC2 195SRbC3 76

            Lepstop2

            L90 740L100 56L110 90L120 170

            2bstop

            SRA 857SRB 270SRA150 380SRA200 260SRA250 90SRA300 75SRA350 52

            CMSTopDM1L SRA 1385

            ATLASMonobjetSR1 1240SR2 790

            33 Calculations 45

            |η | lt 24724 and 247 respectively (for eminusmicrominusτ) Jets were selected with pT gt 20 GeVand |η |lt 45 b-jets were selected if they were tagged by random selection lt 70 and had|η |lt 25 and pT gt 20 GeV

            These jets and leptons were further managed for overlap removal within ∆R lt 04 withnon b-jets removed if ∆R lt 02

            The b-jets and non b-jets were further selected as signal jets if pT gt 35 GeV and |η |lt 28

            These signal jets electrons and muons were finally binned according to the signal regionsin Tables 333435 below

            Table 33 Selection criteria common to all signal regions

            Trigger EmissT

            Nlep 0b-tagged jets ⩾ 2

            EmissT 150 GeV

            |∆φ( jet pmissT )| gtπ5

            mbminT gt175 GeV

            Table 34 Selection criteria for signal regions A

            SRA1 SRA2 SRA3 SRA4anti-kt R=04 jets ⩾ 6 pT gt 808035353535 GeV

            m0b j j lt 225 GeV [50250] GeV

            m1b j j lt 225 GeV [50400] GeV

            min( jet i pmissT ) - gt50 GeV

            τ veto yesEmiss

            T gt150 GeV gt250 GeV gt300 GeV gt 350 GeV

            Table 35 Selection criteria for signal regions C

            SRC1 SRC2 SRC3anti-kt R=04 jets ⩾ 5 pT gt 8080353535 GeV

            |∆φ(bb)| gt02 π

            mbminT gt185 GeV gt200 GeV gt200 GeV

            mbmaxT gt205 GeV gt290 GeV gt325 GeV

            τ veto yesEmiss

            T gt160 GeV gt160 GeV gt215 GeV

            wherembmin

            T =radic

            2pbt Emiss

            T [1minus cos∆φ(pbT pmiss

            T )]gt 175 (314)

            33 Calculations 46

            and pbT is calculated for the b-tagged jet with the smallest ∆φ to the pmiss

            T direction andmbmax

            T is calculated with the pbT for the b-tagged jet with the largest ∆φ to the pmiss

            T direction

            m0b j j is obtained by first selecting the two jets with the highest b-tagging weight- from

            the remaining jets the two jets closest in the η minusφ plane are combined to form a W bosoncandidate this candidate is then combined with the closest of the 2 b-tagged jets in the η minusφ

            plane to form the first top candidate with mass m0b j j A second W boson candidate is formed

            by repeating the procedure with the remaining jets this candidate is combined with thesecond of the selected b-tagged jets to form the second top candidate with mass m1

            b j j Thecuts shown in Appendix1 Table A1 show the cumulative effect of the individual cuts appliedto the data the signal region cuts and the cumulative effects of individual cuts in SignalRegion C

            3332 Lepstop1

            Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

            radics=8 TeV pp collisions using 21 f bminus1 of

            ATLAS data[33]

            The stop decay modes considered were those to a top quark and the lightest supersym-metric particle (LSP) as well as a bottom quark and the lightest chargino where the charginodecays to the LSP by emitting a W boson We should expect the three dark matter models inthis paper with couplings to t and b quarks to show some visibility in this search

            The collider cuts program mimicked the object construction by selecting electrons andmuons with pT gt 10 GeV and |η |lt 247 and 24 respectively and jets with pT gt 20 GeVand |η |lt 10 b-jets with |η |lt 25 and pT gt 25 GeV were selected with random efficiency75 and mistag rate 2 (where non b-jets were mis-classified) We then removed any jetwithin ∆R = 02 of an electron and non overlapping jets with |η |lt 28 were used to removeany electrons and muons within ∆R = 04 of these while non-overlapping jets with |η |lt 25and pT gt 25 GeV were classified as signal jets Non overlapping electrons and muons withpT gt 25 GeV were classified as signal electrons and muons

            Signal selection defined six regions to optimize sensitivity for different stop and LSPmasses as well as sensitivity to two considered stop decay scenarios Three signal regionslabelled SRbC 1-3 where bC is a moniker for b+Chargino and 3 regions labelled SRtN 1-3where tN is a moniker for t+Neutralino were optimised for the scenarios where t1 decaysto the respective particles indicated by the labels Increasing label numbers correspond toincreasingly strict selection criteria

            33 Calculations 47

            The dominant background in all signal regions arises from dileptonic tt events All threeSRtN selectons impose a requirement on the 3-jet mass m j j j of the hadronically decayingtop quark to specifically reject the tt background where both W bosons from the top quarkdecay leponically

            For increasing stop mass and increasing difference between the stop and the LSP therequirents are tightened on Emiss

            T on the ratio EmissT

            radicHT where HT is the scalar sum of the

            momenta of the four selected jets and also tightened on mT

            To further reduce the dileptonic tt background for an event characterised by two one stepdecay chains a and b each producing a missing particle C use is made of the variable

            mT 2 =min

            pCTa + pC

            T b = pmissT

            [max(mTamtb)] (315)

            where mTi is the transverse mass 1 of branch i for a given hypothetical allocation (pCTa pC

            T b)

            of the missing particle momenta Eqn 315 is extended for more general decay chains byamT 2 [81] which is an asymmetric form of mT 2 in which the missing particle is the W bosonin the branch for the lost lepton and the neutrino is the missing particle for the branch withthe observed charged lepton

            ∆φ( jet12 pmissT ) the azimuthal angle between leading or subleading jets and missing mo-

            mentum is used to suppress the background from mostly multijet events with mis-measuredEmiss

            T

            Events with four or more jets are selected where the jets must satisfy |η | lt 25 and pT gt80604025 GeV respectively

            These signal particles were then classified according to the cutflows analysis in Fig36below and compared with the paper in Appendix1 Table A2

            3333 Lepstop2

            Search for direct top squark pair production in final states with two leptons in p pcollisions at

            radics=8TeV with the ATLAS detector[34]

            Three different analysis strategies are used to search for t1 pair production Two targett1 rarr b+ χ

            plusmn1 decay and the three body t1 rarr bW χ0

            1 decay via an off-shell top quark whilst

            1The transverse mass is defined as m2T = 2plep

            T EmissT (1minus cos(∆φ)) of branch i where ∆φ is the azimuthal

            angle between the lepton and the missing transverse momentum

            33 Calculations 48

            Table 36 Signal Regions - Lepstop1

            Requirement SRtN2 SRtN3 SRbC1 SRbC2 SRbC3∆φ( jet1 pmiss

            t )gt - 08 08 08 08∆φ( jet2 pmiss

            T )gt 08 08 08 08 08Emiss

            T [GeV ]gt 200 275 150 160 160Emiss

            T radic

            HT [GeV12 ]gt 13 11 7 8 8

            mT [GeV ]gt 140 200 120 120 120me f f [GeV ]gt - - - 550 700amT 2[GeV ]gt 170 175 - 175 200mT

            T 2[GeV ]gt - 80 - - -m j j j Yes Yes - - -

            one targets t1 rarr t + χ01 to an on shell quark decay mode The analysis is arranged to look

            at complementary mass splittings between χplusmn1 and χ0

            1 smaller and larger than the W bosonmass The smaller splitting provides sensitivity to t1 three body decay

            Object reconstruction was carried out by selecting baseline electronsmuons and τrsquos withpT gt 10 GeV and |η |lt 24724 and 247 respectively Baseline jets and b-jets were selectedwith pT gt 20 GeV and |η |lt 25 and b-jets with 70 efficiency Baseline jets were removedif within ∆R lt 02 of an electron and electrons and muons were removed if within ∆R lt 04of surviving jets Signal electrons required pT gt 10 GeV and signal jets pT gt 20 GeV and|η |lt 25

            The analysis made use of the variable mT 2 [82] the stransverse mass used to mea-sure the masses of pair-produced semi-invisibly decaying heavy particles It is definedas mT 2(pT1 pT2qT ) =

            minqT1+qT2=qT

            max[mT (pT1qT1)mT (pT2qT2)] where mT indicatesthe transverse mass pT1 and pT2 are vectors and qT = qT1 + qT2 The minimisation isperformed over all possible decompositions of qT

            Events were selected if they passed either a single-electron a single-muon a double-electron a double-muon or an electron-muon trigger Events are required to have exactlytwo opposite sign leptons and at least one of the leptons or muons must have a momentumgreater than 25 GeV and the invariant mass (rest mass) of the two leptons is required to begreater than 20 GeV In order to reduce the number of background events containing twoleptons produced by the on-shell decay of the Z boson the invariant mas must be outside711-111 GeV range

            Two additional selections are applied to reduce the number of background events withhigh mT 2 arising from events with large Emiss

            T due to mismeasured jets ∆φb lt 15 and∆φ gt 10 The quantity ∆φb is the azimuthal angle between pll

            T b = pmissT + pl1

            T +Pl2T The

            33 Calculations 49

            vector pllT b is the opposite of the vector sum of all the hadronic activity in the event For WW

            and tt backgrounds it measures the transverse boost of the WW system and for the signalthe transverse boost of the chargino-chargino system The ∆φ variable is the azimuthal anglebetween the pmiss

            T vector and the direction of the closest jet

            By requiring mT 2 gt 40 GeV the tt background is the dominant background in both thesame-flavoured (SF) and different-flavoured (DF) events

            Four signal regions are then defined as shown in Table 37 SR M90 has the loosestselection requiring MT 2gt 90 GeV and no additional requirement on jets and providessensitivity to scenarios with a small difference between the masses of the top-squark andchargino SR M110 and M120 have a loose selection on jets requiring two jets with pT

            gt 20 GeV and mT 2 to be greater than 110 GeV and 120 GeV respectively They providesensitivity to scenarios with small to moderate difference between the invariant masses of thestop quark and the charginos SR M100 has a tighter jet slection and provides sensitivity toscenarios with a large mass difference between the both stop quark and the chargino and thestop quark and the neutralino

            The analysis cut regions are summarised in Fig37 below

            Table 37 Signal Regions Lepstop2

            SR M90 M100 M110 M120pT leading lepton gt 25 GeV

            ∆φ(pmissT closest jet) gt10

            ∆φ(pmissT pll

            T b) lt15mT 2 gt90 GeV gt100 GeV gt110 GeV gt120 GeV

            pT leading jet no selection gt100 GeV gt20 GeV gt20 GeVpT second jet no selection gt50 GeV gt20 GeV gt20 GeV

            To validate the code we ran the cuts shown in Appendix 1 Table A3 with the givenMadgraph simulation and compared to [83] It was not clear what baseline cuts had beenemployed in the ATLAS paper so our results were normalised to the 2 SF electrons beforecomparison Some of the ATLAS cut numbers were less than 3 and we did not produce anyevents passing these cuts but clearly the margin of error at this level was greater than thenumber of ATLAS events

            33 Calculations 50

            3334 2bstop

            Search for direct third-generation squark pair production in final states with miss-ing transverse momentum and two b-jets in

            radics= 8 TeV pp collisions with the ATLAS

            detector[31]

            Two sets of scenarios are considered In the first the coloured sbottom is the onlyproduction process decaying via b1 rarr bχ0

            1 In the second the stop is the only productionprocess decaying via t1 rarr t χplusmn

            1 and the lightest chargino (χplusmn1 ) decays via a virtual W boson

            into the final state χ01 f f prime Both scenarios are characterised by the presence of two jets

            resulting from the hadronisation of b quarks and large missing transverse momentum Theproduction of a mediator in association with top quarks followed by an invisible mediatordecay to dark matter particles has a similar final state as that considered in the ATLAS stopsearches Thus we expect the ATLAS reach to have sensitivity to the three dark mattermodels considered in this thesis

            The collider cuts program mimicked object reconstruction by selecting baseline electronswith pT gt 7 GeV and |η |lt 247 and muons with pT gt 6 GeV and |η |lt 24 Baseline jetswere selected with pT gt 20 GeV and |η |lt 28 We then removed any jets within ∆R lt 02of an electron and any electron within ∆R lt 04 of surviving jets Signal electrons andmuons were required to have pT gt 25 GeV b-jets were selected with 60 tagging efficiencyand |η |lt 25 and pT gt 20 GeV

            Two sets of signal regions are defined to provide sensitivity to the kinematic topologiesassociated with the different mass splittings between the sbottom or the stop and the neu-tralino In all cases the presence of at least one primary vertex (with at least five tracks withpT gt 04 GeV) is required Events are selected with Emiss

            T gt 150 GeV and no electrons ormuons identified in the final state For signal region selections jets with |η | lt 28 are orderedaccording to their pT and 2 jets are required to be b-tagged

            The variables are defined as follows

            bull ∆φmin is the minimal azimuthal angle ∆φ between any of the three leading jets andpmiss

            T

            bull me f f (k) = sumki=1(p jet

            T )i +EmissT where the index refers to the pT ordered list of jets

            bull HT3 is defined as the scalar sum of the pT of the n leading pT ordered jets withoutincluding the 3 leading jets HT3 = sum

            ni=4(p jet

            T )i

            bull mbb is the invariant mass of the two b-tagged jets in the event

            33 Calculations 51

            bull mCT is the contransverse mass [82] and is a kinematical variable that can be usedto measure the masses of pair produced semi-invisibly decaying particles For twoidentical decays of heavy particles into two invisible particles v1 and v2 and twoinvisible particles mCT is defined as m2

            CT (v1v2) = [ET (v1)+ET (v2)]2 minus [pT (v1)minus

            pT (v2)]2 where ET =

            radicp2

            T +m2 The the contransverse mass is invariant under equaland opposite boosts of the parent particles in the transverse plane and is bounded fromabove by an analytical expression of the particle masses For production of sbottompairs the bound is given by mmax

            CT =m2(b)minusm2(χ0

            1 )

            m(b) and for tt events the bound is 135

            GeV A similar equation can be written for stop pairs in term of the stop and charginomasses

            A definition of the signal regions is given in the Table38

            Table 38 Signal Regions 2bstop

            Description SRA SRBEvent cleaning All signal regions

            Lepton veto No emicro after overlap removal with pT gt7(6) GeV for e(micro)Emiss

            T gt 150 GeV gt 250 GeVLeading jet pT ( j1) gt 130 GeV gt 150 GeVSecond jet pT ( j2) gt 50 GeV gt 30 GeVThird jet pT ( j3) veto if gt 50 GeV gt 30 GeV

            ∆φ(pmissT j1) - gt 25

            b-tagging leading 2 jets(pT gt 50 GeV |η | lt 25)

            2nd- and 3rd- leading jets(pT gt 30 GeV|η | lt 25)

            ∆φmin gt 04 gt 04Emiss

            T me f f (k) EmissT me f f (2) gt 025 Emiss

            T me f f (3) gt 04mCT gt 150200250300350 GeV -HT3 - lt 50 GeVmbb gt200 GeV -

            The analysis cuts are summarised in Table A4 of Appendix 1

            3335 ATLASMonobjet

            Search for dark matter in events with heavy quarks and missing transverse momentumin pp collisions with the ATLAS detector[30]

            Events were selected with high missing transverse momentum when produced in associa-tion with one or more jets identified as containing b quarks Final states with top quarks wereselected by requiring a high jet multiplicity and in some cases a single lepton The research

            33 Calculations 52

            studied effective contact operators in the form of a scalar operator Oscalar = sumqmqMN

            lowastqqχχ

            where MNlowast is the effective mass of the coupling and the quark and DM fields are denoted by

            q and χ respectively These are the equivalent of the simplified models in this paper with themediator integrated out

            Object reconstruction selected baseline electrons muons jets and τrsquos with pT gt 20 GeVand |η |lt 2525 and 45 respectively Signal jets were selected by removing any jets within∆R lt 02 of an electron and electrons and muons were selected by removing them if theywere within ∆R lt 04 of surviving jets b-jets were tagged with an efficiency of 60

            Only signal regions SR1 and SR2 were analysed

            The analysis split events into signal regions SR1 and SR2 according to the followingCriteria

            Table 39 Signal Region ATLASmonobjet

            Cut SR1 SR2Jet multiplicity n j 1minus2 3minus4

            bminusjet multiplicity nb gt 0 (60 eff) gt 0 (60 eff)Lepton multiplicity nl 0 0

            EmissT gt300 GeV gt200 GeV

            Jet kinematics pb1T gt100 GeV pb1

            T gt100 GeV p j2T gt100 (60) GeV

            ∆φ( j1EmissT ) gt 10 i = 12 gt 10 i = 14

            Where p jiT (pbi

            T ) represent the transverse momentum of the i-th jet b-tagged jet respec-tively If there is a second b-tagged jet the pT must be greater than 60 GeV

            3336 CMSTop1L

            Search for top-squark pair production in the single-lepton final state in pp collisionsat

            radics=8 TeV[41]

            This search was for pair production of top squarks in events with a single isolatedelectron or muon jets large missing transverse momentum and large transverse mass

            (MT =radic

            2EmissT pl

            T (1minus cos(∆φ))) where plT is the leptonrsquos transverse momentum and ∆(φ)

            is the difference between the azimuthal angles of the lepton and EmissT The 3 models

            considered should have some visibility at certain parameter combinations in this regionThe search focuses on two decay modes of the top squark leading to the two processespp rarr t tlowast rarr tt χ0

            1 χ01 rarr bbW+Wminusχ0

            1 χ01 and pp rarr t tlowast rarr bbχ

            +1 χ

            minus1 rarr bbW+Wminusχ0

            1 χ01 The

            33 Calculations 53

            lightest neutralino χ01 is considered the lightest supersymmetric particle and escapes without

            detection The analysis is based on events where one of the W bosons decays leptonicallyand the other hadronically resulting in one isolated lepton and 4 jets two of which originatefrom b quarks The two neutralinos and the neutrino from the W decay can result in largemissing transverse momentom The kinematical end point of MT is lt the mass of the Wboson However for signal events the presence of LSPs in the final state allows MT to exceedthe mass of the W boson The search is therefore for events with large MT The dominantbackground with large MT arises from tt decays where W bosons decay leptonically withone of the leptons not being identified

            To reduce the dominant tt background use was made of the MWT 2 variable defined as

            the minimum mother particle mass compatible with the transverse momenta and mass shellconstraints [84]

            Object reconstruction required electrons and muons to have pT gt 30 and 25 GeV respand |η |lt 25 and 21 respectively Baseline jets were required to have pT gt 30 and |η |lt 4and b-jets were tagged with an efficiency of 75 for |η |lt 24

            Events were chosen if they contained one isolated electron or muon at least 4 jets and atleast one b-tagged jet and Emiss

            T gt 100 GeV MT gt 160 GeV MWT 2 was required to be greater

            than 200 GeV and the smallest azimuthal angle difference between baseline jets and EmissT

            gt12

            Chapter 4

            Calculation Tools

            41 Summary

            Figure 41 Calculation Tools

            The simplified models were implemented in Feynrules [4] and the model files importedinto MicrOmegas v425 which was used to calculate the log-likelihoods of the astrophysicalobservables MicrOmegas was modified to allow only s-channel decays in both relic densityand gamma ray calculations Direct detection constraints were easy to implement for thescalar and vector models by coding the analytical expression for the spin independent WIMP-nucleon scattering cross section but due to the momentum dependence of the WIMP-nucleon

            42 FeynRules 55

            scattering cross section of the Majorana fermion model LUXCalc v101 [79] was modifiedfor this model to implement momentum dependent cross sections [85] We used MicrOmegasv425 to calculate relic density values and to produce gamma ray spectra for comparisonwith the Fermi-LAT results Parameter scans were carried out using MultiNest v310 withthe parameter ranges given in Table 51 and using astrophysical constraints only Oncethese scans had converged a random sample of 900 points was chosen (weighted by theposterior probability) and then used to calculate collider constraints using the MadgraphMonte Carlo generator This included comparison with mediator production and decay limitsfrom the LHC experiments and the reproduction of searches for dark matter plus other finalstate objects which required dedicated C++ code that reproduces the cut and count analysesperformed by CMS and ATLAS

            42 FeynRules

            FeynRules is a Mathematicareg package that allows the calculation of Feynman rules inmomentum space for any QFT physics model The user needs to provide FeynRules withminimal information to describe the new model contained in the model-file This informationis then used to calculate a set of Fenman rules associated with the Lagrangian The Feynmanrules calculated by the code can then be used to implement new physics models into otherexisting tools such as Monte Carlo (MC) generators via a set of interfaces developed andmaintained by the MC authors

            Feynrules can be run with a base model = SM (or MSSM) and user defined extensions tothe base model

            43 LUXCalc

            LUXCalc is a utility program for calculating and deriving WIMP-nucleon coupling limitsfrom the recent results of the LUX direct search dark matter experiment The program hasbeen used to derive limits on both spin-dependent and spin-independent WIMP-nucleoncouplings and it was found under standard astrophysical assumptions and common otherassumptions that LUX excludes the entire spin-dependent parameter space consistent witha dark matter interpretation of DAMArsquos anomalous signal and also found good agreementwith published LUX results [79]

            We modified the program for the Majorana fermion model to implement momentumdependent cross -sections [85]

            44 Multinest 56

            44 Multinest

            Multinest is an efficient Bayesian Inference Tool which uses a simultaneous ellipsoidalnested sampling method a Monte Carlo technique aimed at an efficient Bayesian Evidencecalculation with posterior inferences as a by product

            Bayes theorem states that

            Pr(θ |DH) =Pr(D|θ H)Pr(θ |H)

            Pr(D|H) (41)

            Here D is the data and H is the Hypothesis Pr(θ |DH) is the posterior probability ofthe parameters Pr(D|θ H) = L(θ) is the likelihood Pr(θ |H) is the prior and Pr(D|H) isthe evidence

            The evidence Pr(D|H) =int

            Pr(θ |DH)Pr(θ |H)d(θ) =int

            L(θ)Pr(θ |H)d(θ) Thus ifwe define the prior volume dX = Pr(θ |H)d(θ) and

            X(λ ) =int

            L(θ)gtλ

            Pr(θ |H)d(θ) (42)

            where the integral extends over the region of parameter space within the iso likelihoodcontour L(θ) = λ then the evidence integral can be written

            int 10 L (X)dX where L (X) the

            inverse of equation 42 and is a monotonically decreasing function of X (the prior volumewhich is 1 for the total sample space and 0 for a single point)

            Thus if one can evaluate likelihoods L (Xi) where 0 lt Xn lt lt X2 lt X1 lt X0 = 1the evidence can be estimated as sumLiwi where wi = 5(Ximinus1 minusXi+1) Ellipsoidal samplingapproximates the iso likelihood contour by a D dimensional covariance matrix of the currentset of active points New points are selected from the prior within this ellipsoidal bounduntil one is obtained that has a likelihood greater than the likelihood of the removed lowestlikelihood point This method is extended for multimodal sampling where there may bemore than one local maximum by clustering points that are well separated and constructingindividual ellipsoidal bounds for each cluster For more details consult [7]

            The results of the analysis are expressed as a data file containing the sample probability-2loglikelihood followed by the parameters of the model for the particular sampling Theprobabilities are normalised to total 10

            45 Madgraph 57

            45 Madgraph

            Madgraph is a general purpose matrix element calculator and Monte Carlo generator Basedon user-supplied Feynman rules (eg the output of Feynrules) the package calculates the tree-level matrix elements for any processes that the user specifies and generates samples of LHCevents including an accurate simulation of the proton collisions themselves (which requires aknowledge of the parton distribution functions that determine the relative proportion of eachparton inside the nucleus) Hadronisation of final state quarks is not possible in Madgraphand hence the code must be used in conjunction with Pythia which performs a parton shower

            The Madgraph output was automatically showered by Pythia (Pythia describes partonradiation as successive parton emissions using Markov chain techniques based on Sudakovform factors and this description is only formally correct for the soft and collinear emissions)A Madgraph simulation is at the level of the fundamental particles (quarks and leptons)Madgraph generates a limited set of matrix element processes at LO representing hard (highenergy) emissions and passes these to Pythia for showering The final state partons in theMadgraph simulation must be matched to the Pythia final state partons to avoid doublecounting

            The output from Pythia is automatically fed through Delphes where basic collider cutscan be performed (calculation of the numbers of particles and jets that would be observed atdifferent energies and locations within the collider) The output of the Delphes simulation is afile containing reconstructed events that can be analysed using the standard ROOT framework(the most popular data analysis package in particle physics) [86]

            The parameters that gave the best fits to the direct [16] and indirect DM experiments [87]and relic density [59] are ranked by posterior probability from the Multinest [7] output

            The parameters in the Madgraph simulations are stored in files in the cards sub directoryand in particular the run card param card and Delphes card record items relevent to thesimulation

            In the run card the number of events the Beam 1 and 2 total energy in GeV (4000 each)and the type of jet matching were modified (to MLM type matching where the final statepartons in Madgraph are clustered according to the KT jet algorithm to find the equivalentparton shower history of the event there is a minimum cutoff scale for the momentum pT

            given by the xqcut parameter) The maxjetflavour (=5 to include b quarks) and xqcut = 30GeV parameters were also modified

            46 Collider Cuts C++ Code 58

            The param card was modified to specify the masses of the new particles and the autodecay modes The Delphes card is overwritten with either the ATLAS or CMS defaultcards which record tracking efficiencies and isolation momentum and energy resolutionscalorimeter details jet finder and b and τ -tagging parameters However the isolation andtagging in this study were done by the collider cuts C++ code since multiple sets of cutswere applied to the same output from Pythia

            When performing simulations of the three dark matter models it was necessary to definethe proton and jets to contain b quarks as this is not the default in Madgraph The processesgenerated included 1 and 2 jets

            46 Collider Cuts C++ Code

            Because of the volume of simulations required for the collider constraints we used C++code (originally developed by Martin White and Sky French and modified by me) that readsthe ROOT files and outputs the numbers of events that would have been observed under thechosen collider experiments had the model been true for the given parameters Thus the sameoutput files can be used for all of the collider cuts and hence no cuts were performed throughthe Delphes cards

            In order to validate the C++ code we replicated the observed events with the papersdescribing the various collider searches (see Appendix)

            Chapter 5

            Majorana Model Results

            51 Bayesian Scans

            To build some intuition we first examined the constraining effect of each observable on itsown We ran separate scans for each constraint turned on and then for all the constraintstogether In Fig 51 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together The paramaters were scanned over the range in Table 51 below

            Table 51 Scanned Ranges

            Parameter mχ [GeV ] mS[GeV ] λt λb λτ

            Range 0minus103 0minus103 10minus5 minus10 10minus5 minus10 10minus5 minus10

            In Fig51 we can see that the preferred DM mass is centered around 35 GeV primarily dueto the influence of the Fermi-LAT gamma ray constraint whilst the relic density constraintallows a wide range of possible mediator and DM masses Direct detection (LUX) constraintssplit the mass regions into a top left and bottom right region and together these split theoverall mass parameter regions into two regions by mediator mass but with DM mass around35 GeV Later studies using a lower excess gamma ray spectrum tend to increase the preferredDM mass up to 50 GeV [76] and [88]

            The marginalised posterior probabilities for all combinations of parameters taken twoat a time are shown in Fig52 These plots have all constraints (Gamma LUX and RelicDensity) applied together The idea of these graphs is to study the posterior probabilities ofcombinations of two paramaters taken at a time (the off-diagonal graphs) The diagonal plotsshow marginalised histograms of a single parameter at a time- this gives an indication of thespread of each parameter The plots show that mS and λb show a high degree of correlationfor mS gt 100 GeV but there is also a range of values of mS around 35 GeV which are

            51 Bayesian Scans 60

            1 0 1 2 3 4log10(mχ)[GeV]

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            Figure 51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether

            51 Bayesian Scans 61

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            log10(λτ)

            Figure 52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter

            52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 62

            possible for λb around 001 λt and λτ show a wide range of possible values for two possibleranges of mS lt 35 GeV and greater than 100 GeV mχ shows a narrow range of possiblevalues around 35 GeV for all values of the other parameters

            52 Best fit Gamma Ray Spectrum for the Majorana FermionDM model

            We can see from the Fig51(a) scan that the maximum log-likelihood Fermi-LAT gammaray spectrum on its own shows values that centre around 35 GeV This is not changed byadding a relic density constraint or a direct detection (LUX) constraint

            The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraintsis shown in Fig53 The parameters for this point were

            Table 52 Best Fit Parameters

            Parameter mχ [GeV ] mS[GeV ] λt λb λτ

            Value 3332 49266 0322371 409990 0008106

            10-1 100 101 102

            E(GeV)

            10

            05

            00

            05

            10

            15

            20

            25

            30

            35

            E2dφd

            E(G

            eVc

            m2ss

            r)

            1e 6

            Best fitData

            Figure 53 Gamma Ray Spectrum

            The relic density (Ωh2) at this point has a value of 0135 but the point is separated onlyslightly in posterior probability from points with a value of 01199 with mS lt 10 GeV (iethe posterior probability is almost the same differing by less than 1)

            To better understand the interplay of the different constraints the three graphs in Fig54show the log likelihood surfaces and individual best fit points (best log likelihoods) for the γ

            and Ω scans seperately and then combined The diagrams each choose a pair of couplings(fixing the remaining parameters) and display the 2d surfaces of minuslog(χ2) (proportional to

            52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 63

            the log likelihoods of the coupling combinations) obtained by varying the pair of couplingsThese graphs do not show marginalised posterior probabilities but rather are based on the-2log likelihood calculations directly from MicrOmegas for the given coupling combinationsWe have termed this value as χ2 for convenience and used the log of this value to give abetter range for the colours These log likelihood values are what drive the MultiNest searchof parameter space for the best parameter combinations

            The graphs show a complex interplay of the different constraints resulting often in acompromise parameter combination The arrow on each graph points to the maximum point We have chosen a single representative set of couplings near the best overall fit point for allthe graphs

            52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 64

            00 05 10 15 20 25 30

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            Both Maximum at mχ=8317 GeV mS=2884GeV The best fit point for these couplings maybe above below or between the individual bestfit points but will on average be between thepoints

            Figure 54 Plots of log likelihoods by individual and combined constraints Masses in GeV

            53 Collider Constraints 65

            53 Collider Constraints

            531 Mediator Decay

            1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof τ leptons in pp collisions [44] Refer also to Chapter 3 for details

            We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

            The results of the calculations are shown in the graph Fig55 and can be compared withthe limits in the table from the paper [44] in Fig 56 The black line is the observed ATLASlimit

            Figure 55 σ lowastBr(σ rarr ττ) versus Mass of Scalar

            0 200 400 600 800

            mS[GeV]

            10

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            ))[pb]

            Observed LimitLikely PointsExcluded Points

            0

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            0 5 10 15 20 25 30 35 40 45

            We can see that only one point here exceeded the limit and the points showed a widespread over σ(bbS)lowastBr(σ rarr ττ) by mS space which is consistent with the spread of pointson the MultiNest scan on the mS by λτ plane causing a spread of branching ratios to the τ

            quark It may seem odd that only one point was excluded here but this point was sampledwith a very low probability having high coupling constants of 597 to the τ quark and 278 tothe t quark and only just exceeds the limit As such it can be explained as a random outlier

            53 Collider Constraints 66

            Figure 56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar

            2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

            This scan calculates the contributions of b-quark associated Higgs boson productionreproduced from [43] Refer to Chapter 3 for details

            We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

            The cross section times branching ratios to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig57 The black line is the CMSLimit

            Figure 57 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar

            0 200 400 600 800

            mS[GeV]

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            53 Collider Constraints 67

            The results of this scan were compared to the limits in [89] with the plot shown inFig58

            Figure 58 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

            We can see that there are a number of exclusions across the range of mS which occuredfor high values of the coupling constants gt 1 for λbλt λtau - these can be seen also on Fig59The excluded parameter combinations represent less than 2 of the total sampled number

            532 Collider Cuts Analyses

            We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude and the results are shown against theposterior probabilities of the parameter combinations in Fig 59

            The circles in Figure 59 show which points were excluded by the collider cuts as well asthe limits on cross sections times branching ratios to each of τ+τminus and bb

            All the excluded points under the heading Collider Cuts showed exclusion by the Lepstop0experiment (Search for direct pair production of the top quark in all hadronic final states

            53 Collider Constraints 68

            0 1 2 3

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            Figure 59 Excluded points from Collider Cuts and σBranching Ratio

            53 Collider Constraints 69

            [32]) Some of these same points were also excluded by Lepstop2 2bstop CMSTop andATLASMonobjet experiments There were expected events at other points but none weresignificant at the 95 exclusion level These significant exclusions occurred mainly wherethe mass of the mediator was over 100 GeV and higher values of the couplings to b and tquarks but there were a few points with mediator mass around 25 GeV The exclusions withmediator mass around 25 GeV were all accompanied by couplings to t quarks of order 10which is probably non-perturbative

            Looking at the exclusions in Fig 59 we can see that these occur mainly in the regionof higher mediator mass gt 100 GeV and lt 400 GeV These exclusions come from theastrophysical constraints (Gamma) which show a strong correlation between λb and mS andhigh values of λt gt 1 However these mass regions do not overlap at the same couplingcombinations as the high likelihood region The green and white circles correspond to theexcluded points in Figs 57 and 58 respectively

            The remaining regions of highest likelihood can be seen to be where 0001 λb lt 01accompanied by λt lt 0001 and mass of the mediator mS lt 10 GeV or 400 GeV lt mS lt1000GeV with mχ sim 35 GeV

            The plot of λt by λb shows the significant regions do not overlap with the collider cutsand cross section constraints since the latter occur at high values of either coupling while thelikely region has low values of these couplings (also accompanied by low couplings to λτ )

            Chapter 6

            Real Scalar Model Results

            61 Bayesian Scans

            To build some intuition we first examined the constraining effect of each observable on itsown We ran seperate scans for each constraint turned on and then for all the constraintstogether In Figure 61 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together

            In Figure 61 we can see that there are is a narrow region which dominates when allconstraints are taken into account This centers on mχ=9 GeV and a range of values for mS

            from 35 GeV to 400 GeV The region is not the highest likelihood region for the Fermi-LATgamma ray spectrum on its own (as can be seen from the individual plot - there is a narrowband to the left of the highest likelihood band) nor for the LUX likelihood or relic densitybut represents a compromise between the three constraints

            The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig62 The diagonal plots show marginalised histograms of a singleparameter at a time

            61 Bayesian Scans 71

            05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

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            Figure 61 Real Scalar Dark Matter - By Individual Constraint and All Together

            61 Bayesian Scans 72

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            Figure 62 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter

            62 Best fit Gamma Ray Spectrum for the Real Scalar DM model 73

            62 Best fit Gamma Ray Spectrum for the Real Scalar DMmodel

            We can see from Fig61 that the maximum log-likelihood Fermi-LAT Gamma ray spectrumon its own shows values of mχ that centre around 35 GeV However the LUX constraintin particular moves the best fit to a point around 9 GeV- the tight LUX constraint onlydisappears for low masses and at that point the proportion of τ+τminus annihilation needs toincrease to fit the constraints

            The gamma ray spectrum at the points of best fit for Fermi-LAT including all constraintsis shown in Fig63 The parameters for these points were respectively and involve decays tomainly τ and b quarks respectively

            Table 61 Best Fit Parameters

            Parameter mχ [GeV ] mS[GeV ] λt λb λτ

            Value 932 3526 000049 0002561 000781

            10-1 100 101 102

            E(GeV)

            10

            05

            00

            05

            10

            15

            20

            25

            30

            35

            E2dφdE

            (GeVc

            m2ss

            r)

            1e 6

            Best fitData

            Figure 63 Gamma Ray Spectrum

            This is not a particularly good fit and it occcurs at a relic density Ωh2 of 335 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

            63 Collider Constraints 74

            63 Collider Constraints

            631 Mediator Decay

            1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44]

            We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

            The results of the scan are shown in the graph Fig64 and can be compared with thelimits in the table from the paper [44] in Fig 56The black line is the observed ATLAS limit

            Figure 64 σ lowastBr(σ rarr ττ) versus Mass of Scalar

            0 200 400 600 800

            mS[GeV]

            8

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            Observed LimitLikely PointsExcluded Points

            050

            100150200250300350

            0 10 20 30 40 50 60

            We can see that there is a narrow band in the region of interest in the σ(bbS)lowastB(S gt ττ)

            by mS plane due to the high degree of correlation between mS and λτ seen in the likelihoodscan Values for mS gt 300Gev are excluded

            2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

            We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinations

            63 Collider Constraints 75

            randomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

            The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig65 The black line is the CMSlimit

            Figure 65 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

            0 200 400 600 800

            mS[GeV]

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            050

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            The results of this scan were compared to the limits in [89] with the plot shown inFig58

            We can see a similar shape for the region of interest to that in Fig64 due to the highcorrelation between mS and λb over the region preferred by astrophysical constraints resultingin branching ratios highly correlated to ms Only one point is excluded with mS gt 500GeV

            632 Collider Cuts Analyses

            We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expected

            63 Collider Constraints 76

            with 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 66

            We can see that collider cuts show some exclusions only for masses mχ below the highlikelihood region but these exclusions do not occur at the same λb and λt combinations asseen from the plot of these two couplings except for a small region at the highest mS andcouplings

            All but one of the collider cut exclusions were excluded by the Lepstop0 experiment (thiswas excluded by 2bstop with high coupling to b quarks of 73) A number of the excludedpoints were also excluded by Lepstop2 2bstop CMSTopDM1L and ATLASMonobJet-(around 30 of the total for each)

            63 Collider Constraints 77

            0 1 2 3

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            Figure 66 Excluded points from Collider Cuts and σBranching Ratio

            Chapter 7

            Real Vector Dark Matter Results

            71 Bayesian Scans

            In the Figure 71 we can see that the preferred DM mass is centered around 35 GeV primarilydue to the influence of the Fermi-LAT gamma ray constraint with relic density having a widerange of possible mediator and DM masses The result of combining all the constraints is todefine a small region around mχ = 8 GeV and mS = 20 GeV which dominates the posteriorprobability

            The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig72 The diagonal plots show marginalised histograms of a singleparameter at a time

            71 Bayesian Scans 79

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            Figure 71 Real Vector Dark Matter - By Individual Constraint and All Together

            71 Bayesian Scans 80

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            b)

            00 05 10 15 20 25 30

            log10(mχ)[GeV]

            5

            4

            3

            2

            1

            0

            1

            log 1

            0(λ

            τ)

            00 05 10 15 20 25 30

            ms[Gev]5 4 3 2 1 0 1

            log10(λt)5 4 3 2 1 0 1

            log10(λb)5 4 3 2 1 0 1

            log10(λτ)

            Figure 72 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter

            72 Best fit Gamma Ray Spectrum for the Real Vector DM model 81

            72 Best fit Gamma Ray Spectrum for the Real Vector DMmodel

            The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraints isshown in Fig73

            Table 71 Best Fit Parameters

            Parameter mχ [GeV ] mS[GeV ] λt λb λτ

            Value 8447 20685 0000022 0000746 0002439

            10-1 100 101 102

            E(GeV)

            10

            05

            00

            05

            10

            15

            20

            25

            30

            35

            E2dφdE

            (GeVc

            m2s

            sr)

            1e 6

            Best fitData

            Figure 73 Gamma Ray Spectrum

            This is not a particularly good fit and it occcurs at a relic density Ωh2 of 025 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

            73 Collider Constraints

            731 Mediator Decay

            1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44] (See Chapter 3 for more details)

            We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

            73 Collider Constraints 82

            The results of the scan are shown in the graph Fig74 and can be compared with thelimits in the table from the paper [44] in Fig 56

            Figure 74 σ lowastBr(σ rarr ττ) versus Mass of Scalar

            0 200 400 600 800

            mS[GeV]

            8

            6

            4

            2

            0

            2

            log 1

            0(σ

            (bbS

            )lowastB

            (Sgtττ

            ))[pb]

            Observed LimitLikely PointsExcluded Points

            0100200300400500600700800

            0 20 40 60 80 100120140

            We can see that the region of highest likelihood is limited to mS lt 100 GeV and a narrowrange of couplings to λτ giving a narrow region of interest which falls well below the limit

            2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

            We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

            The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig75 The black line is the CMSlimit

            The results of this scan were compared to the limits in [89] with the plot shown in Fig58

            We can see that again the region of interest for mS is below 100 GeV and the points fallwell below the limit

            73 Collider Constraints 83

            Figure 75 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

            0 200 400 600 800

            mS[GeV]

            8

            6

            4

            2

            0

            2

            4

            log

            10(σ

            (bS

            +X

            )lowastB

            (Sgt

            bb))

            [pb]

            Observed LimitLikely PointsExcluded Points

            0100200300400500600700800

            0 20 40 60 80 100120140

            732 Collider Cuts Analyses

            We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 76

            We can see from Figure 76 that like the real scalar model the region of highest likelihoodhas λb and λt lt 001 while the excluded points from collider cuts and cross sections areabove 01 Most of the excluded points have mχ gt than the values in the region of highestlikelihood and all points are widely separated in parameter space from this region

            Once again 89 of excluded points were excluded by Lepstop0 with many also beingexcluded by the other experiments (50 by Lepstop2 and 56 by CMSTopDM1L 35 byLepstop2 and 20 by ATLASMonobjet)

            73 Collider Constraints 84

            0 1 2 3

            log10(mχ)[GeV]

            0

            1

            2

            3

            log 1

            0(m

            s)[GeV

            ]Collider Cuts

            σ lowastBr(σgt bS+X)

            σ lowastBr(σgt ττ)

            (a) mχ by mS

            5 4 3 2 1 0 1

            log10(λt)

            0

            1

            2

            3

            log 1

            0(m

            s)[GeV

            ](b) λt by mS

            5 4 3 2 1 0 1

            log10(λb)

            5

            4

            3

            2

            1

            0

            1

            log 1

            0(λ

            t)

            (c) λb by λt

            5 4 3 2 1 0 1

            log10(λb)

            5

            4

            3

            2

            1

            0

            1

            log 1

            0(λ

            τ)

            (d) λb by λτ

            5 4 3 2 1 0 1

            log10(λt)

            5

            4

            3

            2

            1

            0

            1

            log 1

            0(λ

            τ)

            (e) λt by λτ

            5 4 3 2 1 0 1

            log10(λb)

            0

            1

            2

            3

            log 1

            0(m

            s)[GeV

            ]

            (f) λb by mS

            Figure 76 Excluded points from Collider Cuts and σBranching Ratio

            Chapter 8

            Conclusion

            We first conducted Bayesian scans over mass and coupling parameters for three simplifiedmodels - Majorana fermion real scalar and real vector using direct detecion (LUX) indirectdetection (GCE γ rays) and relic density constraints to find the regions of highest likelihoodWe ran these scans one constraint at a time and then together to better understand the interplayof the different constraints on best fit and likely parameter regions

            We also developed and tested C++ collider cut software specifically designed to mimickthe cuts applied in a number of ATLAS and CMS experiments that searched for Emiss

            T inconnection with specific beyond the standard model (BSM) theories of Dark Matter

            We then selected parameter combinations by randomly sampling the scanned resultsbased on the posterior probability distribution of the points and applied collider cuts andcross section times branching ratio calculations to these points to see if there would havebeen any sensitivity from the ATLAS and CMS experiments

            We found that the Majorana fermion model showed the best overall fit to all the constraintswith relic density close to the experimental value of 01199 and a low χ2 error on the gammaspectrum fit The preferred mass mχ was around 35 GeV and mS ranging from lt10 GeV to500 GeV The smaller masses give a better fit for relic density and also are well separatedfrom current collider constraints mainly due the the fact that the exclusions occur mainlywith higher couplings to t quarks (greater than 1) which is consistent with the fact that manyof the experiments were targeting dark matter pair production associated with t quark decaysNevertheless there were some exclusions that did fall in the region sampled by randomselection from the posterior probability distribution

            The preferred parameter regions were also consistent with the observation in [20] thatthe Majorana model showed the best overall fit with similar mass constraints We find thatthe couplings to the SM fermions should all be less that 001

            86

            The scalar and vector DM models give slightly less good fits especially to the currentrelic density estimate of 01199 but again the current collider constraints are well separatedfrom the best fit region and in fact none of the points randomly sampled from the posteriordistribution showed any exclusion The best fit parameter combination favours couplings toSM fermions of less than 001

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            Appendix A

            Validation of Calculation Tools

            Lepstop0Search for direct pair production of the top squarkin all hadronic final states in proton=proton collisions atradic

            s=8 TeV with the ATLAS detector [32]

            Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1) mt1=600 GeV mn1=1 GeVWhere t1 is the top squark and n1 is the neutralino of the MSSM

            94

            Table A1 0 Leptons in the final state

            Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 10000 460 460 100Emiss

            T gt 130GeV 8574 395 451 88Leptonveto 6621 307 285 108Emiss

            T gt 150GeV 6316 293 294 100Jet multiplicityand pT 1099 48 51 94|∆φ( jet pmiss

            T )|gt π5 902 40 42 96gt= 2b jets 396 19 18 106τ veto 396 19 18 106mbmin

            T gt 175GeV 270 13 13 105SRA1 79 4 4 104SRA2 70 3 3 93SRA3 67 3 3 97SRA4 52 2 2 91SRC exactly5 jets 1297 50 60 83SRC |∆φ( jet pmiss

            T )|gt π5 1082 50 50 99SRC gt= 2b jets 430 20 20 97SRC τ veto 430 20 20 97SRC |∆φ(bb)|gt 02π 367 17 17 99SRC1 246 12 11 105SRC2 177 9 8 116SRC3 152 8 7 116

            95

            Lepstop1Search for direct top squark pair production infinal states with one isolated lepton jets and missing trans-verse momentum in

            radics = 8 TeV pp collisions using 21 f bminus1

            of ATLAS data[33]

            Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1)(t1 gt b x+1 )(t1 gt b xminus1 )mt1=650 GeV mn1=1 GeV Where t1 is the top squark n1 is the neutralino xplusmn1 are thecharginos

            96

            Table A2 1 Lepton in the Final state

            Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 100000 99990 99990 100Electron(= 1signal) 11558 11557 11785 984 jets(80604025) 5187 5186 6271 83gt= 1btag 4998 4997 5389 93Emiss

            T gt 100GeV [allSRs] 4131 4131 4393 94Emiss

            T radic

            HT gt 5 [allSRs] 4026 4026 4286 94∆φ( jet2 pmiss

            T )gt 08 [allSRs] 3565 3565 3817 93∆φ( jet1Pmiss

            T )gt 08 [not SRtN2] 3446 3446 3723 93Emiss

            T gt 200GeV (SRtN2) 2027 2027 2186 93Emiss

            T radic

            HT gt 13(SRtN2) 1156 1156 1172 99mT gt 140GeV (SRtN2) 839 839 974 86Emiss

            T gt 275GeV (SRtN3) 948 948 965 98Emiss

            T radic

            HT gt 11(SRtN3) 932 932 937 99mT gt 200GeV (SRtN3) 456 456 566 81Emiss

            T gt 150GeV (SRbC1minusSRbC3) 2803 2803 3080 91Emiss

            T radic

            HT gt 7(SRbC1minusSRbC3) 2742 2742 3017 91mT gt 12GeV (SRbC1minusSRbC3) 1917 1917 2347 82Emiss

            T gt 160GeV (SRbC2SRbC3) 1836 1836 2257 81Emiss

            T radic

            HT gt 8(SRbC2SRbC3) 1788 1788 2175 82me f f gt 550GeV (SRbC2) 1741 1741 2042 85me f f gt 700GeV (SRbC3) 1075 1075 1398 77Muon(= 1signal) 11379 11378 11975 954 jets(80604025) 5264 5263 6046 87gt= 1btag 5101 5100 5711 89Emiss

            T gt 100GeV [allSRs] 4199 4199 4770 88Emiss

            T radic

            HT gt 5 [all SRs] 4115 4115 4653 88∆φ( jet2 pmiss

            T )gt 08 [all SRs] 3622 3622 4077 89∆φ( jet1 pmiss

            T )gt 08 [not SRtN2] 3515 3515 3970 89Emiss

            T gt 200GeV (SRtN2) 2136 2136 2534 84Emiss

            T radic

            HT gt 13(SRtN2) 1261 1261 1441 87mT gt 140GeV (SRtN2) 928 928 1033 90Emiss

            T gt 275GeV (SRtN3) 1036 1036 1274 81Emiss

            T radic

            HT gt 11(SRtN3) 1018 1018 1235 82mT gt 200GeV (SRtN3) 497 497 612 81Emiss

            T gt 150GeV (SRbC1minusSRbC3) 2891 2891 3380 86Emiss

            T radic

            HT gt 7(SRbC1minusSRbC3) 2821 2821 3290 86mT gt 120GeV (SRbC1minusSRbC3) 1977 1977 2290 86Emiss

            T gt 160GeV (SRbC2SRbC3) 1913 1913 2220 86Emiss

            T radic

            HT gt 8(SRbC2SRbC3) 1863 1863 2152 87me f f gt 550GeV (SRbC2) 1826 1826 2116 86me f f gt 700GeV (SRbC3) 1131 1131 1452 78

            97

            Lepstop2Search for direct top squark pair production infinal states with two leptons in

            radics =8 TeV pp collisions using

            20 f bminus1 of ATLAS data[83][34]

            Simulated in Madgraph with p p gt t1 t1(t1 gt b w+ n1)(t1 gt b wminusn1) mt1=180 GeV mn1=60 GeV Where t1 is the top squark n1 is the neutralino b1 is the bottom squark

            Table A3 2 Leptons in the final state

            Cut RawMe Scaled ATLAS MeATLAS PercentNoCuts 100000 324308 199000 1632BaselineCuts 3593 11652 7203 1622SF Signal Leptons 1699 5510 5510 1002OSSF Signal Leptons 1632 5293 5233 101mll gt 20Gev 1553 5037 5028 100leadinglepton pT 1479 4797 5048 95|mll minusmZ|gt 20Gev 1015 3292 3484 94∆φmin gt 10 850 2757 2160 128∆φb lt 15 680 2205 1757 126SRM90 [SF ] 13 42 60 70SRM120 [SF ] 0 0 19 0SRM100 +2 jets [SF ] 0 0 17 0SRM110 +2 jets [SF ] 0 0 1 02DF Signal Leptons 1737 5633 7164 792OSDF Signal Leptons 1670 5416 5396 100mll20Gev 1587 5147 5136 100leadinglepton pT 1501 4868 4931 99∆φmin gt 10 1247 4044 3122 130∆φb lt 15 1010 3276 2558 128SRM90 [SF ] 21 68 95 72SRM120 [SF ] 0 0 23 0SRM100 +2 jets [SF ] 0 0 19 0SRM110 +2 jets [SF ] 0 0 14 0

            98

            2bstopSearch for direct third-generation squark pair pro-duction in final states with missing transverse momentumand two b-jets[31]

            Simulated in Madgraph with p p gt b1 b1(b1 gt b n1)(b1 gt b n1)

            SRAmb1=500 GeV mn1=1 GeV SRBmb1=350 GeV mn1=320 GeVWhere b1 is the bottom squark and n1 is the neutralino

            Table A4 2b jets in the final state

            Cut RawMe Scaled ATLAS MeATLAS PercentSRA Nocuts 100000 173800 1738 100SRA MET gt 80 93407 162341 1606 101SRA Leptonveto 93302 162159 1505 108SRA MET gt 150 79449 138082 1323 104SRA Jet selection 10213 17750 119 149SRA mbb gt 200 8344 14502 96 151SRA mCT gt 150 6535 11358 82 139SRA mCT gt 200 5175 8994 67 134SRA mCT gt 250 3793 6592 51 129SRA mCT gt 300 2487 4322 35 123SRB NoCuts 100000 1624100 16241 100SRB leptonveto 11489 186593 4069 46SRB MET gt 250 2530 41090 757 54SRB Jet selection 63 1023 79 130SRB HT 3 lt 50 51 828 52 159

            99

            CMSTop1LSearch for the Production of Dark Matter in As-sociation with Top Quark Pairs in the Single-lepton FinalState in pp collisions[41]

            Simulated in Madgraph with p p gt t t p1 p1

            Where p1= Majorana fermion Table 3 shows Signal efficiencies and cross sections TheM scale was read from Figure 7 for 90 CL

            Table A5 Signal Efficiencies 90 CL on σ limexp[ f b] on pp gt tt +χχ

            Mχ[GeV ] My Signal Efficiency Quoted Efficiency My σ [ f b] Quoted σ [ f b]1 102 101 63 47

            10 132 101 62 4650 143 120 50 38100 170 146 56 32200 185 173 69 27600 265 240 32 19

            1000 320 276 41 17

            Appendix B

            Branching ratio calculations for narrowwidth approximation

            B1 Code obtained from decayspy in Madgraph

            Br(S rarr bb) = (minus24λ2b m2

            b +6λ2b m2

            s

            radicminus4m2

            bm2S +m4

            S)16πm3S

            Br(S rarr tt) = (6λ2t m2

            S minus24λ2t m2

            t

            radicm4

            S minus4ms2m2t )16πm3

            S

            Br(S rarr τ+

            τminus) = (2λ

            2τ m2

            S minus8λ2τ m2

            τ

            radicm4

            S minus4m2Sm2

            τ)16πm3S

            Br(S rarr χχ) = (2λ2χm2

            S

            radicm4

            S minus4m2Sm2

            χ)32πm3S

            (B1)

            Where

            mS is the mass of the scalar mediator

            mχ is the mass of the Dark Matter particle

            mb is the mass of the b quark

            mt is the mass of the t quark

            mτ is the mass of the τ lepton

            The coupling constants λ follow the same pattern

            • TITLE Collider Constraints applied to Simplified Models of Dark Matter fitted to the Fermi-LAT gamma ray excess using Bayesian Techniques
              • Dedication
              • Declaration
              • Acknowledgements
              • Contents
              • List of Figures
              • List of Tables
                • Chapter 1 Introduction
                • Chapter 2 Review of Physics
                • Chapter 3 Fitting Models to the Observables
                • Chapter 4 Calculation Tools
                • Chapter 5 Majorana Model Results
                • Chapter 6 Real Scalar Model Results
                • Chapter 7 Real Vector Dark Matter Results
                • Chapter 8 Conclusion
                • Bibliography
                • Appendix A Validation of Calculation Tools
                • Appendix B Branching ratio calculations for narrow width approximation

              Contents vii

              7 Real Vector Dark Matter Results 8171 Bayesian Scans 8172 Best fit Gamma Ray Spectrum for the Real Vector DM model 8473 Collider Constraints 84

              731 Mediator Decay 84732 Collider Cuts Analyses 86

              8 Conclusion 89

              Bibliography 91

              Appendix A Validation of Calculation Tools 97

              Appendix B Branching ratio calculations for narrow width approximation 105B1 Code obtained from decayspy in Madgraph 105

              List of Figures

              21 Feynman Diagram of electron interacting with a muon 1122 Weak Interaction Vertices [48] 1523 Higgs Potential [49] 1824 Standard Model Particles and Forces [50] 2425 Bullet Cluster [52] 2526 Galaxy Rotation Curves [54] 2627 WMAP Cosmic Microwave Background Fluctuations [58] 2928 Dark Matter Interactions [60] 2929 Gamma Ray Excess from the Milky Way Center [75] 31210 ATLAS Experiment 31211 CMS Experiment 34

              31 Main Feyman diagrams leading to the cross section for scalar decaying to apair of τ leptons 40

              32 WidthmS vs mS 4033 WidthmS vs λb 4134 WidthmS vs λτ 4135 Main Feyman diagrams leading to the cross section for scalar decaying to a

              pair of b quarks in the presence of at least one b quark 42

              41 Calculation Tools 55

              51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether 62

              52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter 6353 Gamma Ray Spectrum 6454 Plots of log likelihoods by individual and combined constraints Masses in

              GeV 6655 σ lowastBr(σ rarr ττ) versus Mass of Scalar 67

              List of Figures ix

              56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar 6857 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar 6858 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 6959 Excluded points from Collider Cuts and σBranching Ratio 70

              61 Real Scalar Dark Matter - By Individual Constraint and All Together 7462 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter 7563 Gamma Ray Spectrum 7664 σ lowastBr(σ rarr ττ) versus Mass of Scalar 7765 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 7866 Excluded points from Collider Cuts and σBranching Ratio 80

              71 Real Vector Dark Matter - By Individual Constraint and All Together 8272 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter 8373 Gamma Ray Spectrum 8474 σ lowastBr(σ rarr ττ) versus Mass of Scalar 8575 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 8676 Excluded points from Collider Cuts and σBranching Ratio 87

              List of Tables

              21 Quantum numbers of the Higgs field 1922 Weak Quantum numbers of Lepton and Quarks 21

              31 Simplified Models 3532 95 CL by Signal Region 4433 Selection criteria common to all signal regions 4534 Selection criteria for signal regions A 4535 Selection criteria for signal regions C 4536 Signal Regions - Lepstop1 4837 Signal Regions Lepstop2 4938 Signal Regions 2bstop 5139 Signal Region ATLASmonobjet 52

              51 Scanned Ranges 6152 Best Fit Parameters 64

              61 Best Fit Parameters 76

              71 Best Fit Parameters 84

              A1 0 Leptons in the final state 98A2 1 Lepton in the Final state 100A3 2 Leptons in the final state 101A4 2b jets in the final state 102A5 Signal Efficiencies 90 CL on σ lim

              exp[ f b] on pp gt tt +χχ 103

              Chapter 1

              Introduction

              Dark matter (DM) was first postulated over 80 years ago when Swiss astronomer FritzZwicky observed a discrepancy between the amount of light emitted by a cluster of galaxiesand the total mass contained within the cluster inferred from the relative motion of thosegalaxies by a simple application of the theory of Newtonian gravitation The surprising resultof this observation was that the vast majority of the mass in the cluster did not emit lightwhich was contrary to the expectation that most of the mass would be carried by the starsSince that time further observations over a wide range of scales and experimental techniqueshave continued to point to the same result and refine it Some of these observations and otherevidence are discussed in section 22 We now know with certainty that in the entire Universeall of the matter we know about - stars planets gases and other cosmic objects such as blackholes can only account for less than 5 of the mass that we calculate to be there

              A recent phenomenon that has received much attention is the significant deviation frombackground expectations of the Fermi Large Area Telescope(Fermi-LAT) gamma ray flux atthe galactic centre [1] A number of astrophysical explanations have been proposed includingmillisecond pulsars of supernova remnants [2] or burst-like continuous events at the galacticcentre but these are unresolved However it has also been noted that the observed Fermi-LATexcess is consistent with the annihilation of dark matter particles which would naturally beconcentrated at the Galactic centre in a manner consistent with the Navarro-Frenk-Whitedistribution of dark matter [3]

              There are a number of other purely theoretical (particle physics) reasons to postulatethe existence of weakly interacting matter particles that could supply the missing mass andyet remain unobservable Weakly interacting massive particle (WIMPS) have been a majorfocus of Run I and ongoing Run II searches of the Large Hadron Collider (LHC) In spite

              2

              of substantial theoretical and experimental effort the microscopic properties of dark matterparticles are still unknown

              One of the main problems with extracting these properties is the plethora of competingtheoretical models that provide dark matter candidates Another is the very weak interactionof these particles and the lack of experimental observables allowing one to discriminatebetween the competing models

              A particularly fruitful approach is the simplified model approach which makes minimalassumptions about the DM candidates as extensions to the Standard Model interacting weaklywith ordinary matter through a mediator These can be coded as Lagrangian interaction termsusing tools such as FeynRules [4] and Calchep [5] and used to simulate Dark Matter relicdensities indirect detection expectations as well as direct detection expectations throughMicrOmegas [6] This in turn can be run under a Bayesian inference tool (see below)

              Using a Bayesian inference tool such as MultiNest [7][8][9] one can scan the parameterspace of a particular model to find regions of highest probability for that model based onobservations of a range of phenomena including direct detection experiments such as ZEPLIN[10] XENON [11] DEAP [12] ARDM [13] DarkSide [14] PandaX [15] and LUX [16]indirect detection - eg Fermi-LAT gamma ray excess [17] Pamela-positron and antiprotonflux [18] and production at colliders MultiNest calculates the evidence value which is theaverage likelihood of the model weighted by prior probability and a posterior distributionwhich is the distribution of the parameter(s) after taking into account the observed databy scanning over a range of parameters based on prior probabilities This can be used tocompare models on a consistent basis The scans also provide information about the best fitparameters for the overall fit as well as for the seperate experimentsobservations

              The simplified model approach has been taken in a number of papers to study the Fermi-LAT gamma ray excess (see literature review in section 12) and one in particular comparesthese models using a Bayesian inference tool which found that a Majorana fermion DMcandidate interacting via a scalar mediator with arbitrary couplings to third generationfermions showed the best fit [19] A subsequent paper [20] examined the most likelyparameter combinations that simultaneously fit the Planck relic density Fermi-LAT galacticcentre gamma ray AMS-02 positron flux and Pamela antiproton flux data

              There has not as yet been a thorough study of the collider constraints on DM simplifiedmodels that fit the Fermi-LAT galactic centre excess In this thesis we reproduce the

              11 Motivation 3

              previous results of [20] using similar astrophysical constraints and then implement detailedsimulations of collider constraints on DM and mediator production

              In Section 11 we review the motivation for this study and in Section 12 we considerrecent astronomical data which is consistent with the annihilation of DM particles andreview the literature and searches which attempt to explain it with a range of extensions tothe standard model

              In Chapter 2 we review the Standard Model of particle physics (SM) the evidence fordark matter and the ATLAS and CMS experiments at the LHC

              In Chapter 3 we describe the models considered in this paper the observables used inthe Bayesian scans and the collider experiments which provide limits against which theMagraph simulations of the three models can be compared

              In Chapter 4 we review the calculation tools that have been used in this paper

              In Chapters 5 6 and 7 we give the results of the Bayesian scans and collider cutsanalyses for each of the models and in Chapter 8 we summarise our conclusions

              11 Motivation

              The Fermi gamma ray space telescope has over the past few years detected a gamma raysignal from the inner few degrees around the galactic centre corresponding to a regionseveral hundred parsecs in radius The spectrum and angular distribution are compatible withthat from annihilating dark matter particles

              A number of papers have recently attempted to explain the Gamma Ray excess in termsof extensions to the Standard Model taking into account constraints from ATLAS and CMScollider experiments direct detection of dark matter colliding off nucleii in well shieldeddetectors (eg LUX) as well as other indirect measurements (astrophysical observations)These papers take the approach of simplified models which describe the contact interactionsof a mediator beween DM and ordinary matter using an effective field theory approach(EFT) Until recently it has been difficult to discriminate between the results of these paperswhich look at different models and produce different statistical results A recent paper[19] Simplified Dark Matter Models Confront the Gamma Ray Excess used a Bayesianinference tool to compare a variety of simplified models using the Bayesian evidence values

              12 Literature review 4

              calculated for each model to find the most preferred explanation The motivation for thechosen models is discussed in section 121

              A subsequent paper [20] Interpreting the Fermi-LAT gamma ray excess in the simplifiedframework studied the most likely model again using Bayesian inference to find the mostlikely parameters for the (most likely) model The data inputs (observables) to the scanwere the Planck measurement of the dark matter relic density [21] Fermi-LAT gamma rayflux data from the galactic center [17] cosmic positron flux data [22] cosmic anti-protonflux data [18] cosmic microwave background data and direct detection data from the LUXexperiment [16] The result of the scan was a posterior probability density function in theparameter space of the model

              This paper extends the findings in [19] by adding collider constraints to the likelihoodestimations for the three models It is not a simple matter to add collider constraints becauseeach point in parameter space demands a significant amount of computing to calculate theexpected excess amount of observable particles corresponding to the particular experimentgiven the proposed simplified model A sampling method has been employed to calculatethe collider constraints using the probability of the parameters based on the astrophysicalobservables

              12 Literature review

              121 Simplified Models

              A recent paper [23] summarised the requirements and properties that simplified models forLHC dark matter searches should have

              The general principles are

              bull Besides the Standard Model (SM) the Lagrangian should contain a DM candidate thatis absolutely stable or lives long enough to escape the LHC detector and a mediatorthat couples the two sectors

              bull The Lagrangian should contain all terms that are renormalizable consistent withLorentz invariance SM symmetries and DM stability

              bull Additional interactions should not violate exact and approximate global symmetries ofthe SM

              12 Literature review 5

              The examples of models that satisfy these requirements are

              1 Scalar (or pseudo-scalar) s-channel mediator coupled to fermionic dark matter (Diracor Majorana)

              2 Higgs portal DM where the DM is a scalar singlet or fermion singlet under the gaugesymmetries of the SM coupling to a scalar boson which mixes with the Higgs (Thefermion singlet is a specific realisation of 1 above)

              3 DM is a mixture of an electroweak singlet and doublet as in the Minimal Supersym-metric SM (MSSM)

              4 Vector (or Axial vector) s-channel mediator obtained by extending the SM gaugesymmetry by a new U(1)rsquo which is spontaneously broken such that the mediatoracquires mass

              5 t-channel flavoured mediator (if the DM particle is a fermion χ the mediator can bea coloured scalar or vector particle φ -eg assuming a scalar mediator a couplingof the form φ χq requires either χ or φ to carry a flavor index to be consistent withminimal flavour violation (MFV) which postulates the flavour changing neutral currentstructure in the SM is preserved

              Another recent paper [24] summarised the requirements and properties that simplifiedmodels for LHC dark matter searches should have The models studied in this paper arepossible candidates competing with Dirac fermion and complex scalar dark matter andmodels with vector and pseudo-scalar mediators Other models would have an elasticscattering cross section that will remain beyond the reach of direct detection experiments dueto the neutrino floor (the irreducible background produced by neutrino scattering) or arealready close to sensitivity to existing LUX and XENONIT experiments

              A recent paper [25] Scalar Simplified Models for Dark Matter looks at Dirac fermiondark matter mediated by either a new scalar or pseudo-scalar to SM fermions This paperplaces bounds on the coupling combination gχgφ (coupling of DM to mediator times couplingof mediator to SM fermions) calculated from direct detection indirect detection and relicdensity by the mass of DM and mass of mediator Bounds are also calculated for colliderconstraints (mono-jet searches heavy-flavour searches and collider bounds on the 125 GeVHiggs are used to place limits on the monojet and heavy flavor channels which can betranslated into limits on the Higgs coupling to dark matter and experimental measurements

              12 Literature review 6

              of the Higgs width are used to constrain the addition of new channels to Higgs decay) Thepaper models the top loop in a full theory using the Monte Carlo generator MCFM [26]and extends the process to accommodate off-shell mediator production and decay to a darkmatter pair While this paper does not consider the Majorana fermion dark matter model theconstraints on the coupling product may have some relevence to the present study if only asan order of magnitude indicator

              Constraining dark sectors with monojets and dijets [27] studies a simplified model withDirac fermion dark matter and an axial-vector mediator They use two classes of constraint-searches for DM production in events with large amounts of missing transverse energy inassociation with SM particles (eg monojet events) and direct constraints on the mediatorfrom monojet searches to show that for example where the choice of couplings of themediator to quarks and DM gA

              q =gAχ=1 all mediator masses in the range 130 GeV ltMR lt3

              TeV are excluded

              The paper Constraining Dark Sectors at Colliders Beyond the Effective Theory Ap-proach [28] considers Higgs portal models with scalar pseudo-scalar vector and axial-vectormediators between the SM and the dark sector The paper takes account of loop processesand gluon fusion as well as quark-mediator interactions using MCFM Apart from mediatortype the models are characterised by free parameters of mediator width and mass darkmatter mass and the effective coupling combination gχgφ (coupling of DM to mediator timescoupling of mediator to SM fermions) The conclusion is that LHC searches can proberegions of the parameter space that are not able to be explored in direct detection experiments(since the latter measurements suffer from a velocity suppression of the WIMP-nucleoncross-section)

              The paper Constraining the Fermi-LAT excess with multi-jet plus EmissT collider searches

              [29] shows that models with a pseudo-scalar or scalar mediator are best constrained by multi-jet final states with Emiss

              T The analysis was done using POWHEG-BOX-V2 at LO and usesM2

              T variables together with multi-jet binning to exploit differences in signal and backgroundin the different regions of phase space to obtain a more stringent set of constraints on DMproduction This paper largely motivated the current work in that it showed that collidersearches that were not traditionally thought of as dark matter searches were neverthelessoften sensitive to dark matter produced in association with heavy quarks The combinationof searches used in this paper reflects this fact

              12 Literature review 7

              122 Collider Constraints

              In order to constrain this model with collider constraints (direct detection limits) we considerthe following papers by the CMS and ATLAS collaborations which describe searches at theLHC

              ATLAS Experiments

              bull Search for dark matter in events with heavy quarks and missing transverse momentumin p p collisions with the ATLAS detector [30]

              bull Search for direct third-generation squark pair production in final states with missingtransverse momentum and two b-jets in

              radics= 8 TeV pp collisions with the ATLAS

              detector[31]

              bull Search for direct pair production of the top squark in all-hadronic final states inprotonndashproton collisions at

              radics=8 TeV with the ATLAS detector [32]

              bull Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

              radic(s)=8TeV pp collisions using 21 f bminus1 of

              ATLAS data [33]

              bull Search for direct top squark pair production in final states with two leptons inradic

              s = 8TeV pp collisions using 20 f bminus1 of ATLAS data [34]

              bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

              radics=8 TeV [35]

              CMS Experiments

              bull Searches for anomalous tt production in p p collisions atradic

              s=8 TeV [36]

              bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

              radics=8 TeV [37]

              bull Search for the production of dark matter in association with top-quark pairs in thesingle-lepton final state in proton-proton collisions at

              radics = 8 TeV [38]

              bull Search for new physics in monojet events in p p collisions atradic

              s = 8 TeV(CMS) [39]

              bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

              s = 8 TeV [40]

              12 Literature review 8

              bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theSingle-lepton Final State in pp collisions at

              radics=8 TeV [41]

              bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

              s=8 TeV [42]

              bull Search for neutral MSSM Higgs bosons decaying into a pair of bottom quarks [43]

              bull Search for neutral MSSM Higgs bosons decaying into a pair of τ leptons in p p collision[44]

              bull Search for dark matter extra dimensions and unparticles in monojet events in proton-proton collisions at

              radics=8 TeV [45]

              In particular we have implemented Madgraph simulations of certain data analysesapplying to collider experiments described in papers [32][33][34][31][30][41] Theremaining models were not implemented in Madgraph because of inability to validate themodels against published results in the time available We developed C++ analysis code toimplement the collider cuts after showering with Pythia and basic analysis with Delpheswith output to root files The C++ code allowed multiple experimental cuts to be applied tothe same simulation output The C++ code was first checked against the results of the variousexperiments above and then the simulations were rerun using the three modelsrsquo Lagrangians(implemented in FeynRules) described in this paper

              Chapter 2

              Review of Physics

              21 Standard Model

              211 Introduction

              The Standard Model of particle physics (SM) has sometimes been described as the theoryof Almost Everything The SM is a field theory which emerged in the second half of thetwentieth century from quantum mechanics and special relativity which themselves wereonly conceived after 1905 by Einsten Pauli Bohr Dirac and others In the first decades ofthe twenty first century the discovery of the Higgs boson [46][47] completed the pictureand confirmed the last previously unverified part of the SM The SM is a self consistenttheory that describes and classifies all known particles and their interactions (other thangravity) to an astonishing level of accuracy and detail However the theory is incomplete-it does not describe the complete theory of gravitation or account for DM (see below) ordark energy (the accelerating expansion of the universe) Related to these shortcomings aretheoretical deficiencies such as the asymmetry of matter and anti-matter the fact that the SMis inconsistent with general relativity (to the extent that one or both theories break down atspacetime singularities like the Big Bang and black hole event horizons)

              212 Quantum Mechanics

              Quantum Mechanics (QM) describes particles as probability waves which can be thoughtof as packets of energy which can be localised or spread through all space depending on how

              21 Standard Model 10

              accurately one measures the particlersquos momentum However QM is not itself consistent withspecial relativity It was realised too that because of relativity particles could be created anddestroyed and because of the uncertainty principle the theory should describe the probabilityof particles being created and destroyed on arbitrarily short timescales A quantum fieldtheory (QFT) is a way to describe this probability at every point of space and time

              213 Field Theory

              A field theory is described by a mathematical function of space and time called a LagrangianAs in classical mechanics where Lagrangians were introduced by Louis Lagrange at thebeginning of the nineteenth century to summarise dynamical systems the Lagrangian sum-marises the dynamics of space which is permeated by force fields The equations of motionarising from a Lagrangian are determined by finding the contours of the Lagrangian describedby stationary points (ie where the infinitesimal change in the Lagrangian is zero for a smallchange in field) Particles are generally described by singularities in the Lagrangian (wherethe values tend to infinity also called resonances)

              214 Spin and Statistics

              It was also realised that different types of particle needed to be represented by different fieldsin the Lagrangian Diracrsquos discovery that particles with spin 12 quantum numbers couldbe represented by a field obtained by effectively taking the square root of the relativisticequation (Klein Gordon equation) was probably the most significant advance in quantumfield theory It required the introduction of a matrix operators called the gamma matricesor Dirac matrices which helped explain the already known fact that spin half particles orfermions obey Fermi-Dirac statistics ie when two spin half particles are exchanged thesign of the wavefunction must change and hence they cannot occupy the same quantum stateunlike bosons with integer spin that can occupy the same quantum state

              Other particles need to be represented by fields called vector pseudo-vector scalar (as inthe Klein Gordon equation) pseudo-scalar and complex fields A vector field is really fourfields one for each dimension of space plus time and is parameterised by the four gammamatrices γmicro which transform among themselves in a relativistically invariant way (Lorentzinvariance) A pseudo-vector field is a vector field which transforms in the same way but

              21 Standard Model 11

              with additional sign change under parity transformations (a simultaneous reversal of the threespatial directions) and is represented by γmicroγ5 A pseudo-scalar is a scalar which changes signunder parity transformations and a complex scalar field is represented by complex numbers

              215 Feynman Diagrams

              QFT took another giant leap forward when Richard Feynman realised that the propagationof particles described by the field could be described by expanding the so called S-matrixor scattering matrix derived from calculating the probability of initial states transitioning tofinal states The expansion is similar to a Taylor expansion in mathematics allowing for thespecial commuting properties of elements of the field The Feynman diagrams representingindividual terms in the expansion can be interpreted as interactions between the particlecomponents The S-Matrix (or more correctly the non trivial part called the T-Matrix) isderived using the interaction Hamiltonian of the field The Hamiltonian is an operator whoseeigenvalues or characteristic values give the energy quanta of the field The T-matrix canalso be described by a path integral which is an integral over all of quantum space weightedby the exponential of the Lagrangian This is completely equivalent to the Hamiltonianformulation and shows how the Lagrangian arises naturally out of the Hamiltonian (in factthe Lagrangian is the Legendre transformation of the Hamiltonian)

              Figure 21 Feynman Diagram of electron interacting with a muon

              γ

              eminus

              e+

              micro+

              microminus

              The diagram Fig21 is a tree diagram (diagrams with closed loops indicate self in-teractions and are higher order terms in the expansion) and shows an electron (eminus) and ananti-muon (micro+) interacting by exchanging a photon

              21 Standard Model 12

              216 Gauge Symmetries and Quantum Electrodynamics (QED)

              The Dirac equation with its gamma matrices ushered in the use of group theory to describethe transformation of fields (because the multiplication table of the gamma matrices isclosed and described by group theory) It also ushered in the idea of symmetries within theLagrangian- for example consider a Lagrangian

              ψ(ipart minusm)ψ (21)

              The Lagrangian is invariant under a global U(1) transformation which acts on the fieldsand hence the derivatives as follows

              ψ rarr eiqαψ ψ rarr ψeminusiqα partmicroψ rarr eiqα

              partmicroψ (22)

              where qα is a global phase and α is a continuous parameter

              A symmetry leads by Noetherrsquos theorem to a conserved current jmicro = qψγmicroψ and is thebasis of charge conservation since partmicro jmicro = 0 and q =

              intd3x j0(x)

              By promoting this transformation to a local one (ie α rarr α(x) where x = xmicro are thespace time co-ordinates) the transformations become

              ψ rarr eiqα(x)ψ ψ rarr ψeminusiqα(x)partmicroψ rarr eiqα(x)(partmicroψ + iq(partmicroα(x))ψ) (23)

              The derivative transformation introduces an extra term which would spoil the gaugeinvariance This can be solved by introducing a new field the photon field Amicro(x) whichinteracts with the fermion field ψ and transforms under a U(1) transformation

              Amicro rarr Amicro minuspartmicroα(x) (24)

              If we replace normal derivatives partmicro with covariant derivatives Dmicro = (partmicro + iqAmicro(x)) in theLagrangian then the extra term is cancelled and we can now see that the covariant derivativetransforms like ψ - under a local U(1) transform ie

              Dmicro rarr Dmicroψprime = eiqα(x)Dmicroψ (25)

              The additional term iqAmicro(x) can be interpreted in a geometric sense as a connectionwhich facilitates the parallel transport of vectors along a curve in a parallel consistent manner

              21 Standard Model 13

              We must add a gauge-invariant kinetic term to the Lagrangian to describe the dynamicsof the photon field This is given by the electromagnetic field strength tensor

              Fmicroν = partmicroAν minuspartνAmicro (26)

              The full Quantum Electrodynamics (QED) Lagrangian is then for electrons with mass mand charge -e

              LQED = ψ(i Dminusm)ψ minus 14

              Fmicroν(X)Fmicroν(x) (27)

              This Lagrangian is Lorentz and U(1) guage invariant The interaction term is

              Lint =+eψ Aψ = eψγmicro

              ψAmicro = jmicro

              EMAmicro (28)

              where jmicro

              EM is the electromagnetic four current

              217 The Standard Electroweak Model

              The Standard Electroweak (EW) Model was created by Glashow-Weinberg-Salam (also calledthe GWS model) and unifies the electomagnetic and weak interactions of the fundamentalparticles

              The idea of gauge symmetry was extended to the idea of a non-abelian gauge symmetry(essentially the operators defining the symmetry do not commute and so are represented bymatrices- in this case the Pauli matrices for the SU(2) group) - the gauge group is the directproduct G = SU(2)

              otimesU(1) It was known that weak interactions were mediated by Wplusmn

              and Z0 particles where the superscripts are the signs of the electromagnetic charge Thesewere fairly massive but it was also known that through spontaneous symmetry breaking ofa continuous global non-abelian symmetry there would be one massive propagation modefor each spontaneously broken-symmetry generator and massless modes for the unbrokengenerators (the so-called Goldstone bosons)

              This required an extra field to represent the gauge bosons called Bmicro (a vector fieldhaving the 4 spacetime degrees of freedom) With this identification the covariant derivativebecomes

              Dmicro = partmicro minus igAmicro τ

              2minus i

              gprime

              2Y Bmicro (29)

              21 Standard Model 14

              Here Y is the so called weak hypercharge of the field Bmicro which is a quantum numberdepending on particle (lepton or quark) g and gprime are two independent parameters in theGWS theory and the expression Amicro τ implies a sum over 3 components Aa

              micro a=123 and thePauli matrices τa

              This isnrsquot the full story because it had been known since 1956 when Chin Shuing Wuand her collaborators confirmed in an experiment prompted by Lee and Yang that nature isnot ambivalent to so-called parity It had been assumed that the laws of nature would notchange if all spatial coordinates were reversed (a parity transformation) The experimentinvolved the beta decay of cooled 60C atoms (more electrons were emitted in the backwardhemisphere with respect to spin direction of the atoms) This emission involved the weakinteraction and required that the Lagrangian for weak interactions contain terms which aremaximally parity violating- which turn out to be currents of the form

              ψ(1minus γ5)γmicro

              ψ (210)

              The term

              12(1minus γ

              5) (211)

              projects out a left-handed spinor and can be absorbed into the spinor to describe left-handedleptons (and as later discovered quarks) The right handed leptons are blind to the weakinteraction

              The processes describing left-handed current interactions are shown in Fig 22

              Analogously to isospin where the proton and neutron are considered two eigenstates ofthe nucleon we postulate that Bmicro is the gauge boson of an SU(2)transformation coupled toweak SU(2) doublets (

              νe

              eminus

              )

              (ud

              ) (212)

              We may now write the weak SU(2) currents as eg

              jimicro = (ν e)Lγmicro

              τ i

              2

              e

              )L (213)

              21 Standard Model 15

              Figure 22 Weak Interaction Vertices [48]

              where τ i are the Pauli-spin matrices and the subscript L denotes the left-handed componentof the spinor projected out by Eqn211 The third current corresponding to i=3 is called theneutral current as it doesnrsquot change the charge of the particle involved in the interaction

              We could postulate that the electromagnetic current is a linear combination of the leftand right elecromagnetic fields eL = 1

              2(1minus γ5)e and eR = 12(1+ γ5)e

              jemmicro = eLγmicroQeL + eRγmicroQeR (214)

              where Q is the electromagnetic operator and eR is a singlet However this is not invariantunder SU(2)L To construct an invariant current we need to use the SU(2)L doublet 212The current which is an SU(2)L invariant U(1) current is

              jYmicro = (ν e)LγmicroYL

              e

              )L+ eRγmicroYReR (215)

              where hypercharges YL and YR are associated with the U(1)Y symmetry jYmicro is a linearcombination of the electromagnetic current and the weak neutral current thus explaining whythe hypercharge differs for the left and right-handed components since the electromagneticcurrent has the same charge for both components whilst the weak current exists only forleft-handed particles j3

              micro and the third component of weak isospin T 3 allows us to calculate

              21 Standard Model 16

              the relationship Y = 2(QminusT 3) which can be seen by writing jYmicro as a linear combination ofj3micro (the weak neutral current that doesnrsquot change the charge of the particle involved in the

              interaction) and jemmicro (using the convention that the jYmicro is multiplied by 1

              2 to match the samefactor implicit in j3

              micro ) Substituting

              τ3 =

              (1 00 minus1

              )(216)

              into equation 213 for i=3 and subtracting equation 215 and equating to 12 times equation

              214

              we get

              eLγmicroQeL + eRγmicroQeR minus (νLγmicro

              12

              νL minus eLγmicro

              12

              eL) =12

              eRγmicroYReR +12(ν e)LγmicroYL

              e

              )L (217)

              from which we can read out

              YR = 2QYL = 2Q+1 (218)

              and T3(eR) = 0 T3(νL) =12 and T3(eL) =

              12 The latter three identities are implied by

              the fraction 12 inserted into the definition of equation 213

              The Lagrangian kinetic terms of the fermions can then be written

              L =minus14

              FmicroνFmicroν minus 14

              GmicroνGmicroν

              + sumgenerations

              LL(i D)LL + lR(i D)lR + νR(i D)νR

              + sumgenerations

              QL(i D)QL +UR(i D)UR + DR(i D)DR

              (219)

              LL are the left-handed lepton doublets QL are the left-handed quark doublets lRνR theright handed lepton singlets URDR the right handed quark singlets

              The field strength tensors are given by

              Fmicroν = partmicroAν minuspartνAmicro minus ig[Amicro Aν ] (220)

              21 Standard Model 17

              andGmicroν = partmicroBν minuspartνBmicro (221)

              Bmicro is an abelian U(1) gauge field while Amicro is non-abelian and Fmicroν contains the com-mutator of the field through the action of the covariant derivative [Dmicro Dν ] This equationcontains no mass terms terms for the field Amicro for good reason- this would violate gaugeinvariance We know that the gauge bosons of the weak force have substantial mass and theway that this problem has been solved is through the Higgs mechanism

              218 Higgs Mechanism

              To solve the problem of getting gauge invariant mass terms three independent groupspublished papers in 1964 The mechanism was named after Peter Higgs To introduce theconcept consider a complex scalar field φ that permeates all space Consider a potential forthe Higgs field of the form

              minusmicro2φ

              daggerφ +λ (φ dagger

              φ2) (222)

              which has the form of a mexican hat Fig 23 This potential is still U(1) gauge invariant andcan be added to our Lagrangian together with kinetic+field strength terms to give

              L = (Dmicroφ)dagger(Dmicroφ)minusmicro2φ

              daggerφ +λ (φ dagger

              φ2)minus 1

              4FmicroνFmicroν (223)

              It is easily seen that this is invariant to the transformations

              Amicro rarr Amicro minuspartmicroη(x) (224)

              φ(x)rarr eieη(x)φ(x) (225)

              The minimum of this field is not at 0 but in a circular ring around 0 with a vacuum

              expectation value(vev)radic

              micro2

              2λequiv vradic

              2

              We can parameterise φ as v+h(x)radic2

              ei π

              Fπ where h and π are referred to as the Higgs Bosonand Goldstone Boson respectively and are real scalar fields with no vevs Fπ is a constant

              21 Standard Model 18

              Figure 23 Higgs Potential [49]

              Substituting this back into the Lagrangian 223 we get

              minus14

              FmicroνFmicroν minusevAmicropartmicro

              π+e2v2

              2AmicroAmicro +

              12(partmicrohpart

              microhminus2micro2h2)+

              12

              partmicroπpartmicro

              π+(hπinteractions)(226)

              This Lagrangian now seems to describe a theory with a photon Amicro of mass = ev a Higgsboson h with mass =

              radic2micro and a massless Goldstone π

              However there is a term mixing Amicro and part microπ in this expression and this shows Amicro and π

              are not independent normal coordinates and we can not conclude that the third term is a massterm This π minusAmicro mixing can be removed by making a gauge transformation in equations224 and 225 making

              φrarrv+h(x)radic2

              ei π

              Fπminusieη(x) (227)

              and setting πrarr π

              Fπminus eη(x) = 0 This choosing of a point on the circle for the vev is called

              spontaneous symmetry breaking much like a vertical pencil falling and pointing horizontallyin the plane This choice of gauge is called unitary gauge The Goldstone π will thencompletely disappear from the theory and one says that the Goldstone has been eaten to givethe photon mass

              21 Standard Model 19

              This can be confirmed by counting degrees of freedom before and after spontaneoussymmetry breaking The original Lagrangian had four degrees of freedom- two from thereal massless vector field Amicro (massless photons have only two independent polarisationstates) and two from the complex scalar field (one each from the h and π fields) The newLagrangian has one degree of freedom from the h field and 3 degrees of freedom from themassive real vector field Amicro and the π field has disappeared

              The actual Higgs field is postulated to be a complex scalar doublet with four degrees offreedom

              Φ =

              (φ+

              φ0

              )(228)

              which transforms under SU(2) in the fundamental representation and also transforms underU(1) hypercharge with quantum numbers

              Table 21 Quantum numbers of the Higgs field

              T 3 Q Yφ+

              12 1 1

              φ0 minus12 1 0

              We can parameterise the Higgs field in terms of deviations from the vacuum

              Φ(x) =(

              η1(x)+ iη2(x)v+σ(x)+ iη3(x)

              ) (229)

              It can be shown that the three η fields can be transformed away by working in unitarygauge By analogy with the scalar Higgs field example above the classical action will beminimised for a constant value of the Higgs field Φ = Φ0 for which Φ

              dagger0Φ0 = v2 This again

              defines a range of minima and we can pick one arbitarily (this is equivalent to applying anSU(2)timesU(1) transform to the Higgs doublet transforming the top component to 0 and thebottom component to Φ0 = v) which corresponds to a vacuum of zero charge density andlt φ0 gt=+v

              In this gauge we can write the Higgs doublet as

              Φ =

              (φ+

              φ0

              )rarr M

              (0

              v+ H(x)radic2

              ) (230)

              where H(x) is a real Higgs field v is the vacuum expectation and M is an SU(2)timesU(1)matrix The matrix Mdagger converts the isospinor into a down form (an SU(2) rotation) followedby a U(1) rotation to a real number

              21 Standard Model 20

              If we consider the Higgs part of the Lagrangian

              minus14(Fmicroν)

              2 minus 14(Bmicroν)

              2 +(DmicroΦ)dagger(DmicroΦ)minusλ (Φdagger

              Φminus v2)2 (231)

              Substituting from equation 230 into this and noting that

              DmicroΦ = partmicroΦminus igW amicro τ

              aΦminus 1

              2ig

              primeBmicroΦ (232)

              We can express as

              DmicroΦ = (partmicro minus i2

              (gA3

              micro +gprimeBmicro g(A1micro minusA2

              micro)

              g(A1micro +A2

              micro) minusgA3micro +gprimeBmicro

              ))Φ equiv (partmicro minus i

              2Amicro)Φ (233)

              After some calculation the kinetic term is

              (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

              14(v+

              Hradic2)2[A 2]22 (234)

              where the 22 subscript is the index in the matrix

              If we defineWplusmn

              micro =1radic2(A1

              micro∓iA2micro) (235)

              then [A 2]22 is given by

              [A 2]22 =

              (gprimeBmicro +gA3

              micro

              radic2gW+

              microradic2gWminus

              micro gprimeBmicro minusgA3micro

              ) (236)

              We can now substitute this expression for [A 2]22 into equation 234 and get

              (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

              14(v+

              Hradic2)2(2g2Wminus

              micro W+micro +(gprimeBmicro minusgA3micro)

              2) (237)

              This expression contains mass terms for three massive gauge bosons Wplusmn and the combi-nation of vector fields gprimeBmicro minusgA3

              micro where note

              21 Standard Model 21

              Table 22 Weak Quantum numbers of Lepton and Quarks

              T 3 Q YνL

              12 0 -1

              lminusL minus12 -1 -1

              νR 0 0 0lminusR 0 -1 -2UL

              12

              23

              13

              DL minus12 minus1

              313

              UR 0 23

              43

              DR 0 minus13 minus2

              3

              Wminusmicro = (W+

              micro )dagger equivW 1micro minus iW 2

              micro (238)

              Then the mass terms can be written

              12

              v2g2|Wmicro |2 +14

              v2(gprimeBmicro minusgA3micro)

              2 (239)

              W+micro and W microminus can be associated with the W boson and its anti-particle and (minusgprimeBmicro +

              gA3micro) with the Z Boson (after normalisation by

              radicg2 +(gprime

              )2) The combination gprimeA3micro +gBmicro

              is orthogonal to Z and remains massless This can be associated with the Photon (afternormalisation) These predict mass terms for the W and Z bosons mW = vgradic

              2and mZ =

              vradic2

              radicg2 +(gprime

              )2It is again instructive to count the degrees of freedom before and after the Higgs mech-

              anism Before we had a complex doublet Φ with four degrees of freedom one masslessB with two degrees of freedom and three massless W a fields each with two each givinga total of twelve degrees of freedom After spontaneous symmetry breaking we have onereal scalar Higgs field with one degree of freedom three massive weak bosons with nineand one massless photon with two again giving twelve degrees of freedom One says thescalar degrees of freedom have been eaten to give the Wplusmn and Z bosons their longitudinalcomponents (ie mass)

              Table 22 shows T 3 the weak isospin quantum number (which describes how a particletransforms under SU(2)) Y the weak hypercharge (which describes how the particle trans-

              21 Standard Model 22

              forms under U(1)) and Q the electric charge of leptons and quarks The former two enterdirectly into the Lagrangian while the latter can be derived as a function of the former two

              Strictly speaking the right handed neutrino which has no charge may not exist and is notpart of the standard model although some theories postulate a mass for this (hypothetical)particle

              Writing gauge invariant couplings between ELeRφ with the vacuum expectation ofφ = v (Yukawa matrices λ

              i ju and λ

              i jd respectively for the up and down quarks) we get mass

              terms for the quarks (and similarly for the leptons)

              Mass terms for quarks minussumi j[(λi jd Qi

              Lφd jR)+λ

              i ju εab(Qi

              L)aφlowastb u j

              R +hc]

              Mass terms for leptonsminussumi j[(λi jl Li

              Lφ l jR)+λ

              i jν εab(Li

              L)aφlowastb ν

              jR +hc]

              Here hc is the hermitian conjugate of the first terms L and Q are lepton and quarkdoublets l and ν are the lepton singlet and corresponding neutrino d and u are singlet downand up quarks εab is the Levi-Civita (anti-symmetric) tensor

              If we write the quark generations in a basis that diagonalises the Higgs couplings weget a unitary transformations Uu and Ud for the up and down quarks respectively which iscompounded in the boson current (Udagger

              u Ud) called the CKM matrix where off-diagonal termsallow mixing between quark generations (this is also called the mass basis as opposed tothe flavour basis) A similar argument can give masses to right handed sterile neutrinos butthese can never be measured However a lepton analogue of the CKM matrix the PNMSmatrix allows left-handed neutrinos to mix between various generations and oscillate betweenflavours as they propagate through space allowing a calculation of mass differences

              219 Quantum Chromodynamics

              The final plank in SM theory tackling the strong force Quantum Chromodynamics wasproposed by Gell-Mann and Zweig in 1963 Three generations of quarks were proposedwith charges of 23 for uc and t and -13 for ds and b Initially SU(2) isospin was consideredtransforming u and d states but later irreducible representations of SU(3) were found to fit theexperimental data The need for an additional quantum number was motivated by the fact thatthe lightest excited state of the nucleon consisted of sss (the ∆++) To reconcile the baryon

              21 Standard Model 23

              spectrum with spin statistics colour was proposed Quarks transform under the fundamentalor 3 representation of colour symmetry and anti-quarks under 3 The inner product of 3and 3 is an invariant under SU(3) as is a totally anti-symmetric combination ε i jkqiq jqk (andthe anti-quark equivalent) It was discovered though deep inelastic scattering that quarksshowed asymptotic freedom (the binding force grew as the seperation grew) Non-abeliangauge theories showed this same property and colour was identified with the gauge quantumnumbers of the quarks The quanta of the gauge field are called gluons Because the forcesshow asymptotic freedom - ie the interaction becomes weaker at high energies and shorterdistances perturbation theory can be used (ie Feynman diagrams) at these energies but forlonger distances and lower energies lattice simulations must be done

              2110 Full SM Lagrangian

              The full SM can be written

              L =minus14

              BmicroνBmicroν minus 18

              tr(FmicroνFmicroν)minus 12

              tr(GmicroνGmicroν)

              + sumgenerations

              (ν eL)σmicro iDmicro

              (νL

              eL

              )+ eRσ

              micro iDmicroeR + νRσmicro iDmicroνR +hc

              + sumgenerations

              (u dL)σmicro iDmicro

              (uL

              dL

              )+ uRσ

              micro iDmicrouR + dRσmicro iDmicrodR +hc

              minussumi j[(λ

              i jl Li

              Lφ l jR)+λ

              i jν ε

              ab(LiL)aφ

              lowastb ν

              jR +hc]

              minussumi j[(λ

              i jd Qi

              Lφd jR)+λ

              i ju ε

              ab(QiL)aφ

              lowastb u j

              R +hc]

              + (Dmicroφ)dagger(Dmicroφ)minusmh[φφ minus v22]22v2

              (240)

              where σ micro are the extended Pauli matrices

              (1 00 1

              )

              (0 11 0

              )

              (0 minusii 0

              )

              (1 00 minus1

              )

              The first line gives the gauge field strength terms the second the lepton dynamical termsthe third the quark dynamical terms the fourth the lepton (including neutrino) mass termsthe fifth the quark mass terms and the sixth the Higgs terms

              The SU(3) gauge field is given by Gmicroν and which provides the colour quantum numbers

              21 Standard Model 24

              Figure 24 Standard Model Particles and Forces [50]

              Note The covariant derivatives operating on the left-handed leptons contain only theU(1) and SU(2) gauge fields Bmicro Fmicro whereas those operating on left-handed quarks containU(1)SU(2) and SU(3) gauge fields Bmicro Fmicro Gmicro The covariant derivatives operating onright handed electrons and quarks are as above but do not contain the SU(2) gauge fieldsRight handed neutrinos have no gauge fields

              The sums over i j above are over the different generations of leptons and quarks

              The particles and forces that emerge from the SM are shown in Fig 24

              22 Dark Matter 25

              22 Dark Matter

              221 Evidence for the existence of dark matter

              2211 Bullet Cluster of galaxies

              Most of the mass of the cluster as measured by a gravitational lensing map is concentratedaway from the centre of the object The visible matter (as measured by X-ray emission) ismostly present in the centre of the object The interpretation is that the two galaxy clustershave collided perpendicular to the line of sight with the mostly collisionless dark matterpassing straight through and the visible matter interacting and being trapped in the middleIt was estimated that the ratio of actual to visible matter was about 16 [51]

              Figure 25 Bullet Cluster [52]

              2212 Coma Cluster

              The first evidence for dark matter came in the 1930s when Fritz Zwicky observed the Comacluster of galaxies over 1000 galaxies gravitationally bound to one another He observed thevelocities of individual galaxies in the cluster and estimated the amount of matter based onthe average galaxy He calculated that many of the galaxies should be able to escape fromthe cluster unless the mass was about 10 times higher than estimated

              22 Dark Matter 26

              2213 Rotation Curves [53]

              Since those early observations many spiral galaxies have been observed with the rotationcurves calculated by doppler shift with the result that in all cases the outermost stars arerotating at rates fast enough to escape their host galaxies Indeed the rotation curves arealmost flat indicating large amounts of dark matter rotating with the galaxies and holdingthe galaxies together

              Figure 26 Galaxy Rotation Curves [54]

              2214 WIMPS MACHOS

              The question is if the particles that make up most of the mass of the universe are not baryonswhat are they It is clear that the matter does not interact significantly with radiation so mustbe electrically neutral It has also not lost much of its kinetic energy sufficiently to relax intothe disks of galaxies like ordinary matter Massive particles may survive to the present ifthey carry a conserved additive or multiplicative quantum number This is so even if equalnumbers of particles and anti-particles can annihilate if the number densities become too lowto reduce the density further These particles have been dubbed Weakly interacting MassiveParticles (WIMPS) as they interact with ordinary matter only through a coupling strengthtypically encountered in the weak interaction They are predicted to have frozen out ofthe hot plasma and would give the correct relic density based solely on the weak interactioncross section (sometimes called the WIMP Miracle) Another possibility called MassiveCompact Halo Objects (MACHOS)- are brown dwarf stars Jupiter mass objects or remnantsfrom an earlier generation of star formation

              22 Dark Matter 27

              2215 MACHO Collaboration [55]

              In order to estimate the mass of MACHOS a number of experiments have been conductedusing microlensing which makes use of the fact that if a massive object lies in the line ofsight of a much more distant star the light from the more distant star will be lensed and forma ring around the star It is extemely unlikely that the MACHO will pass directly through theline of sight but if there is a near miss the two images will be seperated by a small angleThe signature for microlensing is a time-symmetric brightening of a star as the MACHOpasses near the line of sight called a light curve One can obtain statistical information aboutthe MACHO masses from the shape and duration of the light curves and a large number ofstars (many million) must be monitored The MACHO Collaboration [55] have monitoredover 11 million stars and less than 20 events were detected The conclusion was that thiscould account for a small addition (perhaps up to 20 ) to the visible mass but not enoughto account for Dark Matter

              2216 Big Bang Nucleosynthesis (BBN) [56]

              Another source of evidence for DM comes from the theory of Nucleosynthesis in the fewseconds after the Big Bang In the first few minutes after the big bang the temperatures wereso high that all matter was dissociated but after about 3 minutes the temperature rapidlycooled to about 109 Kelvin Nucleosynthesis could begin when protons and neutrons couldstick together to form Deuterium (Heavy Hydrogen) and isotopes of Helium The ratio ofDeuterium to Hydrogen and Helium is very much a function of the baryon density and alsothe rate of expansion of the universe at the time of Nucleosynthesis There is a small windowwhen the density of Baryons is sufficient to account for the ratios of the various elementsThe rate of expansion of the universe is also a function of the total matter density The ratiosof these lighter elements have been calculated by spectral measurements with Helium some25 of the mass of the universe Deuterium 001 The theory of BBN accounts for theratios of all of the lighter elements if the ratio of Baryon mass to total mass is a few percentsuggesting that the missing matter is some exotic non-baryonic form

              22 Dark Matter 28

              2217 Cosmic Microwave Background [57]

              The discovery of a uniform Microwave Background radiation by Penzias and Wilson in1965 was confirmation of the theory first proposed by Gamow and collaborators that theuniverse should be filled with black body radiation an afterglow of the big bang TheCOBE satelite launched in 1989 confirmed this black body radiation with temperature of2725+-002K with 95 confidence While extremely uniform it found tiny irregularitieswhich could be accounted for by irregularities in the density of the plasma at last scattering(gravitational redshifts or blueshifts called the Sachs Wolfe effect) and temperature andvelocity fluctuations in the plasma The anisotropy of the background has been furtheranalysed by WMAP (the Wilkinson Microwave Anisotropy Probe) launched in 2001 Thefluctuations in the density of the plasma are what account for the seeds of galaxy formationHowever detailed calculations have found that the fluctuations were too small to account forthe observed galaxy distribution without an electrically neutral gravitating matter in additionto ordinary matter that only became electrically neutral at the time of recombination [58]

              In May 2009 the European Space Agency (ESA) launched the Planck sattelite a spacebased observatory to observe the universe at wavelengths between 03mm and 111 mmbroadly covering the far-infrared microwave and high frequency domains The missionrsquosmain goal was to study the cosmic microwave background across the whole sky at greatersensitivity and resolution than ever before Using the information gathered and the standardmodel of cosmology ESA was able to calculate cosmological parameters including the relicdensity of dark matter remaining after the Big Bang to a high degree of precision withinthe framework of the standard model of cosmology The standard model of cosmology canbe described by a relatively small number of parameters including the density of ordinarymatter dark matter and dark energy the speed of cosmic expansion at the present epoch (alsoknown as the Hubble constant) the geometry of the Universe and the relative amount ofthe primordial fluctuations embedded during inflation on different scales and their amplitude[59]

              2218 LUX Experiment - Large Underground Xenon experiment [16]

              The LUX experiment is conducted at the Sanford Underground research facility one mileunderground to minimise the background of cosmic rays The detector is a time projectionchamber allowing a 3D positioning of interactions occurring within its active volume LUX

              22 Dark Matter 29

              Figure 27 WMAP Cosmic Microwave Background Fluctuations [58]

              Figure 28 Dark Matter Interactions [60]

              uses as its target 368 kiliograms of liquefied ultra pure xenon which is a scintillator Interac-tions inside the xenon will create an amount of light proportional to the energy depositedThat light can be collected on arrays of light detectors sensitive to a single photon

              Xenon is very pure and the amount of radiation originating within the target is limitedAlso being three times denser than water it can stop radiation originating from outside thedetector The 3D detector focuses on interactions near the center which is a quiet regionwhere rare dark matter interactions might be detected [16]

              22 Dark Matter 30

              222 Searches for dark matter

              2221 Dark Matter Detection

              Fig 28 illustrates the reactions studied under the different forms of experiment designed todetect DM

              Direct detection experiments are mainly underground experiments with huge detectorsattempting to detect the recoil of dark matter off ordinary matter such as LUX [16] CDMS-II [61] XENON100 [11] COUPP [62] PICASSO [63] ZEPLIN [10] These involveinteractions between DM and SM particles with final states involving DM and SM particles

              Indirect detection involves the search for the annihilation of DM particles in distantastrophysical sources where the DM content is expected to be large Example facilitiesinclude space based experiments (EGRET [64] PAMELA [18] FERMI-LAT [65]) balloonflights (ATIC [66] PEBS [67] PPB-BETS [68]) or atmospheric Cherenkov telescopes(HESS [69]MAGIC [70]) and neutrino telescopes (SuperKamiokande [71] IceCube [72]ANTARES [73] KM3NET [74]) Indirect Detection involves DM particles in the initial stateand SM particles in the final state

              Collider detection involves SM particles in the initial state creating DM particles in thefinal state which can be detected by missing transverse energy and explained by variousmodels extending the SM

              223 Possible signals of dark matter

              224 Gamma Ray Excess at the Centre of the Galaxy [65]

              The center of the galaxy is the most dense and active region and it is no surprise that itproduces intense gamma ray sources When astronomers subtract all known gamma raysources there remains an excess which extends out some 5000 light years from the centerHooper et al find that the annihilation of dark matter particles with a mass between 31 and40 Gev fits the excess to a remarkable degree based on Gamma Ray spectrum distributionand brightness

              23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 31

              Figure 29 Gamma Ray Excess from the Milky Way Center [75]

              23 Background on ATLAS and CMS Experiments at theLarge Hadron collider (LHC)

              The collider constraints in this paper are based on two of the experiments that are part of theLHC in CERN

              Figure 210 ATLAS Experiment

              The LHC accelerates proton beams in opposite directions around a 27 km ring of super-conducting magnets until they are made to collide at an interaction point inside one of the 4experiments placed in the ring An upgrade to the LHC has just been performed providing65 TeV per beam

              23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 32

              231 ATLAS Experiment

              The ATLAS detector consists of concentric cylinders around the interaction point whichmeasure different aspects of the particles that are flung out of the collision point There are 4major parts the Inner Detector the Calorimeters the Muon spectrometer and the magnetsystems

              2311 Inner Detector

              The inner detector extends from a few centimeters to 12 meters from the interaction point andmeasures details of the charged particles and their momentum interacting with the materialIf particles seem to originate from a point other than the interaction point this may indicatethat the particle comes from the decay of a hadron with a bottom quark This is used to b-tagcertain particles

              The inner detector consists of a pixel detector with three concentric layers and threeendcaps containing detecting silicon modules 250 microm thick The pixel detector has over 80million readout channels about 50 percent of the total channels for the experiment All thecomponents are radiation hardened to cope with the extremely high radiation in the proximityof the interaction point

              The next layer is the semi-conductor layer (SCT) similar in concept to the pixel detectorbut with long narrow strips instead of pixels to cover a larger area It has 4 double layers ofsilicon strips with 63 million readout channels

              The outermost component of the inner detector is the Transition Radiation Tracker It is acombination of a straw tracker and a transition radiation detector The detecting elementsare drift tubes (straws) each 4mm in diameter and up to 144 cm long Each straw is filledwith gas which becomes ionised when the radiation passes through The straws are held at-1500V with a thin wire down the center This drives negatively ionised atoms to the centercausing a pulse of electricity in the wire This causes a pattern of hit straws allowing the pathof the particles to be determined Between the straws materials with widely varying indicesof refraction cause transition radiation when particles pass through leaving much strongersignals in some straws Transition radiation is strongest for highly relativistic particles (thelightest particles- electrons and positrons) allowing strong signals to be associated with theseparticles

              23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 33

              2312 Calorimeters

              The inner detector is surrounded by a solenoidal magnet of 2 Tesla (there is also an outertoroidal magnet) but surrounding the inner magnet are the calorimeters to measure theenergy of particles by absorbing them There is an inner electromagnetic calorimeter whichmeasures the angle between the detectorrsquos beam axis and the particle (or more preciselypseudo-rapidity) and its angle in the perpendicular plane to a high degree of accuracy Theabsorbing material is lead and stainless steel and the sampling material liquid argon Thehadron calorimeter measures the energy of particles that interact via the strong force Theenergy absorbing material is scintillating steel

              2313 Muon Specrometer

              The muon spectrometer surrounds the calorimeters and has a set of 1200 chambers tomeasure the tracks of outgoing muons with high precision using a magnetic field andtriggering chambers Very few particles of other types pass through the calorimeters and aresubsequently measured

              2314 Magnets

              The magnets are in two layers the inner solenoidal and outer toroidal which together causecharged particles to curve in the magetic field allowing their momentum to be determinedThe outer toroidal magnet consists of 8 very large air core superconducting magnets whichsupply the field in the muon spectrometer

              232 CMS Experiment

              The CMS Detector is similar in design to the ATLAS Detector with a single solenoidalmagnet which is the largest of its type ever built (length 13m and diameter 7m) and allowsthe tracker and calorimeter detectors to be placed inside the coil in a compact way Thestrong magnetic field gives a high resolution for measuring muons electrons and photons ahigh quality central tracking system and a hermetic hadron calorimeter designed to preventhadrons from escaping

              23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 34

              Figure 211 CMS Experiment

              Chapter 3

              Fitting Models to the Observables

              31 Simplified Models Considered

              In order to extend the findings of previous studies [19] to include collider constraints wefirst performed Bayesian scans [7] over three simplified models real scalar dark matter φ Majorana fermion χ and real vector Xmicro The purpose was to obtain the most probable regionsof each model then to use the resulting posterior samples to investigate whether the LHCexperiments have any sensitivity to the preferred regions

              The three models couple to the mediator with interactions shown in the following table

              Table 31 Simplified Models

              Hypothesis real scalar DM Majorana fermion DM real vector DM

              DM mediator int Lφ sup microφ mφ φ 2S Lχ sup iλχ

              2 χγ5χS LX sup microX mX2 X microXmicroS

              The interactions between the mediator and the standard fermions is assumed to be

              LS sup f f S (31)

              and in line with minimal flavour violation we only consider third generation fermions- ief = b tτ

              For the purposes of these scans we consider the following observables

              32 Observables 36

              32 Observables

              321 Dark Matter Abundance

              We have assumed a single dark matter candidate which freezes out in the early universe witha central value as determined by Planck [21] of

              ΩDMh2 = 01199plusmn 0031 (32)

              h is the reduced hubble constant

              The experimental uncertainty of the background relic density [59] is dominated by atheoretical uncertainty of 50 of the value of the scanned Ωh2 [76] We have added thetheoretical and the experimental uncertainties in quadrature

              SD =radic(05Ωh2)2 + 00312 (33)

              This gives a log likelihood of

              minus05lowast (Ωh2 minus 1199)2

              SD2 minus log(radic

              2πSD) (34)

              322 Gamma Rays from the Galactic Center

              Assuming that the gamma ray flux excess from the galactic centre observed by Fermi-LAT iscaused by self annihilation of dark matter particles the differential flux

              d2Φ

              dEdΩ=

              lt σv gt8πmχ

              2 J(ψ)sumf

              B fdN f

              γ

              dE(35)

              has been calculated using both Micromegas v425 with a Navarro-Frenk-White (NFW) darkmatter galactic distribution function

              ρ(r) = ρ0(rrs)

              minusγ

              (1+ rrs)3minusγ (36)

              with γ = 126 and an angle of 5 to the galactic centre [19]

              32 Observables 37

              Here Φ is the flux dE is the differential energy lt σv gt the velocity averaged annihi-lation cross section of the dark matter across the galactic centre mχ the mass of the darkmatter particle B f =lt σv gt f σv is the annihilation fraction into the f f final state anddN f

              γ dE is the energy distribution of photons produced in the annihilation channel with thefinal state f f The J factor is described below

              The angle ψ to the galactic centre determines the normalisation of the gamma ray fluxthrough the factor

              J(ψ) =int

              losρ(l2 + r⊙2 minus2lr⊙cosψ)dl (37)

              where ρ(r) is the dark matter density at distance r from the centre of the galaxy r⊙=85 kpcis the solar distance from the Galactic center ρ0 is set to reproduce local dark matter densityof 03GeVcm3

              The central values for the gamma ray spectrum were taken from Fig 5 of [77] and thestandard deviations of individual flux points from the error bars on the graph In commonwith [78] we ignore possible segment to segment correlations

              For the Fermi-LAT log likelihood we assumed each point was distributed with indepen-dent Gaussian likelihood with the experimental error for each data point given by the errorbar presented in Fig5 of [76] for each point We also assumed an equal theoretical errorfor each point which when added to the experimental error in quadrature gave an effectivelog likelihood for each point of minus05lowast (giminusdi)

              2

              2lowastσ2i

              where gi are the calculated values and di theexperimental values and σi the experimental errors

              323 Direct Detection - LUX

              The LUX experiment [16] is the most constraining direct search experiment to date for awide range of WIMP models We have used LUXCalc [79] (modified to handle momentumdependent nucleon cross sections) to calculate the likelihood of the expected number ofsignal events for the Majorana fermion model

              The likelihood function is taken as the Poisson distribution

              L(mχ σ | N) = P(N | mχ σ) =(b+micro)Neminus(b+micro)

              N (38)

              32 Observables 38

              where b= expected background of 064 and micro = micro(mχσ ) is the expected number of signalevents calculated for a given model

              micro = MTint

              infin

              0dEφ(E)

              dRdE

              (E) (39)

              where MT is the detector mass times time exposure and φ(E) is a global efficiency factorincorporating trigger efficiencies energy resolution and analysis cuts

              The differential recoil rate of dark matter on nucleii as a function of recoil energy E

              dRdE

              =ρX

              mχmA

              intdvv f (v)

              dσASI

              dER (310)

              where mA is the nucleon mass f (v) is the dark matter velocity distribution and

              dσSIA

              dER= Gχ(q2)

              4micro2A

              Emaxπ[Z f χ

              p +(AminusZ) f χn ]

              2F2A (q) (311)

              where Emax = 2micro2Av2mA Gχ(q2) = q2

              4m2χ

              [24] is an adjustment for a momentum dependantcross section which applies only to the Majorana model

              f χ

              N =λχ

              2m2SgSNN assuming that the relic density is the central value of 1199 We have

              implemented a scaling of the direct search cross section in our code by multiplying gSNN by( Ω

              Ω0)05 as a convenient method of allowing for the fact that the actual relic density enters

              into the calculation of the cross section as a square

              FA(q) is the nucleus form factor and

              microA =mχmA

              (mχ +mA)(312)

              is the reduced WIMP-nucleon mass

              The strength of the dark matter nucleon interaction between dark matter particles andthird generation quarks is

              gSNN =2

              27mN fT G sum

              f=bt

              λ f

              m f (313)

              where fT G = 1minus f NTuminus f N

              Tdminus fTs and f N

              Tu= 02 f N

              Td= 026 fTs = 043 [20]

              33 Calculations 39

              For the real scalar and real vector models where the spin independent cross sections arenot momentum dependent we based the likelihood on the LUX 95 CL from [16] using acomplementary error function Li(d|mχ I) = 05lowastCEr f (dminusx(mχ )

              σ) where x is the LUX limit

              and d the calculated SI cross section We assumed a LUX error σ of 01lowastd lowastradic

              2

              33 Calculations

              We coded the Lagrangians for the three models in Feynrules [4] Calculation of the expectedobservables (dark matter relic density Fermi-LAT gamma ray spectrum and LUX nucleonscattering amplitudes) were performed using Micromegas v425

              331 Mediator Decay

              A number of searches have previously been performed at the LHC which show sensitivity tothe decay of scalar mediators in the form of Higgs bosons (in beyond the standard modeltheories) to bottom quarks top quarks and τ leptons (all third generation fermions) Two inparticular [43 44] provide limits on the cross-sections for the production of the scalar timesthe branching ratios to brsquos and τrsquos In order to check these limits against each of the modelswe ran parameter scans in Madgraph at leading order over the top 900 points from the nestedsampling output from MultiNest [19]

              The two processes were

              1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to apair of τ leptons in pp collisions [44]

              bull generate p p gt b b S where S is the scalar mediator

              The process measures the decay of a scalar in the presence of b-jets that decay to τ+τminus

              leptons Fig31 shows the main diagrams contributing to the process The leftmost diagramis a loop process and the centre and right are tree level however we only got Madgraph togenerated the middle diagram since we are interested in τ leptons and b quarks in the finalstate

              The limit given in [44] only applies in the case of narrow resonances where narrow istaken to mean that the width of the resonance is less than 10 of its mass To test the validity

              33 Calculations 40

              Figure 31 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof τ leptons

              of applying these limits to our models we show in Figures 32 to 34 the percentage widthof the resonance vs mS[GeV ] (pole mass of the scalar mediator) λb and λt (the couplingconstants to the b and t quarks) The probabilities are the posterior probabilities obtainedfrom MultiNest scans of the Majorana model varying the coupling constants and scalarmediator and calculating the decay width of the mediator The results are represented as 2dposterior probability heat maps in the central panels and 1d marginalised distributions on thex and y axes of each graph

              16 18 20 22 24 26 28 30

              log10(mS[GeV])

              001

              002

              003

              004

              005

              Widthm

              S

              00

              04

              08

              12

              0 100 200

              Posterior Probability

              Figure 32 WidthmS vs mS

              The WidthmS was lt 01 at almost all points except where the mass of the scalar wasgreater than 1000 GeV λb was greater than 1 or λτ was greater than 1

              This can be seen from the graphs in Figs 323334

              33 Calculations 41

              4 3 2 1 0

              λb

              001

              002

              003

              004

              005

              WidthmS

              000

              015

              030

              045

              0 100 200

              Posterior Probability

              Figure 33 WidthmS vs λb

              5 4 3 2 1 0

              λτ

              001

              002

              003

              004

              005

              WidthmS

              000

              015

              030

              045

              0 100 200

              Posterior Probability

              Figure 34 WidthmS vs λτ

              The decay widths are calculated in Madgraph at leading order and branching fractionscan be calculated to b t τ and χ (DM) The formulae for the calculations can be found indecayspy which is a subroutine of Madgraph which we have documented in Appendix BB1 Using these we calculated the branching ratio to τ+τminus so that the cross section timesbranching ratio can be compared to the limits in [80] shown in Chapters 56 and 7 in Figs5564 and 74 for each model

              2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

              This scan calculates the contributions of b-quark associated Higgs boson production inthe center and right diagrams of Fig 35 reproduced from [43] The scans did not includePythia and Delphes as only production cross sections were required and the producedparticles were in the final state and required no matching or merging

              The Madgraph processes were

              bull generate p p gt b S where S is the scalar mediator

              bull add process p p gt b S j

              bull add process p p gt b S

              33 Calculations 42

              Figure 35 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof b quarks in the presence of at least one b quark

              bull add process p p gt b S j

              The cross section times branching ratios to bb for mediator production in association withat least one b quark is shown for each model in Chapters 56 and 7 in Figs 5765 and 75

              332 Collider Cuts Analyses

              We generated two parameter scans with parameter combinations from the Multinest outputwhich were the basis of the plots of Fig 6 in [19] The plots in Fig 6 were marginalisedto two parameters each but the raw Multinest output contained 5 parameters plus the priorprobabilities of each combination normalised to total 10

              The first scan was for χ produced with b quarks In order to force the mediator to decayto χ only we had χ in the final state and defined jets to include b quarks in Madgraph Thenwe generated the processes

              bull generate p p gt χ χ j

              bull add process p p gt χ χ j j

              Jet matching was on

              The second scan was for t quarks produced in the final state

              bull generate p p gt χ χ tt

              33 Calculations 43

              No jet matching was required because we did not mix final state jet multiplicities at thematrix element level

              The outputs from these two processes were normalised to 21 f bminus1 and combined

              The results of these scans prior to collider cuts calculations are shown in Figs 5161and 71 These show the posterior probabilities of the various parameter combinations withdifferent constraints applied seperately and then together A more detailed commentary isgiven in each of the sections

              We then sampled 900 points based on posterior probability from these scans and ran thesethrough Madgraph v425 then applied collider cuts with the C++ code using the limits fromthe following six ATLAS and CMS searches shown in Table32 and described below to findany exclusions

              333 Description of Collider Cuts Analyses

              In the following all masses and energies are in GeV and angles in radians unless specificallystated

              3331 Lepstop0

              Search for direct pair production of the top squark in all-hadronic final states in pro-tonndashproton collisions at

              radics=8 TeV with the ATLAS detector[32]

              This analysis searched for direct pair production of the supersymmetric partner to thetop quark using integrated luminosity of 201 f bminus1 The top squark was assumed to de-cay via t rarr t χ0

              1 or t rarr bχ01 or t rarr bχ

              plusmn1 rarr bW (lowast)χ1

              0 where χ01 (χ

              plusmn1 ) denotes the lightest

              neutralino(chargino) in supersymmetric models The search targets a fully hadronic finalstate with four or more jets and large missing momentum The production of a mediator inassociation with top quarks followed by an invisible mediator decay to dark matter particleshas a similar final state to that considered in the ATLAS stop searches Thus we expect theATLAS reach to have sensitivity to the three dark matter models considered in this thesis

              The collider cuts program mimicked the object reconstruction through jetlepton overlapremoval and selected signal electrons muons and τ particles with pT gt 10 GeV and

              33 Calculations 44

              Table 32 95 CL by Signal Region

              Experiment Region Number

              Lepstop0

              SRA1 66SRA2 57SRA3 67SRA4 65SRC1 157SRC2 124SRC3 80

              Lepstop1

              SRtN1-1 857SRtN1-2 498SRtN1-3 389SRtN2 107SRtN3 85SRbC1 832SRbC2 195SRbC3 76

              Lepstop2

              L90 740L100 56L110 90L120 170

              2bstop

              SRA 857SRB 270SRA150 380SRA200 260SRA250 90SRA300 75SRA350 52

              CMSTopDM1L SRA 1385

              ATLASMonobjetSR1 1240SR2 790

              33 Calculations 45

              |η | lt 24724 and 247 respectively (for eminusmicrominusτ) Jets were selected with pT gt 20 GeVand |η |lt 45 b-jets were selected if they were tagged by random selection lt 70 and had|η |lt 25 and pT gt 20 GeV

              These jets and leptons were further managed for overlap removal within ∆R lt 04 withnon b-jets removed if ∆R lt 02

              The b-jets and non b-jets were further selected as signal jets if pT gt 35 GeV and |η |lt 28

              These signal jets electrons and muons were finally binned according to the signal regionsin Tables 333435 below

              Table 33 Selection criteria common to all signal regions

              Trigger EmissT

              Nlep 0b-tagged jets ⩾ 2

              EmissT 150 GeV

              |∆φ( jet pmissT )| gtπ5

              mbminT gt175 GeV

              Table 34 Selection criteria for signal regions A

              SRA1 SRA2 SRA3 SRA4anti-kt R=04 jets ⩾ 6 pT gt 808035353535 GeV

              m0b j j lt 225 GeV [50250] GeV

              m1b j j lt 225 GeV [50400] GeV

              min( jet i pmissT ) - gt50 GeV

              τ veto yesEmiss

              T gt150 GeV gt250 GeV gt300 GeV gt 350 GeV

              Table 35 Selection criteria for signal regions C

              SRC1 SRC2 SRC3anti-kt R=04 jets ⩾ 5 pT gt 8080353535 GeV

              |∆φ(bb)| gt02 π

              mbminT gt185 GeV gt200 GeV gt200 GeV

              mbmaxT gt205 GeV gt290 GeV gt325 GeV

              τ veto yesEmiss

              T gt160 GeV gt160 GeV gt215 GeV

              wherembmin

              T =radic

              2pbt Emiss

              T [1minus cos∆φ(pbT pmiss

              T )]gt 175 (314)

              33 Calculations 46

              and pbT is calculated for the b-tagged jet with the smallest ∆φ to the pmiss

              T direction andmbmax

              T is calculated with the pbT for the b-tagged jet with the largest ∆φ to the pmiss

              T direction

              m0b j j is obtained by first selecting the two jets with the highest b-tagging weight- from

              the remaining jets the two jets closest in the η minusφ plane are combined to form a W bosoncandidate this candidate is then combined with the closest of the 2 b-tagged jets in the η minusφ

              plane to form the first top candidate with mass m0b j j A second W boson candidate is formed

              by repeating the procedure with the remaining jets this candidate is combined with thesecond of the selected b-tagged jets to form the second top candidate with mass m1

              b j j Thecuts shown in Appendix1 Table A1 show the cumulative effect of the individual cuts appliedto the data the signal region cuts and the cumulative effects of individual cuts in SignalRegion C

              3332 Lepstop1

              Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

              radics=8 TeV pp collisions using 21 f bminus1 of

              ATLAS data[33]

              The stop decay modes considered were those to a top quark and the lightest supersym-metric particle (LSP) as well as a bottom quark and the lightest chargino where the charginodecays to the LSP by emitting a W boson We should expect the three dark matter models inthis paper with couplings to t and b quarks to show some visibility in this search

              The collider cuts program mimicked the object construction by selecting electrons andmuons with pT gt 10 GeV and |η |lt 247 and 24 respectively and jets with pT gt 20 GeVand |η |lt 10 b-jets with |η |lt 25 and pT gt 25 GeV were selected with random efficiency75 and mistag rate 2 (where non b-jets were mis-classified) We then removed any jetwithin ∆R = 02 of an electron and non overlapping jets with |η |lt 28 were used to removeany electrons and muons within ∆R = 04 of these while non-overlapping jets with |η |lt 25and pT gt 25 GeV were classified as signal jets Non overlapping electrons and muons withpT gt 25 GeV were classified as signal electrons and muons

              Signal selection defined six regions to optimize sensitivity for different stop and LSPmasses as well as sensitivity to two considered stop decay scenarios Three signal regionslabelled SRbC 1-3 where bC is a moniker for b+Chargino and 3 regions labelled SRtN 1-3where tN is a moniker for t+Neutralino were optimised for the scenarios where t1 decaysto the respective particles indicated by the labels Increasing label numbers correspond toincreasingly strict selection criteria

              33 Calculations 47

              The dominant background in all signal regions arises from dileptonic tt events All threeSRtN selectons impose a requirement on the 3-jet mass m j j j of the hadronically decayingtop quark to specifically reject the tt background where both W bosons from the top quarkdecay leponically

              For increasing stop mass and increasing difference between the stop and the LSP therequirents are tightened on Emiss

              T on the ratio EmissT

              radicHT where HT is the scalar sum of the

              momenta of the four selected jets and also tightened on mT

              To further reduce the dileptonic tt background for an event characterised by two one stepdecay chains a and b each producing a missing particle C use is made of the variable

              mT 2 =min

              pCTa + pC

              T b = pmissT

              [max(mTamtb)] (315)

              where mTi is the transverse mass 1 of branch i for a given hypothetical allocation (pCTa pC

              T b)

              of the missing particle momenta Eqn 315 is extended for more general decay chains byamT 2 [81] which is an asymmetric form of mT 2 in which the missing particle is the W bosonin the branch for the lost lepton and the neutrino is the missing particle for the branch withthe observed charged lepton

              ∆φ( jet12 pmissT ) the azimuthal angle between leading or subleading jets and missing mo-

              mentum is used to suppress the background from mostly multijet events with mis-measuredEmiss

              T

              Events with four or more jets are selected where the jets must satisfy |η | lt 25 and pT gt80604025 GeV respectively

              These signal particles were then classified according to the cutflows analysis in Fig36below and compared with the paper in Appendix1 Table A2

              3333 Lepstop2

              Search for direct top squark pair production in final states with two leptons in p pcollisions at

              radics=8TeV with the ATLAS detector[34]

              Three different analysis strategies are used to search for t1 pair production Two targett1 rarr b+ χ

              plusmn1 decay and the three body t1 rarr bW χ0

              1 decay via an off-shell top quark whilst

              1The transverse mass is defined as m2T = 2plep

              T EmissT (1minus cos(∆φ)) of branch i where ∆φ is the azimuthal

              angle between the lepton and the missing transverse momentum

              33 Calculations 48

              Table 36 Signal Regions - Lepstop1

              Requirement SRtN2 SRtN3 SRbC1 SRbC2 SRbC3∆φ( jet1 pmiss

              t )gt - 08 08 08 08∆φ( jet2 pmiss

              T )gt 08 08 08 08 08Emiss

              T [GeV ]gt 200 275 150 160 160Emiss

              T radic

              HT [GeV12 ]gt 13 11 7 8 8

              mT [GeV ]gt 140 200 120 120 120me f f [GeV ]gt - - - 550 700amT 2[GeV ]gt 170 175 - 175 200mT

              T 2[GeV ]gt - 80 - - -m j j j Yes Yes - - -

              one targets t1 rarr t + χ01 to an on shell quark decay mode The analysis is arranged to look

              at complementary mass splittings between χplusmn1 and χ0

              1 smaller and larger than the W bosonmass The smaller splitting provides sensitivity to t1 three body decay

              Object reconstruction was carried out by selecting baseline electronsmuons and τrsquos withpT gt 10 GeV and |η |lt 24724 and 247 respectively Baseline jets and b-jets were selectedwith pT gt 20 GeV and |η |lt 25 and b-jets with 70 efficiency Baseline jets were removedif within ∆R lt 02 of an electron and electrons and muons were removed if within ∆R lt 04of surviving jets Signal electrons required pT gt 10 GeV and signal jets pT gt 20 GeV and|η |lt 25

              The analysis made use of the variable mT 2 [82] the stransverse mass used to mea-sure the masses of pair-produced semi-invisibly decaying heavy particles It is definedas mT 2(pT1 pT2qT ) =

              minqT1+qT2=qT

              max[mT (pT1qT1)mT (pT2qT2)] where mT indicatesthe transverse mass pT1 and pT2 are vectors and qT = qT1 + qT2 The minimisation isperformed over all possible decompositions of qT

              Events were selected if they passed either a single-electron a single-muon a double-electron a double-muon or an electron-muon trigger Events are required to have exactlytwo opposite sign leptons and at least one of the leptons or muons must have a momentumgreater than 25 GeV and the invariant mass (rest mass) of the two leptons is required to begreater than 20 GeV In order to reduce the number of background events containing twoleptons produced by the on-shell decay of the Z boson the invariant mas must be outside711-111 GeV range

              Two additional selections are applied to reduce the number of background events withhigh mT 2 arising from events with large Emiss

              T due to mismeasured jets ∆φb lt 15 and∆φ gt 10 The quantity ∆φb is the azimuthal angle between pll

              T b = pmissT + pl1

              T +Pl2T The

              33 Calculations 49

              vector pllT b is the opposite of the vector sum of all the hadronic activity in the event For WW

              and tt backgrounds it measures the transverse boost of the WW system and for the signalthe transverse boost of the chargino-chargino system The ∆φ variable is the azimuthal anglebetween the pmiss

              T vector and the direction of the closest jet

              By requiring mT 2 gt 40 GeV the tt background is the dominant background in both thesame-flavoured (SF) and different-flavoured (DF) events

              Four signal regions are then defined as shown in Table 37 SR M90 has the loosestselection requiring MT 2gt 90 GeV and no additional requirement on jets and providessensitivity to scenarios with a small difference between the masses of the top-squark andchargino SR M110 and M120 have a loose selection on jets requiring two jets with pT

              gt 20 GeV and mT 2 to be greater than 110 GeV and 120 GeV respectively They providesensitivity to scenarios with small to moderate difference between the invariant masses of thestop quark and the charginos SR M100 has a tighter jet slection and provides sensitivity toscenarios with a large mass difference between the both stop quark and the chargino and thestop quark and the neutralino

              The analysis cut regions are summarised in Fig37 below

              Table 37 Signal Regions Lepstop2

              SR M90 M100 M110 M120pT leading lepton gt 25 GeV

              ∆φ(pmissT closest jet) gt10

              ∆φ(pmissT pll

              T b) lt15mT 2 gt90 GeV gt100 GeV gt110 GeV gt120 GeV

              pT leading jet no selection gt100 GeV gt20 GeV gt20 GeVpT second jet no selection gt50 GeV gt20 GeV gt20 GeV

              To validate the code we ran the cuts shown in Appendix 1 Table A3 with the givenMadgraph simulation and compared to [83] It was not clear what baseline cuts had beenemployed in the ATLAS paper so our results were normalised to the 2 SF electrons beforecomparison Some of the ATLAS cut numbers were less than 3 and we did not produce anyevents passing these cuts but clearly the margin of error at this level was greater than thenumber of ATLAS events

              33 Calculations 50

              3334 2bstop

              Search for direct third-generation squark pair production in final states with miss-ing transverse momentum and two b-jets in

              radics= 8 TeV pp collisions with the ATLAS

              detector[31]

              Two sets of scenarios are considered In the first the coloured sbottom is the onlyproduction process decaying via b1 rarr bχ0

              1 In the second the stop is the only productionprocess decaying via t1 rarr t χplusmn

              1 and the lightest chargino (χplusmn1 ) decays via a virtual W boson

              into the final state χ01 f f prime Both scenarios are characterised by the presence of two jets

              resulting from the hadronisation of b quarks and large missing transverse momentum Theproduction of a mediator in association with top quarks followed by an invisible mediatordecay to dark matter particles has a similar final state as that considered in the ATLAS stopsearches Thus we expect the ATLAS reach to have sensitivity to the three dark mattermodels considered in this thesis

              The collider cuts program mimicked object reconstruction by selecting baseline electronswith pT gt 7 GeV and |η |lt 247 and muons with pT gt 6 GeV and |η |lt 24 Baseline jetswere selected with pT gt 20 GeV and |η |lt 28 We then removed any jets within ∆R lt 02of an electron and any electron within ∆R lt 04 of surviving jets Signal electrons andmuons were required to have pT gt 25 GeV b-jets were selected with 60 tagging efficiencyand |η |lt 25 and pT gt 20 GeV

              Two sets of signal regions are defined to provide sensitivity to the kinematic topologiesassociated with the different mass splittings between the sbottom or the stop and the neu-tralino In all cases the presence of at least one primary vertex (with at least five tracks withpT gt 04 GeV) is required Events are selected with Emiss

              T gt 150 GeV and no electrons ormuons identified in the final state For signal region selections jets with |η | lt 28 are orderedaccording to their pT and 2 jets are required to be b-tagged

              The variables are defined as follows

              bull ∆φmin is the minimal azimuthal angle ∆φ between any of the three leading jets andpmiss

              T

              bull me f f (k) = sumki=1(p jet

              T )i +EmissT where the index refers to the pT ordered list of jets

              bull HT3 is defined as the scalar sum of the pT of the n leading pT ordered jets withoutincluding the 3 leading jets HT3 = sum

              ni=4(p jet

              T )i

              bull mbb is the invariant mass of the two b-tagged jets in the event

              33 Calculations 51

              bull mCT is the contransverse mass [82] and is a kinematical variable that can be usedto measure the masses of pair produced semi-invisibly decaying particles For twoidentical decays of heavy particles into two invisible particles v1 and v2 and twoinvisible particles mCT is defined as m2

              CT (v1v2) = [ET (v1)+ET (v2)]2 minus [pT (v1)minus

              pT (v2)]2 where ET =

              radicp2

              T +m2 The the contransverse mass is invariant under equaland opposite boosts of the parent particles in the transverse plane and is bounded fromabove by an analytical expression of the particle masses For production of sbottompairs the bound is given by mmax

              CT =m2(b)minusm2(χ0

              1 )

              m(b) and for tt events the bound is 135

              GeV A similar equation can be written for stop pairs in term of the stop and charginomasses

              A definition of the signal regions is given in the Table38

              Table 38 Signal Regions 2bstop

              Description SRA SRBEvent cleaning All signal regions

              Lepton veto No emicro after overlap removal with pT gt7(6) GeV for e(micro)Emiss

              T gt 150 GeV gt 250 GeVLeading jet pT ( j1) gt 130 GeV gt 150 GeVSecond jet pT ( j2) gt 50 GeV gt 30 GeVThird jet pT ( j3) veto if gt 50 GeV gt 30 GeV

              ∆φ(pmissT j1) - gt 25

              b-tagging leading 2 jets(pT gt 50 GeV |η | lt 25)

              2nd- and 3rd- leading jets(pT gt 30 GeV|η | lt 25)

              ∆φmin gt 04 gt 04Emiss

              T me f f (k) EmissT me f f (2) gt 025 Emiss

              T me f f (3) gt 04mCT gt 150200250300350 GeV -HT3 - lt 50 GeVmbb gt200 GeV -

              The analysis cuts are summarised in Table A4 of Appendix 1

              3335 ATLASMonobjet

              Search for dark matter in events with heavy quarks and missing transverse momentumin pp collisions with the ATLAS detector[30]

              Events were selected with high missing transverse momentum when produced in associa-tion with one or more jets identified as containing b quarks Final states with top quarks wereselected by requiring a high jet multiplicity and in some cases a single lepton The research

              33 Calculations 52

              studied effective contact operators in the form of a scalar operator Oscalar = sumqmqMN

              lowastqqχχ

              where MNlowast is the effective mass of the coupling and the quark and DM fields are denoted by

              q and χ respectively These are the equivalent of the simplified models in this paper with themediator integrated out

              Object reconstruction selected baseline electrons muons jets and τrsquos with pT gt 20 GeVand |η |lt 2525 and 45 respectively Signal jets were selected by removing any jets within∆R lt 02 of an electron and electrons and muons were selected by removing them if theywere within ∆R lt 04 of surviving jets b-jets were tagged with an efficiency of 60

              Only signal regions SR1 and SR2 were analysed

              The analysis split events into signal regions SR1 and SR2 according to the followingCriteria

              Table 39 Signal Region ATLASmonobjet

              Cut SR1 SR2Jet multiplicity n j 1minus2 3minus4

              bminusjet multiplicity nb gt 0 (60 eff) gt 0 (60 eff)Lepton multiplicity nl 0 0

              EmissT gt300 GeV gt200 GeV

              Jet kinematics pb1T gt100 GeV pb1

              T gt100 GeV p j2T gt100 (60) GeV

              ∆φ( j1EmissT ) gt 10 i = 12 gt 10 i = 14

              Where p jiT (pbi

              T ) represent the transverse momentum of the i-th jet b-tagged jet respec-tively If there is a second b-tagged jet the pT must be greater than 60 GeV

              3336 CMSTop1L

              Search for top-squark pair production in the single-lepton final state in pp collisionsat

              radics=8 TeV[41]

              This search was for pair production of top squarks in events with a single isolatedelectron or muon jets large missing transverse momentum and large transverse mass

              (MT =radic

              2EmissT pl

              T (1minus cos(∆φ))) where plT is the leptonrsquos transverse momentum and ∆(φ)

              is the difference between the azimuthal angles of the lepton and EmissT The 3 models

              considered should have some visibility at certain parameter combinations in this regionThe search focuses on two decay modes of the top squark leading to the two processespp rarr t tlowast rarr tt χ0

              1 χ01 rarr bbW+Wminusχ0

              1 χ01 and pp rarr t tlowast rarr bbχ

              +1 χ

              minus1 rarr bbW+Wminusχ0

              1 χ01 The

              33 Calculations 53

              lightest neutralino χ01 is considered the lightest supersymmetric particle and escapes without

              detection The analysis is based on events where one of the W bosons decays leptonicallyand the other hadronically resulting in one isolated lepton and 4 jets two of which originatefrom b quarks The two neutralinos and the neutrino from the W decay can result in largemissing transverse momentom The kinematical end point of MT is lt the mass of the Wboson However for signal events the presence of LSPs in the final state allows MT to exceedthe mass of the W boson The search is therefore for events with large MT The dominantbackground with large MT arises from tt decays where W bosons decay leptonically withone of the leptons not being identified

              To reduce the dominant tt background use was made of the MWT 2 variable defined as

              the minimum mother particle mass compatible with the transverse momenta and mass shellconstraints [84]

              Object reconstruction required electrons and muons to have pT gt 30 and 25 GeV respand |η |lt 25 and 21 respectively Baseline jets were required to have pT gt 30 and |η |lt 4and b-jets were tagged with an efficiency of 75 for |η |lt 24

              Events were chosen if they contained one isolated electron or muon at least 4 jets and atleast one b-tagged jet and Emiss

              T gt 100 GeV MT gt 160 GeV MWT 2 was required to be greater

              than 200 GeV and the smallest azimuthal angle difference between baseline jets and EmissT

              gt12

              Chapter 4

              Calculation Tools

              41 Summary

              Figure 41 Calculation Tools

              The simplified models were implemented in Feynrules [4] and the model files importedinto MicrOmegas v425 which was used to calculate the log-likelihoods of the astrophysicalobservables MicrOmegas was modified to allow only s-channel decays in both relic densityand gamma ray calculations Direct detection constraints were easy to implement for thescalar and vector models by coding the analytical expression for the spin independent WIMP-nucleon scattering cross section but due to the momentum dependence of the WIMP-nucleon

              42 FeynRules 55

              scattering cross section of the Majorana fermion model LUXCalc v101 [79] was modifiedfor this model to implement momentum dependent cross sections [85] We used MicrOmegasv425 to calculate relic density values and to produce gamma ray spectra for comparisonwith the Fermi-LAT results Parameter scans were carried out using MultiNest v310 withthe parameter ranges given in Table 51 and using astrophysical constraints only Oncethese scans had converged a random sample of 900 points was chosen (weighted by theposterior probability) and then used to calculate collider constraints using the MadgraphMonte Carlo generator This included comparison with mediator production and decay limitsfrom the LHC experiments and the reproduction of searches for dark matter plus other finalstate objects which required dedicated C++ code that reproduces the cut and count analysesperformed by CMS and ATLAS

              42 FeynRules

              FeynRules is a Mathematicareg package that allows the calculation of Feynman rules inmomentum space for any QFT physics model The user needs to provide FeynRules withminimal information to describe the new model contained in the model-file This informationis then used to calculate a set of Fenman rules associated with the Lagrangian The Feynmanrules calculated by the code can then be used to implement new physics models into otherexisting tools such as Monte Carlo (MC) generators via a set of interfaces developed andmaintained by the MC authors

              Feynrules can be run with a base model = SM (or MSSM) and user defined extensions tothe base model

              43 LUXCalc

              LUXCalc is a utility program for calculating and deriving WIMP-nucleon coupling limitsfrom the recent results of the LUX direct search dark matter experiment The program hasbeen used to derive limits on both spin-dependent and spin-independent WIMP-nucleoncouplings and it was found under standard astrophysical assumptions and common otherassumptions that LUX excludes the entire spin-dependent parameter space consistent witha dark matter interpretation of DAMArsquos anomalous signal and also found good agreementwith published LUX results [79]

              We modified the program for the Majorana fermion model to implement momentumdependent cross -sections [85]

              44 Multinest 56

              44 Multinest

              Multinest is an efficient Bayesian Inference Tool which uses a simultaneous ellipsoidalnested sampling method a Monte Carlo technique aimed at an efficient Bayesian Evidencecalculation with posterior inferences as a by product

              Bayes theorem states that

              Pr(θ |DH) =Pr(D|θ H)Pr(θ |H)

              Pr(D|H) (41)

              Here D is the data and H is the Hypothesis Pr(θ |DH) is the posterior probability ofthe parameters Pr(D|θ H) = L(θ) is the likelihood Pr(θ |H) is the prior and Pr(D|H) isthe evidence

              The evidence Pr(D|H) =int

              Pr(θ |DH)Pr(θ |H)d(θ) =int

              L(θ)Pr(θ |H)d(θ) Thus ifwe define the prior volume dX = Pr(θ |H)d(θ) and

              X(λ ) =int

              L(θ)gtλ

              Pr(θ |H)d(θ) (42)

              where the integral extends over the region of parameter space within the iso likelihoodcontour L(θ) = λ then the evidence integral can be written

              int 10 L (X)dX where L (X) the

              inverse of equation 42 and is a monotonically decreasing function of X (the prior volumewhich is 1 for the total sample space and 0 for a single point)

              Thus if one can evaluate likelihoods L (Xi) where 0 lt Xn lt lt X2 lt X1 lt X0 = 1the evidence can be estimated as sumLiwi where wi = 5(Ximinus1 minusXi+1) Ellipsoidal samplingapproximates the iso likelihood contour by a D dimensional covariance matrix of the currentset of active points New points are selected from the prior within this ellipsoidal bounduntil one is obtained that has a likelihood greater than the likelihood of the removed lowestlikelihood point This method is extended for multimodal sampling where there may bemore than one local maximum by clustering points that are well separated and constructingindividual ellipsoidal bounds for each cluster For more details consult [7]

              The results of the analysis are expressed as a data file containing the sample probability-2loglikelihood followed by the parameters of the model for the particular sampling Theprobabilities are normalised to total 10

              45 Madgraph 57

              45 Madgraph

              Madgraph is a general purpose matrix element calculator and Monte Carlo generator Basedon user-supplied Feynman rules (eg the output of Feynrules) the package calculates the tree-level matrix elements for any processes that the user specifies and generates samples of LHCevents including an accurate simulation of the proton collisions themselves (which requires aknowledge of the parton distribution functions that determine the relative proportion of eachparton inside the nucleus) Hadronisation of final state quarks is not possible in Madgraphand hence the code must be used in conjunction with Pythia which performs a parton shower

              The Madgraph output was automatically showered by Pythia (Pythia describes partonradiation as successive parton emissions using Markov chain techniques based on Sudakovform factors and this description is only formally correct for the soft and collinear emissions)A Madgraph simulation is at the level of the fundamental particles (quarks and leptons)Madgraph generates a limited set of matrix element processes at LO representing hard (highenergy) emissions and passes these to Pythia for showering The final state partons in theMadgraph simulation must be matched to the Pythia final state partons to avoid doublecounting

              The output from Pythia is automatically fed through Delphes where basic collider cutscan be performed (calculation of the numbers of particles and jets that would be observed atdifferent energies and locations within the collider) The output of the Delphes simulation is afile containing reconstructed events that can be analysed using the standard ROOT framework(the most popular data analysis package in particle physics) [86]

              The parameters that gave the best fits to the direct [16] and indirect DM experiments [87]and relic density [59] are ranked by posterior probability from the Multinest [7] output

              The parameters in the Madgraph simulations are stored in files in the cards sub directoryand in particular the run card param card and Delphes card record items relevent to thesimulation

              In the run card the number of events the Beam 1 and 2 total energy in GeV (4000 each)and the type of jet matching were modified (to MLM type matching where the final statepartons in Madgraph are clustered according to the KT jet algorithm to find the equivalentparton shower history of the event there is a minimum cutoff scale for the momentum pT

              given by the xqcut parameter) The maxjetflavour (=5 to include b quarks) and xqcut = 30GeV parameters were also modified

              46 Collider Cuts C++ Code 58

              The param card was modified to specify the masses of the new particles and the autodecay modes The Delphes card is overwritten with either the ATLAS or CMS defaultcards which record tracking efficiencies and isolation momentum and energy resolutionscalorimeter details jet finder and b and τ -tagging parameters However the isolation andtagging in this study were done by the collider cuts C++ code since multiple sets of cutswere applied to the same output from Pythia

              When performing simulations of the three dark matter models it was necessary to definethe proton and jets to contain b quarks as this is not the default in Madgraph The processesgenerated included 1 and 2 jets

              46 Collider Cuts C++ Code

              Because of the volume of simulations required for the collider constraints we used C++code (originally developed by Martin White and Sky French and modified by me) that readsthe ROOT files and outputs the numbers of events that would have been observed under thechosen collider experiments had the model been true for the given parameters Thus the sameoutput files can be used for all of the collider cuts and hence no cuts were performed throughthe Delphes cards

              In order to validate the C++ code we replicated the observed events with the papersdescribing the various collider searches (see Appendix)

              Chapter 5

              Majorana Model Results

              51 Bayesian Scans

              To build some intuition we first examined the constraining effect of each observable on itsown We ran separate scans for each constraint turned on and then for all the constraintstogether In Fig 51 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together The paramaters were scanned over the range in Table 51 below

              Table 51 Scanned Ranges

              Parameter mχ [GeV ] mS[GeV ] λt λb λτ

              Range 0minus103 0minus103 10minus5 minus10 10minus5 minus10 10minus5 minus10

              In Fig51 we can see that the preferred DM mass is centered around 35 GeV primarily dueto the influence of the Fermi-LAT gamma ray constraint whilst the relic density constraintallows a wide range of possible mediator and DM masses Direct detection (LUX) constraintssplit the mass regions into a top left and bottom right region and together these split theoverall mass parameter regions into two regions by mediator mass but with DM mass around35 GeV Later studies using a lower excess gamma ray spectrum tend to increase the preferredDM mass up to 50 GeV [76] and [88]

              The marginalised posterior probabilities for all combinations of parameters taken twoat a time are shown in Fig52 These plots have all constraints (Gamma LUX and RelicDensity) applied together The idea of these graphs is to study the posterior probabilities ofcombinations of two paramaters taken at a time (the off-diagonal graphs) The diagonal plotsshow marginalised histograms of a single parameter at a time- this gives an indication of thespread of each parameter The plots show that mS and λb show a high degree of correlationfor mS gt 100 GeV but there is also a range of values of mS around 35 GeV which are

              51 Bayesian Scans 60

              1 0 1 2 3 4log10(mχ)[GeV]

              1

              0

              1

              2

              3

              4

              log 1

              0(m

              s)[GeV

              ]

              (a) Gamma Only

              1 0 1 2 3 4log10(mχ)[GeV]

              1

              0

              1

              2

              3

              4

              log 1

              0(m

              s)[GeV

              ]

              (b) Relic Density

              1 0 1 2 3 4log10(mχ)[GeV]

              1

              0

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              s)[GeV

              ]

              (c) LUX

              05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

              05

              00

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              35

              log 1

              0(m

              s)[GeV

              ]

              (d) All Constraints

              Figure 51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether

              51 Bayesian Scans 61

              00

              05

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              30

              log 1

              0(m

              χ)[GeV

              ]

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              ms[Gev

              ]

              5

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              t)

              5

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              b)

              00 05 10 15 20 25 30

              log10(mχ)[GeV]

              5

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              2

              1

              0

              1

              log 1

              0(λ

              τ)

              00 05 10 15 20 25 30

              ms[Gev]5 4 3 2 1 0 1

              log10(λt)5 4 3 2 1 0 1

              log10(λb)5 4 3 2 1 0 1

              log10(λτ)

              Figure 52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter

              52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 62

              possible for λb around 001 λt and λτ show a wide range of possible values for two possibleranges of mS lt 35 GeV and greater than 100 GeV mχ shows a narrow range of possiblevalues around 35 GeV for all values of the other parameters

              52 Best fit Gamma Ray Spectrum for the Majorana FermionDM model

              We can see from the Fig51(a) scan that the maximum log-likelihood Fermi-LAT gammaray spectrum on its own shows values that centre around 35 GeV This is not changed byadding a relic density constraint or a direct detection (LUX) constraint

              The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraintsis shown in Fig53 The parameters for this point were

              Table 52 Best Fit Parameters

              Parameter mχ [GeV ] mS[GeV ] λt λb λτ

              Value 3332 49266 0322371 409990 0008106

              10-1 100 101 102

              E(GeV)

              10

              05

              00

              05

              10

              15

              20

              25

              30

              35

              E2dφd

              E(G

              eVc

              m2ss

              r)

              1e 6

              Best fitData

              Figure 53 Gamma Ray Spectrum

              The relic density (Ωh2) at this point has a value of 0135 but the point is separated onlyslightly in posterior probability from points with a value of 01199 with mS lt 10 GeV (iethe posterior probability is almost the same differing by less than 1)

              To better understand the interplay of the different constraints the three graphs in Fig54show the log likelihood surfaces and individual best fit points (best log likelihoods) for the γ

              and Ω scans seperately and then combined The diagrams each choose a pair of couplings(fixing the remaining parameters) and display the 2d surfaces of minuslog(χ2) (proportional to

              52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 63

              the log likelihoods of the coupling combinations) obtained by varying the pair of couplingsThese graphs do not show marginalised posterior probabilities but rather are based on the-2log likelihood calculations directly from MicrOmegas for the given coupling combinationsWe have termed this value as χ2 for convenience and used the log of this value to give abetter range for the colours These log likelihood values are what drive the MultiNest searchof parameter space for the best parameter combinations

              The graphs show a complex interplay of the different constraints resulting often in acompromise parameter combination The arrow on each graph points to the maximum point We have chosen a single representative set of couplings near the best overall fit point for allthe graphs

              52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 64

              00 05 10 15 20 25 30

              log10(mχ)

              00

              05

              10

              15

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              30

              log

              10(m

              S)

              Max

              minuslog10(χ2(Γ)) λt = 487 λτ = 024 λb = 0344

              16

              14

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              8

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              γ Maximum at mχ=416 GeV mS=2188 GeV

              00 05 10 15 20 25 30

              log10(mχ)

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              10(m

              S)

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              minuslog10(χ2(Omega)) λt = 487 λτ = 024 λb = 0344

              28

              24

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              04

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              Ω Maximum at mχ=363 GeV mS=1659 GeV

              00 05 10 15 20 25 30

              log10(mχ)

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              10(m

              S)

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              minuslog10(χ2(Both)) λt = 487 λτ = 024 λb = 0344

              16

              14

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              8

              6

              4

              2

              0

              Both Maximum at mχ=8317 GeV mS=2884GeV The best fit point for these couplings maybe above below or between the individual bestfit points but will on average be between thepoints

              Figure 54 Plots of log likelihoods by individual and combined constraints Masses in GeV

              53 Collider Constraints 65

              53 Collider Constraints

              531 Mediator Decay

              1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof τ leptons in pp collisions [44] Refer also to Chapter 3 for details

              We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

              The results of the calculations are shown in the graph Fig55 and can be compared withthe limits in the table from the paper [44] in Fig 56 The black line is the observed ATLASlimit

              Figure 55 σ lowastBr(σ rarr ττ) versus Mass of Scalar

              0 200 400 600 800

              mS[GeV]

              10

              5

              0

              log 1

              0(σ

              (bbS

              )lowastB

              (Sgtττ

              ))[pb]

              Observed LimitLikely PointsExcluded Points

              0

              20

              40

              60

              80

              100

              120

              0 5 10 15 20 25 30 35 40 45

              We can see that only one point here exceeded the limit and the points showed a widespread over σ(bbS)lowastBr(σ rarr ττ) by mS space which is consistent with the spread of pointson the MultiNest scan on the mS by λτ plane causing a spread of branching ratios to the τ

              quark It may seem odd that only one point was excluded here but this point was sampledwith a very low probability having high coupling constants of 597 to the τ quark and 278 tothe t quark and only just exceeds the limit As such it can be explained as a random outlier

              53 Collider Constraints 66

              Figure 56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar

              2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

              This scan calculates the contributions of b-quark associated Higgs boson productionreproduced from [43] Refer to Chapter 3 for details

              We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

              The cross section times branching ratios to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig57 The black line is the CMSLimit

              Figure 57 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar

              0 200 400 600 800

              mS[GeV]

              15

              10

              5

              0

              5

              log

              10(σ

              (bS

              +X

              )lowastB

              (Sgt

              bb))

              [pb]

              Observed LimitLikely PointsExcluded Points

              0

              20

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              60

              80

              100

              120

              0 50 100 150 200 250

              53 Collider Constraints 67

              The results of this scan were compared to the limits in [89] with the plot shown inFig58

              Figure 58 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

              We can see that there are a number of exclusions across the range of mS which occuredfor high values of the coupling constants gt 1 for λbλt λtau - these can be seen also on Fig59The excluded parameter combinations represent less than 2 of the total sampled number

              532 Collider Cuts Analyses

              We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude and the results are shown against theposterior probabilities of the parameter combinations in Fig 59

              The circles in Figure 59 show which points were excluded by the collider cuts as well asthe limits on cross sections times branching ratios to each of τ+τminus and bb

              All the excluded points under the heading Collider Cuts showed exclusion by the Lepstop0experiment (Search for direct pair production of the top quark in all hadronic final states

              53 Collider Constraints 68

              0 1 2 3

              log10(mχ)[GeV]

              0

              1

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              0(m

              s)[GeV

              ]Collider Cuts

              σ lowastBr(σgt bS+X)

              σ lowastBr(σgt ττ)

              (a) mχ by mS

              6 5 4 3 2 1 0 1 2

              log10(λt)

              0

              1

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              s)[GeV

              ](b) λt by mS

              5 4 3 2 1 0 1

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              t)

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              0

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              s)[GeV

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              (f) λb by mS

              Figure 59 Excluded points from Collider Cuts and σBranching Ratio

              53 Collider Constraints 69

              [32]) Some of these same points were also excluded by Lepstop2 2bstop CMSTop andATLASMonobjet experiments There were expected events at other points but none weresignificant at the 95 exclusion level These significant exclusions occurred mainly wherethe mass of the mediator was over 100 GeV and higher values of the couplings to b and tquarks but there were a few points with mediator mass around 25 GeV The exclusions withmediator mass around 25 GeV were all accompanied by couplings to t quarks of order 10which is probably non-perturbative

              Looking at the exclusions in Fig 59 we can see that these occur mainly in the regionof higher mediator mass gt 100 GeV and lt 400 GeV These exclusions come from theastrophysical constraints (Gamma) which show a strong correlation between λb and mS andhigh values of λt gt 1 However these mass regions do not overlap at the same couplingcombinations as the high likelihood region The green and white circles correspond to theexcluded points in Figs 57 and 58 respectively

              The remaining regions of highest likelihood can be seen to be where 0001 λb lt 01accompanied by λt lt 0001 and mass of the mediator mS lt 10 GeV or 400 GeV lt mS lt1000GeV with mχ sim 35 GeV

              The plot of λt by λb shows the significant regions do not overlap with the collider cutsand cross section constraints since the latter occur at high values of either coupling while thelikely region has low values of these couplings (also accompanied by low couplings to λτ )

              Chapter 6

              Real Scalar Model Results

              61 Bayesian Scans

              To build some intuition we first examined the constraining effect of each observable on itsown We ran seperate scans for each constraint turned on and then for all the constraintstogether In Figure 61 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together

              In Figure 61 we can see that there are is a narrow region which dominates when allconstraints are taken into account This centers on mχ=9 GeV and a range of values for mS

              from 35 GeV to 400 GeV The region is not the highest likelihood region for the Fermi-LATgamma ray spectrum on its own (as can be seen from the individual plot - there is a narrowband to the left of the highest likelihood band) nor for the LUX likelihood or relic densitybut represents a compromise between the three constraints

              The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig62 The diagonal plots show marginalised histograms of a singleparameter at a time

              61 Bayesian Scans 71

              05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

              05

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              Figure 61 Real Scalar Dark Matter - By Individual Constraint and All Together

              61 Bayesian Scans 72

              00

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              Figure 62 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter

              62 Best fit Gamma Ray Spectrum for the Real Scalar DM model 73

              62 Best fit Gamma Ray Spectrum for the Real Scalar DMmodel

              We can see from Fig61 that the maximum log-likelihood Fermi-LAT Gamma ray spectrumon its own shows values of mχ that centre around 35 GeV However the LUX constraintin particular moves the best fit to a point around 9 GeV- the tight LUX constraint onlydisappears for low masses and at that point the proportion of τ+τminus annihilation needs toincrease to fit the constraints

              The gamma ray spectrum at the points of best fit for Fermi-LAT including all constraintsis shown in Fig63 The parameters for these points were respectively and involve decays tomainly τ and b quarks respectively

              Table 61 Best Fit Parameters

              Parameter mχ [GeV ] mS[GeV ] λt λb λτ

              Value 932 3526 000049 0002561 000781

              10-1 100 101 102

              E(GeV)

              10

              05

              00

              05

              10

              15

              20

              25

              30

              35

              E2dφdE

              (GeVc

              m2ss

              r)

              1e 6

              Best fitData

              Figure 63 Gamma Ray Spectrum

              This is not a particularly good fit and it occcurs at a relic density Ωh2 of 335 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

              63 Collider Constraints 74

              63 Collider Constraints

              631 Mediator Decay

              1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44]

              We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

              The results of the scan are shown in the graph Fig64 and can be compared with thelimits in the table from the paper [44] in Fig 56The black line is the observed ATLAS limit

              Figure 64 σ lowastBr(σ rarr ττ) versus Mass of Scalar

              0 200 400 600 800

              mS[GeV]

              8

              6

              4

              2

              0

              2

              4

              log 1

              0(σ

              (bbS

              )lowastB

              (Sgtττ

              ))[pb]

              Observed LimitLikely PointsExcluded Points

              050

              100150200250300350

              0 10 20 30 40 50 60

              We can see that there is a narrow band in the region of interest in the σ(bbS)lowastB(S gt ττ)

              by mS plane due to the high degree of correlation between mS and λτ seen in the likelihoodscan Values for mS gt 300Gev are excluded

              2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

              We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinations

              63 Collider Constraints 75

              randomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

              The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig65 The black line is the CMSlimit

              Figure 65 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

              0 200 400 600 800

              mS[GeV]

              8

              6

              4

              2

              0

              2

              4

              log

              10(σ

              (bS

              +X

              )lowastB

              (Sgt

              bb))

              [pb]

              Observed LimitLikely PointsExcluded Points

              050

              100150200250300350

              0 10 20 30 40 50 60

              The results of this scan were compared to the limits in [89] with the plot shown inFig58

              We can see a similar shape for the region of interest to that in Fig64 due to the highcorrelation between mS and λb over the region preferred by astrophysical constraints resultingin branching ratios highly correlated to ms Only one point is excluded with mS gt 500GeV

              632 Collider Cuts Analyses

              We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expected

              63 Collider Constraints 76

              with 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 66

              We can see that collider cuts show some exclusions only for masses mχ below the highlikelihood region but these exclusions do not occur at the same λb and λt combinations asseen from the plot of these two couplings except for a small region at the highest mS andcouplings

              All but one of the collider cut exclusions were excluded by the Lepstop0 experiment (thiswas excluded by 2bstop with high coupling to b quarks of 73) A number of the excludedpoints were also excluded by Lepstop2 2bstop CMSTopDM1L and ATLASMonobJet-(around 30 of the total for each)

              63 Collider Constraints 77

              0 1 2 3

              log10(mχ)[GeV]

              0

              1

              2

              3

              log 1

              0(m

              s)[GeV

              ]Collider Cuts

              σ lowastBr(σgt bS+X)

              σ lowastBr(σgt ττ)

              (a) mχ by mS

              5 4 3 2 1 0 1

              log10(λt)

              0

              1

              2

              3

              log 1

              0(m

              s)[GeV

              ](b) λt by mS

              5 4 3 2 1 0 1

              log10(λb)

              5

              4

              3

              2

              1

              0

              1

              log 1

              0(λ

              t)

              (c) λb by λt

              5 4 3 2 1 0 1

              log10(λb)

              6

              5

              4

              3

              2

              1

              0

              1

              2

              log 1

              0(λ

              τ)

              (d) λb by λτ

              5 4 3 2 1 0 1

              log10(λt)

              6

              5

              4

              3

              2

              1

              0

              1

              2

              log 1

              0(λ

              τ)

              (e) λt by λτ

              5 4 3 2 1 0 1

              log10(λb)

              0

              1

              2

              3

              log 1

              0(m

              s)[GeV

              ]

              (f) λb by mS

              Figure 66 Excluded points from Collider Cuts and σBranching Ratio

              Chapter 7

              Real Vector Dark Matter Results

              71 Bayesian Scans

              In the Figure 71 we can see that the preferred DM mass is centered around 35 GeV primarilydue to the influence of the Fermi-LAT gamma ray constraint with relic density having a widerange of possible mediator and DM masses The result of combining all the constraints is todefine a small region around mχ = 8 GeV and mS = 20 GeV which dominates the posteriorprobability

              The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig72 The diagonal plots show marginalised histograms of a singleparameter at a time

              71 Bayesian Scans 79

              05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

              1

              0

              1

              2

              3

              4

              log 1

              0(m

              s)[GeV

              ]

              (a) Gamma Only

              05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

              05

              00

              05

              10

              15

              20

              25

              30

              35

              log 1

              0(m

              s)[GeV

              ]

              (b) Relic Density

              1 0 1 2 3 4log10(mχ)[GeV]

              05

              00

              05

              10

              15

              20

              25

              30

              35

              log 1

              0(m

              s)[GeV

              ]

              (c) LUX

              05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

              05

              00

              05

              10

              15

              20

              25

              30

              35

              log 1

              0(m

              s)[GeV

              ]

              (d) All Constraints

              Figure 71 Real Vector Dark Matter - By Individual Constraint and All Together

              71 Bayesian Scans 80

              00

              05

              10

              15

              20

              25

              30

              log 1

              0(m

              χ)[GeV

              ]

              00

              05

              10

              15

              20

              25

              30

              ms[Gev

              ]

              5

              4

              3

              2

              1

              0

              1

              log 1

              0(λ

              t)

              5

              4

              3

              2

              1

              0

              1

              log 1

              0(λ

              b)

              00 05 10 15 20 25 30

              log10(mχ)[GeV]

              5

              4

              3

              2

              1

              0

              1

              log 1

              0(λ

              τ)

              00 05 10 15 20 25 30

              ms[Gev]5 4 3 2 1 0 1

              log10(λt)5 4 3 2 1 0 1

              log10(λb)5 4 3 2 1 0 1

              log10(λτ)

              Figure 72 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter

              72 Best fit Gamma Ray Spectrum for the Real Vector DM model 81

              72 Best fit Gamma Ray Spectrum for the Real Vector DMmodel

              The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraints isshown in Fig73

              Table 71 Best Fit Parameters

              Parameter mχ [GeV ] mS[GeV ] λt λb λτ

              Value 8447 20685 0000022 0000746 0002439

              10-1 100 101 102

              E(GeV)

              10

              05

              00

              05

              10

              15

              20

              25

              30

              35

              E2dφdE

              (GeVc

              m2s

              sr)

              1e 6

              Best fitData

              Figure 73 Gamma Ray Spectrum

              This is not a particularly good fit and it occcurs at a relic density Ωh2 of 025 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

              73 Collider Constraints

              731 Mediator Decay

              1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44] (See Chapter 3 for more details)

              We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

              73 Collider Constraints 82

              The results of the scan are shown in the graph Fig74 and can be compared with thelimits in the table from the paper [44] in Fig 56

              Figure 74 σ lowastBr(σ rarr ττ) versus Mass of Scalar

              0 200 400 600 800

              mS[GeV]

              8

              6

              4

              2

              0

              2

              log 1

              0(σ

              (bbS

              )lowastB

              (Sgtττ

              ))[pb]

              Observed LimitLikely PointsExcluded Points

              0100200300400500600700800

              0 20 40 60 80 100120140

              We can see that the region of highest likelihood is limited to mS lt 100 GeV and a narrowrange of couplings to λτ giving a narrow region of interest which falls well below the limit

              2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

              We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

              The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig75 The black line is the CMSlimit

              The results of this scan were compared to the limits in [89] with the plot shown in Fig58

              We can see that again the region of interest for mS is below 100 GeV and the points fallwell below the limit

              73 Collider Constraints 83

              Figure 75 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

              0 200 400 600 800

              mS[GeV]

              8

              6

              4

              2

              0

              2

              4

              log

              10(σ

              (bS

              +X

              )lowastB

              (Sgt

              bb))

              [pb]

              Observed LimitLikely PointsExcluded Points

              0100200300400500600700800

              0 20 40 60 80 100120140

              732 Collider Cuts Analyses

              We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 76

              We can see from Figure 76 that like the real scalar model the region of highest likelihoodhas λb and λt lt 001 while the excluded points from collider cuts and cross sections areabove 01 Most of the excluded points have mχ gt than the values in the region of highestlikelihood and all points are widely separated in parameter space from this region

              Once again 89 of excluded points were excluded by Lepstop0 with many also beingexcluded by the other experiments (50 by Lepstop2 and 56 by CMSTopDM1L 35 byLepstop2 and 20 by ATLASMonobjet)

              73 Collider Constraints 84

              0 1 2 3

              log10(mχ)[GeV]

              0

              1

              2

              3

              log 1

              0(m

              s)[GeV

              ]Collider Cuts

              σ lowastBr(σgt bS+X)

              σ lowastBr(σgt ττ)

              (a) mχ by mS

              5 4 3 2 1 0 1

              log10(λt)

              0

              1

              2

              3

              log 1

              0(m

              s)[GeV

              ](b) λt by mS

              5 4 3 2 1 0 1

              log10(λb)

              5

              4

              3

              2

              1

              0

              1

              log 1

              0(λ

              t)

              (c) λb by λt

              5 4 3 2 1 0 1

              log10(λb)

              5

              4

              3

              2

              1

              0

              1

              log 1

              0(λ

              τ)

              (d) λb by λτ

              5 4 3 2 1 0 1

              log10(λt)

              5

              4

              3

              2

              1

              0

              1

              log 1

              0(λ

              τ)

              (e) λt by λτ

              5 4 3 2 1 0 1

              log10(λb)

              0

              1

              2

              3

              log 1

              0(m

              s)[GeV

              ]

              (f) λb by mS

              Figure 76 Excluded points from Collider Cuts and σBranching Ratio

              Chapter 8

              Conclusion

              We first conducted Bayesian scans over mass and coupling parameters for three simplifiedmodels - Majorana fermion real scalar and real vector using direct detecion (LUX) indirectdetection (GCE γ rays) and relic density constraints to find the regions of highest likelihoodWe ran these scans one constraint at a time and then together to better understand the interplayof the different constraints on best fit and likely parameter regions

              We also developed and tested C++ collider cut software specifically designed to mimickthe cuts applied in a number of ATLAS and CMS experiments that searched for Emiss

              T inconnection with specific beyond the standard model (BSM) theories of Dark Matter

              We then selected parameter combinations by randomly sampling the scanned resultsbased on the posterior probability distribution of the points and applied collider cuts andcross section times branching ratio calculations to these points to see if there would havebeen any sensitivity from the ATLAS and CMS experiments

              We found that the Majorana fermion model showed the best overall fit to all the constraintswith relic density close to the experimental value of 01199 and a low χ2 error on the gammaspectrum fit The preferred mass mχ was around 35 GeV and mS ranging from lt10 GeV to500 GeV The smaller masses give a better fit for relic density and also are well separatedfrom current collider constraints mainly due the the fact that the exclusions occur mainlywith higher couplings to t quarks (greater than 1) which is consistent with the fact that manyof the experiments were targeting dark matter pair production associated with t quark decaysNevertheless there were some exclusions that did fall in the region sampled by randomselection from the posterior probability distribution

              The preferred parameter regions were also consistent with the observation in [20] thatthe Majorana model showed the best overall fit with similar mass constraints We find thatthe couplings to the SM fermions should all be less that 001

              86

              The scalar and vector DM models give slightly less good fits especially to the currentrelic density estimate of 01199 but again the current collider constraints are well separatedfrom the best fit region and in fact none of the points randomly sampled from the posteriordistribution showed any exclusion The best fit parameter combination favours couplings toSM fermions of less than 001

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              [83] The ATLAS Collaboration Search for direct top squark pair production in final stateswith two leptons in

              radics=8 tev pp collisions using 20 f b1 of atlas data atlaswebcern

              chAtlasGROUPSPHYSICSPAPERSfigaux_25 2013

              [84] Y Bai H-C Cheng J Gallicchio and J Gu Stop the top background of the stopsearch arXiv 12034813 2012

              [85] Jason Kumar and Danny Marfatia Matrix element analyses of dark matter scatteringand annihilation PhysRevD 88(014035) 2013

              [86] I Antcheva et al Root mdash a c++ framework for petabyte data storage statistical analysisand visualization Computer Physics Communications 180(122499-2512) 2009

              [87] Jonathan Feng Mpik website http1bpblogspotcom-U0ltb81JltQUXuBvRV4BbIAAAAAAAAE4kK3x0lQ50d4As1600direct+indirect+collider+imagepng 2005

              [88] Francesca Calore Ilias Cholis and Christoph Weniger Background model systematicsfor the fermi gev excess arXiv 14090042v1 2014

              [89] The CMS Collaboration Search for a higgs boson decaying into a b-quark pair andproduced in association with b quarks in proton-proton collisions at 7 tev PhysLettB207 2013

              Appendix A

              Validation of Calculation Tools

              Lepstop0Search for direct pair production of the top squarkin all hadronic final states in proton=proton collisions atradic

              s=8 TeV with the ATLAS detector [32]

              Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1) mt1=600 GeV mn1=1 GeVWhere t1 is the top squark and n1 is the neutralino of the MSSM

              94

              Table A1 0 Leptons in the final state

              Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 10000 460 460 100Emiss

              T gt 130GeV 8574 395 451 88Leptonveto 6621 307 285 108Emiss

              T gt 150GeV 6316 293 294 100Jet multiplicityand pT 1099 48 51 94|∆φ( jet pmiss

              T )|gt π5 902 40 42 96gt= 2b jets 396 19 18 106τ veto 396 19 18 106mbmin

              T gt 175GeV 270 13 13 105SRA1 79 4 4 104SRA2 70 3 3 93SRA3 67 3 3 97SRA4 52 2 2 91SRC exactly5 jets 1297 50 60 83SRC |∆φ( jet pmiss

              T )|gt π5 1082 50 50 99SRC gt= 2b jets 430 20 20 97SRC τ veto 430 20 20 97SRC |∆φ(bb)|gt 02π 367 17 17 99SRC1 246 12 11 105SRC2 177 9 8 116SRC3 152 8 7 116

              95

              Lepstop1Search for direct top squark pair production infinal states with one isolated lepton jets and missing trans-verse momentum in

              radics = 8 TeV pp collisions using 21 f bminus1

              of ATLAS data[33]

              Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1)(t1 gt b x+1 )(t1 gt b xminus1 )mt1=650 GeV mn1=1 GeV Where t1 is the top squark n1 is the neutralino xplusmn1 are thecharginos

              96

              Table A2 1 Lepton in the Final state

              Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 100000 99990 99990 100Electron(= 1signal) 11558 11557 11785 984 jets(80604025) 5187 5186 6271 83gt= 1btag 4998 4997 5389 93Emiss

              T gt 100GeV [allSRs] 4131 4131 4393 94Emiss

              T radic

              HT gt 5 [allSRs] 4026 4026 4286 94∆φ( jet2 pmiss

              T )gt 08 [allSRs] 3565 3565 3817 93∆φ( jet1Pmiss

              T )gt 08 [not SRtN2] 3446 3446 3723 93Emiss

              T gt 200GeV (SRtN2) 2027 2027 2186 93Emiss

              T radic

              HT gt 13(SRtN2) 1156 1156 1172 99mT gt 140GeV (SRtN2) 839 839 974 86Emiss

              T gt 275GeV (SRtN3) 948 948 965 98Emiss

              T radic

              HT gt 11(SRtN3) 932 932 937 99mT gt 200GeV (SRtN3) 456 456 566 81Emiss

              T gt 150GeV (SRbC1minusSRbC3) 2803 2803 3080 91Emiss

              T radic

              HT gt 7(SRbC1minusSRbC3) 2742 2742 3017 91mT gt 12GeV (SRbC1minusSRbC3) 1917 1917 2347 82Emiss

              T gt 160GeV (SRbC2SRbC3) 1836 1836 2257 81Emiss

              T radic

              HT gt 8(SRbC2SRbC3) 1788 1788 2175 82me f f gt 550GeV (SRbC2) 1741 1741 2042 85me f f gt 700GeV (SRbC3) 1075 1075 1398 77Muon(= 1signal) 11379 11378 11975 954 jets(80604025) 5264 5263 6046 87gt= 1btag 5101 5100 5711 89Emiss

              T gt 100GeV [allSRs] 4199 4199 4770 88Emiss

              T radic

              HT gt 5 [all SRs] 4115 4115 4653 88∆φ( jet2 pmiss

              T )gt 08 [all SRs] 3622 3622 4077 89∆φ( jet1 pmiss

              T )gt 08 [not SRtN2] 3515 3515 3970 89Emiss

              T gt 200GeV (SRtN2) 2136 2136 2534 84Emiss

              T radic

              HT gt 13(SRtN2) 1261 1261 1441 87mT gt 140GeV (SRtN2) 928 928 1033 90Emiss

              T gt 275GeV (SRtN3) 1036 1036 1274 81Emiss

              T radic

              HT gt 11(SRtN3) 1018 1018 1235 82mT gt 200GeV (SRtN3) 497 497 612 81Emiss

              T gt 150GeV (SRbC1minusSRbC3) 2891 2891 3380 86Emiss

              T radic

              HT gt 7(SRbC1minusSRbC3) 2821 2821 3290 86mT gt 120GeV (SRbC1minusSRbC3) 1977 1977 2290 86Emiss

              T gt 160GeV (SRbC2SRbC3) 1913 1913 2220 86Emiss

              T radic

              HT gt 8(SRbC2SRbC3) 1863 1863 2152 87me f f gt 550GeV (SRbC2) 1826 1826 2116 86me f f gt 700GeV (SRbC3) 1131 1131 1452 78

              97

              Lepstop2Search for direct top squark pair production infinal states with two leptons in

              radics =8 TeV pp collisions using

              20 f bminus1 of ATLAS data[83][34]

              Simulated in Madgraph with p p gt t1 t1(t1 gt b w+ n1)(t1 gt b wminusn1) mt1=180 GeV mn1=60 GeV Where t1 is the top squark n1 is the neutralino b1 is the bottom squark

              Table A3 2 Leptons in the final state

              Cut RawMe Scaled ATLAS MeATLAS PercentNoCuts 100000 324308 199000 1632BaselineCuts 3593 11652 7203 1622SF Signal Leptons 1699 5510 5510 1002OSSF Signal Leptons 1632 5293 5233 101mll gt 20Gev 1553 5037 5028 100leadinglepton pT 1479 4797 5048 95|mll minusmZ|gt 20Gev 1015 3292 3484 94∆φmin gt 10 850 2757 2160 128∆φb lt 15 680 2205 1757 126SRM90 [SF ] 13 42 60 70SRM120 [SF ] 0 0 19 0SRM100 +2 jets [SF ] 0 0 17 0SRM110 +2 jets [SF ] 0 0 1 02DF Signal Leptons 1737 5633 7164 792OSDF Signal Leptons 1670 5416 5396 100mll20Gev 1587 5147 5136 100leadinglepton pT 1501 4868 4931 99∆φmin gt 10 1247 4044 3122 130∆φb lt 15 1010 3276 2558 128SRM90 [SF ] 21 68 95 72SRM120 [SF ] 0 0 23 0SRM100 +2 jets [SF ] 0 0 19 0SRM110 +2 jets [SF ] 0 0 14 0

              98

              2bstopSearch for direct third-generation squark pair pro-duction in final states with missing transverse momentumand two b-jets[31]

              Simulated in Madgraph with p p gt b1 b1(b1 gt b n1)(b1 gt b n1)

              SRAmb1=500 GeV mn1=1 GeV SRBmb1=350 GeV mn1=320 GeVWhere b1 is the bottom squark and n1 is the neutralino

              Table A4 2b jets in the final state

              Cut RawMe Scaled ATLAS MeATLAS PercentSRA Nocuts 100000 173800 1738 100SRA MET gt 80 93407 162341 1606 101SRA Leptonveto 93302 162159 1505 108SRA MET gt 150 79449 138082 1323 104SRA Jet selection 10213 17750 119 149SRA mbb gt 200 8344 14502 96 151SRA mCT gt 150 6535 11358 82 139SRA mCT gt 200 5175 8994 67 134SRA mCT gt 250 3793 6592 51 129SRA mCT gt 300 2487 4322 35 123SRB NoCuts 100000 1624100 16241 100SRB leptonveto 11489 186593 4069 46SRB MET gt 250 2530 41090 757 54SRB Jet selection 63 1023 79 130SRB HT 3 lt 50 51 828 52 159

              99

              CMSTop1LSearch for the Production of Dark Matter in As-sociation with Top Quark Pairs in the Single-lepton FinalState in pp collisions[41]

              Simulated in Madgraph with p p gt t t p1 p1

              Where p1= Majorana fermion Table 3 shows Signal efficiencies and cross sections TheM scale was read from Figure 7 for 90 CL

              Table A5 Signal Efficiencies 90 CL on σ limexp[ f b] on pp gt tt +χχ

              Mχ[GeV ] My Signal Efficiency Quoted Efficiency My σ [ f b] Quoted σ [ f b]1 102 101 63 47

              10 132 101 62 4650 143 120 50 38100 170 146 56 32200 185 173 69 27600 265 240 32 19

              1000 320 276 41 17

              Appendix B

              Branching ratio calculations for narrowwidth approximation

              B1 Code obtained from decayspy in Madgraph

              Br(S rarr bb) = (minus24λ2b m2

              b +6λ2b m2

              s

              radicminus4m2

              bm2S +m4

              S)16πm3S

              Br(S rarr tt) = (6λ2t m2

              S minus24λ2t m2

              t

              radicm4

              S minus4ms2m2t )16πm3

              S

              Br(S rarr τ+

              τminus) = (2λ

              2τ m2

              S minus8λ2τ m2

              τ

              radicm4

              S minus4m2Sm2

              τ)16πm3S

              Br(S rarr χχ) = (2λ2χm2

              S

              radicm4

              S minus4m2Sm2

              χ)32πm3S

              (B1)

              Where

              mS is the mass of the scalar mediator

              mχ is the mass of the Dark Matter particle

              mb is the mass of the b quark

              mt is the mass of the t quark

              mτ is the mass of the τ lepton

              The coupling constants λ follow the same pattern

              • TITLE Collider Constraints applied to Simplified Models of Dark Matter fitted to the Fermi-LAT gamma ray excess using Bayesian Techniques
                • Dedication
                • Declaration
                • Acknowledgements
                • Contents
                • List of Figures
                • List of Tables
                  • Chapter 1 Introduction
                  • Chapter 2 Review of Physics
                  • Chapter 3 Fitting Models to the Observables
                  • Chapter 4 Calculation Tools
                  • Chapter 5 Majorana Model Results
                  • Chapter 6 Real Scalar Model Results
                  • Chapter 7 Real Vector Dark Matter Results
                  • Chapter 8 Conclusion
                  • Bibliography
                  • Appendix A Validation of Calculation Tools
                  • Appendix B Branching ratio calculations for narrow width approximation

                List of Figures

                21 Feynman Diagram of electron interacting with a muon 1122 Weak Interaction Vertices [48] 1523 Higgs Potential [49] 1824 Standard Model Particles and Forces [50] 2425 Bullet Cluster [52] 2526 Galaxy Rotation Curves [54] 2627 WMAP Cosmic Microwave Background Fluctuations [58] 2928 Dark Matter Interactions [60] 2929 Gamma Ray Excess from the Milky Way Center [75] 31210 ATLAS Experiment 31211 CMS Experiment 34

                31 Main Feyman diagrams leading to the cross section for scalar decaying to apair of τ leptons 40

                32 WidthmS vs mS 4033 WidthmS vs λb 4134 WidthmS vs λτ 4135 Main Feyman diagrams leading to the cross section for scalar decaying to a

                pair of b quarks in the presence of at least one b quark 42

                41 Calculation Tools 55

                51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether 62

                52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter 6353 Gamma Ray Spectrum 6454 Plots of log likelihoods by individual and combined constraints Masses in

                GeV 6655 σ lowastBr(σ rarr ττ) versus Mass of Scalar 67

                List of Figures ix

                56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar 6857 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar 6858 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 6959 Excluded points from Collider Cuts and σBranching Ratio 70

                61 Real Scalar Dark Matter - By Individual Constraint and All Together 7462 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter 7563 Gamma Ray Spectrum 7664 σ lowastBr(σ rarr ττ) versus Mass of Scalar 7765 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 7866 Excluded points from Collider Cuts and σBranching Ratio 80

                71 Real Vector Dark Matter - By Individual Constraint and All Together 8272 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter 8373 Gamma Ray Spectrum 8474 σ lowastBr(σ rarr ττ) versus Mass of Scalar 8575 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 8676 Excluded points from Collider Cuts and σBranching Ratio 87

                List of Tables

                21 Quantum numbers of the Higgs field 1922 Weak Quantum numbers of Lepton and Quarks 21

                31 Simplified Models 3532 95 CL by Signal Region 4433 Selection criteria common to all signal regions 4534 Selection criteria for signal regions A 4535 Selection criteria for signal regions C 4536 Signal Regions - Lepstop1 4837 Signal Regions Lepstop2 4938 Signal Regions 2bstop 5139 Signal Region ATLASmonobjet 52

                51 Scanned Ranges 6152 Best Fit Parameters 64

                61 Best Fit Parameters 76

                71 Best Fit Parameters 84

                A1 0 Leptons in the final state 98A2 1 Lepton in the Final state 100A3 2 Leptons in the final state 101A4 2b jets in the final state 102A5 Signal Efficiencies 90 CL on σ lim

                exp[ f b] on pp gt tt +χχ 103

                Chapter 1

                Introduction

                Dark matter (DM) was first postulated over 80 years ago when Swiss astronomer FritzZwicky observed a discrepancy between the amount of light emitted by a cluster of galaxiesand the total mass contained within the cluster inferred from the relative motion of thosegalaxies by a simple application of the theory of Newtonian gravitation The surprising resultof this observation was that the vast majority of the mass in the cluster did not emit lightwhich was contrary to the expectation that most of the mass would be carried by the starsSince that time further observations over a wide range of scales and experimental techniqueshave continued to point to the same result and refine it Some of these observations and otherevidence are discussed in section 22 We now know with certainty that in the entire Universeall of the matter we know about - stars planets gases and other cosmic objects such as blackholes can only account for less than 5 of the mass that we calculate to be there

                A recent phenomenon that has received much attention is the significant deviation frombackground expectations of the Fermi Large Area Telescope(Fermi-LAT) gamma ray flux atthe galactic centre [1] A number of astrophysical explanations have been proposed includingmillisecond pulsars of supernova remnants [2] or burst-like continuous events at the galacticcentre but these are unresolved However it has also been noted that the observed Fermi-LATexcess is consistent with the annihilation of dark matter particles which would naturally beconcentrated at the Galactic centre in a manner consistent with the Navarro-Frenk-Whitedistribution of dark matter [3]

                There are a number of other purely theoretical (particle physics) reasons to postulatethe existence of weakly interacting matter particles that could supply the missing mass andyet remain unobservable Weakly interacting massive particle (WIMPS) have been a majorfocus of Run I and ongoing Run II searches of the Large Hadron Collider (LHC) In spite

                2

                of substantial theoretical and experimental effort the microscopic properties of dark matterparticles are still unknown

                One of the main problems with extracting these properties is the plethora of competingtheoretical models that provide dark matter candidates Another is the very weak interactionof these particles and the lack of experimental observables allowing one to discriminatebetween the competing models

                A particularly fruitful approach is the simplified model approach which makes minimalassumptions about the DM candidates as extensions to the Standard Model interacting weaklywith ordinary matter through a mediator These can be coded as Lagrangian interaction termsusing tools such as FeynRules [4] and Calchep [5] and used to simulate Dark Matter relicdensities indirect detection expectations as well as direct detection expectations throughMicrOmegas [6] This in turn can be run under a Bayesian inference tool (see below)

                Using a Bayesian inference tool such as MultiNest [7][8][9] one can scan the parameterspace of a particular model to find regions of highest probability for that model based onobservations of a range of phenomena including direct detection experiments such as ZEPLIN[10] XENON [11] DEAP [12] ARDM [13] DarkSide [14] PandaX [15] and LUX [16]indirect detection - eg Fermi-LAT gamma ray excess [17] Pamela-positron and antiprotonflux [18] and production at colliders MultiNest calculates the evidence value which is theaverage likelihood of the model weighted by prior probability and a posterior distributionwhich is the distribution of the parameter(s) after taking into account the observed databy scanning over a range of parameters based on prior probabilities This can be used tocompare models on a consistent basis The scans also provide information about the best fitparameters for the overall fit as well as for the seperate experimentsobservations

                The simplified model approach has been taken in a number of papers to study the Fermi-LAT gamma ray excess (see literature review in section 12) and one in particular comparesthese models using a Bayesian inference tool which found that a Majorana fermion DMcandidate interacting via a scalar mediator with arbitrary couplings to third generationfermions showed the best fit [19] A subsequent paper [20] examined the most likelyparameter combinations that simultaneously fit the Planck relic density Fermi-LAT galacticcentre gamma ray AMS-02 positron flux and Pamela antiproton flux data

                There has not as yet been a thorough study of the collider constraints on DM simplifiedmodels that fit the Fermi-LAT galactic centre excess In this thesis we reproduce the

                11 Motivation 3

                previous results of [20] using similar astrophysical constraints and then implement detailedsimulations of collider constraints on DM and mediator production

                In Section 11 we review the motivation for this study and in Section 12 we considerrecent astronomical data which is consistent with the annihilation of DM particles andreview the literature and searches which attempt to explain it with a range of extensions tothe standard model

                In Chapter 2 we review the Standard Model of particle physics (SM) the evidence fordark matter and the ATLAS and CMS experiments at the LHC

                In Chapter 3 we describe the models considered in this paper the observables used inthe Bayesian scans and the collider experiments which provide limits against which theMagraph simulations of the three models can be compared

                In Chapter 4 we review the calculation tools that have been used in this paper

                In Chapters 5 6 and 7 we give the results of the Bayesian scans and collider cutsanalyses for each of the models and in Chapter 8 we summarise our conclusions

                11 Motivation

                The Fermi gamma ray space telescope has over the past few years detected a gamma raysignal from the inner few degrees around the galactic centre corresponding to a regionseveral hundred parsecs in radius The spectrum and angular distribution are compatible withthat from annihilating dark matter particles

                A number of papers have recently attempted to explain the Gamma Ray excess in termsof extensions to the Standard Model taking into account constraints from ATLAS and CMScollider experiments direct detection of dark matter colliding off nucleii in well shieldeddetectors (eg LUX) as well as other indirect measurements (astrophysical observations)These papers take the approach of simplified models which describe the contact interactionsof a mediator beween DM and ordinary matter using an effective field theory approach(EFT) Until recently it has been difficult to discriminate between the results of these paperswhich look at different models and produce different statistical results A recent paper[19] Simplified Dark Matter Models Confront the Gamma Ray Excess used a Bayesianinference tool to compare a variety of simplified models using the Bayesian evidence values

                12 Literature review 4

                calculated for each model to find the most preferred explanation The motivation for thechosen models is discussed in section 121

                A subsequent paper [20] Interpreting the Fermi-LAT gamma ray excess in the simplifiedframework studied the most likely model again using Bayesian inference to find the mostlikely parameters for the (most likely) model The data inputs (observables) to the scanwere the Planck measurement of the dark matter relic density [21] Fermi-LAT gamma rayflux data from the galactic center [17] cosmic positron flux data [22] cosmic anti-protonflux data [18] cosmic microwave background data and direct detection data from the LUXexperiment [16] The result of the scan was a posterior probability density function in theparameter space of the model

                This paper extends the findings in [19] by adding collider constraints to the likelihoodestimations for the three models It is not a simple matter to add collider constraints becauseeach point in parameter space demands a significant amount of computing to calculate theexpected excess amount of observable particles corresponding to the particular experimentgiven the proposed simplified model A sampling method has been employed to calculatethe collider constraints using the probability of the parameters based on the astrophysicalobservables

                12 Literature review

                121 Simplified Models

                A recent paper [23] summarised the requirements and properties that simplified models forLHC dark matter searches should have

                The general principles are

                bull Besides the Standard Model (SM) the Lagrangian should contain a DM candidate thatis absolutely stable or lives long enough to escape the LHC detector and a mediatorthat couples the two sectors

                bull The Lagrangian should contain all terms that are renormalizable consistent withLorentz invariance SM symmetries and DM stability

                bull Additional interactions should not violate exact and approximate global symmetries ofthe SM

                12 Literature review 5

                The examples of models that satisfy these requirements are

                1 Scalar (or pseudo-scalar) s-channel mediator coupled to fermionic dark matter (Diracor Majorana)

                2 Higgs portal DM where the DM is a scalar singlet or fermion singlet under the gaugesymmetries of the SM coupling to a scalar boson which mixes with the Higgs (Thefermion singlet is a specific realisation of 1 above)

                3 DM is a mixture of an electroweak singlet and doublet as in the Minimal Supersym-metric SM (MSSM)

                4 Vector (or Axial vector) s-channel mediator obtained by extending the SM gaugesymmetry by a new U(1)rsquo which is spontaneously broken such that the mediatoracquires mass

                5 t-channel flavoured mediator (if the DM particle is a fermion χ the mediator can bea coloured scalar or vector particle φ -eg assuming a scalar mediator a couplingof the form φ χq requires either χ or φ to carry a flavor index to be consistent withminimal flavour violation (MFV) which postulates the flavour changing neutral currentstructure in the SM is preserved

                Another recent paper [24] summarised the requirements and properties that simplifiedmodels for LHC dark matter searches should have The models studied in this paper arepossible candidates competing with Dirac fermion and complex scalar dark matter andmodels with vector and pseudo-scalar mediators Other models would have an elasticscattering cross section that will remain beyond the reach of direct detection experiments dueto the neutrino floor (the irreducible background produced by neutrino scattering) or arealready close to sensitivity to existing LUX and XENONIT experiments

                A recent paper [25] Scalar Simplified Models for Dark Matter looks at Dirac fermiondark matter mediated by either a new scalar or pseudo-scalar to SM fermions This paperplaces bounds on the coupling combination gχgφ (coupling of DM to mediator times couplingof mediator to SM fermions) calculated from direct detection indirect detection and relicdensity by the mass of DM and mass of mediator Bounds are also calculated for colliderconstraints (mono-jet searches heavy-flavour searches and collider bounds on the 125 GeVHiggs are used to place limits on the monojet and heavy flavor channels which can betranslated into limits on the Higgs coupling to dark matter and experimental measurements

                12 Literature review 6

                of the Higgs width are used to constrain the addition of new channels to Higgs decay) Thepaper models the top loop in a full theory using the Monte Carlo generator MCFM [26]and extends the process to accommodate off-shell mediator production and decay to a darkmatter pair While this paper does not consider the Majorana fermion dark matter model theconstraints on the coupling product may have some relevence to the present study if only asan order of magnitude indicator

                Constraining dark sectors with monojets and dijets [27] studies a simplified model withDirac fermion dark matter and an axial-vector mediator They use two classes of constraint-searches for DM production in events with large amounts of missing transverse energy inassociation with SM particles (eg monojet events) and direct constraints on the mediatorfrom monojet searches to show that for example where the choice of couplings of themediator to quarks and DM gA

                q =gAχ=1 all mediator masses in the range 130 GeV ltMR lt3

                TeV are excluded

                The paper Constraining Dark Sectors at Colliders Beyond the Effective Theory Ap-proach [28] considers Higgs portal models with scalar pseudo-scalar vector and axial-vectormediators between the SM and the dark sector The paper takes account of loop processesand gluon fusion as well as quark-mediator interactions using MCFM Apart from mediatortype the models are characterised by free parameters of mediator width and mass darkmatter mass and the effective coupling combination gχgφ (coupling of DM to mediator timescoupling of mediator to SM fermions) The conclusion is that LHC searches can proberegions of the parameter space that are not able to be explored in direct detection experiments(since the latter measurements suffer from a velocity suppression of the WIMP-nucleoncross-section)

                The paper Constraining the Fermi-LAT excess with multi-jet plus EmissT collider searches

                [29] shows that models with a pseudo-scalar or scalar mediator are best constrained by multi-jet final states with Emiss

                T The analysis was done using POWHEG-BOX-V2 at LO and usesM2

                T variables together with multi-jet binning to exploit differences in signal and backgroundin the different regions of phase space to obtain a more stringent set of constraints on DMproduction This paper largely motivated the current work in that it showed that collidersearches that were not traditionally thought of as dark matter searches were neverthelessoften sensitive to dark matter produced in association with heavy quarks The combinationof searches used in this paper reflects this fact

                12 Literature review 7

                122 Collider Constraints

                In order to constrain this model with collider constraints (direct detection limits) we considerthe following papers by the CMS and ATLAS collaborations which describe searches at theLHC

                ATLAS Experiments

                bull Search for dark matter in events with heavy quarks and missing transverse momentumin p p collisions with the ATLAS detector [30]

                bull Search for direct third-generation squark pair production in final states with missingtransverse momentum and two b-jets in

                radics= 8 TeV pp collisions with the ATLAS

                detector[31]

                bull Search for direct pair production of the top squark in all-hadronic final states inprotonndashproton collisions at

                radics=8 TeV with the ATLAS detector [32]

                bull Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

                radic(s)=8TeV pp collisions using 21 f bminus1 of

                ATLAS data [33]

                bull Search for direct top squark pair production in final states with two leptons inradic

                s = 8TeV pp collisions using 20 f bminus1 of ATLAS data [34]

                bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

                radics=8 TeV [35]

                CMS Experiments

                bull Searches for anomalous tt production in p p collisions atradic

                s=8 TeV [36]

                bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

                radics=8 TeV [37]

                bull Search for the production of dark matter in association with top-quark pairs in thesingle-lepton final state in proton-proton collisions at

                radics = 8 TeV [38]

                bull Search for new physics in monojet events in p p collisions atradic

                s = 8 TeV(CMS) [39]

                bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

                s = 8 TeV [40]

                12 Literature review 8

                bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theSingle-lepton Final State in pp collisions at

                radics=8 TeV [41]

                bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

                s=8 TeV [42]

                bull Search for neutral MSSM Higgs bosons decaying into a pair of bottom quarks [43]

                bull Search for neutral MSSM Higgs bosons decaying into a pair of τ leptons in p p collision[44]

                bull Search for dark matter extra dimensions and unparticles in monojet events in proton-proton collisions at

                radics=8 TeV [45]

                In particular we have implemented Madgraph simulations of certain data analysesapplying to collider experiments described in papers [32][33][34][31][30][41] Theremaining models were not implemented in Madgraph because of inability to validate themodels against published results in the time available We developed C++ analysis code toimplement the collider cuts after showering with Pythia and basic analysis with Delpheswith output to root files The C++ code allowed multiple experimental cuts to be applied tothe same simulation output The C++ code was first checked against the results of the variousexperiments above and then the simulations were rerun using the three modelsrsquo Lagrangians(implemented in FeynRules) described in this paper

                Chapter 2

                Review of Physics

                21 Standard Model

                211 Introduction

                The Standard Model of particle physics (SM) has sometimes been described as the theoryof Almost Everything The SM is a field theory which emerged in the second half of thetwentieth century from quantum mechanics and special relativity which themselves wereonly conceived after 1905 by Einsten Pauli Bohr Dirac and others In the first decades ofthe twenty first century the discovery of the Higgs boson [46][47] completed the pictureand confirmed the last previously unverified part of the SM The SM is a self consistenttheory that describes and classifies all known particles and their interactions (other thangravity) to an astonishing level of accuracy and detail However the theory is incomplete-it does not describe the complete theory of gravitation or account for DM (see below) ordark energy (the accelerating expansion of the universe) Related to these shortcomings aretheoretical deficiencies such as the asymmetry of matter and anti-matter the fact that the SMis inconsistent with general relativity (to the extent that one or both theories break down atspacetime singularities like the Big Bang and black hole event horizons)

                212 Quantum Mechanics

                Quantum Mechanics (QM) describes particles as probability waves which can be thoughtof as packets of energy which can be localised or spread through all space depending on how

                21 Standard Model 10

                accurately one measures the particlersquos momentum However QM is not itself consistent withspecial relativity It was realised too that because of relativity particles could be created anddestroyed and because of the uncertainty principle the theory should describe the probabilityof particles being created and destroyed on arbitrarily short timescales A quantum fieldtheory (QFT) is a way to describe this probability at every point of space and time

                213 Field Theory

                A field theory is described by a mathematical function of space and time called a LagrangianAs in classical mechanics where Lagrangians were introduced by Louis Lagrange at thebeginning of the nineteenth century to summarise dynamical systems the Lagrangian sum-marises the dynamics of space which is permeated by force fields The equations of motionarising from a Lagrangian are determined by finding the contours of the Lagrangian describedby stationary points (ie where the infinitesimal change in the Lagrangian is zero for a smallchange in field) Particles are generally described by singularities in the Lagrangian (wherethe values tend to infinity also called resonances)

                214 Spin and Statistics

                It was also realised that different types of particle needed to be represented by different fieldsin the Lagrangian Diracrsquos discovery that particles with spin 12 quantum numbers couldbe represented by a field obtained by effectively taking the square root of the relativisticequation (Klein Gordon equation) was probably the most significant advance in quantumfield theory It required the introduction of a matrix operators called the gamma matricesor Dirac matrices which helped explain the already known fact that spin half particles orfermions obey Fermi-Dirac statistics ie when two spin half particles are exchanged thesign of the wavefunction must change and hence they cannot occupy the same quantum stateunlike bosons with integer spin that can occupy the same quantum state

                Other particles need to be represented by fields called vector pseudo-vector scalar (as inthe Klein Gordon equation) pseudo-scalar and complex fields A vector field is really fourfields one for each dimension of space plus time and is parameterised by the four gammamatrices γmicro which transform among themselves in a relativistically invariant way (Lorentzinvariance) A pseudo-vector field is a vector field which transforms in the same way but

                21 Standard Model 11

                with additional sign change under parity transformations (a simultaneous reversal of the threespatial directions) and is represented by γmicroγ5 A pseudo-scalar is a scalar which changes signunder parity transformations and a complex scalar field is represented by complex numbers

                215 Feynman Diagrams

                QFT took another giant leap forward when Richard Feynman realised that the propagationof particles described by the field could be described by expanding the so called S-matrixor scattering matrix derived from calculating the probability of initial states transitioning tofinal states The expansion is similar to a Taylor expansion in mathematics allowing for thespecial commuting properties of elements of the field The Feynman diagrams representingindividual terms in the expansion can be interpreted as interactions between the particlecomponents The S-Matrix (or more correctly the non trivial part called the T-Matrix) isderived using the interaction Hamiltonian of the field The Hamiltonian is an operator whoseeigenvalues or characteristic values give the energy quanta of the field The T-matrix canalso be described by a path integral which is an integral over all of quantum space weightedby the exponential of the Lagrangian This is completely equivalent to the Hamiltonianformulation and shows how the Lagrangian arises naturally out of the Hamiltonian (in factthe Lagrangian is the Legendre transformation of the Hamiltonian)

                Figure 21 Feynman Diagram of electron interacting with a muon

                γ

                eminus

                e+

                micro+

                microminus

                The diagram Fig21 is a tree diagram (diagrams with closed loops indicate self in-teractions and are higher order terms in the expansion) and shows an electron (eminus) and ananti-muon (micro+) interacting by exchanging a photon

                21 Standard Model 12

                216 Gauge Symmetries and Quantum Electrodynamics (QED)

                The Dirac equation with its gamma matrices ushered in the use of group theory to describethe transformation of fields (because the multiplication table of the gamma matrices isclosed and described by group theory) It also ushered in the idea of symmetries within theLagrangian- for example consider a Lagrangian

                ψ(ipart minusm)ψ (21)

                The Lagrangian is invariant under a global U(1) transformation which acts on the fieldsand hence the derivatives as follows

                ψ rarr eiqαψ ψ rarr ψeminusiqα partmicroψ rarr eiqα

                partmicroψ (22)

                where qα is a global phase and α is a continuous parameter

                A symmetry leads by Noetherrsquos theorem to a conserved current jmicro = qψγmicroψ and is thebasis of charge conservation since partmicro jmicro = 0 and q =

                intd3x j0(x)

                By promoting this transformation to a local one (ie α rarr α(x) where x = xmicro are thespace time co-ordinates) the transformations become

                ψ rarr eiqα(x)ψ ψ rarr ψeminusiqα(x)partmicroψ rarr eiqα(x)(partmicroψ + iq(partmicroα(x))ψ) (23)

                The derivative transformation introduces an extra term which would spoil the gaugeinvariance This can be solved by introducing a new field the photon field Amicro(x) whichinteracts with the fermion field ψ and transforms under a U(1) transformation

                Amicro rarr Amicro minuspartmicroα(x) (24)

                If we replace normal derivatives partmicro with covariant derivatives Dmicro = (partmicro + iqAmicro(x)) in theLagrangian then the extra term is cancelled and we can now see that the covariant derivativetransforms like ψ - under a local U(1) transform ie

                Dmicro rarr Dmicroψprime = eiqα(x)Dmicroψ (25)

                The additional term iqAmicro(x) can be interpreted in a geometric sense as a connectionwhich facilitates the parallel transport of vectors along a curve in a parallel consistent manner

                21 Standard Model 13

                We must add a gauge-invariant kinetic term to the Lagrangian to describe the dynamicsof the photon field This is given by the electromagnetic field strength tensor

                Fmicroν = partmicroAν minuspartνAmicro (26)

                The full Quantum Electrodynamics (QED) Lagrangian is then for electrons with mass mand charge -e

                LQED = ψ(i Dminusm)ψ minus 14

                Fmicroν(X)Fmicroν(x) (27)

                This Lagrangian is Lorentz and U(1) guage invariant The interaction term is

                Lint =+eψ Aψ = eψγmicro

                ψAmicro = jmicro

                EMAmicro (28)

                where jmicro

                EM is the electromagnetic four current

                217 The Standard Electroweak Model

                The Standard Electroweak (EW) Model was created by Glashow-Weinberg-Salam (also calledthe GWS model) and unifies the electomagnetic and weak interactions of the fundamentalparticles

                The idea of gauge symmetry was extended to the idea of a non-abelian gauge symmetry(essentially the operators defining the symmetry do not commute and so are represented bymatrices- in this case the Pauli matrices for the SU(2) group) - the gauge group is the directproduct G = SU(2)

                otimesU(1) It was known that weak interactions were mediated by Wplusmn

                and Z0 particles where the superscripts are the signs of the electromagnetic charge Thesewere fairly massive but it was also known that through spontaneous symmetry breaking ofa continuous global non-abelian symmetry there would be one massive propagation modefor each spontaneously broken-symmetry generator and massless modes for the unbrokengenerators (the so-called Goldstone bosons)

                This required an extra field to represent the gauge bosons called Bmicro (a vector fieldhaving the 4 spacetime degrees of freedom) With this identification the covariant derivativebecomes

                Dmicro = partmicro minus igAmicro τ

                2minus i

                gprime

                2Y Bmicro (29)

                21 Standard Model 14

                Here Y is the so called weak hypercharge of the field Bmicro which is a quantum numberdepending on particle (lepton or quark) g and gprime are two independent parameters in theGWS theory and the expression Amicro τ implies a sum over 3 components Aa

                micro a=123 and thePauli matrices τa

                This isnrsquot the full story because it had been known since 1956 when Chin Shuing Wuand her collaborators confirmed in an experiment prompted by Lee and Yang that nature isnot ambivalent to so-called parity It had been assumed that the laws of nature would notchange if all spatial coordinates were reversed (a parity transformation) The experimentinvolved the beta decay of cooled 60C atoms (more electrons were emitted in the backwardhemisphere with respect to spin direction of the atoms) This emission involved the weakinteraction and required that the Lagrangian for weak interactions contain terms which aremaximally parity violating- which turn out to be currents of the form

                ψ(1minus γ5)γmicro

                ψ (210)

                The term

                12(1minus γ

                5) (211)

                projects out a left-handed spinor and can be absorbed into the spinor to describe left-handedleptons (and as later discovered quarks) The right handed leptons are blind to the weakinteraction

                The processes describing left-handed current interactions are shown in Fig 22

                Analogously to isospin where the proton and neutron are considered two eigenstates ofthe nucleon we postulate that Bmicro is the gauge boson of an SU(2)transformation coupled toweak SU(2) doublets (

                νe

                eminus

                )

                (ud

                ) (212)

                We may now write the weak SU(2) currents as eg

                jimicro = (ν e)Lγmicro

                τ i

                2

                e

                )L (213)

                21 Standard Model 15

                Figure 22 Weak Interaction Vertices [48]

                where τ i are the Pauli-spin matrices and the subscript L denotes the left-handed componentof the spinor projected out by Eqn211 The third current corresponding to i=3 is called theneutral current as it doesnrsquot change the charge of the particle involved in the interaction

                We could postulate that the electromagnetic current is a linear combination of the leftand right elecromagnetic fields eL = 1

                2(1minus γ5)e and eR = 12(1+ γ5)e

                jemmicro = eLγmicroQeL + eRγmicroQeR (214)

                where Q is the electromagnetic operator and eR is a singlet However this is not invariantunder SU(2)L To construct an invariant current we need to use the SU(2)L doublet 212The current which is an SU(2)L invariant U(1) current is

                jYmicro = (ν e)LγmicroYL

                e

                )L+ eRγmicroYReR (215)

                where hypercharges YL and YR are associated with the U(1)Y symmetry jYmicro is a linearcombination of the electromagnetic current and the weak neutral current thus explaining whythe hypercharge differs for the left and right-handed components since the electromagneticcurrent has the same charge for both components whilst the weak current exists only forleft-handed particles j3

                micro and the third component of weak isospin T 3 allows us to calculate

                21 Standard Model 16

                the relationship Y = 2(QminusT 3) which can be seen by writing jYmicro as a linear combination ofj3micro (the weak neutral current that doesnrsquot change the charge of the particle involved in the

                interaction) and jemmicro (using the convention that the jYmicro is multiplied by 1

                2 to match the samefactor implicit in j3

                micro ) Substituting

                τ3 =

                (1 00 minus1

                )(216)

                into equation 213 for i=3 and subtracting equation 215 and equating to 12 times equation

                214

                we get

                eLγmicroQeL + eRγmicroQeR minus (νLγmicro

                12

                νL minus eLγmicro

                12

                eL) =12

                eRγmicroYReR +12(ν e)LγmicroYL

                e

                )L (217)

                from which we can read out

                YR = 2QYL = 2Q+1 (218)

                and T3(eR) = 0 T3(νL) =12 and T3(eL) =

                12 The latter three identities are implied by

                the fraction 12 inserted into the definition of equation 213

                The Lagrangian kinetic terms of the fermions can then be written

                L =minus14

                FmicroνFmicroν minus 14

                GmicroνGmicroν

                + sumgenerations

                LL(i D)LL + lR(i D)lR + νR(i D)νR

                + sumgenerations

                QL(i D)QL +UR(i D)UR + DR(i D)DR

                (219)

                LL are the left-handed lepton doublets QL are the left-handed quark doublets lRνR theright handed lepton singlets URDR the right handed quark singlets

                The field strength tensors are given by

                Fmicroν = partmicroAν minuspartνAmicro minus ig[Amicro Aν ] (220)

                21 Standard Model 17

                andGmicroν = partmicroBν minuspartνBmicro (221)

                Bmicro is an abelian U(1) gauge field while Amicro is non-abelian and Fmicroν contains the com-mutator of the field through the action of the covariant derivative [Dmicro Dν ] This equationcontains no mass terms terms for the field Amicro for good reason- this would violate gaugeinvariance We know that the gauge bosons of the weak force have substantial mass and theway that this problem has been solved is through the Higgs mechanism

                218 Higgs Mechanism

                To solve the problem of getting gauge invariant mass terms three independent groupspublished papers in 1964 The mechanism was named after Peter Higgs To introduce theconcept consider a complex scalar field φ that permeates all space Consider a potential forthe Higgs field of the form

                minusmicro2φ

                daggerφ +λ (φ dagger

                φ2) (222)

                which has the form of a mexican hat Fig 23 This potential is still U(1) gauge invariant andcan be added to our Lagrangian together with kinetic+field strength terms to give

                L = (Dmicroφ)dagger(Dmicroφ)minusmicro2φ

                daggerφ +λ (φ dagger

                φ2)minus 1

                4FmicroνFmicroν (223)

                It is easily seen that this is invariant to the transformations

                Amicro rarr Amicro minuspartmicroη(x) (224)

                φ(x)rarr eieη(x)φ(x) (225)

                The minimum of this field is not at 0 but in a circular ring around 0 with a vacuum

                expectation value(vev)radic

                micro2

                2λequiv vradic

                2

                We can parameterise φ as v+h(x)radic2

                ei π

                Fπ where h and π are referred to as the Higgs Bosonand Goldstone Boson respectively and are real scalar fields with no vevs Fπ is a constant

                21 Standard Model 18

                Figure 23 Higgs Potential [49]

                Substituting this back into the Lagrangian 223 we get

                minus14

                FmicroνFmicroν minusevAmicropartmicro

                π+e2v2

                2AmicroAmicro +

                12(partmicrohpart

                microhminus2micro2h2)+

                12

                partmicroπpartmicro

                π+(hπinteractions)(226)

                This Lagrangian now seems to describe a theory with a photon Amicro of mass = ev a Higgsboson h with mass =

                radic2micro and a massless Goldstone π

                However there is a term mixing Amicro and part microπ in this expression and this shows Amicro and π

                are not independent normal coordinates and we can not conclude that the third term is a massterm This π minusAmicro mixing can be removed by making a gauge transformation in equations224 and 225 making

                φrarrv+h(x)radic2

                ei π

                Fπminusieη(x) (227)

                and setting πrarr π

                Fπminus eη(x) = 0 This choosing of a point on the circle for the vev is called

                spontaneous symmetry breaking much like a vertical pencil falling and pointing horizontallyin the plane This choice of gauge is called unitary gauge The Goldstone π will thencompletely disappear from the theory and one says that the Goldstone has been eaten to givethe photon mass

                21 Standard Model 19

                This can be confirmed by counting degrees of freedom before and after spontaneoussymmetry breaking The original Lagrangian had four degrees of freedom- two from thereal massless vector field Amicro (massless photons have only two independent polarisationstates) and two from the complex scalar field (one each from the h and π fields) The newLagrangian has one degree of freedom from the h field and 3 degrees of freedom from themassive real vector field Amicro and the π field has disappeared

                The actual Higgs field is postulated to be a complex scalar doublet with four degrees offreedom

                Φ =

                (φ+

                φ0

                )(228)

                which transforms under SU(2) in the fundamental representation and also transforms underU(1) hypercharge with quantum numbers

                Table 21 Quantum numbers of the Higgs field

                T 3 Q Yφ+

                12 1 1

                φ0 minus12 1 0

                We can parameterise the Higgs field in terms of deviations from the vacuum

                Φ(x) =(

                η1(x)+ iη2(x)v+σ(x)+ iη3(x)

                ) (229)

                It can be shown that the three η fields can be transformed away by working in unitarygauge By analogy with the scalar Higgs field example above the classical action will beminimised for a constant value of the Higgs field Φ = Φ0 for which Φ

                dagger0Φ0 = v2 This again

                defines a range of minima and we can pick one arbitarily (this is equivalent to applying anSU(2)timesU(1) transform to the Higgs doublet transforming the top component to 0 and thebottom component to Φ0 = v) which corresponds to a vacuum of zero charge density andlt φ0 gt=+v

                In this gauge we can write the Higgs doublet as

                Φ =

                (φ+

                φ0

                )rarr M

                (0

                v+ H(x)radic2

                ) (230)

                where H(x) is a real Higgs field v is the vacuum expectation and M is an SU(2)timesU(1)matrix The matrix Mdagger converts the isospinor into a down form (an SU(2) rotation) followedby a U(1) rotation to a real number

                21 Standard Model 20

                If we consider the Higgs part of the Lagrangian

                minus14(Fmicroν)

                2 minus 14(Bmicroν)

                2 +(DmicroΦ)dagger(DmicroΦ)minusλ (Φdagger

                Φminus v2)2 (231)

                Substituting from equation 230 into this and noting that

                DmicroΦ = partmicroΦminus igW amicro τ

                aΦminus 1

                2ig

                primeBmicroΦ (232)

                We can express as

                DmicroΦ = (partmicro minus i2

                (gA3

                micro +gprimeBmicro g(A1micro minusA2

                micro)

                g(A1micro +A2

                micro) minusgA3micro +gprimeBmicro

                ))Φ equiv (partmicro minus i

                2Amicro)Φ (233)

                After some calculation the kinetic term is

                (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

                14(v+

                Hradic2)2[A 2]22 (234)

                where the 22 subscript is the index in the matrix

                If we defineWplusmn

                micro =1radic2(A1

                micro∓iA2micro) (235)

                then [A 2]22 is given by

                [A 2]22 =

                (gprimeBmicro +gA3

                micro

                radic2gW+

                microradic2gWminus

                micro gprimeBmicro minusgA3micro

                ) (236)

                We can now substitute this expression for [A 2]22 into equation 234 and get

                (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

                14(v+

                Hradic2)2(2g2Wminus

                micro W+micro +(gprimeBmicro minusgA3micro)

                2) (237)

                This expression contains mass terms for three massive gauge bosons Wplusmn and the combi-nation of vector fields gprimeBmicro minusgA3

                micro where note

                21 Standard Model 21

                Table 22 Weak Quantum numbers of Lepton and Quarks

                T 3 Q YνL

                12 0 -1

                lminusL minus12 -1 -1

                νR 0 0 0lminusR 0 -1 -2UL

                12

                23

                13

                DL minus12 minus1

                313

                UR 0 23

                43

                DR 0 minus13 minus2

                3

                Wminusmicro = (W+

                micro )dagger equivW 1micro minus iW 2

                micro (238)

                Then the mass terms can be written

                12

                v2g2|Wmicro |2 +14

                v2(gprimeBmicro minusgA3micro)

                2 (239)

                W+micro and W microminus can be associated with the W boson and its anti-particle and (minusgprimeBmicro +

                gA3micro) with the Z Boson (after normalisation by

                radicg2 +(gprime

                )2) The combination gprimeA3micro +gBmicro

                is orthogonal to Z and remains massless This can be associated with the Photon (afternormalisation) These predict mass terms for the W and Z bosons mW = vgradic

                2and mZ =

                vradic2

                radicg2 +(gprime

                )2It is again instructive to count the degrees of freedom before and after the Higgs mech-

                anism Before we had a complex doublet Φ with four degrees of freedom one masslessB with two degrees of freedom and three massless W a fields each with two each givinga total of twelve degrees of freedom After spontaneous symmetry breaking we have onereal scalar Higgs field with one degree of freedom three massive weak bosons with nineand one massless photon with two again giving twelve degrees of freedom One says thescalar degrees of freedom have been eaten to give the Wplusmn and Z bosons their longitudinalcomponents (ie mass)

                Table 22 shows T 3 the weak isospin quantum number (which describes how a particletransforms under SU(2)) Y the weak hypercharge (which describes how the particle trans-

                21 Standard Model 22

                forms under U(1)) and Q the electric charge of leptons and quarks The former two enterdirectly into the Lagrangian while the latter can be derived as a function of the former two

                Strictly speaking the right handed neutrino which has no charge may not exist and is notpart of the standard model although some theories postulate a mass for this (hypothetical)particle

                Writing gauge invariant couplings between ELeRφ with the vacuum expectation ofφ = v (Yukawa matrices λ

                i ju and λ

                i jd respectively for the up and down quarks) we get mass

                terms for the quarks (and similarly for the leptons)

                Mass terms for quarks minussumi j[(λi jd Qi

                Lφd jR)+λ

                i ju εab(Qi

                L)aφlowastb u j

                R +hc]

                Mass terms for leptonsminussumi j[(λi jl Li

                Lφ l jR)+λ

                i jν εab(Li

                L)aφlowastb ν

                jR +hc]

                Here hc is the hermitian conjugate of the first terms L and Q are lepton and quarkdoublets l and ν are the lepton singlet and corresponding neutrino d and u are singlet downand up quarks εab is the Levi-Civita (anti-symmetric) tensor

                If we write the quark generations in a basis that diagonalises the Higgs couplings weget a unitary transformations Uu and Ud for the up and down quarks respectively which iscompounded in the boson current (Udagger

                u Ud) called the CKM matrix where off-diagonal termsallow mixing between quark generations (this is also called the mass basis as opposed tothe flavour basis) A similar argument can give masses to right handed sterile neutrinos butthese can never be measured However a lepton analogue of the CKM matrix the PNMSmatrix allows left-handed neutrinos to mix between various generations and oscillate betweenflavours as they propagate through space allowing a calculation of mass differences

                219 Quantum Chromodynamics

                The final plank in SM theory tackling the strong force Quantum Chromodynamics wasproposed by Gell-Mann and Zweig in 1963 Three generations of quarks were proposedwith charges of 23 for uc and t and -13 for ds and b Initially SU(2) isospin was consideredtransforming u and d states but later irreducible representations of SU(3) were found to fit theexperimental data The need for an additional quantum number was motivated by the fact thatthe lightest excited state of the nucleon consisted of sss (the ∆++) To reconcile the baryon

                21 Standard Model 23

                spectrum with spin statistics colour was proposed Quarks transform under the fundamentalor 3 representation of colour symmetry and anti-quarks under 3 The inner product of 3and 3 is an invariant under SU(3) as is a totally anti-symmetric combination ε i jkqiq jqk (andthe anti-quark equivalent) It was discovered though deep inelastic scattering that quarksshowed asymptotic freedom (the binding force grew as the seperation grew) Non-abeliangauge theories showed this same property and colour was identified with the gauge quantumnumbers of the quarks The quanta of the gauge field are called gluons Because the forcesshow asymptotic freedom - ie the interaction becomes weaker at high energies and shorterdistances perturbation theory can be used (ie Feynman diagrams) at these energies but forlonger distances and lower energies lattice simulations must be done

                2110 Full SM Lagrangian

                The full SM can be written

                L =minus14

                BmicroνBmicroν minus 18

                tr(FmicroνFmicroν)minus 12

                tr(GmicroνGmicroν)

                + sumgenerations

                (ν eL)σmicro iDmicro

                (νL

                eL

                )+ eRσ

                micro iDmicroeR + νRσmicro iDmicroνR +hc

                + sumgenerations

                (u dL)σmicro iDmicro

                (uL

                dL

                )+ uRσ

                micro iDmicrouR + dRσmicro iDmicrodR +hc

                minussumi j[(λ

                i jl Li

                Lφ l jR)+λ

                i jν ε

                ab(LiL)aφ

                lowastb ν

                jR +hc]

                minussumi j[(λ

                i jd Qi

                Lφd jR)+λ

                i ju ε

                ab(QiL)aφ

                lowastb u j

                R +hc]

                + (Dmicroφ)dagger(Dmicroφ)minusmh[φφ minus v22]22v2

                (240)

                where σ micro are the extended Pauli matrices

                (1 00 1

                )

                (0 11 0

                )

                (0 minusii 0

                )

                (1 00 minus1

                )

                The first line gives the gauge field strength terms the second the lepton dynamical termsthe third the quark dynamical terms the fourth the lepton (including neutrino) mass termsthe fifth the quark mass terms and the sixth the Higgs terms

                The SU(3) gauge field is given by Gmicroν and which provides the colour quantum numbers

                21 Standard Model 24

                Figure 24 Standard Model Particles and Forces [50]

                Note The covariant derivatives operating on the left-handed leptons contain only theU(1) and SU(2) gauge fields Bmicro Fmicro whereas those operating on left-handed quarks containU(1)SU(2) and SU(3) gauge fields Bmicro Fmicro Gmicro The covariant derivatives operating onright handed electrons and quarks are as above but do not contain the SU(2) gauge fieldsRight handed neutrinos have no gauge fields

                The sums over i j above are over the different generations of leptons and quarks

                The particles and forces that emerge from the SM are shown in Fig 24

                22 Dark Matter 25

                22 Dark Matter

                221 Evidence for the existence of dark matter

                2211 Bullet Cluster of galaxies

                Most of the mass of the cluster as measured by a gravitational lensing map is concentratedaway from the centre of the object The visible matter (as measured by X-ray emission) ismostly present in the centre of the object The interpretation is that the two galaxy clustershave collided perpendicular to the line of sight with the mostly collisionless dark matterpassing straight through and the visible matter interacting and being trapped in the middleIt was estimated that the ratio of actual to visible matter was about 16 [51]

                Figure 25 Bullet Cluster [52]

                2212 Coma Cluster

                The first evidence for dark matter came in the 1930s when Fritz Zwicky observed the Comacluster of galaxies over 1000 galaxies gravitationally bound to one another He observed thevelocities of individual galaxies in the cluster and estimated the amount of matter based onthe average galaxy He calculated that many of the galaxies should be able to escape fromthe cluster unless the mass was about 10 times higher than estimated

                22 Dark Matter 26

                2213 Rotation Curves [53]

                Since those early observations many spiral galaxies have been observed with the rotationcurves calculated by doppler shift with the result that in all cases the outermost stars arerotating at rates fast enough to escape their host galaxies Indeed the rotation curves arealmost flat indicating large amounts of dark matter rotating with the galaxies and holdingthe galaxies together

                Figure 26 Galaxy Rotation Curves [54]

                2214 WIMPS MACHOS

                The question is if the particles that make up most of the mass of the universe are not baryonswhat are they It is clear that the matter does not interact significantly with radiation so mustbe electrically neutral It has also not lost much of its kinetic energy sufficiently to relax intothe disks of galaxies like ordinary matter Massive particles may survive to the present ifthey carry a conserved additive or multiplicative quantum number This is so even if equalnumbers of particles and anti-particles can annihilate if the number densities become too lowto reduce the density further These particles have been dubbed Weakly interacting MassiveParticles (WIMPS) as they interact with ordinary matter only through a coupling strengthtypically encountered in the weak interaction They are predicted to have frozen out ofthe hot plasma and would give the correct relic density based solely on the weak interactioncross section (sometimes called the WIMP Miracle) Another possibility called MassiveCompact Halo Objects (MACHOS)- are brown dwarf stars Jupiter mass objects or remnantsfrom an earlier generation of star formation

                22 Dark Matter 27

                2215 MACHO Collaboration [55]

                In order to estimate the mass of MACHOS a number of experiments have been conductedusing microlensing which makes use of the fact that if a massive object lies in the line ofsight of a much more distant star the light from the more distant star will be lensed and forma ring around the star It is extemely unlikely that the MACHO will pass directly through theline of sight but if there is a near miss the two images will be seperated by a small angleThe signature for microlensing is a time-symmetric brightening of a star as the MACHOpasses near the line of sight called a light curve One can obtain statistical information aboutthe MACHO masses from the shape and duration of the light curves and a large number ofstars (many million) must be monitored The MACHO Collaboration [55] have monitoredover 11 million stars and less than 20 events were detected The conclusion was that thiscould account for a small addition (perhaps up to 20 ) to the visible mass but not enoughto account for Dark Matter

                2216 Big Bang Nucleosynthesis (BBN) [56]

                Another source of evidence for DM comes from the theory of Nucleosynthesis in the fewseconds after the Big Bang In the first few minutes after the big bang the temperatures wereso high that all matter was dissociated but after about 3 minutes the temperature rapidlycooled to about 109 Kelvin Nucleosynthesis could begin when protons and neutrons couldstick together to form Deuterium (Heavy Hydrogen) and isotopes of Helium The ratio ofDeuterium to Hydrogen and Helium is very much a function of the baryon density and alsothe rate of expansion of the universe at the time of Nucleosynthesis There is a small windowwhen the density of Baryons is sufficient to account for the ratios of the various elementsThe rate of expansion of the universe is also a function of the total matter density The ratiosof these lighter elements have been calculated by spectral measurements with Helium some25 of the mass of the universe Deuterium 001 The theory of BBN accounts for theratios of all of the lighter elements if the ratio of Baryon mass to total mass is a few percentsuggesting that the missing matter is some exotic non-baryonic form

                22 Dark Matter 28

                2217 Cosmic Microwave Background [57]

                The discovery of a uniform Microwave Background radiation by Penzias and Wilson in1965 was confirmation of the theory first proposed by Gamow and collaborators that theuniverse should be filled with black body radiation an afterglow of the big bang TheCOBE satelite launched in 1989 confirmed this black body radiation with temperature of2725+-002K with 95 confidence While extremely uniform it found tiny irregularitieswhich could be accounted for by irregularities in the density of the plasma at last scattering(gravitational redshifts or blueshifts called the Sachs Wolfe effect) and temperature andvelocity fluctuations in the plasma The anisotropy of the background has been furtheranalysed by WMAP (the Wilkinson Microwave Anisotropy Probe) launched in 2001 Thefluctuations in the density of the plasma are what account for the seeds of galaxy formationHowever detailed calculations have found that the fluctuations were too small to account forthe observed galaxy distribution without an electrically neutral gravitating matter in additionto ordinary matter that only became electrically neutral at the time of recombination [58]

                In May 2009 the European Space Agency (ESA) launched the Planck sattelite a spacebased observatory to observe the universe at wavelengths between 03mm and 111 mmbroadly covering the far-infrared microwave and high frequency domains The missionrsquosmain goal was to study the cosmic microwave background across the whole sky at greatersensitivity and resolution than ever before Using the information gathered and the standardmodel of cosmology ESA was able to calculate cosmological parameters including the relicdensity of dark matter remaining after the Big Bang to a high degree of precision withinthe framework of the standard model of cosmology The standard model of cosmology canbe described by a relatively small number of parameters including the density of ordinarymatter dark matter and dark energy the speed of cosmic expansion at the present epoch (alsoknown as the Hubble constant) the geometry of the Universe and the relative amount ofthe primordial fluctuations embedded during inflation on different scales and their amplitude[59]

                2218 LUX Experiment - Large Underground Xenon experiment [16]

                The LUX experiment is conducted at the Sanford Underground research facility one mileunderground to minimise the background of cosmic rays The detector is a time projectionchamber allowing a 3D positioning of interactions occurring within its active volume LUX

                22 Dark Matter 29

                Figure 27 WMAP Cosmic Microwave Background Fluctuations [58]

                Figure 28 Dark Matter Interactions [60]

                uses as its target 368 kiliograms of liquefied ultra pure xenon which is a scintillator Interac-tions inside the xenon will create an amount of light proportional to the energy depositedThat light can be collected on arrays of light detectors sensitive to a single photon

                Xenon is very pure and the amount of radiation originating within the target is limitedAlso being three times denser than water it can stop radiation originating from outside thedetector The 3D detector focuses on interactions near the center which is a quiet regionwhere rare dark matter interactions might be detected [16]

                22 Dark Matter 30

                222 Searches for dark matter

                2221 Dark Matter Detection

                Fig 28 illustrates the reactions studied under the different forms of experiment designed todetect DM

                Direct detection experiments are mainly underground experiments with huge detectorsattempting to detect the recoil of dark matter off ordinary matter such as LUX [16] CDMS-II [61] XENON100 [11] COUPP [62] PICASSO [63] ZEPLIN [10] These involveinteractions between DM and SM particles with final states involving DM and SM particles

                Indirect detection involves the search for the annihilation of DM particles in distantastrophysical sources where the DM content is expected to be large Example facilitiesinclude space based experiments (EGRET [64] PAMELA [18] FERMI-LAT [65]) balloonflights (ATIC [66] PEBS [67] PPB-BETS [68]) or atmospheric Cherenkov telescopes(HESS [69]MAGIC [70]) and neutrino telescopes (SuperKamiokande [71] IceCube [72]ANTARES [73] KM3NET [74]) Indirect Detection involves DM particles in the initial stateand SM particles in the final state

                Collider detection involves SM particles in the initial state creating DM particles in thefinal state which can be detected by missing transverse energy and explained by variousmodels extending the SM

                223 Possible signals of dark matter

                224 Gamma Ray Excess at the Centre of the Galaxy [65]

                The center of the galaxy is the most dense and active region and it is no surprise that itproduces intense gamma ray sources When astronomers subtract all known gamma raysources there remains an excess which extends out some 5000 light years from the centerHooper et al find that the annihilation of dark matter particles with a mass between 31 and40 Gev fits the excess to a remarkable degree based on Gamma Ray spectrum distributionand brightness

                23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 31

                Figure 29 Gamma Ray Excess from the Milky Way Center [75]

                23 Background on ATLAS and CMS Experiments at theLarge Hadron collider (LHC)

                The collider constraints in this paper are based on two of the experiments that are part of theLHC in CERN

                Figure 210 ATLAS Experiment

                The LHC accelerates proton beams in opposite directions around a 27 km ring of super-conducting magnets until they are made to collide at an interaction point inside one of the 4experiments placed in the ring An upgrade to the LHC has just been performed providing65 TeV per beam

                23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 32

                231 ATLAS Experiment

                The ATLAS detector consists of concentric cylinders around the interaction point whichmeasure different aspects of the particles that are flung out of the collision point There are 4major parts the Inner Detector the Calorimeters the Muon spectrometer and the magnetsystems

                2311 Inner Detector

                The inner detector extends from a few centimeters to 12 meters from the interaction point andmeasures details of the charged particles and their momentum interacting with the materialIf particles seem to originate from a point other than the interaction point this may indicatethat the particle comes from the decay of a hadron with a bottom quark This is used to b-tagcertain particles

                The inner detector consists of a pixel detector with three concentric layers and threeendcaps containing detecting silicon modules 250 microm thick The pixel detector has over 80million readout channels about 50 percent of the total channels for the experiment All thecomponents are radiation hardened to cope with the extremely high radiation in the proximityof the interaction point

                The next layer is the semi-conductor layer (SCT) similar in concept to the pixel detectorbut with long narrow strips instead of pixels to cover a larger area It has 4 double layers ofsilicon strips with 63 million readout channels

                The outermost component of the inner detector is the Transition Radiation Tracker It is acombination of a straw tracker and a transition radiation detector The detecting elementsare drift tubes (straws) each 4mm in diameter and up to 144 cm long Each straw is filledwith gas which becomes ionised when the radiation passes through The straws are held at-1500V with a thin wire down the center This drives negatively ionised atoms to the centercausing a pulse of electricity in the wire This causes a pattern of hit straws allowing the pathof the particles to be determined Between the straws materials with widely varying indicesof refraction cause transition radiation when particles pass through leaving much strongersignals in some straws Transition radiation is strongest for highly relativistic particles (thelightest particles- electrons and positrons) allowing strong signals to be associated with theseparticles

                23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 33

                2312 Calorimeters

                The inner detector is surrounded by a solenoidal magnet of 2 Tesla (there is also an outertoroidal magnet) but surrounding the inner magnet are the calorimeters to measure theenergy of particles by absorbing them There is an inner electromagnetic calorimeter whichmeasures the angle between the detectorrsquos beam axis and the particle (or more preciselypseudo-rapidity) and its angle in the perpendicular plane to a high degree of accuracy Theabsorbing material is lead and stainless steel and the sampling material liquid argon Thehadron calorimeter measures the energy of particles that interact via the strong force Theenergy absorbing material is scintillating steel

                2313 Muon Specrometer

                The muon spectrometer surrounds the calorimeters and has a set of 1200 chambers tomeasure the tracks of outgoing muons with high precision using a magnetic field andtriggering chambers Very few particles of other types pass through the calorimeters and aresubsequently measured

                2314 Magnets

                The magnets are in two layers the inner solenoidal and outer toroidal which together causecharged particles to curve in the magetic field allowing their momentum to be determinedThe outer toroidal magnet consists of 8 very large air core superconducting magnets whichsupply the field in the muon spectrometer

                232 CMS Experiment

                The CMS Detector is similar in design to the ATLAS Detector with a single solenoidalmagnet which is the largest of its type ever built (length 13m and diameter 7m) and allowsthe tracker and calorimeter detectors to be placed inside the coil in a compact way Thestrong magnetic field gives a high resolution for measuring muons electrons and photons ahigh quality central tracking system and a hermetic hadron calorimeter designed to preventhadrons from escaping

                23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 34

                Figure 211 CMS Experiment

                Chapter 3

                Fitting Models to the Observables

                31 Simplified Models Considered

                In order to extend the findings of previous studies [19] to include collider constraints wefirst performed Bayesian scans [7] over three simplified models real scalar dark matter φ Majorana fermion χ and real vector Xmicro The purpose was to obtain the most probable regionsof each model then to use the resulting posterior samples to investigate whether the LHCexperiments have any sensitivity to the preferred regions

                The three models couple to the mediator with interactions shown in the following table

                Table 31 Simplified Models

                Hypothesis real scalar DM Majorana fermion DM real vector DM

                DM mediator int Lφ sup microφ mφ φ 2S Lχ sup iλχ

                2 χγ5χS LX sup microX mX2 X microXmicroS

                The interactions between the mediator and the standard fermions is assumed to be

                LS sup f f S (31)

                and in line with minimal flavour violation we only consider third generation fermions- ief = b tτ

                For the purposes of these scans we consider the following observables

                32 Observables 36

                32 Observables

                321 Dark Matter Abundance

                We have assumed a single dark matter candidate which freezes out in the early universe witha central value as determined by Planck [21] of

                ΩDMh2 = 01199plusmn 0031 (32)

                h is the reduced hubble constant

                The experimental uncertainty of the background relic density [59] is dominated by atheoretical uncertainty of 50 of the value of the scanned Ωh2 [76] We have added thetheoretical and the experimental uncertainties in quadrature

                SD =radic(05Ωh2)2 + 00312 (33)

                This gives a log likelihood of

                minus05lowast (Ωh2 minus 1199)2

                SD2 minus log(radic

                2πSD) (34)

                322 Gamma Rays from the Galactic Center

                Assuming that the gamma ray flux excess from the galactic centre observed by Fermi-LAT iscaused by self annihilation of dark matter particles the differential flux

                d2Φ

                dEdΩ=

                lt σv gt8πmχ

                2 J(ψ)sumf

                B fdN f

                γ

                dE(35)

                has been calculated using both Micromegas v425 with a Navarro-Frenk-White (NFW) darkmatter galactic distribution function

                ρ(r) = ρ0(rrs)

                minusγ

                (1+ rrs)3minusγ (36)

                with γ = 126 and an angle of 5 to the galactic centre [19]

                32 Observables 37

                Here Φ is the flux dE is the differential energy lt σv gt the velocity averaged annihi-lation cross section of the dark matter across the galactic centre mχ the mass of the darkmatter particle B f =lt σv gt f σv is the annihilation fraction into the f f final state anddN f

                γ dE is the energy distribution of photons produced in the annihilation channel with thefinal state f f The J factor is described below

                The angle ψ to the galactic centre determines the normalisation of the gamma ray fluxthrough the factor

                J(ψ) =int

                losρ(l2 + r⊙2 minus2lr⊙cosψ)dl (37)

                where ρ(r) is the dark matter density at distance r from the centre of the galaxy r⊙=85 kpcis the solar distance from the Galactic center ρ0 is set to reproduce local dark matter densityof 03GeVcm3

                The central values for the gamma ray spectrum were taken from Fig 5 of [77] and thestandard deviations of individual flux points from the error bars on the graph In commonwith [78] we ignore possible segment to segment correlations

                For the Fermi-LAT log likelihood we assumed each point was distributed with indepen-dent Gaussian likelihood with the experimental error for each data point given by the errorbar presented in Fig5 of [76] for each point We also assumed an equal theoretical errorfor each point which when added to the experimental error in quadrature gave an effectivelog likelihood for each point of minus05lowast (giminusdi)

                2

                2lowastσ2i

                where gi are the calculated values and di theexperimental values and σi the experimental errors

                323 Direct Detection - LUX

                The LUX experiment [16] is the most constraining direct search experiment to date for awide range of WIMP models We have used LUXCalc [79] (modified to handle momentumdependent nucleon cross sections) to calculate the likelihood of the expected number ofsignal events for the Majorana fermion model

                The likelihood function is taken as the Poisson distribution

                L(mχ σ | N) = P(N | mχ σ) =(b+micro)Neminus(b+micro)

                N (38)

                32 Observables 38

                where b= expected background of 064 and micro = micro(mχσ ) is the expected number of signalevents calculated for a given model

                micro = MTint

                infin

                0dEφ(E)

                dRdE

                (E) (39)

                where MT is the detector mass times time exposure and φ(E) is a global efficiency factorincorporating trigger efficiencies energy resolution and analysis cuts

                The differential recoil rate of dark matter on nucleii as a function of recoil energy E

                dRdE

                =ρX

                mχmA

                intdvv f (v)

                dσASI

                dER (310)

                where mA is the nucleon mass f (v) is the dark matter velocity distribution and

                dσSIA

                dER= Gχ(q2)

                4micro2A

                Emaxπ[Z f χ

                p +(AminusZ) f χn ]

                2F2A (q) (311)

                where Emax = 2micro2Av2mA Gχ(q2) = q2

                4m2χ

                [24] is an adjustment for a momentum dependantcross section which applies only to the Majorana model

                f χ

                N =λχ

                2m2SgSNN assuming that the relic density is the central value of 1199 We have

                implemented a scaling of the direct search cross section in our code by multiplying gSNN by( Ω

                Ω0)05 as a convenient method of allowing for the fact that the actual relic density enters

                into the calculation of the cross section as a square

                FA(q) is the nucleus form factor and

                microA =mχmA

                (mχ +mA)(312)

                is the reduced WIMP-nucleon mass

                The strength of the dark matter nucleon interaction between dark matter particles andthird generation quarks is

                gSNN =2

                27mN fT G sum

                f=bt

                λ f

                m f (313)

                where fT G = 1minus f NTuminus f N

                Tdminus fTs and f N

                Tu= 02 f N

                Td= 026 fTs = 043 [20]

                33 Calculations 39

                For the real scalar and real vector models where the spin independent cross sections arenot momentum dependent we based the likelihood on the LUX 95 CL from [16] using acomplementary error function Li(d|mχ I) = 05lowastCEr f (dminusx(mχ )

                σ) where x is the LUX limit

                and d the calculated SI cross section We assumed a LUX error σ of 01lowastd lowastradic

                2

                33 Calculations

                We coded the Lagrangians for the three models in Feynrules [4] Calculation of the expectedobservables (dark matter relic density Fermi-LAT gamma ray spectrum and LUX nucleonscattering amplitudes) were performed using Micromegas v425

                331 Mediator Decay

                A number of searches have previously been performed at the LHC which show sensitivity tothe decay of scalar mediators in the form of Higgs bosons (in beyond the standard modeltheories) to bottom quarks top quarks and τ leptons (all third generation fermions) Two inparticular [43 44] provide limits on the cross-sections for the production of the scalar timesthe branching ratios to brsquos and τrsquos In order to check these limits against each of the modelswe ran parameter scans in Madgraph at leading order over the top 900 points from the nestedsampling output from MultiNest [19]

                The two processes were

                1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to apair of τ leptons in pp collisions [44]

                bull generate p p gt b b S where S is the scalar mediator

                The process measures the decay of a scalar in the presence of b-jets that decay to τ+τminus

                leptons Fig31 shows the main diagrams contributing to the process The leftmost diagramis a loop process and the centre and right are tree level however we only got Madgraph togenerated the middle diagram since we are interested in τ leptons and b quarks in the finalstate

                The limit given in [44] only applies in the case of narrow resonances where narrow istaken to mean that the width of the resonance is less than 10 of its mass To test the validity

                33 Calculations 40

                Figure 31 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof τ leptons

                of applying these limits to our models we show in Figures 32 to 34 the percentage widthof the resonance vs mS[GeV ] (pole mass of the scalar mediator) λb and λt (the couplingconstants to the b and t quarks) The probabilities are the posterior probabilities obtainedfrom MultiNest scans of the Majorana model varying the coupling constants and scalarmediator and calculating the decay width of the mediator The results are represented as 2dposterior probability heat maps in the central panels and 1d marginalised distributions on thex and y axes of each graph

                16 18 20 22 24 26 28 30

                log10(mS[GeV])

                001

                002

                003

                004

                005

                Widthm

                S

                00

                04

                08

                12

                0 100 200

                Posterior Probability

                Figure 32 WidthmS vs mS

                The WidthmS was lt 01 at almost all points except where the mass of the scalar wasgreater than 1000 GeV λb was greater than 1 or λτ was greater than 1

                This can be seen from the graphs in Figs 323334

                33 Calculations 41

                4 3 2 1 0

                λb

                001

                002

                003

                004

                005

                WidthmS

                000

                015

                030

                045

                0 100 200

                Posterior Probability

                Figure 33 WidthmS vs λb

                5 4 3 2 1 0

                λτ

                001

                002

                003

                004

                005

                WidthmS

                000

                015

                030

                045

                0 100 200

                Posterior Probability

                Figure 34 WidthmS vs λτ

                The decay widths are calculated in Madgraph at leading order and branching fractionscan be calculated to b t τ and χ (DM) The formulae for the calculations can be found indecayspy which is a subroutine of Madgraph which we have documented in Appendix BB1 Using these we calculated the branching ratio to τ+τminus so that the cross section timesbranching ratio can be compared to the limits in [80] shown in Chapters 56 and 7 in Figs5564 and 74 for each model

                2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                This scan calculates the contributions of b-quark associated Higgs boson production inthe center and right diagrams of Fig 35 reproduced from [43] The scans did not includePythia and Delphes as only production cross sections were required and the producedparticles were in the final state and required no matching or merging

                The Madgraph processes were

                bull generate p p gt b S where S is the scalar mediator

                bull add process p p gt b S j

                bull add process p p gt b S

                33 Calculations 42

                Figure 35 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof b quarks in the presence of at least one b quark

                bull add process p p gt b S j

                The cross section times branching ratios to bb for mediator production in association withat least one b quark is shown for each model in Chapters 56 and 7 in Figs 5765 and 75

                332 Collider Cuts Analyses

                We generated two parameter scans with parameter combinations from the Multinest outputwhich were the basis of the plots of Fig 6 in [19] The plots in Fig 6 were marginalisedto two parameters each but the raw Multinest output contained 5 parameters plus the priorprobabilities of each combination normalised to total 10

                The first scan was for χ produced with b quarks In order to force the mediator to decayto χ only we had χ in the final state and defined jets to include b quarks in Madgraph Thenwe generated the processes

                bull generate p p gt χ χ j

                bull add process p p gt χ χ j j

                Jet matching was on

                The second scan was for t quarks produced in the final state

                bull generate p p gt χ χ tt

                33 Calculations 43

                No jet matching was required because we did not mix final state jet multiplicities at thematrix element level

                The outputs from these two processes were normalised to 21 f bminus1 and combined

                The results of these scans prior to collider cuts calculations are shown in Figs 5161and 71 These show the posterior probabilities of the various parameter combinations withdifferent constraints applied seperately and then together A more detailed commentary isgiven in each of the sections

                We then sampled 900 points based on posterior probability from these scans and ran thesethrough Madgraph v425 then applied collider cuts with the C++ code using the limits fromthe following six ATLAS and CMS searches shown in Table32 and described below to findany exclusions

                333 Description of Collider Cuts Analyses

                In the following all masses and energies are in GeV and angles in radians unless specificallystated

                3331 Lepstop0

                Search for direct pair production of the top squark in all-hadronic final states in pro-tonndashproton collisions at

                radics=8 TeV with the ATLAS detector[32]

                This analysis searched for direct pair production of the supersymmetric partner to thetop quark using integrated luminosity of 201 f bminus1 The top squark was assumed to de-cay via t rarr t χ0

                1 or t rarr bχ01 or t rarr bχ

                plusmn1 rarr bW (lowast)χ1

                0 where χ01 (χ

                plusmn1 ) denotes the lightest

                neutralino(chargino) in supersymmetric models The search targets a fully hadronic finalstate with four or more jets and large missing momentum The production of a mediator inassociation with top quarks followed by an invisible mediator decay to dark matter particleshas a similar final state to that considered in the ATLAS stop searches Thus we expect theATLAS reach to have sensitivity to the three dark matter models considered in this thesis

                The collider cuts program mimicked the object reconstruction through jetlepton overlapremoval and selected signal electrons muons and τ particles with pT gt 10 GeV and

                33 Calculations 44

                Table 32 95 CL by Signal Region

                Experiment Region Number

                Lepstop0

                SRA1 66SRA2 57SRA3 67SRA4 65SRC1 157SRC2 124SRC3 80

                Lepstop1

                SRtN1-1 857SRtN1-2 498SRtN1-3 389SRtN2 107SRtN3 85SRbC1 832SRbC2 195SRbC3 76

                Lepstop2

                L90 740L100 56L110 90L120 170

                2bstop

                SRA 857SRB 270SRA150 380SRA200 260SRA250 90SRA300 75SRA350 52

                CMSTopDM1L SRA 1385

                ATLASMonobjetSR1 1240SR2 790

                33 Calculations 45

                |η | lt 24724 and 247 respectively (for eminusmicrominusτ) Jets were selected with pT gt 20 GeVand |η |lt 45 b-jets were selected if they were tagged by random selection lt 70 and had|η |lt 25 and pT gt 20 GeV

                These jets and leptons were further managed for overlap removal within ∆R lt 04 withnon b-jets removed if ∆R lt 02

                The b-jets and non b-jets were further selected as signal jets if pT gt 35 GeV and |η |lt 28

                These signal jets electrons and muons were finally binned according to the signal regionsin Tables 333435 below

                Table 33 Selection criteria common to all signal regions

                Trigger EmissT

                Nlep 0b-tagged jets ⩾ 2

                EmissT 150 GeV

                |∆φ( jet pmissT )| gtπ5

                mbminT gt175 GeV

                Table 34 Selection criteria for signal regions A

                SRA1 SRA2 SRA3 SRA4anti-kt R=04 jets ⩾ 6 pT gt 808035353535 GeV

                m0b j j lt 225 GeV [50250] GeV

                m1b j j lt 225 GeV [50400] GeV

                min( jet i pmissT ) - gt50 GeV

                τ veto yesEmiss

                T gt150 GeV gt250 GeV gt300 GeV gt 350 GeV

                Table 35 Selection criteria for signal regions C

                SRC1 SRC2 SRC3anti-kt R=04 jets ⩾ 5 pT gt 8080353535 GeV

                |∆φ(bb)| gt02 π

                mbminT gt185 GeV gt200 GeV gt200 GeV

                mbmaxT gt205 GeV gt290 GeV gt325 GeV

                τ veto yesEmiss

                T gt160 GeV gt160 GeV gt215 GeV

                wherembmin

                T =radic

                2pbt Emiss

                T [1minus cos∆φ(pbT pmiss

                T )]gt 175 (314)

                33 Calculations 46

                and pbT is calculated for the b-tagged jet with the smallest ∆φ to the pmiss

                T direction andmbmax

                T is calculated with the pbT for the b-tagged jet with the largest ∆φ to the pmiss

                T direction

                m0b j j is obtained by first selecting the two jets with the highest b-tagging weight- from

                the remaining jets the two jets closest in the η minusφ plane are combined to form a W bosoncandidate this candidate is then combined with the closest of the 2 b-tagged jets in the η minusφ

                plane to form the first top candidate with mass m0b j j A second W boson candidate is formed

                by repeating the procedure with the remaining jets this candidate is combined with thesecond of the selected b-tagged jets to form the second top candidate with mass m1

                b j j Thecuts shown in Appendix1 Table A1 show the cumulative effect of the individual cuts appliedto the data the signal region cuts and the cumulative effects of individual cuts in SignalRegion C

                3332 Lepstop1

                Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

                radics=8 TeV pp collisions using 21 f bminus1 of

                ATLAS data[33]

                The stop decay modes considered were those to a top quark and the lightest supersym-metric particle (LSP) as well as a bottom quark and the lightest chargino where the charginodecays to the LSP by emitting a W boson We should expect the three dark matter models inthis paper with couplings to t and b quarks to show some visibility in this search

                The collider cuts program mimicked the object construction by selecting electrons andmuons with pT gt 10 GeV and |η |lt 247 and 24 respectively and jets with pT gt 20 GeVand |η |lt 10 b-jets with |η |lt 25 and pT gt 25 GeV were selected with random efficiency75 and mistag rate 2 (where non b-jets were mis-classified) We then removed any jetwithin ∆R = 02 of an electron and non overlapping jets with |η |lt 28 were used to removeany electrons and muons within ∆R = 04 of these while non-overlapping jets with |η |lt 25and pT gt 25 GeV were classified as signal jets Non overlapping electrons and muons withpT gt 25 GeV were classified as signal electrons and muons

                Signal selection defined six regions to optimize sensitivity for different stop and LSPmasses as well as sensitivity to two considered stop decay scenarios Three signal regionslabelled SRbC 1-3 where bC is a moniker for b+Chargino and 3 regions labelled SRtN 1-3where tN is a moniker for t+Neutralino were optimised for the scenarios where t1 decaysto the respective particles indicated by the labels Increasing label numbers correspond toincreasingly strict selection criteria

                33 Calculations 47

                The dominant background in all signal regions arises from dileptonic tt events All threeSRtN selectons impose a requirement on the 3-jet mass m j j j of the hadronically decayingtop quark to specifically reject the tt background where both W bosons from the top quarkdecay leponically

                For increasing stop mass and increasing difference between the stop and the LSP therequirents are tightened on Emiss

                T on the ratio EmissT

                radicHT where HT is the scalar sum of the

                momenta of the four selected jets and also tightened on mT

                To further reduce the dileptonic tt background for an event characterised by two one stepdecay chains a and b each producing a missing particle C use is made of the variable

                mT 2 =min

                pCTa + pC

                T b = pmissT

                [max(mTamtb)] (315)

                where mTi is the transverse mass 1 of branch i for a given hypothetical allocation (pCTa pC

                T b)

                of the missing particle momenta Eqn 315 is extended for more general decay chains byamT 2 [81] which is an asymmetric form of mT 2 in which the missing particle is the W bosonin the branch for the lost lepton and the neutrino is the missing particle for the branch withthe observed charged lepton

                ∆φ( jet12 pmissT ) the azimuthal angle between leading or subleading jets and missing mo-

                mentum is used to suppress the background from mostly multijet events with mis-measuredEmiss

                T

                Events with four or more jets are selected where the jets must satisfy |η | lt 25 and pT gt80604025 GeV respectively

                These signal particles were then classified according to the cutflows analysis in Fig36below and compared with the paper in Appendix1 Table A2

                3333 Lepstop2

                Search for direct top squark pair production in final states with two leptons in p pcollisions at

                radics=8TeV with the ATLAS detector[34]

                Three different analysis strategies are used to search for t1 pair production Two targett1 rarr b+ χ

                plusmn1 decay and the three body t1 rarr bW χ0

                1 decay via an off-shell top quark whilst

                1The transverse mass is defined as m2T = 2plep

                T EmissT (1minus cos(∆φ)) of branch i where ∆φ is the azimuthal

                angle between the lepton and the missing transverse momentum

                33 Calculations 48

                Table 36 Signal Regions - Lepstop1

                Requirement SRtN2 SRtN3 SRbC1 SRbC2 SRbC3∆φ( jet1 pmiss

                t )gt - 08 08 08 08∆φ( jet2 pmiss

                T )gt 08 08 08 08 08Emiss

                T [GeV ]gt 200 275 150 160 160Emiss

                T radic

                HT [GeV12 ]gt 13 11 7 8 8

                mT [GeV ]gt 140 200 120 120 120me f f [GeV ]gt - - - 550 700amT 2[GeV ]gt 170 175 - 175 200mT

                T 2[GeV ]gt - 80 - - -m j j j Yes Yes - - -

                one targets t1 rarr t + χ01 to an on shell quark decay mode The analysis is arranged to look

                at complementary mass splittings between χplusmn1 and χ0

                1 smaller and larger than the W bosonmass The smaller splitting provides sensitivity to t1 three body decay

                Object reconstruction was carried out by selecting baseline electronsmuons and τrsquos withpT gt 10 GeV and |η |lt 24724 and 247 respectively Baseline jets and b-jets were selectedwith pT gt 20 GeV and |η |lt 25 and b-jets with 70 efficiency Baseline jets were removedif within ∆R lt 02 of an electron and electrons and muons were removed if within ∆R lt 04of surviving jets Signal electrons required pT gt 10 GeV and signal jets pT gt 20 GeV and|η |lt 25

                The analysis made use of the variable mT 2 [82] the stransverse mass used to mea-sure the masses of pair-produced semi-invisibly decaying heavy particles It is definedas mT 2(pT1 pT2qT ) =

                minqT1+qT2=qT

                max[mT (pT1qT1)mT (pT2qT2)] where mT indicatesthe transverse mass pT1 and pT2 are vectors and qT = qT1 + qT2 The minimisation isperformed over all possible decompositions of qT

                Events were selected if they passed either a single-electron a single-muon a double-electron a double-muon or an electron-muon trigger Events are required to have exactlytwo opposite sign leptons and at least one of the leptons or muons must have a momentumgreater than 25 GeV and the invariant mass (rest mass) of the two leptons is required to begreater than 20 GeV In order to reduce the number of background events containing twoleptons produced by the on-shell decay of the Z boson the invariant mas must be outside711-111 GeV range

                Two additional selections are applied to reduce the number of background events withhigh mT 2 arising from events with large Emiss

                T due to mismeasured jets ∆φb lt 15 and∆φ gt 10 The quantity ∆φb is the azimuthal angle between pll

                T b = pmissT + pl1

                T +Pl2T The

                33 Calculations 49

                vector pllT b is the opposite of the vector sum of all the hadronic activity in the event For WW

                and tt backgrounds it measures the transverse boost of the WW system and for the signalthe transverse boost of the chargino-chargino system The ∆φ variable is the azimuthal anglebetween the pmiss

                T vector and the direction of the closest jet

                By requiring mT 2 gt 40 GeV the tt background is the dominant background in both thesame-flavoured (SF) and different-flavoured (DF) events

                Four signal regions are then defined as shown in Table 37 SR M90 has the loosestselection requiring MT 2gt 90 GeV and no additional requirement on jets and providessensitivity to scenarios with a small difference between the masses of the top-squark andchargino SR M110 and M120 have a loose selection on jets requiring two jets with pT

                gt 20 GeV and mT 2 to be greater than 110 GeV and 120 GeV respectively They providesensitivity to scenarios with small to moderate difference between the invariant masses of thestop quark and the charginos SR M100 has a tighter jet slection and provides sensitivity toscenarios with a large mass difference between the both stop quark and the chargino and thestop quark and the neutralino

                The analysis cut regions are summarised in Fig37 below

                Table 37 Signal Regions Lepstop2

                SR M90 M100 M110 M120pT leading lepton gt 25 GeV

                ∆φ(pmissT closest jet) gt10

                ∆φ(pmissT pll

                T b) lt15mT 2 gt90 GeV gt100 GeV gt110 GeV gt120 GeV

                pT leading jet no selection gt100 GeV gt20 GeV gt20 GeVpT second jet no selection gt50 GeV gt20 GeV gt20 GeV

                To validate the code we ran the cuts shown in Appendix 1 Table A3 with the givenMadgraph simulation and compared to [83] It was not clear what baseline cuts had beenemployed in the ATLAS paper so our results were normalised to the 2 SF electrons beforecomparison Some of the ATLAS cut numbers were less than 3 and we did not produce anyevents passing these cuts but clearly the margin of error at this level was greater than thenumber of ATLAS events

                33 Calculations 50

                3334 2bstop

                Search for direct third-generation squark pair production in final states with miss-ing transverse momentum and two b-jets in

                radics= 8 TeV pp collisions with the ATLAS

                detector[31]

                Two sets of scenarios are considered In the first the coloured sbottom is the onlyproduction process decaying via b1 rarr bχ0

                1 In the second the stop is the only productionprocess decaying via t1 rarr t χplusmn

                1 and the lightest chargino (χplusmn1 ) decays via a virtual W boson

                into the final state χ01 f f prime Both scenarios are characterised by the presence of two jets

                resulting from the hadronisation of b quarks and large missing transverse momentum Theproduction of a mediator in association with top quarks followed by an invisible mediatordecay to dark matter particles has a similar final state as that considered in the ATLAS stopsearches Thus we expect the ATLAS reach to have sensitivity to the three dark mattermodels considered in this thesis

                The collider cuts program mimicked object reconstruction by selecting baseline electronswith pT gt 7 GeV and |η |lt 247 and muons with pT gt 6 GeV and |η |lt 24 Baseline jetswere selected with pT gt 20 GeV and |η |lt 28 We then removed any jets within ∆R lt 02of an electron and any electron within ∆R lt 04 of surviving jets Signal electrons andmuons were required to have pT gt 25 GeV b-jets were selected with 60 tagging efficiencyand |η |lt 25 and pT gt 20 GeV

                Two sets of signal regions are defined to provide sensitivity to the kinematic topologiesassociated with the different mass splittings between the sbottom or the stop and the neu-tralino In all cases the presence of at least one primary vertex (with at least five tracks withpT gt 04 GeV) is required Events are selected with Emiss

                T gt 150 GeV and no electrons ormuons identified in the final state For signal region selections jets with |η | lt 28 are orderedaccording to their pT and 2 jets are required to be b-tagged

                The variables are defined as follows

                bull ∆φmin is the minimal azimuthal angle ∆φ between any of the three leading jets andpmiss

                T

                bull me f f (k) = sumki=1(p jet

                T )i +EmissT where the index refers to the pT ordered list of jets

                bull HT3 is defined as the scalar sum of the pT of the n leading pT ordered jets withoutincluding the 3 leading jets HT3 = sum

                ni=4(p jet

                T )i

                bull mbb is the invariant mass of the two b-tagged jets in the event

                33 Calculations 51

                bull mCT is the contransverse mass [82] and is a kinematical variable that can be usedto measure the masses of pair produced semi-invisibly decaying particles For twoidentical decays of heavy particles into two invisible particles v1 and v2 and twoinvisible particles mCT is defined as m2

                CT (v1v2) = [ET (v1)+ET (v2)]2 minus [pT (v1)minus

                pT (v2)]2 where ET =

                radicp2

                T +m2 The the contransverse mass is invariant under equaland opposite boosts of the parent particles in the transverse plane and is bounded fromabove by an analytical expression of the particle masses For production of sbottompairs the bound is given by mmax

                CT =m2(b)minusm2(χ0

                1 )

                m(b) and for tt events the bound is 135

                GeV A similar equation can be written for stop pairs in term of the stop and charginomasses

                A definition of the signal regions is given in the Table38

                Table 38 Signal Regions 2bstop

                Description SRA SRBEvent cleaning All signal regions

                Lepton veto No emicro after overlap removal with pT gt7(6) GeV for e(micro)Emiss

                T gt 150 GeV gt 250 GeVLeading jet pT ( j1) gt 130 GeV gt 150 GeVSecond jet pT ( j2) gt 50 GeV gt 30 GeVThird jet pT ( j3) veto if gt 50 GeV gt 30 GeV

                ∆φ(pmissT j1) - gt 25

                b-tagging leading 2 jets(pT gt 50 GeV |η | lt 25)

                2nd- and 3rd- leading jets(pT gt 30 GeV|η | lt 25)

                ∆φmin gt 04 gt 04Emiss

                T me f f (k) EmissT me f f (2) gt 025 Emiss

                T me f f (3) gt 04mCT gt 150200250300350 GeV -HT3 - lt 50 GeVmbb gt200 GeV -

                The analysis cuts are summarised in Table A4 of Appendix 1

                3335 ATLASMonobjet

                Search for dark matter in events with heavy quarks and missing transverse momentumin pp collisions with the ATLAS detector[30]

                Events were selected with high missing transverse momentum when produced in associa-tion with one or more jets identified as containing b quarks Final states with top quarks wereselected by requiring a high jet multiplicity and in some cases a single lepton The research

                33 Calculations 52

                studied effective contact operators in the form of a scalar operator Oscalar = sumqmqMN

                lowastqqχχ

                where MNlowast is the effective mass of the coupling and the quark and DM fields are denoted by

                q and χ respectively These are the equivalent of the simplified models in this paper with themediator integrated out

                Object reconstruction selected baseline electrons muons jets and τrsquos with pT gt 20 GeVand |η |lt 2525 and 45 respectively Signal jets were selected by removing any jets within∆R lt 02 of an electron and electrons and muons were selected by removing them if theywere within ∆R lt 04 of surviving jets b-jets were tagged with an efficiency of 60

                Only signal regions SR1 and SR2 were analysed

                The analysis split events into signal regions SR1 and SR2 according to the followingCriteria

                Table 39 Signal Region ATLASmonobjet

                Cut SR1 SR2Jet multiplicity n j 1minus2 3minus4

                bminusjet multiplicity nb gt 0 (60 eff) gt 0 (60 eff)Lepton multiplicity nl 0 0

                EmissT gt300 GeV gt200 GeV

                Jet kinematics pb1T gt100 GeV pb1

                T gt100 GeV p j2T gt100 (60) GeV

                ∆φ( j1EmissT ) gt 10 i = 12 gt 10 i = 14

                Where p jiT (pbi

                T ) represent the transverse momentum of the i-th jet b-tagged jet respec-tively If there is a second b-tagged jet the pT must be greater than 60 GeV

                3336 CMSTop1L

                Search for top-squark pair production in the single-lepton final state in pp collisionsat

                radics=8 TeV[41]

                This search was for pair production of top squarks in events with a single isolatedelectron or muon jets large missing transverse momentum and large transverse mass

                (MT =radic

                2EmissT pl

                T (1minus cos(∆φ))) where plT is the leptonrsquos transverse momentum and ∆(φ)

                is the difference between the azimuthal angles of the lepton and EmissT The 3 models

                considered should have some visibility at certain parameter combinations in this regionThe search focuses on two decay modes of the top squark leading to the two processespp rarr t tlowast rarr tt χ0

                1 χ01 rarr bbW+Wminusχ0

                1 χ01 and pp rarr t tlowast rarr bbχ

                +1 χ

                minus1 rarr bbW+Wminusχ0

                1 χ01 The

                33 Calculations 53

                lightest neutralino χ01 is considered the lightest supersymmetric particle and escapes without

                detection The analysis is based on events where one of the W bosons decays leptonicallyand the other hadronically resulting in one isolated lepton and 4 jets two of which originatefrom b quarks The two neutralinos and the neutrino from the W decay can result in largemissing transverse momentom The kinematical end point of MT is lt the mass of the Wboson However for signal events the presence of LSPs in the final state allows MT to exceedthe mass of the W boson The search is therefore for events with large MT The dominantbackground with large MT arises from tt decays where W bosons decay leptonically withone of the leptons not being identified

                To reduce the dominant tt background use was made of the MWT 2 variable defined as

                the minimum mother particle mass compatible with the transverse momenta and mass shellconstraints [84]

                Object reconstruction required electrons and muons to have pT gt 30 and 25 GeV respand |η |lt 25 and 21 respectively Baseline jets were required to have pT gt 30 and |η |lt 4and b-jets were tagged with an efficiency of 75 for |η |lt 24

                Events were chosen if they contained one isolated electron or muon at least 4 jets and atleast one b-tagged jet and Emiss

                T gt 100 GeV MT gt 160 GeV MWT 2 was required to be greater

                than 200 GeV and the smallest azimuthal angle difference between baseline jets and EmissT

                gt12

                Chapter 4

                Calculation Tools

                41 Summary

                Figure 41 Calculation Tools

                The simplified models were implemented in Feynrules [4] and the model files importedinto MicrOmegas v425 which was used to calculate the log-likelihoods of the astrophysicalobservables MicrOmegas was modified to allow only s-channel decays in both relic densityand gamma ray calculations Direct detection constraints were easy to implement for thescalar and vector models by coding the analytical expression for the spin independent WIMP-nucleon scattering cross section but due to the momentum dependence of the WIMP-nucleon

                42 FeynRules 55

                scattering cross section of the Majorana fermion model LUXCalc v101 [79] was modifiedfor this model to implement momentum dependent cross sections [85] We used MicrOmegasv425 to calculate relic density values and to produce gamma ray spectra for comparisonwith the Fermi-LAT results Parameter scans were carried out using MultiNest v310 withthe parameter ranges given in Table 51 and using astrophysical constraints only Oncethese scans had converged a random sample of 900 points was chosen (weighted by theposterior probability) and then used to calculate collider constraints using the MadgraphMonte Carlo generator This included comparison with mediator production and decay limitsfrom the LHC experiments and the reproduction of searches for dark matter plus other finalstate objects which required dedicated C++ code that reproduces the cut and count analysesperformed by CMS and ATLAS

                42 FeynRules

                FeynRules is a Mathematicareg package that allows the calculation of Feynman rules inmomentum space for any QFT physics model The user needs to provide FeynRules withminimal information to describe the new model contained in the model-file This informationis then used to calculate a set of Fenman rules associated with the Lagrangian The Feynmanrules calculated by the code can then be used to implement new physics models into otherexisting tools such as Monte Carlo (MC) generators via a set of interfaces developed andmaintained by the MC authors

                Feynrules can be run with a base model = SM (or MSSM) and user defined extensions tothe base model

                43 LUXCalc

                LUXCalc is a utility program for calculating and deriving WIMP-nucleon coupling limitsfrom the recent results of the LUX direct search dark matter experiment The program hasbeen used to derive limits on both spin-dependent and spin-independent WIMP-nucleoncouplings and it was found under standard astrophysical assumptions and common otherassumptions that LUX excludes the entire spin-dependent parameter space consistent witha dark matter interpretation of DAMArsquos anomalous signal and also found good agreementwith published LUX results [79]

                We modified the program for the Majorana fermion model to implement momentumdependent cross -sections [85]

                44 Multinest 56

                44 Multinest

                Multinest is an efficient Bayesian Inference Tool which uses a simultaneous ellipsoidalnested sampling method a Monte Carlo technique aimed at an efficient Bayesian Evidencecalculation with posterior inferences as a by product

                Bayes theorem states that

                Pr(θ |DH) =Pr(D|θ H)Pr(θ |H)

                Pr(D|H) (41)

                Here D is the data and H is the Hypothesis Pr(θ |DH) is the posterior probability ofthe parameters Pr(D|θ H) = L(θ) is the likelihood Pr(θ |H) is the prior and Pr(D|H) isthe evidence

                The evidence Pr(D|H) =int

                Pr(θ |DH)Pr(θ |H)d(θ) =int

                L(θ)Pr(θ |H)d(θ) Thus ifwe define the prior volume dX = Pr(θ |H)d(θ) and

                X(λ ) =int

                L(θ)gtλ

                Pr(θ |H)d(θ) (42)

                where the integral extends over the region of parameter space within the iso likelihoodcontour L(θ) = λ then the evidence integral can be written

                int 10 L (X)dX where L (X) the

                inverse of equation 42 and is a monotonically decreasing function of X (the prior volumewhich is 1 for the total sample space and 0 for a single point)

                Thus if one can evaluate likelihoods L (Xi) where 0 lt Xn lt lt X2 lt X1 lt X0 = 1the evidence can be estimated as sumLiwi where wi = 5(Ximinus1 minusXi+1) Ellipsoidal samplingapproximates the iso likelihood contour by a D dimensional covariance matrix of the currentset of active points New points are selected from the prior within this ellipsoidal bounduntil one is obtained that has a likelihood greater than the likelihood of the removed lowestlikelihood point This method is extended for multimodal sampling where there may bemore than one local maximum by clustering points that are well separated and constructingindividual ellipsoidal bounds for each cluster For more details consult [7]

                The results of the analysis are expressed as a data file containing the sample probability-2loglikelihood followed by the parameters of the model for the particular sampling Theprobabilities are normalised to total 10

                45 Madgraph 57

                45 Madgraph

                Madgraph is a general purpose matrix element calculator and Monte Carlo generator Basedon user-supplied Feynman rules (eg the output of Feynrules) the package calculates the tree-level matrix elements for any processes that the user specifies and generates samples of LHCevents including an accurate simulation of the proton collisions themselves (which requires aknowledge of the parton distribution functions that determine the relative proportion of eachparton inside the nucleus) Hadronisation of final state quarks is not possible in Madgraphand hence the code must be used in conjunction with Pythia which performs a parton shower

                The Madgraph output was automatically showered by Pythia (Pythia describes partonradiation as successive parton emissions using Markov chain techniques based on Sudakovform factors and this description is only formally correct for the soft and collinear emissions)A Madgraph simulation is at the level of the fundamental particles (quarks and leptons)Madgraph generates a limited set of matrix element processes at LO representing hard (highenergy) emissions and passes these to Pythia for showering The final state partons in theMadgraph simulation must be matched to the Pythia final state partons to avoid doublecounting

                The output from Pythia is automatically fed through Delphes where basic collider cutscan be performed (calculation of the numbers of particles and jets that would be observed atdifferent energies and locations within the collider) The output of the Delphes simulation is afile containing reconstructed events that can be analysed using the standard ROOT framework(the most popular data analysis package in particle physics) [86]

                The parameters that gave the best fits to the direct [16] and indirect DM experiments [87]and relic density [59] are ranked by posterior probability from the Multinest [7] output

                The parameters in the Madgraph simulations are stored in files in the cards sub directoryand in particular the run card param card and Delphes card record items relevent to thesimulation

                In the run card the number of events the Beam 1 and 2 total energy in GeV (4000 each)and the type of jet matching were modified (to MLM type matching where the final statepartons in Madgraph are clustered according to the KT jet algorithm to find the equivalentparton shower history of the event there is a minimum cutoff scale for the momentum pT

                given by the xqcut parameter) The maxjetflavour (=5 to include b quarks) and xqcut = 30GeV parameters were also modified

                46 Collider Cuts C++ Code 58

                The param card was modified to specify the masses of the new particles and the autodecay modes The Delphes card is overwritten with either the ATLAS or CMS defaultcards which record tracking efficiencies and isolation momentum and energy resolutionscalorimeter details jet finder and b and τ -tagging parameters However the isolation andtagging in this study were done by the collider cuts C++ code since multiple sets of cutswere applied to the same output from Pythia

                When performing simulations of the three dark matter models it was necessary to definethe proton and jets to contain b quarks as this is not the default in Madgraph The processesgenerated included 1 and 2 jets

                46 Collider Cuts C++ Code

                Because of the volume of simulations required for the collider constraints we used C++code (originally developed by Martin White and Sky French and modified by me) that readsthe ROOT files and outputs the numbers of events that would have been observed under thechosen collider experiments had the model been true for the given parameters Thus the sameoutput files can be used for all of the collider cuts and hence no cuts were performed throughthe Delphes cards

                In order to validate the C++ code we replicated the observed events with the papersdescribing the various collider searches (see Appendix)

                Chapter 5

                Majorana Model Results

                51 Bayesian Scans

                To build some intuition we first examined the constraining effect of each observable on itsown We ran separate scans for each constraint turned on and then for all the constraintstogether In Fig 51 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together The paramaters were scanned over the range in Table 51 below

                Table 51 Scanned Ranges

                Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                Range 0minus103 0minus103 10minus5 minus10 10minus5 minus10 10minus5 minus10

                In Fig51 we can see that the preferred DM mass is centered around 35 GeV primarily dueto the influence of the Fermi-LAT gamma ray constraint whilst the relic density constraintallows a wide range of possible mediator and DM masses Direct detection (LUX) constraintssplit the mass regions into a top left and bottom right region and together these split theoverall mass parameter regions into two regions by mediator mass but with DM mass around35 GeV Later studies using a lower excess gamma ray spectrum tend to increase the preferredDM mass up to 50 GeV [76] and [88]

                The marginalised posterior probabilities for all combinations of parameters taken twoat a time are shown in Fig52 These plots have all constraints (Gamma LUX and RelicDensity) applied together The idea of these graphs is to study the posterior probabilities ofcombinations of two paramaters taken at a time (the off-diagonal graphs) The diagonal plotsshow marginalised histograms of a single parameter at a time- this gives an indication of thespread of each parameter The plots show that mS and λb show a high degree of correlationfor mS gt 100 GeV but there is also a range of values of mS around 35 GeV which are

                51 Bayesian Scans 60

                1 0 1 2 3 4log10(mχ)[GeV]

                1

                0

                1

                2

                3

                4

                log 1

                0(m

                s)[GeV

                ]

                (a) Gamma Only

                1 0 1 2 3 4log10(mχ)[GeV]

                1

                0

                1

                2

                3

                4

                log 1

                0(m

                s)[GeV

                ]

                (b) Relic Density

                1 0 1 2 3 4log10(mχ)[GeV]

                1

                0

                1

                2

                3

                4

                log 1

                0(m

                s)[GeV

                ]

                (c) LUX

                05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                05

                00

                05

                10

                15

                20

                25

                30

                35

                log 1

                0(m

                s)[GeV

                ]

                (d) All Constraints

                Figure 51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether

                51 Bayesian Scans 61

                00

                05

                10

                15

                20

                25

                30

                log 1

                0(m

                χ)[GeV

                ]

                00

                05

                10

                15

                20

                25

                30

                ms[Gev

                ]

                5

                4

                3

                2

                1

                0

                1

                log 1

                0(λ

                t)

                5

                4

                3

                2

                1

                0

                1

                log 1

                0(λ

                b)

                00 05 10 15 20 25 30

                log10(mχ)[GeV]

                5

                4

                3

                2

                1

                0

                1

                log 1

                0(λ

                τ)

                00 05 10 15 20 25 30

                ms[Gev]5 4 3 2 1 0 1

                log10(λt)5 4 3 2 1 0 1

                log10(λb)5 4 3 2 1 0 1

                log10(λτ)

                Figure 52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter

                52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 62

                possible for λb around 001 λt and λτ show a wide range of possible values for two possibleranges of mS lt 35 GeV and greater than 100 GeV mχ shows a narrow range of possiblevalues around 35 GeV for all values of the other parameters

                52 Best fit Gamma Ray Spectrum for the Majorana FermionDM model

                We can see from the Fig51(a) scan that the maximum log-likelihood Fermi-LAT gammaray spectrum on its own shows values that centre around 35 GeV This is not changed byadding a relic density constraint or a direct detection (LUX) constraint

                The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraintsis shown in Fig53 The parameters for this point were

                Table 52 Best Fit Parameters

                Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                Value 3332 49266 0322371 409990 0008106

                10-1 100 101 102

                E(GeV)

                10

                05

                00

                05

                10

                15

                20

                25

                30

                35

                E2dφd

                E(G

                eVc

                m2ss

                r)

                1e 6

                Best fitData

                Figure 53 Gamma Ray Spectrum

                The relic density (Ωh2) at this point has a value of 0135 but the point is separated onlyslightly in posterior probability from points with a value of 01199 with mS lt 10 GeV (iethe posterior probability is almost the same differing by less than 1)

                To better understand the interplay of the different constraints the three graphs in Fig54show the log likelihood surfaces and individual best fit points (best log likelihoods) for the γ

                and Ω scans seperately and then combined The diagrams each choose a pair of couplings(fixing the remaining parameters) and display the 2d surfaces of minuslog(χ2) (proportional to

                52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 63

                the log likelihoods of the coupling combinations) obtained by varying the pair of couplingsThese graphs do not show marginalised posterior probabilities but rather are based on the-2log likelihood calculations directly from MicrOmegas for the given coupling combinationsWe have termed this value as χ2 for convenience and used the log of this value to give abetter range for the colours These log likelihood values are what drive the MultiNest searchof parameter space for the best parameter combinations

                The graphs show a complex interplay of the different constraints resulting often in acompromise parameter combination The arrow on each graph points to the maximum point We have chosen a single representative set of couplings near the best overall fit point for allthe graphs

                52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 64

                00 05 10 15 20 25 30

                log10(mχ)

                00

                05

                10

                15

                20

                25

                30

                log

                10(m

                S)

                Max

                minuslog10(χ2(Γ)) λt = 487 λτ = 024 λb = 0344

                16

                14

                12

                10

                8

                6

                4

                2

                0

                γ Maximum at mχ=416 GeV mS=2188 GeV

                00 05 10 15 20 25 30

                log10(mχ)

                00

                05

                10

                15

                20

                25

                30

                log

                10(m

                S)

                Max

                minuslog10(χ2(Omega)) λt = 487 λτ = 024 λb = 0344

                28

                24

                20

                16

                12

                08

                04

                00

                04

                Ω Maximum at mχ=363 GeV mS=1659 GeV

                00 05 10 15 20 25 30

                log10(mχ)

                00

                05

                10

                15

                20

                25

                30

                log

                10(m

                S)

                Max

                minuslog10(χ2(Both)) λt = 487 λτ = 024 λb = 0344

                16

                14

                12

                10

                8

                6

                4

                2

                0

                Both Maximum at mχ=8317 GeV mS=2884GeV The best fit point for these couplings maybe above below or between the individual bestfit points but will on average be between thepoints

                Figure 54 Plots of log likelihoods by individual and combined constraints Masses in GeV

                53 Collider Constraints 65

                53 Collider Constraints

                531 Mediator Decay

                1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof τ leptons in pp collisions [44] Refer also to Chapter 3 for details

                We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

                The results of the calculations are shown in the graph Fig55 and can be compared withthe limits in the table from the paper [44] in Fig 56 The black line is the observed ATLASlimit

                Figure 55 σ lowastBr(σ rarr ττ) versus Mass of Scalar

                0 200 400 600 800

                mS[GeV]

                10

                5

                0

                log 1

                0(σ

                (bbS

                )lowastB

                (Sgtττ

                ))[pb]

                Observed LimitLikely PointsExcluded Points

                0

                20

                40

                60

                80

                100

                120

                0 5 10 15 20 25 30 35 40 45

                We can see that only one point here exceeded the limit and the points showed a widespread over σ(bbS)lowastBr(σ rarr ττ) by mS space which is consistent with the spread of pointson the MultiNest scan on the mS by λτ plane causing a spread of branching ratios to the τ

                quark It may seem odd that only one point was excluded here but this point was sampledwith a very low probability having high coupling constants of 597 to the τ quark and 278 tothe t quark and only just exceeds the limit As such it can be explained as a random outlier

                53 Collider Constraints 66

                Figure 56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar

                2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                This scan calculates the contributions of b-quark associated Higgs boson productionreproduced from [43] Refer to Chapter 3 for details

                We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

                The cross section times branching ratios to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig57 The black line is the CMSLimit

                Figure 57 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar

                0 200 400 600 800

                mS[GeV]

                15

                10

                5

                0

                5

                log

                10(σ

                (bS

                +X

                )lowastB

                (Sgt

                bb))

                [pb]

                Observed LimitLikely PointsExcluded Points

                0

                20

                40

                60

                80

                100

                120

                0 50 100 150 200 250

                53 Collider Constraints 67

                The results of this scan were compared to the limits in [89] with the plot shown inFig58

                Figure 58 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

                We can see that there are a number of exclusions across the range of mS which occuredfor high values of the coupling constants gt 1 for λbλt λtau - these can be seen also on Fig59The excluded parameter combinations represent less than 2 of the total sampled number

                532 Collider Cuts Analyses

                We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude and the results are shown against theposterior probabilities of the parameter combinations in Fig 59

                The circles in Figure 59 show which points were excluded by the collider cuts as well asthe limits on cross sections times branching ratios to each of τ+τminus and bb

                All the excluded points under the heading Collider Cuts showed exclusion by the Lepstop0experiment (Search for direct pair production of the top quark in all hadronic final states

                53 Collider Constraints 68

                0 1 2 3

                log10(mχ)[GeV]

                0

                1

                2

                3

                log 1

                0(m

                s)[GeV

                ]Collider Cuts

                σ lowastBr(σgt bS+X)

                σ lowastBr(σgt ττ)

                (a) mχ by mS

                6 5 4 3 2 1 0 1 2

                log10(λt)

                0

                1

                2

                3

                log 1

                0(m

                s)[GeV

                ](b) λt by mS

                5 4 3 2 1 0 1

                log10(λb)

                6

                5

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                2

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                1

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                0(λ

                t)

                (c) λb by λt

                5 4 3 2 1 0 1

                log10(λb)

                6

                5

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                2

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                1

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                0(λ

                τ)

                (d) λb by λτ

                6 5 4 3 2 1 0 1 2

                log10(λt)

                6

                5

                4

                3

                2

                1

                0

                1

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                log 1

                0(λ

                τ)

                (e) λt by λτ

                5 4 3 2 1 0 1

                log10(λb)

                0

                1

                2

                3

                log 1

                0(m

                s)[GeV

                ]

                (f) λb by mS

                Figure 59 Excluded points from Collider Cuts and σBranching Ratio

                53 Collider Constraints 69

                [32]) Some of these same points were also excluded by Lepstop2 2bstop CMSTop andATLASMonobjet experiments There were expected events at other points but none weresignificant at the 95 exclusion level These significant exclusions occurred mainly wherethe mass of the mediator was over 100 GeV and higher values of the couplings to b and tquarks but there were a few points with mediator mass around 25 GeV The exclusions withmediator mass around 25 GeV were all accompanied by couplings to t quarks of order 10which is probably non-perturbative

                Looking at the exclusions in Fig 59 we can see that these occur mainly in the regionof higher mediator mass gt 100 GeV and lt 400 GeV These exclusions come from theastrophysical constraints (Gamma) which show a strong correlation between λb and mS andhigh values of λt gt 1 However these mass regions do not overlap at the same couplingcombinations as the high likelihood region The green and white circles correspond to theexcluded points in Figs 57 and 58 respectively

                The remaining regions of highest likelihood can be seen to be where 0001 λb lt 01accompanied by λt lt 0001 and mass of the mediator mS lt 10 GeV or 400 GeV lt mS lt1000GeV with mχ sim 35 GeV

                The plot of λt by λb shows the significant regions do not overlap with the collider cutsand cross section constraints since the latter occur at high values of either coupling while thelikely region has low values of these couplings (also accompanied by low couplings to λτ )

                Chapter 6

                Real Scalar Model Results

                61 Bayesian Scans

                To build some intuition we first examined the constraining effect of each observable on itsown We ran seperate scans for each constraint turned on and then for all the constraintstogether In Figure 61 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together

                In Figure 61 we can see that there are is a narrow region which dominates when allconstraints are taken into account This centers on mχ=9 GeV and a range of values for mS

                from 35 GeV to 400 GeV The region is not the highest likelihood region for the Fermi-LATgamma ray spectrum on its own (as can be seen from the individual plot - there is a narrowband to the left of the highest likelihood band) nor for the LUX likelihood or relic densitybut represents a compromise between the three constraints

                The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig62 The diagonal plots show marginalised histograms of a singleparameter at a time

                61 Bayesian Scans 71

                05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                05

                00

                05

                10

                15

                20

                25

                30

                35

                log 1

                0(m

                s)[GeV

                ]

                (a) Gamma Only

                05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                05

                00

                05

                10

                15

                20

                25

                30

                35

                log 1

                0(m

                s)[GeV

                ]

                (b) Relic Density

                05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                05

                00

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                35

                log 1

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                s)[GeV

                ]

                (c) LUX

                05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                05

                00

                05

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                15

                20

                25

                30

                35

                log 1

                0(m

                s)[GeV

                ]

                (d) All Constraints

                Figure 61 Real Scalar Dark Matter - By Individual Constraint and All Together

                61 Bayesian Scans 72

                00

                05

                10

                15

                20

                25

                30

                log 1

                0(m

                χ)[GeV

                ]

                00

                05

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                30

                ms[Gev

                ]

                5

                4

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                t)

                5

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                0(λ

                b)

                00 05 10 15 20 25 30

                log10(mχ)[GeV]

                5

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                2

                1

                0

                1

                log 1

                0(λ

                τ)

                00 05 10 15 20 25 30

                ms[Gev]5 4 3 2 1 0 1

                log10(λt)5 4 3 2 1 0 1

                log10(λb)5 4 3 2 1 0 1

                log10(λτ)

                Figure 62 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter

                62 Best fit Gamma Ray Spectrum for the Real Scalar DM model 73

                62 Best fit Gamma Ray Spectrum for the Real Scalar DMmodel

                We can see from Fig61 that the maximum log-likelihood Fermi-LAT Gamma ray spectrumon its own shows values of mχ that centre around 35 GeV However the LUX constraintin particular moves the best fit to a point around 9 GeV- the tight LUX constraint onlydisappears for low masses and at that point the proportion of τ+τminus annihilation needs toincrease to fit the constraints

                The gamma ray spectrum at the points of best fit for Fermi-LAT including all constraintsis shown in Fig63 The parameters for these points were respectively and involve decays tomainly τ and b quarks respectively

                Table 61 Best Fit Parameters

                Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                Value 932 3526 000049 0002561 000781

                10-1 100 101 102

                E(GeV)

                10

                05

                00

                05

                10

                15

                20

                25

                30

                35

                E2dφdE

                (GeVc

                m2ss

                r)

                1e 6

                Best fitData

                Figure 63 Gamma Ray Spectrum

                This is not a particularly good fit and it occcurs at a relic density Ωh2 of 335 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

                63 Collider Constraints 74

                63 Collider Constraints

                631 Mediator Decay

                1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44]

                We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

                The results of the scan are shown in the graph Fig64 and can be compared with thelimits in the table from the paper [44] in Fig 56The black line is the observed ATLAS limit

                Figure 64 σ lowastBr(σ rarr ττ) versus Mass of Scalar

                0 200 400 600 800

                mS[GeV]

                8

                6

                4

                2

                0

                2

                4

                log 1

                0(σ

                (bbS

                )lowastB

                (Sgtττ

                ))[pb]

                Observed LimitLikely PointsExcluded Points

                050

                100150200250300350

                0 10 20 30 40 50 60

                We can see that there is a narrow band in the region of interest in the σ(bbS)lowastB(S gt ττ)

                by mS plane due to the high degree of correlation between mS and λτ seen in the likelihoodscan Values for mS gt 300Gev are excluded

                2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinations

                63 Collider Constraints 75

                randomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

                The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig65 The black line is the CMSlimit

                Figure 65 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

                0 200 400 600 800

                mS[GeV]

                8

                6

                4

                2

                0

                2

                4

                log

                10(σ

                (bS

                +X

                )lowastB

                (Sgt

                bb))

                [pb]

                Observed LimitLikely PointsExcluded Points

                050

                100150200250300350

                0 10 20 30 40 50 60

                The results of this scan were compared to the limits in [89] with the plot shown inFig58

                We can see a similar shape for the region of interest to that in Fig64 due to the highcorrelation between mS and λb over the region preferred by astrophysical constraints resultingin branching ratios highly correlated to ms Only one point is excluded with mS gt 500GeV

                632 Collider Cuts Analyses

                We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expected

                63 Collider Constraints 76

                with 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 66

                We can see that collider cuts show some exclusions only for masses mχ below the highlikelihood region but these exclusions do not occur at the same λb and λt combinations asseen from the plot of these two couplings except for a small region at the highest mS andcouplings

                All but one of the collider cut exclusions were excluded by the Lepstop0 experiment (thiswas excluded by 2bstop with high coupling to b quarks of 73) A number of the excludedpoints were also excluded by Lepstop2 2bstop CMSTopDM1L and ATLASMonobJet-(around 30 of the total for each)

                63 Collider Constraints 77

                0 1 2 3

                log10(mχ)[GeV]

                0

                1

                2

                3

                log 1

                0(m

                s)[GeV

                ]Collider Cuts

                σ lowastBr(σgt bS+X)

                σ lowastBr(σgt ττ)

                (a) mχ by mS

                5 4 3 2 1 0 1

                log10(λt)

                0

                1

                2

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                0(m

                s)[GeV

                ](b) λt by mS

                5 4 3 2 1 0 1

                log10(λb)

                5

                4

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                2

                1

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                1

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                0(λ

                t)

                (c) λb by λt

                5 4 3 2 1 0 1

                log10(λb)

                6

                5

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                2

                1

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                log 1

                0(λ

                τ)

                (d) λb by λτ

                5 4 3 2 1 0 1

                log10(λt)

                6

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                1

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                0(λ

                τ)

                (e) λt by λτ

                5 4 3 2 1 0 1

                log10(λb)

                0

                1

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                3

                log 1

                0(m

                s)[GeV

                ]

                (f) λb by mS

                Figure 66 Excluded points from Collider Cuts and σBranching Ratio

                Chapter 7

                Real Vector Dark Matter Results

                71 Bayesian Scans

                In the Figure 71 we can see that the preferred DM mass is centered around 35 GeV primarilydue to the influence of the Fermi-LAT gamma ray constraint with relic density having a widerange of possible mediator and DM masses The result of combining all the constraints is todefine a small region around mχ = 8 GeV and mS = 20 GeV which dominates the posteriorprobability

                The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig72 The diagonal plots show marginalised histograms of a singleparameter at a time

                71 Bayesian Scans 79

                05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                1

                0

                1

                2

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                log 1

                0(m

                s)[GeV

                ]

                (a) Gamma Only

                05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                05

                00

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                log 1

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                s)[GeV

                ]

                (b) Relic Density

                1 0 1 2 3 4log10(mχ)[GeV]

                05

                00

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                log 1

                0(m

                s)[GeV

                ]

                (c) LUX

                05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                05

                00

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                35

                log 1

                0(m

                s)[GeV

                ]

                (d) All Constraints

                Figure 71 Real Vector Dark Matter - By Individual Constraint and All Together

                71 Bayesian Scans 80

                00

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                30

                log 1

                0(m

                χ)[GeV

                ]

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                ms[Gev

                ]

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                t)

                5

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                log 1

                0(λ

                b)

                00 05 10 15 20 25 30

                log10(mχ)[GeV]

                5

                4

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                2

                1

                0

                1

                log 1

                0(λ

                τ)

                00 05 10 15 20 25 30

                ms[Gev]5 4 3 2 1 0 1

                log10(λt)5 4 3 2 1 0 1

                log10(λb)5 4 3 2 1 0 1

                log10(λτ)

                Figure 72 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter

                72 Best fit Gamma Ray Spectrum for the Real Vector DM model 81

                72 Best fit Gamma Ray Spectrum for the Real Vector DMmodel

                The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraints isshown in Fig73

                Table 71 Best Fit Parameters

                Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                Value 8447 20685 0000022 0000746 0002439

                10-1 100 101 102

                E(GeV)

                10

                05

                00

                05

                10

                15

                20

                25

                30

                35

                E2dφdE

                (GeVc

                m2s

                sr)

                1e 6

                Best fitData

                Figure 73 Gamma Ray Spectrum

                This is not a particularly good fit and it occcurs at a relic density Ωh2 of 025 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

                73 Collider Constraints

                731 Mediator Decay

                1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44] (See Chapter 3 for more details)

                We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

                73 Collider Constraints 82

                The results of the scan are shown in the graph Fig74 and can be compared with thelimits in the table from the paper [44] in Fig 56

                Figure 74 σ lowastBr(σ rarr ττ) versus Mass of Scalar

                0 200 400 600 800

                mS[GeV]

                8

                6

                4

                2

                0

                2

                log 1

                0(σ

                (bbS

                )lowastB

                (Sgtττ

                ))[pb]

                Observed LimitLikely PointsExcluded Points

                0100200300400500600700800

                0 20 40 60 80 100120140

                We can see that the region of highest likelihood is limited to mS lt 100 GeV and a narrowrange of couplings to λτ giving a narrow region of interest which falls well below the limit

                2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

                The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig75 The black line is the CMSlimit

                The results of this scan were compared to the limits in [89] with the plot shown in Fig58

                We can see that again the region of interest for mS is below 100 GeV and the points fallwell below the limit

                73 Collider Constraints 83

                Figure 75 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

                0 200 400 600 800

                mS[GeV]

                8

                6

                4

                2

                0

                2

                4

                log

                10(σ

                (bS

                +X

                )lowastB

                (Sgt

                bb))

                [pb]

                Observed LimitLikely PointsExcluded Points

                0100200300400500600700800

                0 20 40 60 80 100120140

                732 Collider Cuts Analyses

                We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 76

                We can see from Figure 76 that like the real scalar model the region of highest likelihoodhas λb and λt lt 001 while the excluded points from collider cuts and cross sections areabove 01 Most of the excluded points have mχ gt than the values in the region of highestlikelihood and all points are widely separated in parameter space from this region

                Once again 89 of excluded points were excluded by Lepstop0 with many also beingexcluded by the other experiments (50 by Lepstop2 and 56 by CMSTopDM1L 35 byLepstop2 and 20 by ATLASMonobjet)

                73 Collider Constraints 84

                0 1 2 3

                log10(mχ)[GeV]

                0

                1

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                3

                log 1

                0(m

                s)[GeV

                ]Collider Cuts

                σ lowastBr(σgt bS+X)

                σ lowastBr(σgt ττ)

                (a) mχ by mS

                5 4 3 2 1 0 1

                log10(λt)

                0

                1

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                0(m

                s)[GeV

                ](b) λt by mS

                5 4 3 2 1 0 1

                log10(λb)

                5

                4

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                1

                0

                1

                log 1

                0(λ

                t)

                (c) λb by λt

                5 4 3 2 1 0 1

                log10(λb)

                5

                4

                3

                2

                1

                0

                1

                log 1

                0(λ

                τ)

                (d) λb by λτ

                5 4 3 2 1 0 1

                log10(λt)

                5

                4

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                2

                1

                0

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                0(λ

                τ)

                (e) λt by λτ

                5 4 3 2 1 0 1

                log10(λb)

                0

                1

                2

                3

                log 1

                0(m

                s)[GeV

                ]

                (f) λb by mS

                Figure 76 Excluded points from Collider Cuts and σBranching Ratio

                Chapter 8

                Conclusion

                We first conducted Bayesian scans over mass and coupling parameters for three simplifiedmodels - Majorana fermion real scalar and real vector using direct detecion (LUX) indirectdetection (GCE γ rays) and relic density constraints to find the regions of highest likelihoodWe ran these scans one constraint at a time and then together to better understand the interplayof the different constraints on best fit and likely parameter regions

                We also developed and tested C++ collider cut software specifically designed to mimickthe cuts applied in a number of ATLAS and CMS experiments that searched for Emiss

                T inconnection with specific beyond the standard model (BSM) theories of Dark Matter

                We then selected parameter combinations by randomly sampling the scanned resultsbased on the posterior probability distribution of the points and applied collider cuts andcross section times branching ratio calculations to these points to see if there would havebeen any sensitivity from the ATLAS and CMS experiments

                We found that the Majorana fermion model showed the best overall fit to all the constraintswith relic density close to the experimental value of 01199 and a low χ2 error on the gammaspectrum fit The preferred mass mχ was around 35 GeV and mS ranging from lt10 GeV to500 GeV The smaller masses give a better fit for relic density and also are well separatedfrom current collider constraints mainly due the the fact that the exclusions occur mainlywith higher couplings to t quarks (greater than 1) which is consistent with the fact that manyof the experiments were targeting dark matter pair production associated with t quark decaysNevertheless there were some exclusions that did fall in the region sampled by randomselection from the posterior probability distribution

                The preferred parameter regions were also consistent with the observation in [20] thatthe Majorana model showed the best overall fit with similar mass constraints We find thatthe couplings to the SM fermions should all be less that 001

                86

                The scalar and vector DM models give slightly less good fits especially to the currentrelic density estimate of 01199 but again the current collider constraints are well separatedfrom the best fit region and in fact none of the points randomly sampled from the posteriordistribution showed any exclusion The best fit parameter combination favours couplings toSM fermions of less than 001

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                Appendix A

                Validation of Calculation Tools

                Lepstop0Search for direct pair production of the top squarkin all hadronic final states in proton=proton collisions atradic

                s=8 TeV with the ATLAS detector [32]

                Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1) mt1=600 GeV mn1=1 GeVWhere t1 is the top squark and n1 is the neutralino of the MSSM

                94

                Table A1 0 Leptons in the final state

                Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 10000 460 460 100Emiss

                T gt 130GeV 8574 395 451 88Leptonveto 6621 307 285 108Emiss

                T gt 150GeV 6316 293 294 100Jet multiplicityand pT 1099 48 51 94|∆φ( jet pmiss

                T )|gt π5 902 40 42 96gt= 2b jets 396 19 18 106τ veto 396 19 18 106mbmin

                T gt 175GeV 270 13 13 105SRA1 79 4 4 104SRA2 70 3 3 93SRA3 67 3 3 97SRA4 52 2 2 91SRC exactly5 jets 1297 50 60 83SRC |∆φ( jet pmiss

                T )|gt π5 1082 50 50 99SRC gt= 2b jets 430 20 20 97SRC τ veto 430 20 20 97SRC |∆φ(bb)|gt 02π 367 17 17 99SRC1 246 12 11 105SRC2 177 9 8 116SRC3 152 8 7 116

                95

                Lepstop1Search for direct top squark pair production infinal states with one isolated lepton jets and missing trans-verse momentum in

                radics = 8 TeV pp collisions using 21 f bminus1

                of ATLAS data[33]

                Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1)(t1 gt b x+1 )(t1 gt b xminus1 )mt1=650 GeV mn1=1 GeV Where t1 is the top squark n1 is the neutralino xplusmn1 are thecharginos

                96

                Table A2 1 Lepton in the Final state

                Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 100000 99990 99990 100Electron(= 1signal) 11558 11557 11785 984 jets(80604025) 5187 5186 6271 83gt= 1btag 4998 4997 5389 93Emiss

                T gt 100GeV [allSRs] 4131 4131 4393 94Emiss

                T radic

                HT gt 5 [allSRs] 4026 4026 4286 94∆φ( jet2 pmiss

                T )gt 08 [allSRs] 3565 3565 3817 93∆φ( jet1Pmiss

                T )gt 08 [not SRtN2] 3446 3446 3723 93Emiss

                T gt 200GeV (SRtN2) 2027 2027 2186 93Emiss

                T radic

                HT gt 13(SRtN2) 1156 1156 1172 99mT gt 140GeV (SRtN2) 839 839 974 86Emiss

                T gt 275GeV (SRtN3) 948 948 965 98Emiss

                T radic

                HT gt 11(SRtN3) 932 932 937 99mT gt 200GeV (SRtN3) 456 456 566 81Emiss

                T gt 150GeV (SRbC1minusSRbC3) 2803 2803 3080 91Emiss

                T radic

                HT gt 7(SRbC1minusSRbC3) 2742 2742 3017 91mT gt 12GeV (SRbC1minusSRbC3) 1917 1917 2347 82Emiss

                T gt 160GeV (SRbC2SRbC3) 1836 1836 2257 81Emiss

                T radic

                HT gt 8(SRbC2SRbC3) 1788 1788 2175 82me f f gt 550GeV (SRbC2) 1741 1741 2042 85me f f gt 700GeV (SRbC3) 1075 1075 1398 77Muon(= 1signal) 11379 11378 11975 954 jets(80604025) 5264 5263 6046 87gt= 1btag 5101 5100 5711 89Emiss

                T gt 100GeV [allSRs] 4199 4199 4770 88Emiss

                T radic

                HT gt 5 [all SRs] 4115 4115 4653 88∆φ( jet2 pmiss

                T )gt 08 [all SRs] 3622 3622 4077 89∆φ( jet1 pmiss

                T )gt 08 [not SRtN2] 3515 3515 3970 89Emiss

                T gt 200GeV (SRtN2) 2136 2136 2534 84Emiss

                T radic

                HT gt 13(SRtN2) 1261 1261 1441 87mT gt 140GeV (SRtN2) 928 928 1033 90Emiss

                T gt 275GeV (SRtN3) 1036 1036 1274 81Emiss

                T radic

                HT gt 11(SRtN3) 1018 1018 1235 82mT gt 200GeV (SRtN3) 497 497 612 81Emiss

                T gt 150GeV (SRbC1minusSRbC3) 2891 2891 3380 86Emiss

                T radic

                HT gt 7(SRbC1minusSRbC3) 2821 2821 3290 86mT gt 120GeV (SRbC1minusSRbC3) 1977 1977 2290 86Emiss

                T gt 160GeV (SRbC2SRbC3) 1913 1913 2220 86Emiss

                T radic

                HT gt 8(SRbC2SRbC3) 1863 1863 2152 87me f f gt 550GeV (SRbC2) 1826 1826 2116 86me f f gt 700GeV (SRbC3) 1131 1131 1452 78

                97

                Lepstop2Search for direct top squark pair production infinal states with two leptons in

                radics =8 TeV pp collisions using

                20 f bminus1 of ATLAS data[83][34]

                Simulated in Madgraph with p p gt t1 t1(t1 gt b w+ n1)(t1 gt b wminusn1) mt1=180 GeV mn1=60 GeV Where t1 is the top squark n1 is the neutralino b1 is the bottom squark

                Table A3 2 Leptons in the final state

                Cut RawMe Scaled ATLAS MeATLAS PercentNoCuts 100000 324308 199000 1632BaselineCuts 3593 11652 7203 1622SF Signal Leptons 1699 5510 5510 1002OSSF Signal Leptons 1632 5293 5233 101mll gt 20Gev 1553 5037 5028 100leadinglepton pT 1479 4797 5048 95|mll minusmZ|gt 20Gev 1015 3292 3484 94∆φmin gt 10 850 2757 2160 128∆φb lt 15 680 2205 1757 126SRM90 [SF ] 13 42 60 70SRM120 [SF ] 0 0 19 0SRM100 +2 jets [SF ] 0 0 17 0SRM110 +2 jets [SF ] 0 0 1 02DF Signal Leptons 1737 5633 7164 792OSDF Signal Leptons 1670 5416 5396 100mll20Gev 1587 5147 5136 100leadinglepton pT 1501 4868 4931 99∆φmin gt 10 1247 4044 3122 130∆φb lt 15 1010 3276 2558 128SRM90 [SF ] 21 68 95 72SRM120 [SF ] 0 0 23 0SRM100 +2 jets [SF ] 0 0 19 0SRM110 +2 jets [SF ] 0 0 14 0

                98

                2bstopSearch for direct third-generation squark pair pro-duction in final states with missing transverse momentumand two b-jets[31]

                Simulated in Madgraph with p p gt b1 b1(b1 gt b n1)(b1 gt b n1)

                SRAmb1=500 GeV mn1=1 GeV SRBmb1=350 GeV mn1=320 GeVWhere b1 is the bottom squark and n1 is the neutralino

                Table A4 2b jets in the final state

                Cut RawMe Scaled ATLAS MeATLAS PercentSRA Nocuts 100000 173800 1738 100SRA MET gt 80 93407 162341 1606 101SRA Leptonveto 93302 162159 1505 108SRA MET gt 150 79449 138082 1323 104SRA Jet selection 10213 17750 119 149SRA mbb gt 200 8344 14502 96 151SRA mCT gt 150 6535 11358 82 139SRA mCT gt 200 5175 8994 67 134SRA mCT gt 250 3793 6592 51 129SRA mCT gt 300 2487 4322 35 123SRB NoCuts 100000 1624100 16241 100SRB leptonveto 11489 186593 4069 46SRB MET gt 250 2530 41090 757 54SRB Jet selection 63 1023 79 130SRB HT 3 lt 50 51 828 52 159

                99

                CMSTop1LSearch for the Production of Dark Matter in As-sociation with Top Quark Pairs in the Single-lepton FinalState in pp collisions[41]

                Simulated in Madgraph with p p gt t t p1 p1

                Where p1= Majorana fermion Table 3 shows Signal efficiencies and cross sections TheM scale was read from Figure 7 for 90 CL

                Table A5 Signal Efficiencies 90 CL on σ limexp[ f b] on pp gt tt +χχ

                Mχ[GeV ] My Signal Efficiency Quoted Efficiency My σ [ f b] Quoted σ [ f b]1 102 101 63 47

                10 132 101 62 4650 143 120 50 38100 170 146 56 32200 185 173 69 27600 265 240 32 19

                1000 320 276 41 17

                Appendix B

                Branching ratio calculations for narrowwidth approximation

                B1 Code obtained from decayspy in Madgraph

                Br(S rarr bb) = (minus24λ2b m2

                b +6λ2b m2

                s

                radicminus4m2

                bm2S +m4

                S)16πm3S

                Br(S rarr tt) = (6λ2t m2

                S minus24λ2t m2

                t

                radicm4

                S minus4ms2m2t )16πm3

                S

                Br(S rarr τ+

                τminus) = (2λ

                2τ m2

                S minus8λ2τ m2

                τ

                radicm4

                S minus4m2Sm2

                τ)16πm3S

                Br(S rarr χχ) = (2λ2χm2

                S

                radicm4

                S minus4m2Sm2

                χ)32πm3S

                (B1)

                Where

                mS is the mass of the scalar mediator

                mχ is the mass of the Dark Matter particle

                mb is the mass of the b quark

                mt is the mass of the t quark

                mτ is the mass of the τ lepton

                The coupling constants λ follow the same pattern

                • TITLE Collider Constraints applied to Simplified Models of Dark Matter fitted to the Fermi-LAT gamma ray excess using Bayesian Techniques
                  • Dedication
                  • Declaration
                  • Acknowledgements
                  • Contents
                  • List of Figures
                  • List of Tables
                    • Chapter 1 Introduction
                    • Chapter 2 Review of Physics
                    • Chapter 3 Fitting Models to the Observables
                    • Chapter 4 Calculation Tools
                    • Chapter 5 Majorana Model Results
                    • Chapter 6 Real Scalar Model Results
                    • Chapter 7 Real Vector Dark Matter Results
                    • Chapter 8 Conclusion
                    • Bibliography
                    • Appendix A Validation of Calculation Tools
                    • Appendix B Branching ratio calculations for narrow width approximation

                  List of Figures ix

                  56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar 6857 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar 6858 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 6959 Excluded points from Collider Cuts and σBranching Ratio 70

                  61 Real Scalar Dark Matter - By Individual Constraint and All Together 7462 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter 7563 Gamma Ray Spectrum 7664 σ lowastBr(σ rarr ττ) versus Mass of Scalar 7765 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 7866 Excluded points from Collider Cuts and σBranching Ratio 80

                  71 Real Vector Dark Matter - By Individual Constraint and All Together 8272 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter 8373 Gamma Ray Spectrum 8474 σ lowastBr(σ rarr ττ) versus Mass of Scalar 8575 σ lowastBr(σ rarr bS+X) versus Mass of Scalar 8676 Excluded points from Collider Cuts and σBranching Ratio 87

                  List of Tables

                  21 Quantum numbers of the Higgs field 1922 Weak Quantum numbers of Lepton and Quarks 21

                  31 Simplified Models 3532 95 CL by Signal Region 4433 Selection criteria common to all signal regions 4534 Selection criteria for signal regions A 4535 Selection criteria for signal regions C 4536 Signal Regions - Lepstop1 4837 Signal Regions Lepstop2 4938 Signal Regions 2bstop 5139 Signal Region ATLASmonobjet 52

                  51 Scanned Ranges 6152 Best Fit Parameters 64

                  61 Best Fit Parameters 76

                  71 Best Fit Parameters 84

                  A1 0 Leptons in the final state 98A2 1 Lepton in the Final state 100A3 2 Leptons in the final state 101A4 2b jets in the final state 102A5 Signal Efficiencies 90 CL on σ lim

                  exp[ f b] on pp gt tt +χχ 103

                  Chapter 1

                  Introduction

                  Dark matter (DM) was first postulated over 80 years ago when Swiss astronomer FritzZwicky observed a discrepancy between the amount of light emitted by a cluster of galaxiesand the total mass contained within the cluster inferred from the relative motion of thosegalaxies by a simple application of the theory of Newtonian gravitation The surprising resultof this observation was that the vast majority of the mass in the cluster did not emit lightwhich was contrary to the expectation that most of the mass would be carried by the starsSince that time further observations over a wide range of scales and experimental techniqueshave continued to point to the same result and refine it Some of these observations and otherevidence are discussed in section 22 We now know with certainty that in the entire Universeall of the matter we know about - stars planets gases and other cosmic objects such as blackholes can only account for less than 5 of the mass that we calculate to be there

                  A recent phenomenon that has received much attention is the significant deviation frombackground expectations of the Fermi Large Area Telescope(Fermi-LAT) gamma ray flux atthe galactic centre [1] A number of astrophysical explanations have been proposed includingmillisecond pulsars of supernova remnants [2] or burst-like continuous events at the galacticcentre but these are unresolved However it has also been noted that the observed Fermi-LATexcess is consistent with the annihilation of dark matter particles which would naturally beconcentrated at the Galactic centre in a manner consistent with the Navarro-Frenk-Whitedistribution of dark matter [3]

                  There are a number of other purely theoretical (particle physics) reasons to postulatethe existence of weakly interacting matter particles that could supply the missing mass andyet remain unobservable Weakly interacting massive particle (WIMPS) have been a majorfocus of Run I and ongoing Run II searches of the Large Hadron Collider (LHC) In spite

                  2

                  of substantial theoretical and experimental effort the microscopic properties of dark matterparticles are still unknown

                  One of the main problems with extracting these properties is the plethora of competingtheoretical models that provide dark matter candidates Another is the very weak interactionof these particles and the lack of experimental observables allowing one to discriminatebetween the competing models

                  A particularly fruitful approach is the simplified model approach which makes minimalassumptions about the DM candidates as extensions to the Standard Model interacting weaklywith ordinary matter through a mediator These can be coded as Lagrangian interaction termsusing tools such as FeynRules [4] and Calchep [5] and used to simulate Dark Matter relicdensities indirect detection expectations as well as direct detection expectations throughMicrOmegas [6] This in turn can be run under a Bayesian inference tool (see below)

                  Using a Bayesian inference tool such as MultiNest [7][8][9] one can scan the parameterspace of a particular model to find regions of highest probability for that model based onobservations of a range of phenomena including direct detection experiments such as ZEPLIN[10] XENON [11] DEAP [12] ARDM [13] DarkSide [14] PandaX [15] and LUX [16]indirect detection - eg Fermi-LAT gamma ray excess [17] Pamela-positron and antiprotonflux [18] and production at colliders MultiNest calculates the evidence value which is theaverage likelihood of the model weighted by prior probability and a posterior distributionwhich is the distribution of the parameter(s) after taking into account the observed databy scanning over a range of parameters based on prior probabilities This can be used tocompare models on a consistent basis The scans also provide information about the best fitparameters for the overall fit as well as for the seperate experimentsobservations

                  The simplified model approach has been taken in a number of papers to study the Fermi-LAT gamma ray excess (see literature review in section 12) and one in particular comparesthese models using a Bayesian inference tool which found that a Majorana fermion DMcandidate interacting via a scalar mediator with arbitrary couplings to third generationfermions showed the best fit [19] A subsequent paper [20] examined the most likelyparameter combinations that simultaneously fit the Planck relic density Fermi-LAT galacticcentre gamma ray AMS-02 positron flux and Pamela antiproton flux data

                  There has not as yet been a thorough study of the collider constraints on DM simplifiedmodels that fit the Fermi-LAT galactic centre excess In this thesis we reproduce the

                  11 Motivation 3

                  previous results of [20] using similar astrophysical constraints and then implement detailedsimulations of collider constraints on DM and mediator production

                  In Section 11 we review the motivation for this study and in Section 12 we considerrecent astronomical data which is consistent with the annihilation of DM particles andreview the literature and searches which attempt to explain it with a range of extensions tothe standard model

                  In Chapter 2 we review the Standard Model of particle physics (SM) the evidence fordark matter and the ATLAS and CMS experiments at the LHC

                  In Chapter 3 we describe the models considered in this paper the observables used inthe Bayesian scans and the collider experiments which provide limits against which theMagraph simulations of the three models can be compared

                  In Chapter 4 we review the calculation tools that have been used in this paper

                  In Chapters 5 6 and 7 we give the results of the Bayesian scans and collider cutsanalyses for each of the models and in Chapter 8 we summarise our conclusions

                  11 Motivation

                  The Fermi gamma ray space telescope has over the past few years detected a gamma raysignal from the inner few degrees around the galactic centre corresponding to a regionseveral hundred parsecs in radius The spectrum and angular distribution are compatible withthat from annihilating dark matter particles

                  A number of papers have recently attempted to explain the Gamma Ray excess in termsof extensions to the Standard Model taking into account constraints from ATLAS and CMScollider experiments direct detection of dark matter colliding off nucleii in well shieldeddetectors (eg LUX) as well as other indirect measurements (astrophysical observations)These papers take the approach of simplified models which describe the contact interactionsof a mediator beween DM and ordinary matter using an effective field theory approach(EFT) Until recently it has been difficult to discriminate between the results of these paperswhich look at different models and produce different statistical results A recent paper[19] Simplified Dark Matter Models Confront the Gamma Ray Excess used a Bayesianinference tool to compare a variety of simplified models using the Bayesian evidence values

                  12 Literature review 4

                  calculated for each model to find the most preferred explanation The motivation for thechosen models is discussed in section 121

                  A subsequent paper [20] Interpreting the Fermi-LAT gamma ray excess in the simplifiedframework studied the most likely model again using Bayesian inference to find the mostlikely parameters for the (most likely) model The data inputs (observables) to the scanwere the Planck measurement of the dark matter relic density [21] Fermi-LAT gamma rayflux data from the galactic center [17] cosmic positron flux data [22] cosmic anti-protonflux data [18] cosmic microwave background data and direct detection data from the LUXexperiment [16] The result of the scan was a posterior probability density function in theparameter space of the model

                  This paper extends the findings in [19] by adding collider constraints to the likelihoodestimations for the three models It is not a simple matter to add collider constraints becauseeach point in parameter space demands a significant amount of computing to calculate theexpected excess amount of observable particles corresponding to the particular experimentgiven the proposed simplified model A sampling method has been employed to calculatethe collider constraints using the probability of the parameters based on the astrophysicalobservables

                  12 Literature review

                  121 Simplified Models

                  A recent paper [23] summarised the requirements and properties that simplified models forLHC dark matter searches should have

                  The general principles are

                  bull Besides the Standard Model (SM) the Lagrangian should contain a DM candidate thatis absolutely stable or lives long enough to escape the LHC detector and a mediatorthat couples the two sectors

                  bull The Lagrangian should contain all terms that are renormalizable consistent withLorentz invariance SM symmetries and DM stability

                  bull Additional interactions should not violate exact and approximate global symmetries ofthe SM

                  12 Literature review 5

                  The examples of models that satisfy these requirements are

                  1 Scalar (or pseudo-scalar) s-channel mediator coupled to fermionic dark matter (Diracor Majorana)

                  2 Higgs portal DM where the DM is a scalar singlet or fermion singlet under the gaugesymmetries of the SM coupling to a scalar boson which mixes with the Higgs (Thefermion singlet is a specific realisation of 1 above)

                  3 DM is a mixture of an electroweak singlet and doublet as in the Minimal Supersym-metric SM (MSSM)

                  4 Vector (or Axial vector) s-channel mediator obtained by extending the SM gaugesymmetry by a new U(1)rsquo which is spontaneously broken such that the mediatoracquires mass

                  5 t-channel flavoured mediator (if the DM particle is a fermion χ the mediator can bea coloured scalar or vector particle φ -eg assuming a scalar mediator a couplingof the form φ χq requires either χ or φ to carry a flavor index to be consistent withminimal flavour violation (MFV) which postulates the flavour changing neutral currentstructure in the SM is preserved

                  Another recent paper [24] summarised the requirements and properties that simplifiedmodels for LHC dark matter searches should have The models studied in this paper arepossible candidates competing with Dirac fermion and complex scalar dark matter andmodels with vector and pseudo-scalar mediators Other models would have an elasticscattering cross section that will remain beyond the reach of direct detection experiments dueto the neutrino floor (the irreducible background produced by neutrino scattering) or arealready close to sensitivity to existing LUX and XENONIT experiments

                  A recent paper [25] Scalar Simplified Models for Dark Matter looks at Dirac fermiondark matter mediated by either a new scalar or pseudo-scalar to SM fermions This paperplaces bounds on the coupling combination gχgφ (coupling of DM to mediator times couplingof mediator to SM fermions) calculated from direct detection indirect detection and relicdensity by the mass of DM and mass of mediator Bounds are also calculated for colliderconstraints (mono-jet searches heavy-flavour searches and collider bounds on the 125 GeVHiggs are used to place limits on the monojet and heavy flavor channels which can betranslated into limits on the Higgs coupling to dark matter and experimental measurements

                  12 Literature review 6

                  of the Higgs width are used to constrain the addition of new channels to Higgs decay) Thepaper models the top loop in a full theory using the Monte Carlo generator MCFM [26]and extends the process to accommodate off-shell mediator production and decay to a darkmatter pair While this paper does not consider the Majorana fermion dark matter model theconstraints on the coupling product may have some relevence to the present study if only asan order of magnitude indicator

                  Constraining dark sectors with monojets and dijets [27] studies a simplified model withDirac fermion dark matter and an axial-vector mediator They use two classes of constraint-searches for DM production in events with large amounts of missing transverse energy inassociation with SM particles (eg monojet events) and direct constraints on the mediatorfrom monojet searches to show that for example where the choice of couplings of themediator to quarks and DM gA

                  q =gAχ=1 all mediator masses in the range 130 GeV ltMR lt3

                  TeV are excluded

                  The paper Constraining Dark Sectors at Colliders Beyond the Effective Theory Ap-proach [28] considers Higgs portal models with scalar pseudo-scalar vector and axial-vectormediators between the SM and the dark sector The paper takes account of loop processesand gluon fusion as well as quark-mediator interactions using MCFM Apart from mediatortype the models are characterised by free parameters of mediator width and mass darkmatter mass and the effective coupling combination gχgφ (coupling of DM to mediator timescoupling of mediator to SM fermions) The conclusion is that LHC searches can proberegions of the parameter space that are not able to be explored in direct detection experiments(since the latter measurements suffer from a velocity suppression of the WIMP-nucleoncross-section)

                  The paper Constraining the Fermi-LAT excess with multi-jet plus EmissT collider searches

                  [29] shows that models with a pseudo-scalar or scalar mediator are best constrained by multi-jet final states with Emiss

                  T The analysis was done using POWHEG-BOX-V2 at LO and usesM2

                  T variables together with multi-jet binning to exploit differences in signal and backgroundin the different regions of phase space to obtain a more stringent set of constraints on DMproduction This paper largely motivated the current work in that it showed that collidersearches that were not traditionally thought of as dark matter searches were neverthelessoften sensitive to dark matter produced in association with heavy quarks The combinationof searches used in this paper reflects this fact

                  12 Literature review 7

                  122 Collider Constraints

                  In order to constrain this model with collider constraints (direct detection limits) we considerthe following papers by the CMS and ATLAS collaborations which describe searches at theLHC

                  ATLAS Experiments

                  bull Search for dark matter in events with heavy quarks and missing transverse momentumin p p collisions with the ATLAS detector [30]

                  bull Search for direct third-generation squark pair production in final states with missingtransverse momentum and two b-jets in

                  radics= 8 TeV pp collisions with the ATLAS

                  detector[31]

                  bull Search for direct pair production of the top squark in all-hadronic final states inprotonndashproton collisions at

                  radics=8 TeV with the ATLAS detector [32]

                  bull Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

                  radic(s)=8TeV pp collisions using 21 f bminus1 of

                  ATLAS data [33]

                  bull Search for direct top squark pair production in final states with two leptons inradic

                  s = 8TeV pp collisions using 20 f bminus1 of ATLAS data [34]

                  bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

                  radics=8 TeV [35]

                  CMS Experiments

                  bull Searches for anomalous tt production in p p collisions atradic

                  s=8 TeV [36]

                  bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

                  radics=8 TeV [37]

                  bull Search for the production of dark matter in association with top-quark pairs in thesingle-lepton final state in proton-proton collisions at

                  radics = 8 TeV [38]

                  bull Search for new physics in monojet events in p p collisions atradic

                  s = 8 TeV(CMS) [39]

                  bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

                  s = 8 TeV [40]

                  12 Literature review 8

                  bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theSingle-lepton Final State in pp collisions at

                  radics=8 TeV [41]

                  bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

                  s=8 TeV [42]

                  bull Search for neutral MSSM Higgs bosons decaying into a pair of bottom quarks [43]

                  bull Search for neutral MSSM Higgs bosons decaying into a pair of τ leptons in p p collision[44]

                  bull Search for dark matter extra dimensions and unparticles in monojet events in proton-proton collisions at

                  radics=8 TeV [45]

                  In particular we have implemented Madgraph simulations of certain data analysesapplying to collider experiments described in papers [32][33][34][31][30][41] Theremaining models were not implemented in Madgraph because of inability to validate themodels against published results in the time available We developed C++ analysis code toimplement the collider cuts after showering with Pythia and basic analysis with Delpheswith output to root files The C++ code allowed multiple experimental cuts to be applied tothe same simulation output The C++ code was first checked against the results of the variousexperiments above and then the simulations were rerun using the three modelsrsquo Lagrangians(implemented in FeynRules) described in this paper

                  Chapter 2

                  Review of Physics

                  21 Standard Model

                  211 Introduction

                  The Standard Model of particle physics (SM) has sometimes been described as the theoryof Almost Everything The SM is a field theory which emerged in the second half of thetwentieth century from quantum mechanics and special relativity which themselves wereonly conceived after 1905 by Einsten Pauli Bohr Dirac and others In the first decades ofthe twenty first century the discovery of the Higgs boson [46][47] completed the pictureand confirmed the last previously unverified part of the SM The SM is a self consistenttheory that describes and classifies all known particles and their interactions (other thangravity) to an astonishing level of accuracy and detail However the theory is incomplete-it does not describe the complete theory of gravitation or account for DM (see below) ordark energy (the accelerating expansion of the universe) Related to these shortcomings aretheoretical deficiencies such as the asymmetry of matter and anti-matter the fact that the SMis inconsistent with general relativity (to the extent that one or both theories break down atspacetime singularities like the Big Bang and black hole event horizons)

                  212 Quantum Mechanics

                  Quantum Mechanics (QM) describes particles as probability waves which can be thoughtof as packets of energy which can be localised or spread through all space depending on how

                  21 Standard Model 10

                  accurately one measures the particlersquos momentum However QM is not itself consistent withspecial relativity It was realised too that because of relativity particles could be created anddestroyed and because of the uncertainty principle the theory should describe the probabilityof particles being created and destroyed on arbitrarily short timescales A quantum fieldtheory (QFT) is a way to describe this probability at every point of space and time

                  213 Field Theory

                  A field theory is described by a mathematical function of space and time called a LagrangianAs in classical mechanics where Lagrangians were introduced by Louis Lagrange at thebeginning of the nineteenth century to summarise dynamical systems the Lagrangian sum-marises the dynamics of space which is permeated by force fields The equations of motionarising from a Lagrangian are determined by finding the contours of the Lagrangian describedby stationary points (ie where the infinitesimal change in the Lagrangian is zero for a smallchange in field) Particles are generally described by singularities in the Lagrangian (wherethe values tend to infinity also called resonances)

                  214 Spin and Statistics

                  It was also realised that different types of particle needed to be represented by different fieldsin the Lagrangian Diracrsquos discovery that particles with spin 12 quantum numbers couldbe represented by a field obtained by effectively taking the square root of the relativisticequation (Klein Gordon equation) was probably the most significant advance in quantumfield theory It required the introduction of a matrix operators called the gamma matricesor Dirac matrices which helped explain the already known fact that spin half particles orfermions obey Fermi-Dirac statistics ie when two spin half particles are exchanged thesign of the wavefunction must change and hence they cannot occupy the same quantum stateunlike bosons with integer spin that can occupy the same quantum state

                  Other particles need to be represented by fields called vector pseudo-vector scalar (as inthe Klein Gordon equation) pseudo-scalar and complex fields A vector field is really fourfields one for each dimension of space plus time and is parameterised by the four gammamatrices γmicro which transform among themselves in a relativistically invariant way (Lorentzinvariance) A pseudo-vector field is a vector field which transforms in the same way but

                  21 Standard Model 11

                  with additional sign change under parity transformations (a simultaneous reversal of the threespatial directions) and is represented by γmicroγ5 A pseudo-scalar is a scalar which changes signunder parity transformations and a complex scalar field is represented by complex numbers

                  215 Feynman Diagrams

                  QFT took another giant leap forward when Richard Feynman realised that the propagationof particles described by the field could be described by expanding the so called S-matrixor scattering matrix derived from calculating the probability of initial states transitioning tofinal states The expansion is similar to a Taylor expansion in mathematics allowing for thespecial commuting properties of elements of the field The Feynman diagrams representingindividual terms in the expansion can be interpreted as interactions between the particlecomponents The S-Matrix (or more correctly the non trivial part called the T-Matrix) isderived using the interaction Hamiltonian of the field The Hamiltonian is an operator whoseeigenvalues or characteristic values give the energy quanta of the field The T-matrix canalso be described by a path integral which is an integral over all of quantum space weightedby the exponential of the Lagrangian This is completely equivalent to the Hamiltonianformulation and shows how the Lagrangian arises naturally out of the Hamiltonian (in factthe Lagrangian is the Legendre transformation of the Hamiltonian)

                  Figure 21 Feynman Diagram of electron interacting with a muon

                  γ

                  eminus

                  e+

                  micro+

                  microminus

                  The diagram Fig21 is a tree diagram (diagrams with closed loops indicate self in-teractions and are higher order terms in the expansion) and shows an electron (eminus) and ananti-muon (micro+) interacting by exchanging a photon

                  21 Standard Model 12

                  216 Gauge Symmetries and Quantum Electrodynamics (QED)

                  The Dirac equation with its gamma matrices ushered in the use of group theory to describethe transformation of fields (because the multiplication table of the gamma matrices isclosed and described by group theory) It also ushered in the idea of symmetries within theLagrangian- for example consider a Lagrangian

                  ψ(ipart minusm)ψ (21)

                  The Lagrangian is invariant under a global U(1) transformation which acts on the fieldsand hence the derivatives as follows

                  ψ rarr eiqαψ ψ rarr ψeminusiqα partmicroψ rarr eiqα

                  partmicroψ (22)

                  where qα is a global phase and α is a continuous parameter

                  A symmetry leads by Noetherrsquos theorem to a conserved current jmicro = qψγmicroψ and is thebasis of charge conservation since partmicro jmicro = 0 and q =

                  intd3x j0(x)

                  By promoting this transformation to a local one (ie α rarr α(x) where x = xmicro are thespace time co-ordinates) the transformations become

                  ψ rarr eiqα(x)ψ ψ rarr ψeminusiqα(x)partmicroψ rarr eiqα(x)(partmicroψ + iq(partmicroα(x))ψ) (23)

                  The derivative transformation introduces an extra term which would spoil the gaugeinvariance This can be solved by introducing a new field the photon field Amicro(x) whichinteracts with the fermion field ψ and transforms under a U(1) transformation

                  Amicro rarr Amicro minuspartmicroα(x) (24)

                  If we replace normal derivatives partmicro with covariant derivatives Dmicro = (partmicro + iqAmicro(x)) in theLagrangian then the extra term is cancelled and we can now see that the covariant derivativetransforms like ψ - under a local U(1) transform ie

                  Dmicro rarr Dmicroψprime = eiqα(x)Dmicroψ (25)

                  The additional term iqAmicro(x) can be interpreted in a geometric sense as a connectionwhich facilitates the parallel transport of vectors along a curve in a parallel consistent manner

                  21 Standard Model 13

                  We must add a gauge-invariant kinetic term to the Lagrangian to describe the dynamicsof the photon field This is given by the electromagnetic field strength tensor

                  Fmicroν = partmicroAν minuspartνAmicro (26)

                  The full Quantum Electrodynamics (QED) Lagrangian is then for electrons with mass mand charge -e

                  LQED = ψ(i Dminusm)ψ minus 14

                  Fmicroν(X)Fmicroν(x) (27)

                  This Lagrangian is Lorentz and U(1) guage invariant The interaction term is

                  Lint =+eψ Aψ = eψγmicro

                  ψAmicro = jmicro

                  EMAmicro (28)

                  where jmicro

                  EM is the electromagnetic four current

                  217 The Standard Electroweak Model

                  The Standard Electroweak (EW) Model was created by Glashow-Weinberg-Salam (also calledthe GWS model) and unifies the electomagnetic and weak interactions of the fundamentalparticles

                  The idea of gauge symmetry was extended to the idea of a non-abelian gauge symmetry(essentially the operators defining the symmetry do not commute and so are represented bymatrices- in this case the Pauli matrices for the SU(2) group) - the gauge group is the directproduct G = SU(2)

                  otimesU(1) It was known that weak interactions were mediated by Wplusmn

                  and Z0 particles where the superscripts are the signs of the electromagnetic charge Thesewere fairly massive but it was also known that through spontaneous symmetry breaking ofa continuous global non-abelian symmetry there would be one massive propagation modefor each spontaneously broken-symmetry generator and massless modes for the unbrokengenerators (the so-called Goldstone bosons)

                  This required an extra field to represent the gauge bosons called Bmicro (a vector fieldhaving the 4 spacetime degrees of freedom) With this identification the covariant derivativebecomes

                  Dmicro = partmicro minus igAmicro τ

                  2minus i

                  gprime

                  2Y Bmicro (29)

                  21 Standard Model 14

                  Here Y is the so called weak hypercharge of the field Bmicro which is a quantum numberdepending on particle (lepton or quark) g and gprime are two independent parameters in theGWS theory and the expression Amicro τ implies a sum over 3 components Aa

                  micro a=123 and thePauli matrices τa

                  This isnrsquot the full story because it had been known since 1956 when Chin Shuing Wuand her collaborators confirmed in an experiment prompted by Lee and Yang that nature isnot ambivalent to so-called parity It had been assumed that the laws of nature would notchange if all spatial coordinates were reversed (a parity transformation) The experimentinvolved the beta decay of cooled 60C atoms (more electrons were emitted in the backwardhemisphere with respect to spin direction of the atoms) This emission involved the weakinteraction and required that the Lagrangian for weak interactions contain terms which aremaximally parity violating- which turn out to be currents of the form

                  ψ(1minus γ5)γmicro

                  ψ (210)

                  The term

                  12(1minus γ

                  5) (211)

                  projects out a left-handed spinor and can be absorbed into the spinor to describe left-handedleptons (and as later discovered quarks) The right handed leptons are blind to the weakinteraction

                  The processes describing left-handed current interactions are shown in Fig 22

                  Analogously to isospin where the proton and neutron are considered two eigenstates ofthe nucleon we postulate that Bmicro is the gauge boson of an SU(2)transformation coupled toweak SU(2) doublets (

                  νe

                  eminus

                  )

                  (ud

                  ) (212)

                  We may now write the weak SU(2) currents as eg

                  jimicro = (ν e)Lγmicro

                  τ i

                  2

                  e

                  )L (213)

                  21 Standard Model 15

                  Figure 22 Weak Interaction Vertices [48]

                  where τ i are the Pauli-spin matrices and the subscript L denotes the left-handed componentof the spinor projected out by Eqn211 The third current corresponding to i=3 is called theneutral current as it doesnrsquot change the charge of the particle involved in the interaction

                  We could postulate that the electromagnetic current is a linear combination of the leftand right elecromagnetic fields eL = 1

                  2(1minus γ5)e and eR = 12(1+ γ5)e

                  jemmicro = eLγmicroQeL + eRγmicroQeR (214)

                  where Q is the electromagnetic operator and eR is a singlet However this is not invariantunder SU(2)L To construct an invariant current we need to use the SU(2)L doublet 212The current which is an SU(2)L invariant U(1) current is

                  jYmicro = (ν e)LγmicroYL

                  e

                  )L+ eRγmicroYReR (215)

                  where hypercharges YL and YR are associated with the U(1)Y symmetry jYmicro is a linearcombination of the electromagnetic current and the weak neutral current thus explaining whythe hypercharge differs for the left and right-handed components since the electromagneticcurrent has the same charge for both components whilst the weak current exists only forleft-handed particles j3

                  micro and the third component of weak isospin T 3 allows us to calculate

                  21 Standard Model 16

                  the relationship Y = 2(QminusT 3) which can be seen by writing jYmicro as a linear combination ofj3micro (the weak neutral current that doesnrsquot change the charge of the particle involved in the

                  interaction) and jemmicro (using the convention that the jYmicro is multiplied by 1

                  2 to match the samefactor implicit in j3

                  micro ) Substituting

                  τ3 =

                  (1 00 minus1

                  )(216)

                  into equation 213 for i=3 and subtracting equation 215 and equating to 12 times equation

                  214

                  we get

                  eLγmicroQeL + eRγmicroQeR minus (νLγmicro

                  12

                  νL minus eLγmicro

                  12

                  eL) =12

                  eRγmicroYReR +12(ν e)LγmicroYL

                  e

                  )L (217)

                  from which we can read out

                  YR = 2QYL = 2Q+1 (218)

                  and T3(eR) = 0 T3(νL) =12 and T3(eL) =

                  12 The latter three identities are implied by

                  the fraction 12 inserted into the definition of equation 213

                  The Lagrangian kinetic terms of the fermions can then be written

                  L =minus14

                  FmicroνFmicroν minus 14

                  GmicroνGmicroν

                  + sumgenerations

                  LL(i D)LL + lR(i D)lR + νR(i D)νR

                  + sumgenerations

                  QL(i D)QL +UR(i D)UR + DR(i D)DR

                  (219)

                  LL are the left-handed lepton doublets QL are the left-handed quark doublets lRνR theright handed lepton singlets URDR the right handed quark singlets

                  The field strength tensors are given by

                  Fmicroν = partmicroAν minuspartνAmicro minus ig[Amicro Aν ] (220)

                  21 Standard Model 17

                  andGmicroν = partmicroBν minuspartνBmicro (221)

                  Bmicro is an abelian U(1) gauge field while Amicro is non-abelian and Fmicroν contains the com-mutator of the field through the action of the covariant derivative [Dmicro Dν ] This equationcontains no mass terms terms for the field Amicro for good reason- this would violate gaugeinvariance We know that the gauge bosons of the weak force have substantial mass and theway that this problem has been solved is through the Higgs mechanism

                  218 Higgs Mechanism

                  To solve the problem of getting gauge invariant mass terms three independent groupspublished papers in 1964 The mechanism was named after Peter Higgs To introduce theconcept consider a complex scalar field φ that permeates all space Consider a potential forthe Higgs field of the form

                  minusmicro2φ

                  daggerφ +λ (φ dagger

                  φ2) (222)

                  which has the form of a mexican hat Fig 23 This potential is still U(1) gauge invariant andcan be added to our Lagrangian together with kinetic+field strength terms to give

                  L = (Dmicroφ)dagger(Dmicroφ)minusmicro2φ

                  daggerφ +λ (φ dagger

                  φ2)minus 1

                  4FmicroνFmicroν (223)

                  It is easily seen that this is invariant to the transformations

                  Amicro rarr Amicro minuspartmicroη(x) (224)

                  φ(x)rarr eieη(x)φ(x) (225)

                  The minimum of this field is not at 0 but in a circular ring around 0 with a vacuum

                  expectation value(vev)radic

                  micro2

                  2λequiv vradic

                  2

                  We can parameterise φ as v+h(x)radic2

                  ei π

                  Fπ where h and π are referred to as the Higgs Bosonand Goldstone Boson respectively and are real scalar fields with no vevs Fπ is a constant

                  21 Standard Model 18

                  Figure 23 Higgs Potential [49]

                  Substituting this back into the Lagrangian 223 we get

                  minus14

                  FmicroνFmicroν minusevAmicropartmicro

                  π+e2v2

                  2AmicroAmicro +

                  12(partmicrohpart

                  microhminus2micro2h2)+

                  12

                  partmicroπpartmicro

                  π+(hπinteractions)(226)

                  This Lagrangian now seems to describe a theory with a photon Amicro of mass = ev a Higgsboson h with mass =

                  radic2micro and a massless Goldstone π

                  However there is a term mixing Amicro and part microπ in this expression and this shows Amicro and π

                  are not independent normal coordinates and we can not conclude that the third term is a massterm This π minusAmicro mixing can be removed by making a gauge transformation in equations224 and 225 making

                  φrarrv+h(x)radic2

                  ei π

                  Fπminusieη(x) (227)

                  and setting πrarr π

                  Fπminus eη(x) = 0 This choosing of a point on the circle for the vev is called

                  spontaneous symmetry breaking much like a vertical pencil falling and pointing horizontallyin the plane This choice of gauge is called unitary gauge The Goldstone π will thencompletely disappear from the theory and one says that the Goldstone has been eaten to givethe photon mass

                  21 Standard Model 19

                  This can be confirmed by counting degrees of freedom before and after spontaneoussymmetry breaking The original Lagrangian had four degrees of freedom- two from thereal massless vector field Amicro (massless photons have only two independent polarisationstates) and two from the complex scalar field (one each from the h and π fields) The newLagrangian has one degree of freedom from the h field and 3 degrees of freedom from themassive real vector field Amicro and the π field has disappeared

                  The actual Higgs field is postulated to be a complex scalar doublet with four degrees offreedom

                  Φ =

                  (φ+

                  φ0

                  )(228)

                  which transforms under SU(2) in the fundamental representation and also transforms underU(1) hypercharge with quantum numbers

                  Table 21 Quantum numbers of the Higgs field

                  T 3 Q Yφ+

                  12 1 1

                  φ0 minus12 1 0

                  We can parameterise the Higgs field in terms of deviations from the vacuum

                  Φ(x) =(

                  η1(x)+ iη2(x)v+σ(x)+ iη3(x)

                  ) (229)

                  It can be shown that the three η fields can be transformed away by working in unitarygauge By analogy with the scalar Higgs field example above the classical action will beminimised for a constant value of the Higgs field Φ = Φ0 for which Φ

                  dagger0Φ0 = v2 This again

                  defines a range of minima and we can pick one arbitarily (this is equivalent to applying anSU(2)timesU(1) transform to the Higgs doublet transforming the top component to 0 and thebottom component to Φ0 = v) which corresponds to a vacuum of zero charge density andlt φ0 gt=+v

                  In this gauge we can write the Higgs doublet as

                  Φ =

                  (φ+

                  φ0

                  )rarr M

                  (0

                  v+ H(x)radic2

                  ) (230)

                  where H(x) is a real Higgs field v is the vacuum expectation and M is an SU(2)timesU(1)matrix The matrix Mdagger converts the isospinor into a down form (an SU(2) rotation) followedby a U(1) rotation to a real number

                  21 Standard Model 20

                  If we consider the Higgs part of the Lagrangian

                  minus14(Fmicroν)

                  2 minus 14(Bmicroν)

                  2 +(DmicroΦ)dagger(DmicroΦ)minusλ (Φdagger

                  Φminus v2)2 (231)

                  Substituting from equation 230 into this and noting that

                  DmicroΦ = partmicroΦminus igW amicro τ

                  aΦminus 1

                  2ig

                  primeBmicroΦ (232)

                  We can express as

                  DmicroΦ = (partmicro minus i2

                  (gA3

                  micro +gprimeBmicro g(A1micro minusA2

                  micro)

                  g(A1micro +A2

                  micro) minusgA3micro +gprimeBmicro

                  ))Φ equiv (partmicro minus i

                  2Amicro)Φ (233)

                  After some calculation the kinetic term is

                  (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

                  14(v+

                  Hradic2)2[A 2]22 (234)

                  where the 22 subscript is the index in the matrix

                  If we defineWplusmn

                  micro =1radic2(A1

                  micro∓iA2micro) (235)

                  then [A 2]22 is given by

                  [A 2]22 =

                  (gprimeBmicro +gA3

                  micro

                  radic2gW+

                  microradic2gWminus

                  micro gprimeBmicro minusgA3micro

                  ) (236)

                  We can now substitute this expression for [A 2]22 into equation 234 and get

                  (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

                  14(v+

                  Hradic2)2(2g2Wminus

                  micro W+micro +(gprimeBmicro minusgA3micro)

                  2) (237)

                  This expression contains mass terms for three massive gauge bosons Wplusmn and the combi-nation of vector fields gprimeBmicro minusgA3

                  micro where note

                  21 Standard Model 21

                  Table 22 Weak Quantum numbers of Lepton and Quarks

                  T 3 Q YνL

                  12 0 -1

                  lminusL minus12 -1 -1

                  νR 0 0 0lminusR 0 -1 -2UL

                  12

                  23

                  13

                  DL minus12 minus1

                  313

                  UR 0 23

                  43

                  DR 0 minus13 minus2

                  3

                  Wminusmicro = (W+

                  micro )dagger equivW 1micro minus iW 2

                  micro (238)

                  Then the mass terms can be written

                  12

                  v2g2|Wmicro |2 +14

                  v2(gprimeBmicro minusgA3micro)

                  2 (239)

                  W+micro and W microminus can be associated with the W boson and its anti-particle and (minusgprimeBmicro +

                  gA3micro) with the Z Boson (after normalisation by

                  radicg2 +(gprime

                  )2) The combination gprimeA3micro +gBmicro

                  is orthogonal to Z and remains massless This can be associated with the Photon (afternormalisation) These predict mass terms for the W and Z bosons mW = vgradic

                  2and mZ =

                  vradic2

                  radicg2 +(gprime

                  )2It is again instructive to count the degrees of freedom before and after the Higgs mech-

                  anism Before we had a complex doublet Φ with four degrees of freedom one masslessB with two degrees of freedom and three massless W a fields each with two each givinga total of twelve degrees of freedom After spontaneous symmetry breaking we have onereal scalar Higgs field with one degree of freedom three massive weak bosons with nineand one massless photon with two again giving twelve degrees of freedom One says thescalar degrees of freedom have been eaten to give the Wplusmn and Z bosons their longitudinalcomponents (ie mass)

                  Table 22 shows T 3 the weak isospin quantum number (which describes how a particletransforms under SU(2)) Y the weak hypercharge (which describes how the particle trans-

                  21 Standard Model 22

                  forms under U(1)) and Q the electric charge of leptons and quarks The former two enterdirectly into the Lagrangian while the latter can be derived as a function of the former two

                  Strictly speaking the right handed neutrino which has no charge may not exist and is notpart of the standard model although some theories postulate a mass for this (hypothetical)particle

                  Writing gauge invariant couplings between ELeRφ with the vacuum expectation ofφ = v (Yukawa matrices λ

                  i ju and λ

                  i jd respectively for the up and down quarks) we get mass

                  terms for the quarks (and similarly for the leptons)

                  Mass terms for quarks minussumi j[(λi jd Qi

                  Lφd jR)+λ

                  i ju εab(Qi

                  L)aφlowastb u j

                  R +hc]

                  Mass terms for leptonsminussumi j[(λi jl Li

                  Lφ l jR)+λ

                  i jν εab(Li

                  L)aφlowastb ν

                  jR +hc]

                  Here hc is the hermitian conjugate of the first terms L and Q are lepton and quarkdoublets l and ν are the lepton singlet and corresponding neutrino d and u are singlet downand up quarks εab is the Levi-Civita (anti-symmetric) tensor

                  If we write the quark generations in a basis that diagonalises the Higgs couplings weget a unitary transformations Uu and Ud for the up and down quarks respectively which iscompounded in the boson current (Udagger

                  u Ud) called the CKM matrix where off-diagonal termsallow mixing between quark generations (this is also called the mass basis as opposed tothe flavour basis) A similar argument can give masses to right handed sterile neutrinos butthese can never be measured However a lepton analogue of the CKM matrix the PNMSmatrix allows left-handed neutrinos to mix between various generations and oscillate betweenflavours as they propagate through space allowing a calculation of mass differences

                  219 Quantum Chromodynamics

                  The final plank in SM theory tackling the strong force Quantum Chromodynamics wasproposed by Gell-Mann and Zweig in 1963 Three generations of quarks were proposedwith charges of 23 for uc and t and -13 for ds and b Initially SU(2) isospin was consideredtransforming u and d states but later irreducible representations of SU(3) were found to fit theexperimental data The need for an additional quantum number was motivated by the fact thatthe lightest excited state of the nucleon consisted of sss (the ∆++) To reconcile the baryon

                  21 Standard Model 23

                  spectrum with spin statistics colour was proposed Quarks transform under the fundamentalor 3 representation of colour symmetry and anti-quarks under 3 The inner product of 3and 3 is an invariant under SU(3) as is a totally anti-symmetric combination ε i jkqiq jqk (andthe anti-quark equivalent) It was discovered though deep inelastic scattering that quarksshowed asymptotic freedom (the binding force grew as the seperation grew) Non-abeliangauge theories showed this same property and colour was identified with the gauge quantumnumbers of the quarks The quanta of the gauge field are called gluons Because the forcesshow asymptotic freedom - ie the interaction becomes weaker at high energies and shorterdistances perturbation theory can be used (ie Feynman diagrams) at these energies but forlonger distances and lower energies lattice simulations must be done

                  2110 Full SM Lagrangian

                  The full SM can be written

                  L =minus14

                  BmicroνBmicroν minus 18

                  tr(FmicroνFmicroν)minus 12

                  tr(GmicroνGmicroν)

                  + sumgenerations

                  (ν eL)σmicro iDmicro

                  (νL

                  eL

                  )+ eRσ

                  micro iDmicroeR + νRσmicro iDmicroνR +hc

                  + sumgenerations

                  (u dL)σmicro iDmicro

                  (uL

                  dL

                  )+ uRσ

                  micro iDmicrouR + dRσmicro iDmicrodR +hc

                  minussumi j[(λ

                  i jl Li

                  Lφ l jR)+λ

                  i jν ε

                  ab(LiL)aφ

                  lowastb ν

                  jR +hc]

                  minussumi j[(λ

                  i jd Qi

                  Lφd jR)+λ

                  i ju ε

                  ab(QiL)aφ

                  lowastb u j

                  R +hc]

                  + (Dmicroφ)dagger(Dmicroφ)minusmh[φφ minus v22]22v2

                  (240)

                  where σ micro are the extended Pauli matrices

                  (1 00 1

                  )

                  (0 11 0

                  )

                  (0 minusii 0

                  )

                  (1 00 minus1

                  )

                  The first line gives the gauge field strength terms the second the lepton dynamical termsthe third the quark dynamical terms the fourth the lepton (including neutrino) mass termsthe fifth the quark mass terms and the sixth the Higgs terms

                  The SU(3) gauge field is given by Gmicroν and which provides the colour quantum numbers

                  21 Standard Model 24

                  Figure 24 Standard Model Particles and Forces [50]

                  Note The covariant derivatives operating on the left-handed leptons contain only theU(1) and SU(2) gauge fields Bmicro Fmicro whereas those operating on left-handed quarks containU(1)SU(2) and SU(3) gauge fields Bmicro Fmicro Gmicro The covariant derivatives operating onright handed electrons and quarks are as above but do not contain the SU(2) gauge fieldsRight handed neutrinos have no gauge fields

                  The sums over i j above are over the different generations of leptons and quarks

                  The particles and forces that emerge from the SM are shown in Fig 24

                  22 Dark Matter 25

                  22 Dark Matter

                  221 Evidence for the existence of dark matter

                  2211 Bullet Cluster of galaxies

                  Most of the mass of the cluster as measured by a gravitational lensing map is concentratedaway from the centre of the object The visible matter (as measured by X-ray emission) ismostly present in the centre of the object The interpretation is that the two galaxy clustershave collided perpendicular to the line of sight with the mostly collisionless dark matterpassing straight through and the visible matter interacting and being trapped in the middleIt was estimated that the ratio of actual to visible matter was about 16 [51]

                  Figure 25 Bullet Cluster [52]

                  2212 Coma Cluster

                  The first evidence for dark matter came in the 1930s when Fritz Zwicky observed the Comacluster of galaxies over 1000 galaxies gravitationally bound to one another He observed thevelocities of individual galaxies in the cluster and estimated the amount of matter based onthe average galaxy He calculated that many of the galaxies should be able to escape fromthe cluster unless the mass was about 10 times higher than estimated

                  22 Dark Matter 26

                  2213 Rotation Curves [53]

                  Since those early observations many spiral galaxies have been observed with the rotationcurves calculated by doppler shift with the result that in all cases the outermost stars arerotating at rates fast enough to escape their host galaxies Indeed the rotation curves arealmost flat indicating large amounts of dark matter rotating with the galaxies and holdingthe galaxies together

                  Figure 26 Galaxy Rotation Curves [54]

                  2214 WIMPS MACHOS

                  The question is if the particles that make up most of the mass of the universe are not baryonswhat are they It is clear that the matter does not interact significantly with radiation so mustbe electrically neutral It has also not lost much of its kinetic energy sufficiently to relax intothe disks of galaxies like ordinary matter Massive particles may survive to the present ifthey carry a conserved additive or multiplicative quantum number This is so even if equalnumbers of particles and anti-particles can annihilate if the number densities become too lowto reduce the density further These particles have been dubbed Weakly interacting MassiveParticles (WIMPS) as they interact with ordinary matter only through a coupling strengthtypically encountered in the weak interaction They are predicted to have frozen out ofthe hot plasma and would give the correct relic density based solely on the weak interactioncross section (sometimes called the WIMP Miracle) Another possibility called MassiveCompact Halo Objects (MACHOS)- are brown dwarf stars Jupiter mass objects or remnantsfrom an earlier generation of star formation

                  22 Dark Matter 27

                  2215 MACHO Collaboration [55]

                  In order to estimate the mass of MACHOS a number of experiments have been conductedusing microlensing which makes use of the fact that if a massive object lies in the line ofsight of a much more distant star the light from the more distant star will be lensed and forma ring around the star It is extemely unlikely that the MACHO will pass directly through theline of sight but if there is a near miss the two images will be seperated by a small angleThe signature for microlensing is a time-symmetric brightening of a star as the MACHOpasses near the line of sight called a light curve One can obtain statistical information aboutthe MACHO masses from the shape and duration of the light curves and a large number ofstars (many million) must be monitored The MACHO Collaboration [55] have monitoredover 11 million stars and less than 20 events were detected The conclusion was that thiscould account for a small addition (perhaps up to 20 ) to the visible mass but not enoughto account for Dark Matter

                  2216 Big Bang Nucleosynthesis (BBN) [56]

                  Another source of evidence for DM comes from the theory of Nucleosynthesis in the fewseconds after the Big Bang In the first few minutes after the big bang the temperatures wereso high that all matter was dissociated but after about 3 minutes the temperature rapidlycooled to about 109 Kelvin Nucleosynthesis could begin when protons and neutrons couldstick together to form Deuterium (Heavy Hydrogen) and isotopes of Helium The ratio ofDeuterium to Hydrogen and Helium is very much a function of the baryon density and alsothe rate of expansion of the universe at the time of Nucleosynthesis There is a small windowwhen the density of Baryons is sufficient to account for the ratios of the various elementsThe rate of expansion of the universe is also a function of the total matter density The ratiosof these lighter elements have been calculated by spectral measurements with Helium some25 of the mass of the universe Deuterium 001 The theory of BBN accounts for theratios of all of the lighter elements if the ratio of Baryon mass to total mass is a few percentsuggesting that the missing matter is some exotic non-baryonic form

                  22 Dark Matter 28

                  2217 Cosmic Microwave Background [57]

                  The discovery of a uniform Microwave Background radiation by Penzias and Wilson in1965 was confirmation of the theory first proposed by Gamow and collaborators that theuniverse should be filled with black body radiation an afterglow of the big bang TheCOBE satelite launched in 1989 confirmed this black body radiation with temperature of2725+-002K with 95 confidence While extremely uniform it found tiny irregularitieswhich could be accounted for by irregularities in the density of the plasma at last scattering(gravitational redshifts or blueshifts called the Sachs Wolfe effect) and temperature andvelocity fluctuations in the plasma The anisotropy of the background has been furtheranalysed by WMAP (the Wilkinson Microwave Anisotropy Probe) launched in 2001 Thefluctuations in the density of the plasma are what account for the seeds of galaxy formationHowever detailed calculations have found that the fluctuations were too small to account forthe observed galaxy distribution without an electrically neutral gravitating matter in additionto ordinary matter that only became electrically neutral at the time of recombination [58]

                  In May 2009 the European Space Agency (ESA) launched the Planck sattelite a spacebased observatory to observe the universe at wavelengths between 03mm and 111 mmbroadly covering the far-infrared microwave and high frequency domains The missionrsquosmain goal was to study the cosmic microwave background across the whole sky at greatersensitivity and resolution than ever before Using the information gathered and the standardmodel of cosmology ESA was able to calculate cosmological parameters including the relicdensity of dark matter remaining after the Big Bang to a high degree of precision withinthe framework of the standard model of cosmology The standard model of cosmology canbe described by a relatively small number of parameters including the density of ordinarymatter dark matter and dark energy the speed of cosmic expansion at the present epoch (alsoknown as the Hubble constant) the geometry of the Universe and the relative amount ofthe primordial fluctuations embedded during inflation on different scales and their amplitude[59]

                  2218 LUX Experiment - Large Underground Xenon experiment [16]

                  The LUX experiment is conducted at the Sanford Underground research facility one mileunderground to minimise the background of cosmic rays The detector is a time projectionchamber allowing a 3D positioning of interactions occurring within its active volume LUX

                  22 Dark Matter 29

                  Figure 27 WMAP Cosmic Microwave Background Fluctuations [58]

                  Figure 28 Dark Matter Interactions [60]

                  uses as its target 368 kiliograms of liquefied ultra pure xenon which is a scintillator Interac-tions inside the xenon will create an amount of light proportional to the energy depositedThat light can be collected on arrays of light detectors sensitive to a single photon

                  Xenon is very pure and the amount of radiation originating within the target is limitedAlso being three times denser than water it can stop radiation originating from outside thedetector The 3D detector focuses on interactions near the center which is a quiet regionwhere rare dark matter interactions might be detected [16]

                  22 Dark Matter 30

                  222 Searches for dark matter

                  2221 Dark Matter Detection

                  Fig 28 illustrates the reactions studied under the different forms of experiment designed todetect DM

                  Direct detection experiments are mainly underground experiments with huge detectorsattempting to detect the recoil of dark matter off ordinary matter such as LUX [16] CDMS-II [61] XENON100 [11] COUPP [62] PICASSO [63] ZEPLIN [10] These involveinteractions between DM and SM particles with final states involving DM and SM particles

                  Indirect detection involves the search for the annihilation of DM particles in distantastrophysical sources where the DM content is expected to be large Example facilitiesinclude space based experiments (EGRET [64] PAMELA [18] FERMI-LAT [65]) balloonflights (ATIC [66] PEBS [67] PPB-BETS [68]) or atmospheric Cherenkov telescopes(HESS [69]MAGIC [70]) and neutrino telescopes (SuperKamiokande [71] IceCube [72]ANTARES [73] KM3NET [74]) Indirect Detection involves DM particles in the initial stateand SM particles in the final state

                  Collider detection involves SM particles in the initial state creating DM particles in thefinal state which can be detected by missing transverse energy and explained by variousmodels extending the SM

                  223 Possible signals of dark matter

                  224 Gamma Ray Excess at the Centre of the Galaxy [65]

                  The center of the galaxy is the most dense and active region and it is no surprise that itproduces intense gamma ray sources When astronomers subtract all known gamma raysources there remains an excess which extends out some 5000 light years from the centerHooper et al find that the annihilation of dark matter particles with a mass between 31 and40 Gev fits the excess to a remarkable degree based on Gamma Ray spectrum distributionand brightness

                  23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 31

                  Figure 29 Gamma Ray Excess from the Milky Way Center [75]

                  23 Background on ATLAS and CMS Experiments at theLarge Hadron collider (LHC)

                  The collider constraints in this paper are based on two of the experiments that are part of theLHC in CERN

                  Figure 210 ATLAS Experiment

                  The LHC accelerates proton beams in opposite directions around a 27 km ring of super-conducting magnets until they are made to collide at an interaction point inside one of the 4experiments placed in the ring An upgrade to the LHC has just been performed providing65 TeV per beam

                  23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 32

                  231 ATLAS Experiment

                  The ATLAS detector consists of concentric cylinders around the interaction point whichmeasure different aspects of the particles that are flung out of the collision point There are 4major parts the Inner Detector the Calorimeters the Muon spectrometer and the magnetsystems

                  2311 Inner Detector

                  The inner detector extends from a few centimeters to 12 meters from the interaction point andmeasures details of the charged particles and their momentum interacting with the materialIf particles seem to originate from a point other than the interaction point this may indicatethat the particle comes from the decay of a hadron with a bottom quark This is used to b-tagcertain particles

                  The inner detector consists of a pixel detector with three concentric layers and threeendcaps containing detecting silicon modules 250 microm thick The pixel detector has over 80million readout channels about 50 percent of the total channels for the experiment All thecomponents are radiation hardened to cope with the extremely high radiation in the proximityof the interaction point

                  The next layer is the semi-conductor layer (SCT) similar in concept to the pixel detectorbut with long narrow strips instead of pixels to cover a larger area It has 4 double layers ofsilicon strips with 63 million readout channels

                  The outermost component of the inner detector is the Transition Radiation Tracker It is acombination of a straw tracker and a transition radiation detector The detecting elementsare drift tubes (straws) each 4mm in diameter and up to 144 cm long Each straw is filledwith gas which becomes ionised when the radiation passes through The straws are held at-1500V with a thin wire down the center This drives negatively ionised atoms to the centercausing a pulse of electricity in the wire This causes a pattern of hit straws allowing the pathof the particles to be determined Between the straws materials with widely varying indicesof refraction cause transition radiation when particles pass through leaving much strongersignals in some straws Transition radiation is strongest for highly relativistic particles (thelightest particles- electrons and positrons) allowing strong signals to be associated with theseparticles

                  23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 33

                  2312 Calorimeters

                  The inner detector is surrounded by a solenoidal magnet of 2 Tesla (there is also an outertoroidal magnet) but surrounding the inner magnet are the calorimeters to measure theenergy of particles by absorbing them There is an inner electromagnetic calorimeter whichmeasures the angle between the detectorrsquos beam axis and the particle (or more preciselypseudo-rapidity) and its angle in the perpendicular plane to a high degree of accuracy Theabsorbing material is lead and stainless steel and the sampling material liquid argon Thehadron calorimeter measures the energy of particles that interact via the strong force Theenergy absorbing material is scintillating steel

                  2313 Muon Specrometer

                  The muon spectrometer surrounds the calorimeters and has a set of 1200 chambers tomeasure the tracks of outgoing muons with high precision using a magnetic field andtriggering chambers Very few particles of other types pass through the calorimeters and aresubsequently measured

                  2314 Magnets

                  The magnets are in two layers the inner solenoidal and outer toroidal which together causecharged particles to curve in the magetic field allowing their momentum to be determinedThe outer toroidal magnet consists of 8 very large air core superconducting magnets whichsupply the field in the muon spectrometer

                  232 CMS Experiment

                  The CMS Detector is similar in design to the ATLAS Detector with a single solenoidalmagnet which is the largest of its type ever built (length 13m and diameter 7m) and allowsthe tracker and calorimeter detectors to be placed inside the coil in a compact way Thestrong magnetic field gives a high resolution for measuring muons electrons and photons ahigh quality central tracking system and a hermetic hadron calorimeter designed to preventhadrons from escaping

                  23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 34

                  Figure 211 CMS Experiment

                  Chapter 3

                  Fitting Models to the Observables

                  31 Simplified Models Considered

                  In order to extend the findings of previous studies [19] to include collider constraints wefirst performed Bayesian scans [7] over three simplified models real scalar dark matter φ Majorana fermion χ and real vector Xmicro The purpose was to obtain the most probable regionsof each model then to use the resulting posterior samples to investigate whether the LHCexperiments have any sensitivity to the preferred regions

                  The three models couple to the mediator with interactions shown in the following table

                  Table 31 Simplified Models

                  Hypothesis real scalar DM Majorana fermion DM real vector DM

                  DM mediator int Lφ sup microφ mφ φ 2S Lχ sup iλχ

                  2 χγ5χS LX sup microX mX2 X microXmicroS

                  The interactions between the mediator and the standard fermions is assumed to be

                  LS sup f f S (31)

                  and in line with minimal flavour violation we only consider third generation fermions- ief = b tτ

                  For the purposes of these scans we consider the following observables

                  32 Observables 36

                  32 Observables

                  321 Dark Matter Abundance

                  We have assumed a single dark matter candidate which freezes out in the early universe witha central value as determined by Planck [21] of

                  ΩDMh2 = 01199plusmn 0031 (32)

                  h is the reduced hubble constant

                  The experimental uncertainty of the background relic density [59] is dominated by atheoretical uncertainty of 50 of the value of the scanned Ωh2 [76] We have added thetheoretical and the experimental uncertainties in quadrature

                  SD =radic(05Ωh2)2 + 00312 (33)

                  This gives a log likelihood of

                  minus05lowast (Ωh2 minus 1199)2

                  SD2 minus log(radic

                  2πSD) (34)

                  322 Gamma Rays from the Galactic Center

                  Assuming that the gamma ray flux excess from the galactic centre observed by Fermi-LAT iscaused by self annihilation of dark matter particles the differential flux

                  d2Φ

                  dEdΩ=

                  lt σv gt8πmχ

                  2 J(ψ)sumf

                  B fdN f

                  γ

                  dE(35)

                  has been calculated using both Micromegas v425 with a Navarro-Frenk-White (NFW) darkmatter galactic distribution function

                  ρ(r) = ρ0(rrs)

                  minusγ

                  (1+ rrs)3minusγ (36)

                  with γ = 126 and an angle of 5 to the galactic centre [19]

                  32 Observables 37

                  Here Φ is the flux dE is the differential energy lt σv gt the velocity averaged annihi-lation cross section of the dark matter across the galactic centre mχ the mass of the darkmatter particle B f =lt σv gt f σv is the annihilation fraction into the f f final state anddN f

                  γ dE is the energy distribution of photons produced in the annihilation channel with thefinal state f f The J factor is described below

                  The angle ψ to the galactic centre determines the normalisation of the gamma ray fluxthrough the factor

                  J(ψ) =int

                  losρ(l2 + r⊙2 minus2lr⊙cosψ)dl (37)

                  where ρ(r) is the dark matter density at distance r from the centre of the galaxy r⊙=85 kpcis the solar distance from the Galactic center ρ0 is set to reproduce local dark matter densityof 03GeVcm3

                  The central values for the gamma ray spectrum were taken from Fig 5 of [77] and thestandard deviations of individual flux points from the error bars on the graph In commonwith [78] we ignore possible segment to segment correlations

                  For the Fermi-LAT log likelihood we assumed each point was distributed with indepen-dent Gaussian likelihood with the experimental error for each data point given by the errorbar presented in Fig5 of [76] for each point We also assumed an equal theoretical errorfor each point which when added to the experimental error in quadrature gave an effectivelog likelihood for each point of minus05lowast (giminusdi)

                  2

                  2lowastσ2i

                  where gi are the calculated values and di theexperimental values and σi the experimental errors

                  323 Direct Detection - LUX

                  The LUX experiment [16] is the most constraining direct search experiment to date for awide range of WIMP models We have used LUXCalc [79] (modified to handle momentumdependent nucleon cross sections) to calculate the likelihood of the expected number ofsignal events for the Majorana fermion model

                  The likelihood function is taken as the Poisson distribution

                  L(mχ σ | N) = P(N | mχ σ) =(b+micro)Neminus(b+micro)

                  N (38)

                  32 Observables 38

                  where b= expected background of 064 and micro = micro(mχσ ) is the expected number of signalevents calculated for a given model

                  micro = MTint

                  infin

                  0dEφ(E)

                  dRdE

                  (E) (39)

                  where MT is the detector mass times time exposure and φ(E) is a global efficiency factorincorporating trigger efficiencies energy resolution and analysis cuts

                  The differential recoil rate of dark matter on nucleii as a function of recoil energy E

                  dRdE

                  =ρX

                  mχmA

                  intdvv f (v)

                  dσASI

                  dER (310)

                  where mA is the nucleon mass f (v) is the dark matter velocity distribution and

                  dσSIA

                  dER= Gχ(q2)

                  4micro2A

                  Emaxπ[Z f χ

                  p +(AminusZ) f χn ]

                  2F2A (q) (311)

                  where Emax = 2micro2Av2mA Gχ(q2) = q2

                  4m2χ

                  [24] is an adjustment for a momentum dependantcross section which applies only to the Majorana model

                  f χ

                  N =λχ

                  2m2SgSNN assuming that the relic density is the central value of 1199 We have

                  implemented a scaling of the direct search cross section in our code by multiplying gSNN by( Ω

                  Ω0)05 as a convenient method of allowing for the fact that the actual relic density enters

                  into the calculation of the cross section as a square

                  FA(q) is the nucleus form factor and

                  microA =mχmA

                  (mχ +mA)(312)

                  is the reduced WIMP-nucleon mass

                  The strength of the dark matter nucleon interaction between dark matter particles andthird generation quarks is

                  gSNN =2

                  27mN fT G sum

                  f=bt

                  λ f

                  m f (313)

                  where fT G = 1minus f NTuminus f N

                  Tdminus fTs and f N

                  Tu= 02 f N

                  Td= 026 fTs = 043 [20]

                  33 Calculations 39

                  For the real scalar and real vector models where the spin independent cross sections arenot momentum dependent we based the likelihood on the LUX 95 CL from [16] using acomplementary error function Li(d|mχ I) = 05lowastCEr f (dminusx(mχ )

                  σ) where x is the LUX limit

                  and d the calculated SI cross section We assumed a LUX error σ of 01lowastd lowastradic

                  2

                  33 Calculations

                  We coded the Lagrangians for the three models in Feynrules [4] Calculation of the expectedobservables (dark matter relic density Fermi-LAT gamma ray spectrum and LUX nucleonscattering amplitudes) were performed using Micromegas v425

                  331 Mediator Decay

                  A number of searches have previously been performed at the LHC which show sensitivity tothe decay of scalar mediators in the form of Higgs bosons (in beyond the standard modeltheories) to bottom quarks top quarks and τ leptons (all third generation fermions) Two inparticular [43 44] provide limits on the cross-sections for the production of the scalar timesthe branching ratios to brsquos and τrsquos In order to check these limits against each of the modelswe ran parameter scans in Madgraph at leading order over the top 900 points from the nestedsampling output from MultiNest [19]

                  The two processes were

                  1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to apair of τ leptons in pp collisions [44]

                  bull generate p p gt b b S where S is the scalar mediator

                  The process measures the decay of a scalar in the presence of b-jets that decay to τ+τminus

                  leptons Fig31 shows the main diagrams contributing to the process The leftmost diagramis a loop process and the centre and right are tree level however we only got Madgraph togenerated the middle diagram since we are interested in τ leptons and b quarks in the finalstate

                  The limit given in [44] only applies in the case of narrow resonances where narrow istaken to mean that the width of the resonance is less than 10 of its mass To test the validity

                  33 Calculations 40

                  Figure 31 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof τ leptons

                  of applying these limits to our models we show in Figures 32 to 34 the percentage widthof the resonance vs mS[GeV ] (pole mass of the scalar mediator) λb and λt (the couplingconstants to the b and t quarks) The probabilities are the posterior probabilities obtainedfrom MultiNest scans of the Majorana model varying the coupling constants and scalarmediator and calculating the decay width of the mediator The results are represented as 2dposterior probability heat maps in the central panels and 1d marginalised distributions on thex and y axes of each graph

                  16 18 20 22 24 26 28 30

                  log10(mS[GeV])

                  001

                  002

                  003

                  004

                  005

                  Widthm

                  S

                  00

                  04

                  08

                  12

                  0 100 200

                  Posterior Probability

                  Figure 32 WidthmS vs mS

                  The WidthmS was lt 01 at almost all points except where the mass of the scalar wasgreater than 1000 GeV λb was greater than 1 or λτ was greater than 1

                  This can be seen from the graphs in Figs 323334

                  33 Calculations 41

                  4 3 2 1 0

                  λb

                  001

                  002

                  003

                  004

                  005

                  WidthmS

                  000

                  015

                  030

                  045

                  0 100 200

                  Posterior Probability

                  Figure 33 WidthmS vs λb

                  5 4 3 2 1 0

                  λτ

                  001

                  002

                  003

                  004

                  005

                  WidthmS

                  000

                  015

                  030

                  045

                  0 100 200

                  Posterior Probability

                  Figure 34 WidthmS vs λτ

                  The decay widths are calculated in Madgraph at leading order and branching fractionscan be calculated to b t τ and χ (DM) The formulae for the calculations can be found indecayspy which is a subroutine of Madgraph which we have documented in Appendix BB1 Using these we calculated the branching ratio to τ+τminus so that the cross section timesbranching ratio can be compared to the limits in [80] shown in Chapters 56 and 7 in Figs5564 and 74 for each model

                  2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                  This scan calculates the contributions of b-quark associated Higgs boson production inthe center and right diagrams of Fig 35 reproduced from [43] The scans did not includePythia and Delphes as only production cross sections were required and the producedparticles were in the final state and required no matching or merging

                  The Madgraph processes were

                  bull generate p p gt b S where S is the scalar mediator

                  bull add process p p gt b S j

                  bull add process p p gt b S

                  33 Calculations 42

                  Figure 35 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof b quarks in the presence of at least one b quark

                  bull add process p p gt b S j

                  The cross section times branching ratios to bb for mediator production in association withat least one b quark is shown for each model in Chapters 56 and 7 in Figs 5765 and 75

                  332 Collider Cuts Analyses

                  We generated two parameter scans with parameter combinations from the Multinest outputwhich were the basis of the plots of Fig 6 in [19] The plots in Fig 6 were marginalisedto two parameters each but the raw Multinest output contained 5 parameters plus the priorprobabilities of each combination normalised to total 10

                  The first scan was for χ produced with b quarks In order to force the mediator to decayto χ only we had χ in the final state and defined jets to include b quarks in Madgraph Thenwe generated the processes

                  bull generate p p gt χ χ j

                  bull add process p p gt χ χ j j

                  Jet matching was on

                  The second scan was for t quarks produced in the final state

                  bull generate p p gt χ χ tt

                  33 Calculations 43

                  No jet matching was required because we did not mix final state jet multiplicities at thematrix element level

                  The outputs from these two processes were normalised to 21 f bminus1 and combined

                  The results of these scans prior to collider cuts calculations are shown in Figs 5161and 71 These show the posterior probabilities of the various parameter combinations withdifferent constraints applied seperately and then together A more detailed commentary isgiven in each of the sections

                  We then sampled 900 points based on posterior probability from these scans and ran thesethrough Madgraph v425 then applied collider cuts with the C++ code using the limits fromthe following six ATLAS and CMS searches shown in Table32 and described below to findany exclusions

                  333 Description of Collider Cuts Analyses

                  In the following all masses and energies are in GeV and angles in radians unless specificallystated

                  3331 Lepstop0

                  Search for direct pair production of the top squark in all-hadronic final states in pro-tonndashproton collisions at

                  radics=8 TeV with the ATLAS detector[32]

                  This analysis searched for direct pair production of the supersymmetric partner to thetop quark using integrated luminosity of 201 f bminus1 The top squark was assumed to de-cay via t rarr t χ0

                  1 or t rarr bχ01 or t rarr bχ

                  plusmn1 rarr bW (lowast)χ1

                  0 where χ01 (χ

                  plusmn1 ) denotes the lightest

                  neutralino(chargino) in supersymmetric models The search targets a fully hadronic finalstate with four or more jets and large missing momentum The production of a mediator inassociation with top quarks followed by an invisible mediator decay to dark matter particleshas a similar final state to that considered in the ATLAS stop searches Thus we expect theATLAS reach to have sensitivity to the three dark matter models considered in this thesis

                  The collider cuts program mimicked the object reconstruction through jetlepton overlapremoval and selected signal electrons muons and τ particles with pT gt 10 GeV and

                  33 Calculations 44

                  Table 32 95 CL by Signal Region

                  Experiment Region Number

                  Lepstop0

                  SRA1 66SRA2 57SRA3 67SRA4 65SRC1 157SRC2 124SRC3 80

                  Lepstop1

                  SRtN1-1 857SRtN1-2 498SRtN1-3 389SRtN2 107SRtN3 85SRbC1 832SRbC2 195SRbC3 76

                  Lepstop2

                  L90 740L100 56L110 90L120 170

                  2bstop

                  SRA 857SRB 270SRA150 380SRA200 260SRA250 90SRA300 75SRA350 52

                  CMSTopDM1L SRA 1385

                  ATLASMonobjetSR1 1240SR2 790

                  33 Calculations 45

                  |η | lt 24724 and 247 respectively (for eminusmicrominusτ) Jets were selected with pT gt 20 GeVand |η |lt 45 b-jets were selected if they were tagged by random selection lt 70 and had|η |lt 25 and pT gt 20 GeV

                  These jets and leptons were further managed for overlap removal within ∆R lt 04 withnon b-jets removed if ∆R lt 02

                  The b-jets and non b-jets were further selected as signal jets if pT gt 35 GeV and |η |lt 28

                  These signal jets electrons and muons were finally binned according to the signal regionsin Tables 333435 below

                  Table 33 Selection criteria common to all signal regions

                  Trigger EmissT

                  Nlep 0b-tagged jets ⩾ 2

                  EmissT 150 GeV

                  |∆φ( jet pmissT )| gtπ5

                  mbminT gt175 GeV

                  Table 34 Selection criteria for signal regions A

                  SRA1 SRA2 SRA3 SRA4anti-kt R=04 jets ⩾ 6 pT gt 808035353535 GeV

                  m0b j j lt 225 GeV [50250] GeV

                  m1b j j lt 225 GeV [50400] GeV

                  min( jet i pmissT ) - gt50 GeV

                  τ veto yesEmiss

                  T gt150 GeV gt250 GeV gt300 GeV gt 350 GeV

                  Table 35 Selection criteria for signal regions C

                  SRC1 SRC2 SRC3anti-kt R=04 jets ⩾ 5 pT gt 8080353535 GeV

                  |∆φ(bb)| gt02 π

                  mbminT gt185 GeV gt200 GeV gt200 GeV

                  mbmaxT gt205 GeV gt290 GeV gt325 GeV

                  τ veto yesEmiss

                  T gt160 GeV gt160 GeV gt215 GeV

                  wherembmin

                  T =radic

                  2pbt Emiss

                  T [1minus cos∆φ(pbT pmiss

                  T )]gt 175 (314)

                  33 Calculations 46

                  and pbT is calculated for the b-tagged jet with the smallest ∆φ to the pmiss

                  T direction andmbmax

                  T is calculated with the pbT for the b-tagged jet with the largest ∆φ to the pmiss

                  T direction

                  m0b j j is obtained by first selecting the two jets with the highest b-tagging weight- from

                  the remaining jets the two jets closest in the η minusφ plane are combined to form a W bosoncandidate this candidate is then combined with the closest of the 2 b-tagged jets in the η minusφ

                  plane to form the first top candidate with mass m0b j j A second W boson candidate is formed

                  by repeating the procedure with the remaining jets this candidate is combined with thesecond of the selected b-tagged jets to form the second top candidate with mass m1

                  b j j Thecuts shown in Appendix1 Table A1 show the cumulative effect of the individual cuts appliedto the data the signal region cuts and the cumulative effects of individual cuts in SignalRegion C

                  3332 Lepstop1

                  Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

                  radics=8 TeV pp collisions using 21 f bminus1 of

                  ATLAS data[33]

                  The stop decay modes considered were those to a top quark and the lightest supersym-metric particle (LSP) as well as a bottom quark and the lightest chargino where the charginodecays to the LSP by emitting a W boson We should expect the three dark matter models inthis paper with couplings to t and b quarks to show some visibility in this search

                  The collider cuts program mimicked the object construction by selecting electrons andmuons with pT gt 10 GeV and |η |lt 247 and 24 respectively and jets with pT gt 20 GeVand |η |lt 10 b-jets with |η |lt 25 and pT gt 25 GeV were selected with random efficiency75 and mistag rate 2 (where non b-jets were mis-classified) We then removed any jetwithin ∆R = 02 of an electron and non overlapping jets with |η |lt 28 were used to removeany electrons and muons within ∆R = 04 of these while non-overlapping jets with |η |lt 25and pT gt 25 GeV were classified as signal jets Non overlapping electrons and muons withpT gt 25 GeV were classified as signal electrons and muons

                  Signal selection defined six regions to optimize sensitivity for different stop and LSPmasses as well as sensitivity to two considered stop decay scenarios Three signal regionslabelled SRbC 1-3 where bC is a moniker for b+Chargino and 3 regions labelled SRtN 1-3where tN is a moniker for t+Neutralino were optimised for the scenarios where t1 decaysto the respective particles indicated by the labels Increasing label numbers correspond toincreasingly strict selection criteria

                  33 Calculations 47

                  The dominant background in all signal regions arises from dileptonic tt events All threeSRtN selectons impose a requirement on the 3-jet mass m j j j of the hadronically decayingtop quark to specifically reject the tt background where both W bosons from the top quarkdecay leponically

                  For increasing stop mass and increasing difference between the stop and the LSP therequirents are tightened on Emiss

                  T on the ratio EmissT

                  radicHT where HT is the scalar sum of the

                  momenta of the four selected jets and also tightened on mT

                  To further reduce the dileptonic tt background for an event characterised by two one stepdecay chains a and b each producing a missing particle C use is made of the variable

                  mT 2 =min

                  pCTa + pC

                  T b = pmissT

                  [max(mTamtb)] (315)

                  where mTi is the transverse mass 1 of branch i for a given hypothetical allocation (pCTa pC

                  T b)

                  of the missing particle momenta Eqn 315 is extended for more general decay chains byamT 2 [81] which is an asymmetric form of mT 2 in which the missing particle is the W bosonin the branch for the lost lepton and the neutrino is the missing particle for the branch withthe observed charged lepton

                  ∆φ( jet12 pmissT ) the azimuthal angle between leading or subleading jets and missing mo-

                  mentum is used to suppress the background from mostly multijet events with mis-measuredEmiss

                  T

                  Events with four or more jets are selected where the jets must satisfy |η | lt 25 and pT gt80604025 GeV respectively

                  These signal particles were then classified according to the cutflows analysis in Fig36below and compared with the paper in Appendix1 Table A2

                  3333 Lepstop2

                  Search for direct top squark pair production in final states with two leptons in p pcollisions at

                  radics=8TeV with the ATLAS detector[34]

                  Three different analysis strategies are used to search for t1 pair production Two targett1 rarr b+ χ

                  plusmn1 decay and the three body t1 rarr bW χ0

                  1 decay via an off-shell top quark whilst

                  1The transverse mass is defined as m2T = 2plep

                  T EmissT (1minus cos(∆φ)) of branch i where ∆φ is the azimuthal

                  angle between the lepton and the missing transverse momentum

                  33 Calculations 48

                  Table 36 Signal Regions - Lepstop1

                  Requirement SRtN2 SRtN3 SRbC1 SRbC2 SRbC3∆φ( jet1 pmiss

                  t )gt - 08 08 08 08∆φ( jet2 pmiss

                  T )gt 08 08 08 08 08Emiss

                  T [GeV ]gt 200 275 150 160 160Emiss

                  T radic

                  HT [GeV12 ]gt 13 11 7 8 8

                  mT [GeV ]gt 140 200 120 120 120me f f [GeV ]gt - - - 550 700amT 2[GeV ]gt 170 175 - 175 200mT

                  T 2[GeV ]gt - 80 - - -m j j j Yes Yes - - -

                  one targets t1 rarr t + χ01 to an on shell quark decay mode The analysis is arranged to look

                  at complementary mass splittings between χplusmn1 and χ0

                  1 smaller and larger than the W bosonmass The smaller splitting provides sensitivity to t1 three body decay

                  Object reconstruction was carried out by selecting baseline electronsmuons and τrsquos withpT gt 10 GeV and |η |lt 24724 and 247 respectively Baseline jets and b-jets were selectedwith pT gt 20 GeV and |η |lt 25 and b-jets with 70 efficiency Baseline jets were removedif within ∆R lt 02 of an electron and electrons and muons were removed if within ∆R lt 04of surviving jets Signal electrons required pT gt 10 GeV and signal jets pT gt 20 GeV and|η |lt 25

                  The analysis made use of the variable mT 2 [82] the stransverse mass used to mea-sure the masses of pair-produced semi-invisibly decaying heavy particles It is definedas mT 2(pT1 pT2qT ) =

                  minqT1+qT2=qT

                  max[mT (pT1qT1)mT (pT2qT2)] where mT indicatesthe transverse mass pT1 and pT2 are vectors and qT = qT1 + qT2 The minimisation isperformed over all possible decompositions of qT

                  Events were selected if they passed either a single-electron a single-muon a double-electron a double-muon or an electron-muon trigger Events are required to have exactlytwo opposite sign leptons and at least one of the leptons or muons must have a momentumgreater than 25 GeV and the invariant mass (rest mass) of the two leptons is required to begreater than 20 GeV In order to reduce the number of background events containing twoleptons produced by the on-shell decay of the Z boson the invariant mas must be outside711-111 GeV range

                  Two additional selections are applied to reduce the number of background events withhigh mT 2 arising from events with large Emiss

                  T due to mismeasured jets ∆φb lt 15 and∆φ gt 10 The quantity ∆φb is the azimuthal angle between pll

                  T b = pmissT + pl1

                  T +Pl2T The

                  33 Calculations 49

                  vector pllT b is the opposite of the vector sum of all the hadronic activity in the event For WW

                  and tt backgrounds it measures the transverse boost of the WW system and for the signalthe transverse boost of the chargino-chargino system The ∆φ variable is the azimuthal anglebetween the pmiss

                  T vector and the direction of the closest jet

                  By requiring mT 2 gt 40 GeV the tt background is the dominant background in both thesame-flavoured (SF) and different-flavoured (DF) events

                  Four signal regions are then defined as shown in Table 37 SR M90 has the loosestselection requiring MT 2gt 90 GeV and no additional requirement on jets and providessensitivity to scenarios with a small difference between the masses of the top-squark andchargino SR M110 and M120 have a loose selection on jets requiring two jets with pT

                  gt 20 GeV and mT 2 to be greater than 110 GeV and 120 GeV respectively They providesensitivity to scenarios with small to moderate difference between the invariant masses of thestop quark and the charginos SR M100 has a tighter jet slection and provides sensitivity toscenarios with a large mass difference between the both stop quark and the chargino and thestop quark and the neutralino

                  The analysis cut regions are summarised in Fig37 below

                  Table 37 Signal Regions Lepstop2

                  SR M90 M100 M110 M120pT leading lepton gt 25 GeV

                  ∆φ(pmissT closest jet) gt10

                  ∆φ(pmissT pll

                  T b) lt15mT 2 gt90 GeV gt100 GeV gt110 GeV gt120 GeV

                  pT leading jet no selection gt100 GeV gt20 GeV gt20 GeVpT second jet no selection gt50 GeV gt20 GeV gt20 GeV

                  To validate the code we ran the cuts shown in Appendix 1 Table A3 with the givenMadgraph simulation and compared to [83] It was not clear what baseline cuts had beenemployed in the ATLAS paper so our results were normalised to the 2 SF electrons beforecomparison Some of the ATLAS cut numbers were less than 3 and we did not produce anyevents passing these cuts but clearly the margin of error at this level was greater than thenumber of ATLAS events

                  33 Calculations 50

                  3334 2bstop

                  Search for direct third-generation squark pair production in final states with miss-ing transverse momentum and two b-jets in

                  radics= 8 TeV pp collisions with the ATLAS

                  detector[31]

                  Two sets of scenarios are considered In the first the coloured sbottom is the onlyproduction process decaying via b1 rarr bχ0

                  1 In the second the stop is the only productionprocess decaying via t1 rarr t χplusmn

                  1 and the lightest chargino (χplusmn1 ) decays via a virtual W boson

                  into the final state χ01 f f prime Both scenarios are characterised by the presence of two jets

                  resulting from the hadronisation of b quarks and large missing transverse momentum Theproduction of a mediator in association with top quarks followed by an invisible mediatordecay to dark matter particles has a similar final state as that considered in the ATLAS stopsearches Thus we expect the ATLAS reach to have sensitivity to the three dark mattermodels considered in this thesis

                  The collider cuts program mimicked object reconstruction by selecting baseline electronswith pT gt 7 GeV and |η |lt 247 and muons with pT gt 6 GeV and |η |lt 24 Baseline jetswere selected with pT gt 20 GeV and |η |lt 28 We then removed any jets within ∆R lt 02of an electron and any electron within ∆R lt 04 of surviving jets Signal electrons andmuons were required to have pT gt 25 GeV b-jets were selected with 60 tagging efficiencyand |η |lt 25 and pT gt 20 GeV

                  Two sets of signal regions are defined to provide sensitivity to the kinematic topologiesassociated with the different mass splittings between the sbottom or the stop and the neu-tralino In all cases the presence of at least one primary vertex (with at least five tracks withpT gt 04 GeV) is required Events are selected with Emiss

                  T gt 150 GeV and no electrons ormuons identified in the final state For signal region selections jets with |η | lt 28 are orderedaccording to their pT and 2 jets are required to be b-tagged

                  The variables are defined as follows

                  bull ∆φmin is the minimal azimuthal angle ∆φ between any of the three leading jets andpmiss

                  T

                  bull me f f (k) = sumki=1(p jet

                  T )i +EmissT where the index refers to the pT ordered list of jets

                  bull HT3 is defined as the scalar sum of the pT of the n leading pT ordered jets withoutincluding the 3 leading jets HT3 = sum

                  ni=4(p jet

                  T )i

                  bull mbb is the invariant mass of the two b-tagged jets in the event

                  33 Calculations 51

                  bull mCT is the contransverse mass [82] and is a kinematical variable that can be usedto measure the masses of pair produced semi-invisibly decaying particles For twoidentical decays of heavy particles into two invisible particles v1 and v2 and twoinvisible particles mCT is defined as m2

                  CT (v1v2) = [ET (v1)+ET (v2)]2 minus [pT (v1)minus

                  pT (v2)]2 where ET =

                  radicp2

                  T +m2 The the contransverse mass is invariant under equaland opposite boosts of the parent particles in the transverse plane and is bounded fromabove by an analytical expression of the particle masses For production of sbottompairs the bound is given by mmax

                  CT =m2(b)minusm2(χ0

                  1 )

                  m(b) and for tt events the bound is 135

                  GeV A similar equation can be written for stop pairs in term of the stop and charginomasses

                  A definition of the signal regions is given in the Table38

                  Table 38 Signal Regions 2bstop

                  Description SRA SRBEvent cleaning All signal regions

                  Lepton veto No emicro after overlap removal with pT gt7(6) GeV for e(micro)Emiss

                  T gt 150 GeV gt 250 GeVLeading jet pT ( j1) gt 130 GeV gt 150 GeVSecond jet pT ( j2) gt 50 GeV gt 30 GeVThird jet pT ( j3) veto if gt 50 GeV gt 30 GeV

                  ∆φ(pmissT j1) - gt 25

                  b-tagging leading 2 jets(pT gt 50 GeV |η | lt 25)

                  2nd- and 3rd- leading jets(pT gt 30 GeV|η | lt 25)

                  ∆φmin gt 04 gt 04Emiss

                  T me f f (k) EmissT me f f (2) gt 025 Emiss

                  T me f f (3) gt 04mCT gt 150200250300350 GeV -HT3 - lt 50 GeVmbb gt200 GeV -

                  The analysis cuts are summarised in Table A4 of Appendix 1

                  3335 ATLASMonobjet

                  Search for dark matter in events with heavy quarks and missing transverse momentumin pp collisions with the ATLAS detector[30]

                  Events were selected with high missing transverse momentum when produced in associa-tion with one or more jets identified as containing b quarks Final states with top quarks wereselected by requiring a high jet multiplicity and in some cases a single lepton The research

                  33 Calculations 52

                  studied effective contact operators in the form of a scalar operator Oscalar = sumqmqMN

                  lowastqqχχ

                  where MNlowast is the effective mass of the coupling and the quark and DM fields are denoted by

                  q and χ respectively These are the equivalent of the simplified models in this paper with themediator integrated out

                  Object reconstruction selected baseline electrons muons jets and τrsquos with pT gt 20 GeVand |η |lt 2525 and 45 respectively Signal jets were selected by removing any jets within∆R lt 02 of an electron and electrons and muons were selected by removing them if theywere within ∆R lt 04 of surviving jets b-jets were tagged with an efficiency of 60

                  Only signal regions SR1 and SR2 were analysed

                  The analysis split events into signal regions SR1 and SR2 according to the followingCriteria

                  Table 39 Signal Region ATLASmonobjet

                  Cut SR1 SR2Jet multiplicity n j 1minus2 3minus4

                  bminusjet multiplicity nb gt 0 (60 eff) gt 0 (60 eff)Lepton multiplicity nl 0 0

                  EmissT gt300 GeV gt200 GeV

                  Jet kinematics pb1T gt100 GeV pb1

                  T gt100 GeV p j2T gt100 (60) GeV

                  ∆φ( j1EmissT ) gt 10 i = 12 gt 10 i = 14

                  Where p jiT (pbi

                  T ) represent the transverse momentum of the i-th jet b-tagged jet respec-tively If there is a second b-tagged jet the pT must be greater than 60 GeV

                  3336 CMSTop1L

                  Search for top-squark pair production in the single-lepton final state in pp collisionsat

                  radics=8 TeV[41]

                  This search was for pair production of top squarks in events with a single isolatedelectron or muon jets large missing transverse momentum and large transverse mass

                  (MT =radic

                  2EmissT pl

                  T (1minus cos(∆φ))) where plT is the leptonrsquos transverse momentum and ∆(φ)

                  is the difference between the azimuthal angles of the lepton and EmissT The 3 models

                  considered should have some visibility at certain parameter combinations in this regionThe search focuses on two decay modes of the top squark leading to the two processespp rarr t tlowast rarr tt χ0

                  1 χ01 rarr bbW+Wminusχ0

                  1 χ01 and pp rarr t tlowast rarr bbχ

                  +1 χ

                  minus1 rarr bbW+Wminusχ0

                  1 χ01 The

                  33 Calculations 53

                  lightest neutralino χ01 is considered the lightest supersymmetric particle and escapes without

                  detection The analysis is based on events where one of the W bosons decays leptonicallyand the other hadronically resulting in one isolated lepton and 4 jets two of which originatefrom b quarks The two neutralinos and the neutrino from the W decay can result in largemissing transverse momentom The kinematical end point of MT is lt the mass of the Wboson However for signal events the presence of LSPs in the final state allows MT to exceedthe mass of the W boson The search is therefore for events with large MT The dominantbackground with large MT arises from tt decays where W bosons decay leptonically withone of the leptons not being identified

                  To reduce the dominant tt background use was made of the MWT 2 variable defined as

                  the minimum mother particle mass compatible with the transverse momenta and mass shellconstraints [84]

                  Object reconstruction required electrons and muons to have pT gt 30 and 25 GeV respand |η |lt 25 and 21 respectively Baseline jets were required to have pT gt 30 and |η |lt 4and b-jets were tagged with an efficiency of 75 for |η |lt 24

                  Events were chosen if they contained one isolated electron or muon at least 4 jets and atleast one b-tagged jet and Emiss

                  T gt 100 GeV MT gt 160 GeV MWT 2 was required to be greater

                  than 200 GeV and the smallest azimuthal angle difference between baseline jets and EmissT

                  gt12

                  Chapter 4

                  Calculation Tools

                  41 Summary

                  Figure 41 Calculation Tools

                  The simplified models were implemented in Feynrules [4] and the model files importedinto MicrOmegas v425 which was used to calculate the log-likelihoods of the astrophysicalobservables MicrOmegas was modified to allow only s-channel decays in both relic densityand gamma ray calculations Direct detection constraints were easy to implement for thescalar and vector models by coding the analytical expression for the spin independent WIMP-nucleon scattering cross section but due to the momentum dependence of the WIMP-nucleon

                  42 FeynRules 55

                  scattering cross section of the Majorana fermion model LUXCalc v101 [79] was modifiedfor this model to implement momentum dependent cross sections [85] We used MicrOmegasv425 to calculate relic density values and to produce gamma ray spectra for comparisonwith the Fermi-LAT results Parameter scans were carried out using MultiNest v310 withthe parameter ranges given in Table 51 and using astrophysical constraints only Oncethese scans had converged a random sample of 900 points was chosen (weighted by theposterior probability) and then used to calculate collider constraints using the MadgraphMonte Carlo generator This included comparison with mediator production and decay limitsfrom the LHC experiments and the reproduction of searches for dark matter plus other finalstate objects which required dedicated C++ code that reproduces the cut and count analysesperformed by CMS and ATLAS

                  42 FeynRules

                  FeynRules is a Mathematicareg package that allows the calculation of Feynman rules inmomentum space for any QFT physics model The user needs to provide FeynRules withminimal information to describe the new model contained in the model-file This informationis then used to calculate a set of Fenman rules associated with the Lagrangian The Feynmanrules calculated by the code can then be used to implement new physics models into otherexisting tools such as Monte Carlo (MC) generators via a set of interfaces developed andmaintained by the MC authors

                  Feynrules can be run with a base model = SM (or MSSM) and user defined extensions tothe base model

                  43 LUXCalc

                  LUXCalc is a utility program for calculating and deriving WIMP-nucleon coupling limitsfrom the recent results of the LUX direct search dark matter experiment The program hasbeen used to derive limits on both spin-dependent and spin-independent WIMP-nucleoncouplings and it was found under standard astrophysical assumptions and common otherassumptions that LUX excludes the entire spin-dependent parameter space consistent witha dark matter interpretation of DAMArsquos anomalous signal and also found good agreementwith published LUX results [79]

                  We modified the program for the Majorana fermion model to implement momentumdependent cross -sections [85]

                  44 Multinest 56

                  44 Multinest

                  Multinest is an efficient Bayesian Inference Tool which uses a simultaneous ellipsoidalnested sampling method a Monte Carlo technique aimed at an efficient Bayesian Evidencecalculation with posterior inferences as a by product

                  Bayes theorem states that

                  Pr(θ |DH) =Pr(D|θ H)Pr(θ |H)

                  Pr(D|H) (41)

                  Here D is the data and H is the Hypothesis Pr(θ |DH) is the posterior probability ofthe parameters Pr(D|θ H) = L(θ) is the likelihood Pr(θ |H) is the prior and Pr(D|H) isthe evidence

                  The evidence Pr(D|H) =int

                  Pr(θ |DH)Pr(θ |H)d(θ) =int

                  L(θ)Pr(θ |H)d(θ) Thus ifwe define the prior volume dX = Pr(θ |H)d(θ) and

                  X(λ ) =int

                  L(θ)gtλ

                  Pr(θ |H)d(θ) (42)

                  where the integral extends over the region of parameter space within the iso likelihoodcontour L(θ) = λ then the evidence integral can be written

                  int 10 L (X)dX where L (X) the

                  inverse of equation 42 and is a monotonically decreasing function of X (the prior volumewhich is 1 for the total sample space and 0 for a single point)

                  Thus if one can evaluate likelihoods L (Xi) where 0 lt Xn lt lt X2 lt X1 lt X0 = 1the evidence can be estimated as sumLiwi where wi = 5(Ximinus1 minusXi+1) Ellipsoidal samplingapproximates the iso likelihood contour by a D dimensional covariance matrix of the currentset of active points New points are selected from the prior within this ellipsoidal bounduntil one is obtained that has a likelihood greater than the likelihood of the removed lowestlikelihood point This method is extended for multimodal sampling where there may bemore than one local maximum by clustering points that are well separated and constructingindividual ellipsoidal bounds for each cluster For more details consult [7]

                  The results of the analysis are expressed as a data file containing the sample probability-2loglikelihood followed by the parameters of the model for the particular sampling Theprobabilities are normalised to total 10

                  45 Madgraph 57

                  45 Madgraph

                  Madgraph is a general purpose matrix element calculator and Monte Carlo generator Basedon user-supplied Feynman rules (eg the output of Feynrules) the package calculates the tree-level matrix elements for any processes that the user specifies and generates samples of LHCevents including an accurate simulation of the proton collisions themselves (which requires aknowledge of the parton distribution functions that determine the relative proportion of eachparton inside the nucleus) Hadronisation of final state quarks is not possible in Madgraphand hence the code must be used in conjunction with Pythia which performs a parton shower

                  The Madgraph output was automatically showered by Pythia (Pythia describes partonradiation as successive parton emissions using Markov chain techniques based on Sudakovform factors and this description is only formally correct for the soft and collinear emissions)A Madgraph simulation is at the level of the fundamental particles (quarks and leptons)Madgraph generates a limited set of matrix element processes at LO representing hard (highenergy) emissions and passes these to Pythia for showering The final state partons in theMadgraph simulation must be matched to the Pythia final state partons to avoid doublecounting

                  The output from Pythia is automatically fed through Delphes where basic collider cutscan be performed (calculation of the numbers of particles and jets that would be observed atdifferent energies and locations within the collider) The output of the Delphes simulation is afile containing reconstructed events that can be analysed using the standard ROOT framework(the most popular data analysis package in particle physics) [86]

                  The parameters that gave the best fits to the direct [16] and indirect DM experiments [87]and relic density [59] are ranked by posterior probability from the Multinest [7] output

                  The parameters in the Madgraph simulations are stored in files in the cards sub directoryand in particular the run card param card and Delphes card record items relevent to thesimulation

                  In the run card the number of events the Beam 1 and 2 total energy in GeV (4000 each)and the type of jet matching were modified (to MLM type matching where the final statepartons in Madgraph are clustered according to the KT jet algorithm to find the equivalentparton shower history of the event there is a minimum cutoff scale for the momentum pT

                  given by the xqcut parameter) The maxjetflavour (=5 to include b quarks) and xqcut = 30GeV parameters were also modified

                  46 Collider Cuts C++ Code 58

                  The param card was modified to specify the masses of the new particles and the autodecay modes The Delphes card is overwritten with either the ATLAS or CMS defaultcards which record tracking efficiencies and isolation momentum and energy resolutionscalorimeter details jet finder and b and τ -tagging parameters However the isolation andtagging in this study were done by the collider cuts C++ code since multiple sets of cutswere applied to the same output from Pythia

                  When performing simulations of the three dark matter models it was necessary to definethe proton and jets to contain b quarks as this is not the default in Madgraph The processesgenerated included 1 and 2 jets

                  46 Collider Cuts C++ Code

                  Because of the volume of simulations required for the collider constraints we used C++code (originally developed by Martin White and Sky French and modified by me) that readsthe ROOT files and outputs the numbers of events that would have been observed under thechosen collider experiments had the model been true for the given parameters Thus the sameoutput files can be used for all of the collider cuts and hence no cuts were performed throughthe Delphes cards

                  In order to validate the C++ code we replicated the observed events with the papersdescribing the various collider searches (see Appendix)

                  Chapter 5

                  Majorana Model Results

                  51 Bayesian Scans

                  To build some intuition we first examined the constraining effect of each observable on itsown We ran separate scans for each constraint turned on and then for all the constraintstogether In Fig 51 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together The paramaters were scanned over the range in Table 51 below

                  Table 51 Scanned Ranges

                  Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                  Range 0minus103 0minus103 10minus5 minus10 10minus5 minus10 10minus5 minus10

                  In Fig51 we can see that the preferred DM mass is centered around 35 GeV primarily dueto the influence of the Fermi-LAT gamma ray constraint whilst the relic density constraintallows a wide range of possible mediator and DM masses Direct detection (LUX) constraintssplit the mass regions into a top left and bottom right region and together these split theoverall mass parameter regions into two regions by mediator mass but with DM mass around35 GeV Later studies using a lower excess gamma ray spectrum tend to increase the preferredDM mass up to 50 GeV [76] and [88]

                  The marginalised posterior probabilities for all combinations of parameters taken twoat a time are shown in Fig52 These plots have all constraints (Gamma LUX and RelicDensity) applied together The idea of these graphs is to study the posterior probabilities ofcombinations of two paramaters taken at a time (the off-diagonal graphs) The diagonal plotsshow marginalised histograms of a single parameter at a time- this gives an indication of thespread of each parameter The plots show that mS and λb show a high degree of correlationfor mS gt 100 GeV but there is also a range of values of mS around 35 GeV which are

                  51 Bayesian Scans 60

                  1 0 1 2 3 4log10(mχ)[GeV]

                  1

                  0

                  1

                  2

                  3

                  4

                  log 1

                  0(m

                  s)[GeV

                  ]

                  (a) Gamma Only

                  1 0 1 2 3 4log10(mχ)[GeV]

                  1

                  0

                  1

                  2

                  3

                  4

                  log 1

                  0(m

                  s)[GeV

                  ]

                  (b) Relic Density

                  1 0 1 2 3 4log10(mχ)[GeV]

                  1

                  0

                  1

                  2

                  3

                  4

                  log 1

                  0(m

                  s)[GeV

                  ]

                  (c) LUX

                  05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                  05

                  00

                  05

                  10

                  15

                  20

                  25

                  30

                  35

                  log 1

                  0(m

                  s)[GeV

                  ]

                  (d) All Constraints

                  Figure 51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether

                  51 Bayesian Scans 61

                  00

                  05

                  10

                  15

                  20

                  25

                  30

                  log 1

                  0(m

                  χ)[GeV

                  ]

                  00

                  05

                  10

                  15

                  20

                  25

                  30

                  ms[Gev

                  ]

                  5

                  4

                  3

                  2

                  1

                  0

                  1

                  log 1

                  0(λ

                  t)

                  5

                  4

                  3

                  2

                  1

                  0

                  1

                  log 1

                  0(λ

                  b)

                  00 05 10 15 20 25 30

                  log10(mχ)[GeV]

                  5

                  4

                  3

                  2

                  1

                  0

                  1

                  log 1

                  0(λ

                  τ)

                  00 05 10 15 20 25 30

                  ms[Gev]5 4 3 2 1 0 1

                  log10(λt)5 4 3 2 1 0 1

                  log10(λb)5 4 3 2 1 0 1

                  log10(λτ)

                  Figure 52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter

                  52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 62

                  possible for λb around 001 λt and λτ show a wide range of possible values for two possibleranges of mS lt 35 GeV and greater than 100 GeV mχ shows a narrow range of possiblevalues around 35 GeV for all values of the other parameters

                  52 Best fit Gamma Ray Spectrum for the Majorana FermionDM model

                  We can see from the Fig51(a) scan that the maximum log-likelihood Fermi-LAT gammaray spectrum on its own shows values that centre around 35 GeV This is not changed byadding a relic density constraint or a direct detection (LUX) constraint

                  The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraintsis shown in Fig53 The parameters for this point were

                  Table 52 Best Fit Parameters

                  Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                  Value 3332 49266 0322371 409990 0008106

                  10-1 100 101 102

                  E(GeV)

                  10

                  05

                  00

                  05

                  10

                  15

                  20

                  25

                  30

                  35

                  E2dφd

                  E(G

                  eVc

                  m2ss

                  r)

                  1e 6

                  Best fitData

                  Figure 53 Gamma Ray Spectrum

                  The relic density (Ωh2) at this point has a value of 0135 but the point is separated onlyslightly in posterior probability from points with a value of 01199 with mS lt 10 GeV (iethe posterior probability is almost the same differing by less than 1)

                  To better understand the interplay of the different constraints the three graphs in Fig54show the log likelihood surfaces and individual best fit points (best log likelihoods) for the γ

                  and Ω scans seperately and then combined The diagrams each choose a pair of couplings(fixing the remaining parameters) and display the 2d surfaces of minuslog(χ2) (proportional to

                  52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 63

                  the log likelihoods of the coupling combinations) obtained by varying the pair of couplingsThese graphs do not show marginalised posterior probabilities but rather are based on the-2log likelihood calculations directly from MicrOmegas for the given coupling combinationsWe have termed this value as χ2 for convenience and used the log of this value to give abetter range for the colours These log likelihood values are what drive the MultiNest searchof parameter space for the best parameter combinations

                  The graphs show a complex interplay of the different constraints resulting often in acompromise parameter combination The arrow on each graph points to the maximum point We have chosen a single representative set of couplings near the best overall fit point for allthe graphs

                  52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 64

                  00 05 10 15 20 25 30

                  log10(mχ)

                  00

                  05

                  10

                  15

                  20

                  25

                  30

                  log

                  10(m

                  S)

                  Max

                  minuslog10(χ2(Γ)) λt = 487 λτ = 024 λb = 0344

                  16

                  14

                  12

                  10

                  8

                  6

                  4

                  2

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                  γ Maximum at mχ=416 GeV mS=2188 GeV

                  00 05 10 15 20 25 30

                  log10(mχ)

                  00

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                  10(m

                  S)

                  Max

                  minuslog10(χ2(Omega)) λt = 487 λτ = 024 λb = 0344

                  28

                  24

                  20

                  16

                  12

                  08

                  04

                  00

                  04

                  Ω Maximum at mχ=363 GeV mS=1659 GeV

                  00 05 10 15 20 25 30

                  log10(mχ)

                  00

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                  10(m

                  S)

                  Max

                  minuslog10(χ2(Both)) λt = 487 λτ = 024 λb = 0344

                  16

                  14

                  12

                  10

                  8

                  6

                  4

                  2

                  0

                  Both Maximum at mχ=8317 GeV mS=2884GeV The best fit point for these couplings maybe above below or between the individual bestfit points but will on average be between thepoints

                  Figure 54 Plots of log likelihoods by individual and combined constraints Masses in GeV

                  53 Collider Constraints 65

                  53 Collider Constraints

                  531 Mediator Decay

                  1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof τ leptons in pp collisions [44] Refer also to Chapter 3 for details

                  We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

                  The results of the calculations are shown in the graph Fig55 and can be compared withthe limits in the table from the paper [44] in Fig 56 The black line is the observed ATLASlimit

                  Figure 55 σ lowastBr(σ rarr ττ) versus Mass of Scalar

                  0 200 400 600 800

                  mS[GeV]

                  10

                  5

                  0

                  log 1

                  0(σ

                  (bbS

                  )lowastB

                  (Sgtττ

                  ))[pb]

                  Observed LimitLikely PointsExcluded Points

                  0

                  20

                  40

                  60

                  80

                  100

                  120

                  0 5 10 15 20 25 30 35 40 45

                  We can see that only one point here exceeded the limit and the points showed a widespread over σ(bbS)lowastBr(σ rarr ττ) by mS space which is consistent with the spread of pointson the MultiNest scan on the mS by λτ plane causing a spread of branching ratios to the τ

                  quark It may seem odd that only one point was excluded here but this point was sampledwith a very low probability having high coupling constants of 597 to the τ quark and 278 tothe t quark and only just exceeds the limit As such it can be explained as a random outlier

                  53 Collider Constraints 66

                  Figure 56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar

                  2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                  This scan calculates the contributions of b-quark associated Higgs boson productionreproduced from [43] Refer to Chapter 3 for details

                  We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

                  The cross section times branching ratios to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig57 The black line is the CMSLimit

                  Figure 57 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar

                  0 200 400 600 800

                  mS[GeV]

                  15

                  10

                  5

                  0

                  5

                  log

                  10(σ

                  (bS

                  +X

                  )lowastB

                  (Sgt

                  bb))

                  [pb]

                  Observed LimitLikely PointsExcluded Points

                  0

                  20

                  40

                  60

                  80

                  100

                  120

                  0 50 100 150 200 250

                  53 Collider Constraints 67

                  The results of this scan were compared to the limits in [89] with the plot shown inFig58

                  Figure 58 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

                  We can see that there are a number of exclusions across the range of mS which occuredfor high values of the coupling constants gt 1 for λbλt λtau - these can be seen also on Fig59The excluded parameter combinations represent less than 2 of the total sampled number

                  532 Collider Cuts Analyses

                  We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude and the results are shown against theposterior probabilities of the parameter combinations in Fig 59

                  The circles in Figure 59 show which points were excluded by the collider cuts as well asthe limits on cross sections times branching ratios to each of τ+τminus and bb

                  All the excluded points under the heading Collider Cuts showed exclusion by the Lepstop0experiment (Search for direct pair production of the top quark in all hadronic final states

                  53 Collider Constraints 68

                  0 1 2 3

                  log10(mχ)[GeV]

                  0

                  1

                  2

                  3

                  log 1

                  0(m

                  s)[GeV

                  ]Collider Cuts

                  σ lowastBr(σgt bS+X)

                  σ lowastBr(σgt ττ)

                  (a) mχ by mS

                  6 5 4 3 2 1 0 1 2

                  log10(λt)

                  0

                  1

                  2

                  3

                  log 1

                  0(m

                  s)[GeV

                  ](b) λt by mS

                  5 4 3 2 1 0 1

                  log10(λb)

                  6

                  5

                  4

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                  2

                  1

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                  1

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                  0(λ

                  t)

                  (c) λb by λt

                  5 4 3 2 1 0 1

                  log10(λb)

                  6

                  5

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                  τ)

                  (d) λb by λτ

                  6 5 4 3 2 1 0 1 2

                  log10(λt)

                  6

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                  τ)

                  (e) λt by λτ

                  5 4 3 2 1 0 1

                  log10(λb)

                  0

                  1

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                  3

                  log 1

                  0(m

                  s)[GeV

                  ]

                  (f) λb by mS

                  Figure 59 Excluded points from Collider Cuts and σBranching Ratio

                  53 Collider Constraints 69

                  [32]) Some of these same points were also excluded by Lepstop2 2bstop CMSTop andATLASMonobjet experiments There were expected events at other points but none weresignificant at the 95 exclusion level These significant exclusions occurred mainly wherethe mass of the mediator was over 100 GeV and higher values of the couplings to b and tquarks but there were a few points with mediator mass around 25 GeV The exclusions withmediator mass around 25 GeV were all accompanied by couplings to t quarks of order 10which is probably non-perturbative

                  Looking at the exclusions in Fig 59 we can see that these occur mainly in the regionof higher mediator mass gt 100 GeV and lt 400 GeV These exclusions come from theastrophysical constraints (Gamma) which show a strong correlation between λb and mS andhigh values of λt gt 1 However these mass regions do not overlap at the same couplingcombinations as the high likelihood region The green and white circles correspond to theexcluded points in Figs 57 and 58 respectively

                  The remaining regions of highest likelihood can be seen to be where 0001 λb lt 01accompanied by λt lt 0001 and mass of the mediator mS lt 10 GeV or 400 GeV lt mS lt1000GeV with mχ sim 35 GeV

                  The plot of λt by λb shows the significant regions do not overlap with the collider cutsand cross section constraints since the latter occur at high values of either coupling while thelikely region has low values of these couplings (also accompanied by low couplings to λτ )

                  Chapter 6

                  Real Scalar Model Results

                  61 Bayesian Scans

                  To build some intuition we first examined the constraining effect of each observable on itsown We ran seperate scans for each constraint turned on and then for all the constraintstogether In Figure 61 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together

                  In Figure 61 we can see that there are is a narrow region which dominates when allconstraints are taken into account This centers on mχ=9 GeV and a range of values for mS

                  from 35 GeV to 400 GeV The region is not the highest likelihood region for the Fermi-LATgamma ray spectrum on its own (as can be seen from the individual plot - there is a narrowband to the left of the highest likelihood band) nor for the LUX likelihood or relic densitybut represents a compromise between the three constraints

                  The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig62 The diagonal plots show marginalised histograms of a singleparameter at a time

                  61 Bayesian Scans 71

                  05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                  05

                  00

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                  35

                  log 1

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                  s)[GeV

                  ]

                  (a) Gamma Only

                  05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                  05

                  00

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                  s)[GeV

                  ]

                  (b) Relic Density

                  05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                  05

                  00

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                  s)[GeV

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                  (c) LUX

                  05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                  05

                  00

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                  log 1

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                  s)[GeV

                  ]

                  (d) All Constraints

                  Figure 61 Real Scalar Dark Matter - By Individual Constraint and All Together

                  61 Bayesian Scans 72

                  00

                  05

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                  30

                  log 1

                  0(m

                  χ)[GeV

                  ]

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                  ]

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                  t)

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                  b)

                  00 05 10 15 20 25 30

                  log10(mχ)[GeV]

                  5

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                  0(λ

                  τ)

                  00 05 10 15 20 25 30

                  ms[Gev]5 4 3 2 1 0 1

                  log10(λt)5 4 3 2 1 0 1

                  log10(λb)5 4 3 2 1 0 1

                  log10(λτ)

                  Figure 62 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter

                  62 Best fit Gamma Ray Spectrum for the Real Scalar DM model 73

                  62 Best fit Gamma Ray Spectrum for the Real Scalar DMmodel

                  We can see from Fig61 that the maximum log-likelihood Fermi-LAT Gamma ray spectrumon its own shows values of mχ that centre around 35 GeV However the LUX constraintin particular moves the best fit to a point around 9 GeV- the tight LUX constraint onlydisappears for low masses and at that point the proportion of τ+τminus annihilation needs toincrease to fit the constraints

                  The gamma ray spectrum at the points of best fit for Fermi-LAT including all constraintsis shown in Fig63 The parameters for these points were respectively and involve decays tomainly τ and b quarks respectively

                  Table 61 Best Fit Parameters

                  Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                  Value 932 3526 000049 0002561 000781

                  10-1 100 101 102

                  E(GeV)

                  10

                  05

                  00

                  05

                  10

                  15

                  20

                  25

                  30

                  35

                  E2dφdE

                  (GeVc

                  m2ss

                  r)

                  1e 6

                  Best fitData

                  Figure 63 Gamma Ray Spectrum

                  This is not a particularly good fit and it occcurs at a relic density Ωh2 of 335 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

                  63 Collider Constraints 74

                  63 Collider Constraints

                  631 Mediator Decay

                  1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44]

                  We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

                  The results of the scan are shown in the graph Fig64 and can be compared with thelimits in the table from the paper [44] in Fig 56The black line is the observed ATLAS limit

                  Figure 64 σ lowastBr(σ rarr ττ) versus Mass of Scalar

                  0 200 400 600 800

                  mS[GeV]

                  8

                  6

                  4

                  2

                  0

                  2

                  4

                  log 1

                  0(σ

                  (bbS

                  )lowastB

                  (Sgtττ

                  ))[pb]

                  Observed LimitLikely PointsExcluded Points

                  050

                  100150200250300350

                  0 10 20 30 40 50 60

                  We can see that there is a narrow band in the region of interest in the σ(bbS)lowastB(S gt ττ)

                  by mS plane due to the high degree of correlation between mS and λτ seen in the likelihoodscan Values for mS gt 300Gev are excluded

                  2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                  We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinations

                  63 Collider Constraints 75

                  randomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

                  The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig65 The black line is the CMSlimit

                  Figure 65 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

                  0 200 400 600 800

                  mS[GeV]

                  8

                  6

                  4

                  2

                  0

                  2

                  4

                  log

                  10(σ

                  (bS

                  +X

                  )lowastB

                  (Sgt

                  bb))

                  [pb]

                  Observed LimitLikely PointsExcluded Points

                  050

                  100150200250300350

                  0 10 20 30 40 50 60

                  The results of this scan were compared to the limits in [89] with the plot shown inFig58

                  We can see a similar shape for the region of interest to that in Fig64 due to the highcorrelation between mS and λb over the region preferred by astrophysical constraints resultingin branching ratios highly correlated to ms Only one point is excluded with mS gt 500GeV

                  632 Collider Cuts Analyses

                  We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expected

                  63 Collider Constraints 76

                  with 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 66

                  We can see that collider cuts show some exclusions only for masses mχ below the highlikelihood region but these exclusions do not occur at the same λb and λt combinations asseen from the plot of these two couplings except for a small region at the highest mS andcouplings

                  All but one of the collider cut exclusions were excluded by the Lepstop0 experiment (thiswas excluded by 2bstop with high coupling to b quarks of 73) A number of the excludedpoints were also excluded by Lepstop2 2bstop CMSTopDM1L and ATLASMonobJet-(around 30 of the total for each)

                  63 Collider Constraints 77

                  0 1 2 3

                  log10(mχ)[GeV]

                  0

                  1

                  2

                  3

                  log 1

                  0(m

                  s)[GeV

                  ]Collider Cuts

                  σ lowastBr(σgt bS+X)

                  σ lowastBr(σgt ττ)

                  (a) mχ by mS

                  5 4 3 2 1 0 1

                  log10(λt)

                  0

                  1

                  2

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                  0(m

                  s)[GeV

                  ](b) λt by mS

                  5 4 3 2 1 0 1

                  log10(λb)

                  5

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                  t)

                  (c) λb by λt

                  5 4 3 2 1 0 1

                  log10(λb)

                  6

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                  τ)

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                  5 4 3 2 1 0 1

                  log10(λt)

                  6

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                  τ)

                  (e) λt by λτ

                  5 4 3 2 1 0 1

                  log10(λb)

                  0

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                  3

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                  0(m

                  s)[GeV

                  ]

                  (f) λb by mS

                  Figure 66 Excluded points from Collider Cuts and σBranching Ratio

                  Chapter 7

                  Real Vector Dark Matter Results

                  71 Bayesian Scans

                  In the Figure 71 we can see that the preferred DM mass is centered around 35 GeV primarilydue to the influence of the Fermi-LAT gamma ray constraint with relic density having a widerange of possible mediator and DM masses The result of combining all the constraints is todefine a small region around mχ = 8 GeV and mS = 20 GeV which dominates the posteriorprobability

                  The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig72 The diagonal plots show marginalised histograms of a singleparameter at a time

                  71 Bayesian Scans 79

                  05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                  1

                  0

                  1

                  2

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                  0(m

                  s)[GeV

                  ]

                  (a) Gamma Only

                  05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                  05

                  00

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                  s)[GeV

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                  (b) Relic Density

                  1 0 1 2 3 4log10(mχ)[GeV]

                  05

                  00

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                  s)[GeV

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                  (c) LUX

                  05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                  05

                  00

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                  log 1

                  0(m

                  s)[GeV

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                  (d) All Constraints

                  Figure 71 Real Vector Dark Matter - By Individual Constraint and All Together

                  71 Bayesian Scans 80

                  00

                  05

                  10

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                  30

                  log 1

                  0(m

                  χ)[GeV

                  ]

                  00

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                  t)

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                  b)

                  00 05 10 15 20 25 30

                  log10(mχ)[GeV]

                  5

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                  0(λ

                  τ)

                  00 05 10 15 20 25 30

                  ms[Gev]5 4 3 2 1 0 1

                  log10(λt)5 4 3 2 1 0 1

                  log10(λb)5 4 3 2 1 0 1

                  log10(λτ)

                  Figure 72 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter

                  72 Best fit Gamma Ray Spectrum for the Real Vector DM model 81

                  72 Best fit Gamma Ray Spectrum for the Real Vector DMmodel

                  The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraints isshown in Fig73

                  Table 71 Best Fit Parameters

                  Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                  Value 8447 20685 0000022 0000746 0002439

                  10-1 100 101 102

                  E(GeV)

                  10

                  05

                  00

                  05

                  10

                  15

                  20

                  25

                  30

                  35

                  E2dφdE

                  (GeVc

                  m2s

                  sr)

                  1e 6

                  Best fitData

                  Figure 73 Gamma Ray Spectrum

                  This is not a particularly good fit and it occcurs at a relic density Ωh2 of 025 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

                  73 Collider Constraints

                  731 Mediator Decay

                  1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44] (See Chapter 3 for more details)

                  We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

                  73 Collider Constraints 82

                  The results of the scan are shown in the graph Fig74 and can be compared with thelimits in the table from the paper [44] in Fig 56

                  Figure 74 σ lowastBr(σ rarr ττ) versus Mass of Scalar

                  0 200 400 600 800

                  mS[GeV]

                  8

                  6

                  4

                  2

                  0

                  2

                  log 1

                  0(σ

                  (bbS

                  )lowastB

                  (Sgtττ

                  ))[pb]

                  Observed LimitLikely PointsExcluded Points

                  0100200300400500600700800

                  0 20 40 60 80 100120140

                  We can see that the region of highest likelihood is limited to mS lt 100 GeV and a narrowrange of couplings to λτ giving a narrow region of interest which falls well below the limit

                  2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                  We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

                  The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig75 The black line is the CMSlimit

                  The results of this scan were compared to the limits in [89] with the plot shown in Fig58

                  We can see that again the region of interest for mS is below 100 GeV and the points fallwell below the limit

                  73 Collider Constraints 83

                  Figure 75 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

                  0 200 400 600 800

                  mS[GeV]

                  8

                  6

                  4

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                  10(σ

                  (bS

                  +X

                  )lowastB

                  (Sgt

                  bb))

                  [pb]

                  Observed LimitLikely PointsExcluded Points

                  0100200300400500600700800

                  0 20 40 60 80 100120140

                  732 Collider Cuts Analyses

                  We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 76

                  We can see from Figure 76 that like the real scalar model the region of highest likelihoodhas λb and λt lt 001 while the excluded points from collider cuts and cross sections areabove 01 Most of the excluded points have mχ gt than the values in the region of highestlikelihood and all points are widely separated in parameter space from this region

                  Once again 89 of excluded points were excluded by Lepstop0 with many also beingexcluded by the other experiments (50 by Lepstop2 and 56 by CMSTopDM1L 35 byLepstop2 and 20 by ATLASMonobjet)

                  73 Collider Constraints 84

                  0 1 2 3

                  log10(mχ)[GeV]

                  0

                  1

                  2

                  3

                  log 1

                  0(m

                  s)[GeV

                  ]Collider Cuts

                  σ lowastBr(σgt bS+X)

                  σ lowastBr(σgt ττ)

                  (a) mχ by mS

                  5 4 3 2 1 0 1

                  log10(λt)

                  0

                  1

                  2

                  3

                  log 1

                  0(m

                  s)[GeV

                  ](b) λt by mS

                  5 4 3 2 1 0 1

                  log10(λb)

                  5

                  4

                  3

                  2

                  1

                  0

                  1

                  log 1

                  0(λ

                  t)

                  (c) λb by λt

                  5 4 3 2 1 0 1

                  log10(λb)

                  5

                  4

                  3

                  2

                  1

                  0

                  1

                  log 1

                  0(λ

                  τ)

                  (d) λb by λτ

                  5 4 3 2 1 0 1

                  log10(λt)

                  5

                  4

                  3

                  2

                  1

                  0

                  1

                  log 1

                  0(λ

                  τ)

                  (e) λt by λτ

                  5 4 3 2 1 0 1

                  log10(λb)

                  0

                  1

                  2

                  3

                  log 1

                  0(m

                  s)[GeV

                  ]

                  (f) λb by mS

                  Figure 76 Excluded points from Collider Cuts and σBranching Ratio

                  Chapter 8

                  Conclusion

                  We first conducted Bayesian scans over mass and coupling parameters for three simplifiedmodels - Majorana fermion real scalar and real vector using direct detecion (LUX) indirectdetection (GCE γ rays) and relic density constraints to find the regions of highest likelihoodWe ran these scans one constraint at a time and then together to better understand the interplayof the different constraints on best fit and likely parameter regions

                  We also developed and tested C++ collider cut software specifically designed to mimickthe cuts applied in a number of ATLAS and CMS experiments that searched for Emiss

                  T inconnection with specific beyond the standard model (BSM) theories of Dark Matter

                  We then selected parameter combinations by randomly sampling the scanned resultsbased on the posterior probability distribution of the points and applied collider cuts andcross section times branching ratio calculations to these points to see if there would havebeen any sensitivity from the ATLAS and CMS experiments

                  We found that the Majorana fermion model showed the best overall fit to all the constraintswith relic density close to the experimental value of 01199 and a low χ2 error on the gammaspectrum fit The preferred mass mχ was around 35 GeV and mS ranging from lt10 GeV to500 GeV The smaller masses give a better fit for relic density and also are well separatedfrom current collider constraints mainly due the the fact that the exclusions occur mainlywith higher couplings to t quarks (greater than 1) which is consistent with the fact that manyof the experiments were targeting dark matter pair production associated with t quark decaysNevertheless there were some exclusions that did fall in the region sampled by randomselection from the posterior probability distribution

                  The preferred parameter regions were also consistent with the observation in [20] thatthe Majorana model showed the best overall fit with similar mass constraints We find thatthe couplings to the SM fermions should all be less that 001

                  86

                  The scalar and vector DM models give slightly less good fits especially to the currentrelic density estimate of 01199 but again the current collider constraints are well separatedfrom the best fit region and in fact none of the points randomly sampled from the posteriordistribution showed any exclusion The best fit parameter combination favours couplings toSM fermions of less than 001

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                  Appendix A

                  Validation of Calculation Tools

                  Lepstop0Search for direct pair production of the top squarkin all hadronic final states in proton=proton collisions atradic

                  s=8 TeV with the ATLAS detector [32]

                  Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1) mt1=600 GeV mn1=1 GeVWhere t1 is the top squark and n1 is the neutralino of the MSSM

                  94

                  Table A1 0 Leptons in the final state

                  Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 10000 460 460 100Emiss

                  T gt 130GeV 8574 395 451 88Leptonveto 6621 307 285 108Emiss

                  T gt 150GeV 6316 293 294 100Jet multiplicityand pT 1099 48 51 94|∆φ( jet pmiss

                  T )|gt π5 902 40 42 96gt= 2b jets 396 19 18 106τ veto 396 19 18 106mbmin

                  T gt 175GeV 270 13 13 105SRA1 79 4 4 104SRA2 70 3 3 93SRA3 67 3 3 97SRA4 52 2 2 91SRC exactly5 jets 1297 50 60 83SRC |∆φ( jet pmiss

                  T )|gt π5 1082 50 50 99SRC gt= 2b jets 430 20 20 97SRC τ veto 430 20 20 97SRC |∆φ(bb)|gt 02π 367 17 17 99SRC1 246 12 11 105SRC2 177 9 8 116SRC3 152 8 7 116

                  95

                  Lepstop1Search for direct top squark pair production infinal states with one isolated lepton jets and missing trans-verse momentum in

                  radics = 8 TeV pp collisions using 21 f bminus1

                  of ATLAS data[33]

                  Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1)(t1 gt b x+1 )(t1 gt b xminus1 )mt1=650 GeV mn1=1 GeV Where t1 is the top squark n1 is the neutralino xplusmn1 are thecharginos

                  96

                  Table A2 1 Lepton in the Final state

                  Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 100000 99990 99990 100Electron(= 1signal) 11558 11557 11785 984 jets(80604025) 5187 5186 6271 83gt= 1btag 4998 4997 5389 93Emiss

                  T gt 100GeV [allSRs] 4131 4131 4393 94Emiss

                  T radic

                  HT gt 5 [allSRs] 4026 4026 4286 94∆φ( jet2 pmiss

                  T )gt 08 [allSRs] 3565 3565 3817 93∆φ( jet1Pmiss

                  T )gt 08 [not SRtN2] 3446 3446 3723 93Emiss

                  T gt 200GeV (SRtN2) 2027 2027 2186 93Emiss

                  T radic

                  HT gt 13(SRtN2) 1156 1156 1172 99mT gt 140GeV (SRtN2) 839 839 974 86Emiss

                  T gt 275GeV (SRtN3) 948 948 965 98Emiss

                  T radic

                  HT gt 11(SRtN3) 932 932 937 99mT gt 200GeV (SRtN3) 456 456 566 81Emiss

                  T gt 150GeV (SRbC1minusSRbC3) 2803 2803 3080 91Emiss

                  T radic

                  HT gt 7(SRbC1minusSRbC3) 2742 2742 3017 91mT gt 12GeV (SRbC1minusSRbC3) 1917 1917 2347 82Emiss

                  T gt 160GeV (SRbC2SRbC3) 1836 1836 2257 81Emiss

                  T radic

                  HT gt 8(SRbC2SRbC3) 1788 1788 2175 82me f f gt 550GeV (SRbC2) 1741 1741 2042 85me f f gt 700GeV (SRbC3) 1075 1075 1398 77Muon(= 1signal) 11379 11378 11975 954 jets(80604025) 5264 5263 6046 87gt= 1btag 5101 5100 5711 89Emiss

                  T gt 100GeV [allSRs] 4199 4199 4770 88Emiss

                  T radic

                  HT gt 5 [all SRs] 4115 4115 4653 88∆φ( jet2 pmiss

                  T )gt 08 [all SRs] 3622 3622 4077 89∆φ( jet1 pmiss

                  T )gt 08 [not SRtN2] 3515 3515 3970 89Emiss

                  T gt 200GeV (SRtN2) 2136 2136 2534 84Emiss

                  T radic

                  HT gt 13(SRtN2) 1261 1261 1441 87mT gt 140GeV (SRtN2) 928 928 1033 90Emiss

                  T gt 275GeV (SRtN3) 1036 1036 1274 81Emiss

                  T radic

                  HT gt 11(SRtN3) 1018 1018 1235 82mT gt 200GeV (SRtN3) 497 497 612 81Emiss

                  T gt 150GeV (SRbC1minusSRbC3) 2891 2891 3380 86Emiss

                  T radic

                  HT gt 7(SRbC1minusSRbC3) 2821 2821 3290 86mT gt 120GeV (SRbC1minusSRbC3) 1977 1977 2290 86Emiss

                  T gt 160GeV (SRbC2SRbC3) 1913 1913 2220 86Emiss

                  T radic

                  HT gt 8(SRbC2SRbC3) 1863 1863 2152 87me f f gt 550GeV (SRbC2) 1826 1826 2116 86me f f gt 700GeV (SRbC3) 1131 1131 1452 78

                  97

                  Lepstop2Search for direct top squark pair production infinal states with two leptons in

                  radics =8 TeV pp collisions using

                  20 f bminus1 of ATLAS data[83][34]

                  Simulated in Madgraph with p p gt t1 t1(t1 gt b w+ n1)(t1 gt b wminusn1) mt1=180 GeV mn1=60 GeV Where t1 is the top squark n1 is the neutralino b1 is the bottom squark

                  Table A3 2 Leptons in the final state

                  Cut RawMe Scaled ATLAS MeATLAS PercentNoCuts 100000 324308 199000 1632BaselineCuts 3593 11652 7203 1622SF Signal Leptons 1699 5510 5510 1002OSSF Signal Leptons 1632 5293 5233 101mll gt 20Gev 1553 5037 5028 100leadinglepton pT 1479 4797 5048 95|mll minusmZ|gt 20Gev 1015 3292 3484 94∆φmin gt 10 850 2757 2160 128∆φb lt 15 680 2205 1757 126SRM90 [SF ] 13 42 60 70SRM120 [SF ] 0 0 19 0SRM100 +2 jets [SF ] 0 0 17 0SRM110 +2 jets [SF ] 0 0 1 02DF Signal Leptons 1737 5633 7164 792OSDF Signal Leptons 1670 5416 5396 100mll20Gev 1587 5147 5136 100leadinglepton pT 1501 4868 4931 99∆φmin gt 10 1247 4044 3122 130∆φb lt 15 1010 3276 2558 128SRM90 [SF ] 21 68 95 72SRM120 [SF ] 0 0 23 0SRM100 +2 jets [SF ] 0 0 19 0SRM110 +2 jets [SF ] 0 0 14 0

                  98

                  2bstopSearch for direct third-generation squark pair pro-duction in final states with missing transverse momentumand two b-jets[31]

                  Simulated in Madgraph with p p gt b1 b1(b1 gt b n1)(b1 gt b n1)

                  SRAmb1=500 GeV mn1=1 GeV SRBmb1=350 GeV mn1=320 GeVWhere b1 is the bottom squark and n1 is the neutralino

                  Table A4 2b jets in the final state

                  Cut RawMe Scaled ATLAS MeATLAS PercentSRA Nocuts 100000 173800 1738 100SRA MET gt 80 93407 162341 1606 101SRA Leptonveto 93302 162159 1505 108SRA MET gt 150 79449 138082 1323 104SRA Jet selection 10213 17750 119 149SRA mbb gt 200 8344 14502 96 151SRA mCT gt 150 6535 11358 82 139SRA mCT gt 200 5175 8994 67 134SRA mCT gt 250 3793 6592 51 129SRA mCT gt 300 2487 4322 35 123SRB NoCuts 100000 1624100 16241 100SRB leptonveto 11489 186593 4069 46SRB MET gt 250 2530 41090 757 54SRB Jet selection 63 1023 79 130SRB HT 3 lt 50 51 828 52 159

                  99

                  CMSTop1LSearch for the Production of Dark Matter in As-sociation with Top Quark Pairs in the Single-lepton FinalState in pp collisions[41]

                  Simulated in Madgraph with p p gt t t p1 p1

                  Where p1= Majorana fermion Table 3 shows Signal efficiencies and cross sections TheM scale was read from Figure 7 for 90 CL

                  Table A5 Signal Efficiencies 90 CL on σ limexp[ f b] on pp gt tt +χχ

                  Mχ[GeV ] My Signal Efficiency Quoted Efficiency My σ [ f b] Quoted σ [ f b]1 102 101 63 47

                  10 132 101 62 4650 143 120 50 38100 170 146 56 32200 185 173 69 27600 265 240 32 19

                  1000 320 276 41 17

                  Appendix B

                  Branching ratio calculations for narrowwidth approximation

                  B1 Code obtained from decayspy in Madgraph

                  Br(S rarr bb) = (minus24λ2b m2

                  b +6λ2b m2

                  s

                  radicminus4m2

                  bm2S +m4

                  S)16πm3S

                  Br(S rarr tt) = (6λ2t m2

                  S minus24λ2t m2

                  t

                  radicm4

                  S minus4ms2m2t )16πm3

                  S

                  Br(S rarr τ+

                  τminus) = (2λ

                  2τ m2

                  S minus8λ2τ m2

                  τ

                  radicm4

                  S minus4m2Sm2

                  τ)16πm3S

                  Br(S rarr χχ) = (2λ2χm2

                  S

                  radicm4

                  S minus4m2Sm2

                  χ)32πm3S

                  (B1)

                  Where

                  mS is the mass of the scalar mediator

                  mχ is the mass of the Dark Matter particle

                  mb is the mass of the b quark

                  mt is the mass of the t quark

                  mτ is the mass of the τ lepton

                  The coupling constants λ follow the same pattern

                  • TITLE Collider Constraints applied to Simplified Models of Dark Matter fitted to the Fermi-LAT gamma ray excess using Bayesian Techniques
                    • Dedication
                    • Declaration
                    • Acknowledgements
                    • Contents
                    • List of Figures
                    • List of Tables
                      • Chapter 1 Introduction
                      • Chapter 2 Review of Physics
                      • Chapter 3 Fitting Models to the Observables
                      • Chapter 4 Calculation Tools
                      • Chapter 5 Majorana Model Results
                      • Chapter 6 Real Scalar Model Results
                      • Chapter 7 Real Vector Dark Matter Results
                      • Chapter 8 Conclusion
                      • Bibliography
                      • Appendix A Validation of Calculation Tools
                      • Appendix B Branching ratio calculations for narrow width approximation

                    List of Tables

                    21 Quantum numbers of the Higgs field 1922 Weak Quantum numbers of Lepton and Quarks 21

                    31 Simplified Models 3532 95 CL by Signal Region 4433 Selection criteria common to all signal regions 4534 Selection criteria for signal regions A 4535 Selection criteria for signal regions C 4536 Signal Regions - Lepstop1 4837 Signal Regions Lepstop2 4938 Signal Regions 2bstop 5139 Signal Region ATLASmonobjet 52

                    51 Scanned Ranges 6152 Best Fit Parameters 64

                    61 Best Fit Parameters 76

                    71 Best Fit Parameters 84

                    A1 0 Leptons in the final state 98A2 1 Lepton in the Final state 100A3 2 Leptons in the final state 101A4 2b jets in the final state 102A5 Signal Efficiencies 90 CL on σ lim

                    exp[ f b] on pp gt tt +χχ 103

                    Chapter 1

                    Introduction

                    Dark matter (DM) was first postulated over 80 years ago when Swiss astronomer FritzZwicky observed a discrepancy between the amount of light emitted by a cluster of galaxiesand the total mass contained within the cluster inferred from the relative motion of thosegalaxies by a simple application of the theory of Newtonian gravitation The surprising resultof this observation was that the vast majority of the mass in the cluster did not emit lightwhich was contrary to the expectation that most of the mass would be carried by the starsSince that time further observations over a wide range of scales and experimental techniqueshave continued to point to the same result and refine it Some of these observations and otherevidence are discussed in section 22 We now know with certainty that in the entire Universeall of the matter we know about - stars planets gases and other cosmic objects such as blackholes can only account for less than 5 of the mass that we calculate to be there

                    A recent phenomenon that has received much attention is the significant deviation frombackground expectations of the Fermi Large Area Telescope(Fermi-LAT) gamma ray flux atthe galactic centre [1] A number of astrophysical explanations have been proposed includingmillisecond pulsars of supernova remnants [2] or burst-like continuous events at the galacticcentre but these are unresolved However it has also been noted that the observed Fermi-LATexcess is consistent with the annihilation of dark matter particles which would naturally beconcentrated at the Galactic centre in a manner consistent with the Navarro-Frenk-Whitedistribution of dark matter [3]

                    There are a number of other purely theoretical (particle physics) reasons to postulatethe existence of weakly interacting matter particles that could supply the missing mass andyet remain unobservable Weakly interacting massive particle (WIMPS) have been a majorfocus of Run I and ongoing Run II searches of the Large Hadron Collider (LHC) In spite

                    2

                    of substantial theoretical and experimental effort the microscopic properties of dark matterparticles are still unknown

                    One of the main problems with extracting these properties is the plethora of competingtheoretical models that provide dark matter candidates Another is the very weak interactionof these particles and the lack of experimental observables allowing one to discriminatebetween the competing models

                    A particularly fruitful approach is the simplified model approach which makes minimalassumptions about the DM candidates as extensions to the Standard Model interacting weaklywith ordinary matter through a mediator These can be coded as Lagrangian interaction termsusing tools such as FeynRules [4] and Calchep [5] and used to simulate Dark Matter relicdensities indirect detection expectations as well as direct detection expectations throughMicrOmegas [6] This in turn can be run under a Bayesian inference tool (see below)

                    Using a Bayesian inference tool such as MultiNest [7][8][9] one can scan the parameterspace of a particular model to find regions of highest probability for that model based onobservations of a range of phenomena including direct detection experiments such as ZEPLIN[10] XENON [11] DEAP [12] ARDM [13] DarkSide [14] PandaX [15] and LUX [16]indirect detection - eg Fermi-LAT gamma ray excess [17] Pamela-positron and antiprotonflux [18] and production at colliders MultiNest calculates the evidence value which is theaverage likelihood of the model weighted by prior probability and a posterior distributionwhich is the distribution of the parameter(s) after taking into account the observed databy scanning over a range of parameters based on prior probabilities This can be used tocompare models on a consistent basis The scans also provide information about the best fitparameters for the overall fit as well as for the seperate experimentsobservations

                    The simplified model approach has been taken in a number of papers to study the Fermi-LAT gamma ray excess (see literature review in section 12) and one in particular comparesthese models using a Bayesian inference tool which found that a Majorana fermion DMcandidate interacting via a scalar mediator with arbitrary couplings to third generationfermions showed the best fit [19] A subsequent paper [20] examined the most likelyparameter combinations that simultaneously fit the Planck relic density Fermi-LAT galacticcentre gamma ray AMS-02 positron flux and Pamela antiproton flux data

                    There has not as yet been a thorough study of the collider constraints on DM simplifiedmodels that fit the Fermi-LAT galactic centre excess In this thesis we reproduce the

                    11 Motivation 3

                    previous results of [20] using similar astrophysical constraints and then implement detailedsimulations of collider constraints on DM and mediator production

                    In Section 11 we review the motivation for this study and in Section 12 we considerrecent astronomical data which is consistent with the annihilation of DM particles andreview the literature and searches which attempt to explain it with a range of extensions tothe standard model

                    In Chapter 2 we review the Standard Model of particle physics (SM) the evidence fordark matter and the ATLAS and CMS experiments at the LHC

                    In Chapter 3 we describe the models considered in this paper the observables used inthe Bayesian scans and the collider experiments which provide limits against which theMagraph simulations of the three models can be compared

                    In Chapter 4 we review the calculation tools that have been used in this paper

                    In Chapters 5 6 and 7 we give the results of the Bayesian scans and collider cutsanalyses for each of the models and in Chapter 8 we summarise our conclusions

                    11 Motivation

                    The Fermi gamma ray space telescope has over the past few years detected a gamma raysignal from the inner few degrees around the galactic centre corresponding to a regionseveral hundred parsecs in radius The spectrum and angular distribution are compatible withthat from annihilating dark matter particles

                    A number of papers have recently attempted to explain the Gamma Ray excess in termsof extensions to the Standard Model taking into account constraints from ATLAS and CMScollider experiments direct detection of dark matter colliding off nucleii in well shieldeddetectors (eg LUX) as well as other indirect measurements (astrophysical observations)These papers take the approach of simplified models which describe the contact interactionsof a mediator beween DM and ordinary matter using an effective field theory approach(EFT) Until recently it has been difficult to discriminate between the results of these paperswhich look at different models and produce different statistical results A recent paper[19] Simplified Dark Matter Models Confront the Gamma Ray Excess used a Bayesianinference tool to compare a variety of simplified models using the Bayesian evidence values

                    12 Literature review 4

                    calculated for each model to find the most preferred explanation The motivation for thechosen models is discussed in section 121

                    A subsequent paper [20] Interpreting the Fermi-LAT gamma ray excess in the simplifiedframework studied the most likely model again using Bayesian inference to find the mostlikely parameters for the (most likely) model The data inputs (observables) to the scanwere the Planck measurement of the dark matter relic density [21] Fermi-LAT gamma rayflux data from the galactic center [17] cosmic positron flux data [22] cosmic anti-protonflux data [18] cosmic microwave background data and direct detection data from the LUXexperiment [16] The result of the scan was a posterior probability density function in theparameter space of the model

                    This paper extends the findings in [19] by adding collider constraints to the likelihoodestimations for the three models It is not a simple matter to add collider constraints becauseeach point in parameter space demands a significant amount of computing to calculate theexpected excess amount of observable particles corresponding to the particular experimentgiven the proposed simplified model A sampling method has been employed to calculatethe collider constraints using the probability of the parameters based on the astrophysicalobservables

                    12 Literature review

                    121 Simplified Models

                    A recent paper [23] summarised the requirements and properties that simplified models forLHC dark matter searches should have

                    The general principles are

                    bull Besides the Standard Model (SM) the Lagrangian should contain a DM candidate thatis absolutely stable or lives long enough to escape the LHC detector and a mediatorthat couples the two sectors

                    bull The Lagrangian should contain all terms that are renormalizable consistent withLorentz invariance SM symmetries and DM stability

                    bull Additional interactions should not violate exact and approximate global symmetries ofthe SM

                    12 Literature review 5

                    The examples of models that satisfy these requirements are

                    1 Scalar (or pseudo-scalar) s-channel mediator coupled to fermionic dark matter (Diracor Majorana)

                    2 Higgs portal DM where the DM is a scalar singlet or fermion singlet under the gaugesymmetries of the SM coupling to a scalar boson which mixes with the Higgs (Thefermion singlet is a specific realisation of 1 above)

                    3 DM is a mixture of an electroweak singlet and doublet as in the Minimal Supersym-metric SM (MSSM)

                    4 Vector (or Axial vector) s-channel mediator obtained by extending the SM gaugesymmetry by a new U(1)rsquo which is spontaneously broken such that the mediatoracquires mass

                    5 t-channel flavoured mediator (if the DM particle is a fermion χ the mediator can bea coloured scalar or vector particle φ -eg assuming a scalar mediator a couplingof the form φ χq requires either χ or φ to carry a flavor index to be consistent withminimal flavour violation (MFV) which postulates the flavour changing neutral currentstructure in the SM is preserved

                    Another recent paper [24] summarised the requirements and properties that simplifiedmodels for LHC dark matter searches should have The models studied in this paper arepossible candidates competing with Dirac fermion and complex scalar dark matter andmodels with vector and pseudo-scalar mediators Other models would have an elasticscattering cross section that will remain beyond the reach of direct detection experiments dueto the neutrino floor (the irreducible background produced by neutrino scattering) or arealready close to sensitivity to existing LUX and XENONIT experiments

                    A recent paper [25] Scalar Simplified Models for Dark Matter looks at Dirac fermiondark matter mediated by either a new scalar or pseudo-scalar to SM fermions This paperplaces bounds on the coupling combination gχgφ (coupling of DM to mediator times couplingof mediator to SM fermions) calculated from direct detection indirect detection and relicdensity by the mass of DM and mass of mediator Bounds are also calculated for colliderconstraints (mono-jet searches heavy-flavour searches and collider bounds on the 125 GeVHiggs are used to place limits on the monojet and heavy flavor channels which can betranslated into limits on the Higgs coupling to dark matter and experimental measurements

                    12 Literature review 6

                    of the Higgs width are used to constrain the addition of new channels to Higgs decay) Thepaper models the top loop in a full theory using the Monte Carlo generator MCFM [26]and extends the process to accommodate off-shell mediator production and decay to a darkmatter pair While this paper does not consider the Majorana fermion dark matter model theconstraints on the coupling product may have some relevence to the present study if only asan order of magnitude indicator

                    Constraining dark sectors with monojets and dijets [27] studies a simplified model withDirac fermion dark matter and an axial-vector mediator They use two classes of constraint-searches for DM production in events with large amounts of missing transverse energy inassociation with SM particles (eg monojet events) and direct constraints on the mediatorfrom monojet searches to show that for example where the choice of couplings of themediator to quarks and DM gA

                    q =gAχ=1 all mediator masses in the range 130 GeV ltMR lt3

                    TeV are excluded

                    The paper Constraining Dark Sectors at Colliders Beyond the Effective Theory Ap-proach [28] considers Higgs portal models with scalar pseudo-scalar vector and axial-vectormediators between the SM and the dark sector The paper takes account of loop processesand gluon fusion as well as quark-mediator interactions using MCFM Apart from mediatortype the models are characterised by free parameters of mediator width and mass darkmatter mass and the effective coupling combination gχgφ (coupling of DM to mediator timescoupling of mediator to SM fermions) The conclusion is that LHC searches can proberegions of the parameter space that are not able to be explored in direct detection experiments(since the latter measurements suffer from a velocity suppression of the WIMP-nucleoncross-section)

                    The paper Constraining the Fermi-LAT excess with multi-jet plus EmissT collider searches

                    [29] shows that models with a pseudo-scalar or scalar mediator are best constrained by multi-jet final states with Emiss

                    T The analysis was done using POWHEG-BOX-V2 at LO and usesM2

                    T variables together with multi-jet binning to exploit differences in signal and backgroundin the different regions of phase space to obtain a more stringent set of constraints on DMproduction This paper largely motivated the current work in that it showed that collidersearches that were not traditionally thought of as dark matter searches were neverthelessoften sensitive to dark matter produced in association with heavy quarks The combinationof searches used in this paper reflects this fact

                    12 Literature review 7

                    122 Collider Constraints

                    In order to constrain this model with collider constraints (direct detection limits) we considerthe following papers by the CMS and ATLAS collaborations which describe searches at theLHC

                    ATLAS Experiments

                    bull Search for dark matter in events with heavy quarks and missing transverse momentumin p p collisions with the ATLAS detector [30]

                    bull Search for direct third-generation squark pair production in final states with missingtransverse momentum and two b-jets in

                    radics= 8 TeV pp collisions with the ATLAS

                    detector[31]

                    bull Search for direct pair production of the top squark in all-hadronic final states inprotonndashproton collisions at

                    radics=8 TeV with the ATLAS detector [32]

                    bull Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

                    radic(s)=8TeV pp collisions using 21 f bminus1 of

                    ATLAS data [33]

                    bull Search for direct top squark pair production in final states with two leptons inradic

                    s = 8TeV pp collisions using 20 f bminus1 of ATLAS data [34]

                    bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

                    radics=8 TeV [35]

                    CMS Experiments

                    bull Searches for anomalous tt production in p p collisions atradic

                    s=8 TeV [36]

                    bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

                    radics=8 TeV [37]

                    bull Search for the production of dark matter in association with top-quark pairs in thesingle-lepton final state in proton-proton collisions at

                    radics = 8 TeV [38]

                    bull Search for new physics in monojet events in p p collisions atradic

                    s = 8 TeV(CMS) [39]

                    bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

                    s = 8 TeV [40]

                    12 Literature review 8

                    bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theSingle-lepton Final State in pp collisions at

                    radics=8 TeV [41]

                    bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

                    s=8 TeV [42]

                    bull Search for neutral MSSM Higgs bosons decaying into a pair of bottom quarks [43]

                    bull Search for neutral MSSM Higgs bosons decaying into a pair of τ leptons in p p collision[44]

                    bull Search for dark matter extra dimensions and unparticles in monojet events in proton-proton collisions at

                    radics=8 TeV [45]

                    In particular we have implemented Madgraph simulations of certain data analysesapplying to collider experiments described in papers [32][33][34][31][30][41] Theremaining models were not implemented in Madgraph because of inability to validate themodels against published results in the time available We developed C++ analysis code toimplement the collider cuts after showering with Pythia and basic analysis with Delpheswith output to root files The C++ code allowed multiple experimental cuts to be applied tothe same simulation output The C++ code was first checked against the results of the variousexperiments above and then the simulations were rerun using the three modelsrsquo Lagrangians(implemented in FeynRules) described in this paper

                    Chapter 2

                    Review of Physics

                    21 Standard Model

                    211 Introduction

                    The Standard Model of particle physics (SM) has sometimes been described as the theoryof Almost Everything The SM is a field theory which emerged in the second half of thetwentieth century from quantum mechanics and special relativity which themselves wereonly conceived after 1905 by Einsten Pauli Bohr Dirac and others In the first decades ofthe twenty first century the discovery of the Higgs boson [46][47] completed the pictureand confirmed the last previously unverified part of the SM The SM is a self consistenttheory that describes and classifies all known particles and their interactions (other thangravity) to an astonishing level of accuracy and detail However the theory is incomplete-it does not describe the complete theory of gravitation or account for DM (see below) ordark energy (the accelerating expansion of the universe) Related to these shortcomings aretheoretical deficiencies such as the asymmetry of matter and anti-matter the fact that the SMis inconsistent with general relativity (to the extent that one or both theories break down atspacetime singularities like the Big Bang and black hole event horizons)

                    212 Quantum Mechanics

                    Quantum Mechanics (QM) describes particles as probability waves which can be thoughtof as packets of energy which can be localised or spread through all space depending on how

                    21 Standard Model 10

                    accurately one measures the particlersquos momentum However QM is not itself consistent withspecial relativity It was realised too that because of relativity particles could be created anddestroyed and because of the uncertainty principle the theory should describe the probabilityof particles being created and destroyed on arbitrarily short timescales A quantum fieldtheory (QFT) is a way to describe this probability at every point of space and time

                    213 Field Theory

                    A field theory is described by a mathematical function of space and time called a LagrangianAs in classical mechanics where Lagrangians were introduced by Louis Lagrange at thebeginning of the nineteenth century to summarise dynamical systems the Lagrangian sum-marises the dynamics of space which is permeated by force fields The equations of motionarising from a Lagrangian are determined by finding the contours of the Lagrangian describedby stationary points (ie where the infinitesimal change in the Lagrangian is zero for a smallchange in field) Particles are generally described by singularities in the Lagrangian (wherethe values tend to infinity also called resonances)

                    214 Spin and Statistics

                    It was also realised that different types of particle needed to be represented by different fieldsin the Lagrangian Diracrsquos discovery that particles with spin 12 quantum numbers couldbe represented by a field obtained by effectively taking the square root of the relativisticequation (Klein Gordon equation) was probably the most significant advance in quantumfield theory It required the introduction of a matrix operators called the gamma matricesor Dirac matrices which helped explain the already known fact that spin half particles orfermions obey Fermi-Dirac statistics ie when two spin half particles are exchanged thesign of the wavefunction must change and hence they cannot occupy the same quantum stateunlike bosons with integer spin that can occupy the same quantum state

                    Other particles need to be represented by fields called vector pseudo-vector scalar (as inthe Klein Gordon equation) pseudo-scalar and complex fields A vector field is really fourfields one for each dimension of space plus time and is parameterised by the four gammamatrices γmicro which transform among themselves in a relativistically invariant way (Lorentzinvariance) A pseudo-vector field is a vector field which transforms in the same way but

                    21 Standard Model 11

                    with additional sign change under parity transformations (a simultaneous reversal of the threespatial directions) and is represented by γmicroγ5 A pseudo-scalar is a scalar which changes signunder parity transformations and a complex scalar field is represented by complex numbers

                    215 Feynman Diagrams

                    QFT took another giant leap forward when Richard Feynman realised that the propagationof particles described by the field could be described by expanding the so called S-matrixor scattering matrix derived from calculating the probability of initial states transitioning tofinal states The expansion is similar to a Taylor expansion in mathematics allowing for thespecial commuting properties of elements of the field The Feynman diagrams representingindividual terms in the expansion can be interpreted as interactions between the particlecomponents The S-Matrix (or more correctly the non trivial part called the T-Matrix) isderived using the interaction Hamiltonian of the field The Hamiltonian is an operator whoseeigenvalues or characteristic values give the energy quanta of the field The T-matrix canalso be described by a path integral which is an integral over all of quantum space weightedby the exponential of the Lagrangian This is completely equivalent to the Hamiltonianformulation and shows how the Lagrangian arises naturally out of the Hamiltonian (in factthe Lagrangian is the Legendre transformation of the Hamiltonian)

                    Figure 21 Feynman Diagram of electron interacting with a muon

                    γ

                    eminus

                    e+

                    micro+

                    microminus

                    The diagram Fig21 is a tree diagram (diagrams with closed loops indicate self in-teractions and are higher order terms in the expansion) and shows an electron (eminus) and ananti-muon (micro+) interacting by exchanging a photon

                    21 Standard Model 12

                    216 Gauge Symmetries and Quantum Electrodynamics (QED)

                    The Dirac equation with its gamma matrices ushered in the use of group theory to describethe transformation of fields (because the multiplication table of the gamma matrices isclosed and described by group theory) It also ushered in the idea of symmetries within theLagrangian- for example consider a Lagrangian

                    ψ(ipart minusm)ψ (21)

                    The Lagrangian is invariant under a global U(1) transformation which acts on the fieldsand hence the derivatives as follows

                    ψ rarr eiqαψ ψ rarr ψeminusiqα partmicroψ rarr eiqα

                    partmicroψ (22)

                    where qα is a global phase and α is a continuous parameter

                    A symmetry leads by Noetherrsquos theorem to a conserved current jmicro = qψγmicroψ and is thebasis of charge conservation since partmicro jmicro = 0 and q =

                    intd3x j0(x)

                    By promoting this transformation to a local one (ie α rarr α(x) where x = xmicro are thespace time co-ordinates) the transformations become

                    ψ rarr eiqα(x)ψ ψ rarr ψeminusiqα(x)partmicroψ rarr eiqα(x)(partmicroψ + iq(partmicroα(x))ψ) (23)

                    The derivative transformation introduces an extra term which would spoil the gaugeinvariance This can be solved by introducing a new field the photon field Amicro(x) whichinteracts with the fermion field ψ and transforms under a U(1) transformation

                    Amicro rarr Amicro minuspartmicroα(x) (24)

                    If we replace normal derivatives partmicro with covariant derivatives Dmicro = (partmicro + iqAmicro(x)) in theLagrangian then the extra term is cancelled and we can now see that the covariant derivativetransforms like ψ - under a local U(1) transform ie

                    Dmicro rarr Dmicroψprime = eiqα(x)Dmicroψ (25)

                    The additional term iqAmicro(x) can be interpreted in a geometric sense as a connectionwhich facilitates the parallel transport of vectors along a curve in a parallel consistent manner

                    21 Standard Model 13

                    We must add a gauge-invariant kinetic term to the Lagrangian to describe the dynamicsof the photon field This is given by the electromagnetic field strength tensor

                    Fmicroν = partmicroAν minuspartνAmicro (26)

                    The full Quantum Electrodynamics (QED) Lagrangian is then for electrons with mass mand charge -e

                    LQED = ψ(i Dminusm)ψ minus 14

                    Fmicroν(X)Fmicroν(x) (27)

                    This Lagrangian is Lorentz and U(1) guage invariant The interaction term is

                    Lint =+eψ Aψ = eψγmicro

                    ψAmicro = jmicro

                    EMAmicro (28)

                    where jmicro

                    EM is the electromagnetic four current

                    217 The Standard Electroweak Model

                    The Standard Electroweak (EW) Model was created by Glashow-Weinberg-Salam (also calledthe GWS model) and unifies the electomagnetic and weak interactions of the fundamentalparticles

                    The idea of gauge symmetry was extended to the idea of a non-abelian gauge symmetry(essentially the operators defining the symmetry do not commute and so are represented bymatrices- in this case the Pauli matrices for the SU(2) group) - the gauge group is the directproduct G = SU(2)

                    otimesU(1) It was known that weak interactions were mediated by Wplusmn

                    and Z0 particles where the superscripts are the signs of the electromagnetic charge Thesewere fairly massive but it was also known that through spontaneous symmetry breaking ofa continuous global non-abelian symmetry there would be one massive propagation modefor each spontaneously broken-symmetry generator and massless modes for the unbrokengenerators (the so-called Goldstone bosons)

                    This required an extra field to represent the gauge bosons called Bmicro (a vector fieldhaving the 4 spacetime degrees of freedom) With this identification the covariant derivativebecomes

                    Dmicro = partmicro minus igAmicro τ

                    2minus i

                    gprime

                    2Y Bmicro (29)

                    21 Standard Model 14

                    Here Y is the so called weak hypercharge of the field Bmicro which is a quantum numberdepending on particle (lepton or quark) g and gprime are two independent parameters in theGWS theory and the expression Amicro τ implies a sum over 3 components Aa

                    micro a=123 and thePauli matrices τa

                    This isnrsquot the full story because it had been known since 1956 when Chin Shuing Wuand her collaborators confirmed in an experiment prompted by Lee and Yang that nature isnot ambivalent to so-called parity It had been assumed that the laws of nature would notchange if all spatial coordinates were reversed (a parity transformation) The experimentinvolved the beta decay of cooled 60C atoms (more electrons were emitted in the backwardhemisphere with respect to spin direction of the atoms) This emission involved the weakinteraction and required that the Lagrangian for weak interactions contain terms which aremaximally parity violating- which turn out to be currents of the form

                    ψ(1minus γ5)γmicro

                    ψ (210)

                    The term

                    12(1minus γ

                    5) (211)

                    projects out a left-handed spinor and can be absorbed into the spinor to describe left-handedleptons (and as later discovered quarks) The right handed leptons are blind to the weakinteraction

                    The processes describing left-handed current interactions are shown in Fig 22

                    Analogously to isospin where the proton and neutron are considered two eigenstates ofthe nucleon we postulate that Bmicro is the gauge boson of an SU(2)transformation coupled toweak SU(2) doublets (

                    νe

                    eminus

                    )

                    (ud

                    ) (212)

                    We may now write the weak SU(2) currents as eg

                    jimicro = (ν e)Lγmicro

                    τ i

                    2

                    e

                    )L (213)

                    21 Standard Model 15

                    Figure 22 Weak Interaction Vertices [48]

                    where τ i are the Pauli-spin matrices and the subscript L denotes the left-handed componentof the spinor projected out by Eqn211 The third current corresponding to i=3 is called theneutral current as it doesnrsquot change the charge of the particle involved in the interaction

                    We could postulate that the electromagnetic current is a linear combination of the leftand right elecromagnetic fields eL = 1

                    2(1minus γ5)e and eR = 12(1+ γ5)e

                    jemmicro = eLγmicroQeL + eRγmicroQeR (214)

                    where Q is the electromagnetic operator and eR is a singlet However this is not invariantunder SU(2)L To construct an invariant current we need to use the SU(2)L doublet 212The current which is an SU(2)L invariant U(1) current is

                    jYmicro = (ν e)LγmicroYL

                    e

                    )L+ eRγmicroYReR (215)

                    where hypercharges YL and YR are associated with the U(1)Y symmetry jYmicro is a linearcombination of the electromagnetic current and the weak neutral current thus explaining whythe hypercharge differs for the left and right-handed components since the electromagneticcurrent has the same charge for both components whilst the weak current exists only forleft-handed particles j3

                    micro and the third component of weak isospin T 3 allows us to calculate

                    21 Standard Model 16

                    the relationship Y = 2(QminusT 3) which can be seen by writing jYmicro as a linear combination ofj3micro (the weak neutral current that doesnrsquot change the charge of the particle involved in the

                    interaction) and jemmicro (using the convention that the jYmicro is multiplied by 1

                    2 to match the samefactor implicit in j3

                    micro ) Substituting

                    τ3 =

                    (1 00 minus1

                    )(216)

                    into equation 213 for i=3 and subtracting equation 215 and equating to 12 times equation

                    214

                    we get

                    eLγmicroQeL + eRγmicroQeR minus (νLγmicro

                    12

                    νL minus eLγmicro

                    12

                    eL) =12

                    eRγmicroYReR +12(ν e)LγmicroYL

                    e

                    )L (217)

                    from which we can read out

                    YR = 2QYL = 2Q+1 (218)

                    and T3(eR) = 0 T3(νL) =12 and T3(eL) =

                    12 The latter three identities are implied by

                    the fraction 12 inserted into the definition of equation 213

                    The Lagrangian kinetic terms of the fermions can then be written

                    L =minus14

                    FmicroνFmicroν minus 14

                    GmicroνGmicroν

                    + sumgenerations

                    LL(i D)LL + lR(i D)lR + νR(i D)νR

                    + sumgenerations

                    QL(i D)QL +UR(i D)UR + DR(i D)DR

                    (219)

                    LL are the left-handed lepton doublets QL are the left-handed quark doublets lRνR theright handed lepton singlets URDR the right handed quark singlets

                    The field strength tensors are given by

                    Fmicroν = partmicroAν minuspartνAmicro minus ig[Amicro Aν ] (220)

                    21 Standard Model 17

                    andGmicroν = partmicroBν minuspartνBmicro (221)

                    Bmicro is an abelian U(1) gauge field while Amicro is non-abelian and Fmicroν contains the com-mutator of the field through the action of the covariant derivative [Dmicro Dν ] This equationcontains no mass terms terms for the field Amicro for good reason- this would violate gaugeinvariance We know that the gauge bosons of the weak force have substantial mass and theway that this problem has been solved is through the Higgs mechanism

                    218 Higgs Mechanism

                    To solve the problem of getting gauge invariant mass terms three independent groupspublished papers in 1964 The mechanism was named after Peter Higgs To introduce theconcept consider a complex scalar field φ that permeates all space Consider a potential forthe Higgs field of the form

                    minusmicro2φ

                    daggerφ +λ (φ dagger

                    φ2) (222)

                    which has the form of a mexican hat Fig 23 This potential is still U(1) gauge invariant andcan be added to our Lagrangian together with kinetic+field strength terms to give

                    L = (Dmicroφ)dagger(Dmicroφ)minusmicro2φ

                    daggerφ +λ (φ dagger

                    φ2)minus 1

                    4FmicroνFmicroν (223)

                    It is easily seen that this is invariant to the transformations

                    Amicro rarr Amicro minuspartmicroη(x) (224)

                    φ(x)rarr eieη(x)φ(x) (225)

                    The minimum of this field is not at 0 but in a circular ring around 0 with a vacuum

                    expectation value(vev)radic

                    micro2

                    2λequiv vradic

                    2

                    We can parameterise φ as v+h(x)radic2

                    ei π

                    Fπ where h and π are referred to as the Higgs Bosonand Goldstone Boson respectively and are real scalar fields with no vevs Fπ is a constant

                    21 Standard Model 18

                    Figure 23 Higgs Potential [49]

                    Substituting this back into the Lagrangian 223 we get

                    minus14

                    FmicroνFmicroν minusevAmicropartmicro

                    π+e2v2

                    2AmicroAmicro +

                    12(partmicrohpart

                    microhminus2micro2h2)+

                    12

                    partmicroπpartmicro

                    π+(hπinteractions)(226)

                    This Lagrangian now seems to describe a theory with a photon Amicro of mass = ev a Higgsboson h with mass =

                    radic2micro and a massless Goldstone π

                    However there is a term mixing Amicro and part microπ in this expression and this shows Amicro and π

                    are not independent normal coordinates and we can not conclude that the third term is a massterm This π minusAmicro mixing can be removed by making a gauge transformation in equations224 and 225 making

                    φrarrv+h(x)radic2

                    ei π

                    Fπminusieη(x) (227)

                    and setting πrarr π

                    Fπminus eη(x) = 0 This choosing of a point on the circle for the vev is called

                    spontaneous symmetry breaking much like a vertical pencil falling and pointing horizontallyin the plane This choice of gauge is called unitary gauge The Goldstone π will thencompletely disappear from the theory and one says that the Goldstone has been eaten to givethe photon mass

                    21 Standard Model 19

                    This can be confirmed by counting degrees of freedom before and after spontaneoussymmetry breaking The original Lagrangian had four degrees of freedom- two from thereal massless vector field Amicro (massless photons have only two independent polarisationstates) and two from the complex scalar field (one each from the h and π fields) The newLagrangian has one degree of freedom from the h field and 3 degrees of freedom from themassive real vector field Amicro and the π field has disappeared

                    The actual Higgs field is postulated to be a complex scalar doublet with four degrees offreedom

                    Φ =

                    (φ+

                    φ0

                    )(228)

                    which transforms under SU(2) in the fundamental representation and also transforms underU(1) hypercharge with quantum numbers

                    Table 21 Quantum numbers of the Higgs field

                    T 3 Q Yφ+

                    12 1 1

                    φ0 minus12 1 0

                    We can parameterise the Higgs field in terms of deviations from the vacuum

                    Φ(x) =(

                    η1(x)+ iη2(x)v+σ(x)+ iη3(x)

                    ) (229)

                    It can be shown that the three η fields can be transformed away by working in unitarygauge By analogy with the scalar Higgs field example above the classical action will beminimised for a constant value of the Higgs field Φ = Φ0 for which Φ

                    dagger0Φ0 = v2 This again

                    defines a range of minima and we can pick one arbitarily (this is equivalent to applying anSU(2)timesU(1) transform to the Higgs doublet transforming the top component to 0 and thebottom component to Φ0 = v) which corresponds to a vacuum of zero charge density andlt φ0 gt=+v

                    In this gauge we can write the Higgs doublet as

                    Φ =

                    (φ+

                    φ0

                    )rarr M

                    (0

                    v+ H(x)radic2

                    ) (230)

                    where H(x) is a real Higgs field v is the vacuum expectation and M is an SU(2)timesU(1)matrix The matrix Mdagger converts the isospinor into a down form (an SU(2) rotation) followedby a U(1) rotation to a real number

                    21 Standard Model 20

                    If we consider the Higgs part of the Lagrangian

                    minus14(Fmicroν)

                    2 minus 14(Bmicroν)

                    2 +(DmicroΦ)dagger(DmicroΦ)minusλ (Φdagger

                    Φminus v2)2 (231)

                    Substituting from equation 230 into this and noting that

                    DmicroΦ = partmicroΦminus igW amicro τ

                    aΦminus 1

                    2ig

                    primeBmicroΦ (232)

                    We can express as

                    DmicroΦ = (partmicro minus i2

                    (gA3

                    micro +gprimeBmicro g(A1micro minusA2

                    micro)

                    g(A1micro +A2

                    micro) minusgA3micro +gprimeBmicro

                    ))Φ equiv (partmicro minus i

                    2Amicro)Φ (233)

                    After some calculation the kinetic term is

                    (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

                    14(v+

                    Hradic2)2[A 2]22 (234)

                    where the 22 subscript is the index in the matrix

                    If we defineWplusmn

                    micro =1radic2(A1

                    micro∓iA2micro) (235)

                    then [A 2]22 is given by

                    [A 2]22 =

                    (gprimeBmicro +gA3

                    micro

                    radic2gW+

                    microradic2gWminus

                    micro gprimeBmicro minusgA3micro

                    ) (236)

                    We can now substitute this expression for [A 2]22 into equation 234 and get

                    (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

                    14(v+

                    Hradic2)2(2g2Wminus

                    micro W+micro +(gprimeBmicro minusgA3micro)

                    2) (237)

                    This expression contains mass terms for three massive gauge bosons Wplusmn and the combi-nation of vector fields gprimeBmicro minusgA3

                    micro where note

                    21 Standard Model 21

                    Table 22 Weak Quantum numbers of Lepton and Quarks

                    T 3 Q YνL

                    12 0 -1

                    lminusL minus12 -1 -1

                    νR 0 0 0lminusR 0 -1 -2UL

                    12

                    23

                    13

                    DL minus12 minus1

                    313

                    UR 0 23

                    43

                    DR 0 minus13 minus2

                    3

                    Wminusmicro = (W+

                    micro )dagger equivW 1micro minus iW 2

                    micro (238)

                    Then the mass terms can be written

                    12

                    v2g2|Wmicro |2 +14

                    v2(gprimeBmicro minusgA3micro)

                    2 (239)

                    W+micro and W microminus can be associated with the W boson and its anti-particle and (minusgprimeBmicro +

                    gA3micro) with the Z Boson (after normalisation by

                    radicg2 +(gprime

                    )2) The combination gprimeA3micro +gBmicro

                    is orthogonal to Z and remains massless This can be associated with the Photon (afternormalisation) These predict mass terms for the W and Z bosons mW = vgradic

                    2and mZ =

                    vradic2

                    radicg2 +(gprime

                    )2It is again instructive to count the degrees of freedom before and after the Higgs mech-

                    anism Before we had a complex doublet Φ with four degrees of freedom one masslessB with two degrees of freedom and three massless W a fields each with two each givinga total of twelve degrees of freedom After spontaneous symmetry breaking we have onereal scalar Higgs field with one degree of freedom three massive weak bosons with nineand one massless photon with two again giving twelve degrees of freedom One says thescalar degrees of freedom have been eaten to give the Wplusmn and Z bosons their longitudinalcomponents (ie mass)

                    Table 22 shows T 3 the weak isospin quantum number (which describes how a particletransforms under SU(2)) Y the weak hypercharge (which describes how the particle trans-

                    21 Standard Model 22

                    forms under U(1)) and Q the electric charge of leptons and quarks The former two enterdirectly into the Lagrangian while the latter can be derived as a function of the former two

                    Strictly speaking the right handed neutrino which has no charge may not exist and is notpart of the standard model although some theories postulate a mass for this (hypothetical)particle

                    Writing gauge invariant couplings between ELeRφ with the vacuum expectation ofφ = v (Yukawa matrices λ

                    i ju and λ

                    i jd respectively for the up and down quarks) we get mass

                    terms for the quarks (and similarly for the leptons)

                    Mass terms for quarks minussumi j[(λi jd Qi

                    Lφd jR)+λ

                    i ju εab(Qi

                    L)aφlowastb u j

                    R +hc]

                    Mass terms for leptonsminussumi j[(λi jl Li

                    Lφ l jR)+λ

                    i jν εab(Li

                    L)aφlowastb ν

                    jR +hc]

                    Here hc is the hermitian conjugate of the first terms L and Q are lepton and quarkdoublets l and ν are the lepton singlet and corresponding neutrino d and u are singlet downand up quarks εab is the Levi-Civita (anti-symmetric) tensor

                    If we write the quark generations in a basis that diagonalises the Higgs couplings weget a unitary transformations Uu and Ud for the up and down quarks respectively which iscompounded in the boson current (Udagger

                    u Ud) called the CKM matrix where off-diagonal termsallow mixing between quark generations (this is also called the mass basis as opposed tothe flavour basis) A similar argument can give masses to right handed sterile neutrinos butthese can never be measured However a lepton analogue of the CKM matrix the PNMSmatrix allows left-handed neutrinos to mix between various generations and oscillate betweenflavours as they propagate through space allowing a calculation of mass differences

                    219 Quantum Chromodynamics

                    The final plank in SM theory tackling the strong force Quantum Chromodynamics wasproposed by Gell-Mann and Zweig in 1963 Three generations of quarks were proposedwith charges of 23 for uc and t and -13 for ds and b Initially SU(2) isospin was consideredtransforming u and d states but later irreducible representations of SU(3) were found to fit theexperimental data The need for an additional quantum number was motivated by the fact thatthe lightest excited state of the nucleon consisted of sss (the ∆++) To reconcile the baryon

                    21 Standard Model 23

                    spectrum with spin statistics colour was proposed Quarks transform under the fundamentalor 3 representation of colour symmetry and anti-quarks under 3 The inner product of 3and 3 is an invariant under SU(3) as is a totally anti-symmetric combination ε i jkqiq jqk (andthe anti-quark equivalent) It was discovered though deep inelastic scattering that quarksshowed asymptotic freedom (the binding force grew as the seperation grew) Non-abeliangauge theories showed this same property and colour was identified with the gauge quantumnumbers of the quarks The quanta of the gauge field are called gluons Because the forcesshow asymptotic freedom - ie the interaction becomes weaker at high energies and shorterdistances perturbation theory can be used (ie Feynman diagrams) at these energies but forlonger distances and lower energies lattice simulations must be done

                    2110 Full SM Lagrangian

                    The full SM can be written

                    L =minus14

                    BmicroνBmicroν minus 18

                    tr(FmicroνFmicroν)minus 12

                    tr(GmicroνGmicroν)

                    + sumgenerations

                    (ν eL)σmicro iDmicro

                    (νL

                    eL

                    )+ eRσ

                    micro iDmicroeR + νRσmicro iDmicroνR +hc

                    + sumgenerations

                    (u dL)σmicro iDmicro

                    (uL

                    dL

                    )+ uRσ

                    micro iDmicrouR + dRσmicro iDmicrodR +hc

                    minussumi j[(λ

                    i jl Li

                    Lφ l jR)+λ

                    i jν ε

                    ab(LiL)aφ

                    lowastb ν

                    jR +hc]

                    minussumi j[(λ

                    i jd Qi

                    Lφd jR)+λ

                    i ju ε

                    ab(QiL)aφ

                    lowastb u j

                    R +hc]

                    + (Dmicroφ)dagger(Dmicroφ)minusmh[φφ minus v22]22v2

                    (240)

                    where σ micro are the extended Pauli matrices

                    (1 00 1

                    )

                    (0 11 0

                    )

                    (0 minusii 0

                    )

                    (1 00 minus1

                    )

                    The first line gives the gauge field strength terms the second the lepton dynamical termsthe third the quark dynamical terms the fourth the lepton (including neutrino) mass termsthe fifth the quark mass terms and the sixth the Higgs terms

                    The SU(3) gauge field is given by Gmicroν and which provides the colour quantum numbers

                    21 Standard Model 24

                    Figure 24 Standard Model Particles and Forces [50]

                    Note The covariant derivatives operating on the left-handed leptons contain only theU(1) and SU(2) gauge fields Bmicro Fmicro whereas those operating on left-handed quarks containU(1)SU(2) and SU(3) gauge fields Bmicro Fmicro Gmicro The covariant derivatives operating onright handed electrons and quarks are as above but do not contain the SU(2) gauge fieldsRight handed neutrinos have no gauge fields

                    The sums over i j above are over the different generations of leptons and quarks

                    The particles and forces that emerge from the SM are shown in Fig 24

                    22 Dark Matter 25

                    22 Dark Matter

                    221 Evidence for the existence of dark matter

                    2211 Bullet Cluster of galaxies

                    Most of the mass of the cluster as measured by a gravitational lensing map is concentratedaway from the centre of the object The visible matter (as measured by X-ray emission) ismostly present in the centre of the object The interpretation is that the two galaxy clustershave collided perpendicular to the line of sight with the mostly collisionless dark matterpassing straight through and the visible matter interacting and being trapped in the middleIt was estimated that the ratio of actual to visible matter was about 16 [51]

                    Figure 25 Bullet Cluster [52]

                    2212 Coma Cluster

                    The first evidence for dark matter came in the 1930s when Fritz Zwicky observed the Comacluster of galaxies over 1000 galaxies gravitationally bound to one another He observed thevelocities of individual galaxies in the cluster and estimated the amount of matter based onthe average galaxy He calculated that many of the galaxies should be able to escape fromthe cluster unless the mass was about 10 times higher than estimated

                    22 Dark Matter 26

                    2213 Rotation Curves [53]

                    Since those early observations many spiral galaxies have been observed with the rotationcurves calculated by doppler shift with the result that in all cases the outermost stars arerotating at rates fast enough to escape their host galaxies Indeed the rotation curves arealmost flat indicating large amounts of dark matter rotating with the galaxies and holdingthe galaxies together

                    Figure 26 Galaxy Rotation Curves [54]

                    2214 WIMPS MACHOS

                    The question is if the particles that make up most of the mass of the universe are not baryonswhat are they It is clear that the matter does not interact significantly with radiation so mustbe electrically neutral It has also not lost much of its kinetic energy sufficiently to relax intothe disks of galaxies like ordinary matter Massive particles may survive to the present ifthey carry a conserved additive or multiplicative quantum number This is so even if equalnumbers of particles and anti-particles can annihilate if the number densities become too lowto reduce the density further These particles have been dubbed Weakly interacting MassiveParticles (WIMPS) as they interact with ordinary matter only through a coupling strengthtypically encountered in the weak interaction They are predicted to have frozen out ofthe hot plasma and would give the correct relic density based solely on the weak interactioncross section (sometimes called the WIMP Miracle) Another possibility called MassiveCompact Halo Objects (MACHOS)- are brown dwarf stars Jupiter mass objects or remnantsfrom an earlier generation of star formation

                    22 Dark Matter 27

                    2215 MACHO Collaboration [55]

                    In order to estimate the mass of MACHOS a number of experiments have been conductedusing microlensing which makes use of the fact that if a massive object lies in the line ofsight of a much more distant star the light from the more distant star will be lensed and forma ring around the star It is extemely unlikely that the MACHO will pass directly through theline of sight but if there is a near miss the two images will be seperated by a small angleThe signature for microlensing is a time-symmetric brightening of a star as the MACHOpasses near the line of sight called a light curve One can obtain statistical information aboutthe MACHO masses from the shape and duration of the light curves and a large number ofstars (many million) must be monitored The MACHO Collaboration [55] have monitoredover 11 million stars and less than 20 events were detected The conclusion was that thiscould account for a small addition (perhaps up to 20 ) to the visible mass but not enoughto account for Dark Matter

                    2216 Big Bang Nucleosynthesis (BBN) [56]

                    Another source of evidence for DM comes from the theory of Nucleosynthesis in the fewseconds after the Big Bang In the first few minutes after the big bang the temperatures wereso high that all matter was dissociated but after about 3 minutes the temperature rapidlycooled to about 109 Kelvin Nucleosynthesis could begin when protons and neutrons couldstick together to form Deuterium (Heavy Hydrogen) and isotopes of Helium The ratio ofDeuterium to Hydrogen and Helium is very much a function of the baryon density and alsothe rate of expansion of the universe at the time of Nucleosynthesis There is a small windowwhen the density of Baryons is sufficient to account for the ratios of the various elementsThe rate of expansion of the universe is also a function of the total matter density The ratiosof these lighter elements have been calculated by spectral measurements with Helium some25 of the mass of the universe Deuterium 001 The theory of BBN accounts for theratios of all of the lighter elements if the ratio of Baryon mass to total mass is a few percentsuggesting that the missing matter is some exotic non-baryonic form

                    22 Dark Matter 28

                    2217 Cosmic Microwave Background [57]

                    The discovery of a uniform Microwave Background radiation by Penzias and Wilson in1965 was confirmation of the theory first proposed by Gamow and collaborators that theuniverse should be filled with black body radiation an afterglow of the big bang TheCOBE satelite launched in 1989 confirmed this black body radiation with temperature of2725+-002K with 95 confidence While extremely uniform it found tiny irregularitieswhich could be accounted for by irregularities in the density of the plasma at last scattering(gravitational redshifts or blueshifts called the Sachs Wolfe effect) and temperature andvelocity fluctuations in the plasma The anisotropy of the background has been furtheranalysed by WMAP (the Wilkinson Microwave Anisotropy Probe) launched in 2001 Thefluctuations in the density of the plasma are what account for the seeds of galaxy formationHowever detailed calculations have found that the fluctuations were too small to account forthe observed galaxy distribution without an electrically neutral gravitating matter in additionto ordinary matter that only became electrically neutral at the time of recombination [58]

                    In May 2009 the European Space Agency (ESA) launched the Planck sattelite a spacebased observatory to observe the universe at wavelengths between 03mm and 111 mmbroadly covering the far-infrared microwave and high frequency domains The missionrsquosmain goal was to study the cosmic microwave background across the whole sky at greatersensitivity and resolution than ever before Using the information gathered and the standardmodel of cosmology ESA was able to calculate cosmological parameters including the relicdensity of dark matter remaining after the Big Bang to a high degree of precision withinthe framework of the standard model of cosmology The standard model of cosmology canbe described by a relatively small number of parameters including the density of ordinarymatter dark matter and dark energy the speed of cosmic expansion at the present epoch (alsoknown as the Hubble constant) the geometry of the Universe and the relative amount ofthe primordial fluctuations embedded during inflation on different scales and their amplitude[59]

                    2218 LUX Experiment - Large Underground Xenon experiment [16]

                    The LUX experiment is conducted at the Sanford Underground research facility one mileunderground to minimise the background of cosmic rays The detector is a time projectionchamber allowing a 3D positioning of interactions occurring within its active volume LUX

                    22 Dark Matter 29

                    Figure 27 WMAP Cosmic Microwave Background Fluctuations [58]

                    Figure 28 Dark Matter Interactions [60]

                    uses as its target 368 kiliograms of liquefied ultra pure xenon which is a scintillator Interac-tions inside the xenon will create an amount of light proportional to the energy depositedThat light can be collected on arrays of light detectors sensitive to a single photon

                    Xenon is very pure and the amount of radiation originating within the target is limitedAlso being three times denser than water it can stop radiation originating from outside thedetector The 3D detector focuses on interactions near the center which is a quiet regionwhere rare dark matter interactions might be detected [16]

                    22 Dark Matter 30

                    222 Searches for dark matter

                    2221 Dark Matter Detection

                    Fig 28 illustrates the reactions studied under the different forms of experiment designed todetect DM

                    Direct detection experiments are mainly underground experiments with huge detectorsattempting to detect the recoil of dark matter off ordinary matter such as LUX [16] CDMS-II [61] XENON100 [11] COUPP [62] PICASSO [63] ZEPLIN [10] These involveinteractions between DM and SM particles with final states involving DM and SM particles

                    Indirect detection involves the search for the annihilation of DM particles in distantastrophysical sources where the DM content is expected to be large Example facilitiesinclude space based experiments (EGRET [64] PAMELA [18] FERMI-LAT [65]) balloonflights (ATIC [66] PEBS [67] PPB-BETS [68]) or atmospheric Cherenkov telescopes(HESS [69]MAGIC [70]) and neutrino telescopes (SuperKamiokande [71] IceCube [72]ANTARES [73] KM3NET [74]) Indirect Detection involves DM particles in the initial stateand SM particles in the final state

                    Collider detection involves SM particles in the initial state creating DM particles in thefinal state which can be detected by missing transverse energy and explained by variousmodels extending the SM

                    223 Possible signals of dark matter

                    224 Gamma Ray Excess at the Centre of the Galaxy [65]

                    The center of the galaxy is the most dense and active region and it is no surprise that itproduces intense gamma ray sources When astronomers subtract all known gamma raysources there remains an excess which extends out some 5000 light years from the centerHooper et al find that the annihilation of dark matter particles with a mass between 31 and40 Gev fits the excess to a remarkable degree based on Gamma Ray spectrum distributionand brightness

                    23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 31

                    Figure 29 Gamma Ray Excess from the Milky Way Center [75]

                    23 Background on ATLAS and CMS Experiments at theLarge Hadron collider (LHC)

                    The collider constraints in this paper are based on two of the experiments that are part of theLHC in CERN

                    Figure 210 ATLAS Experiment

                    The LHC accelerates proton beams in opposite directions around a 27 km ring of super-conducting magnets until they are made to collide at an interaction point inside one of the 4experiments placed in the ring An upgrade to the LHC has just been performed providing65 TeV per beam

                    23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 32

                    231 ATLAS Experiment

                    The ATLAS detector consists of concentric cylinders around the interaction point whichmeasure different aspects of the particles that are flung out of the collision point There are 4major parts the Inner Detector the Calorimeters the Muon spectrometer and the magnetsystems

                    2311 Inner Detector

                    The inner detector extends from a few centimeters to 12 meters from the interaction point andmeasures details of the charged particles and their momentum interacting with the materialIf particles seem to originate from a point other than the interaction point this may indicatethat the particle comes from the decay of a hadron with a bottom quark This is used to b-tagcertain particles

                    The inner detector consists of a pixel detector with three concentric layers and threeendcaps containing detecting silicon modules 250 microm thick The pixel detector has over 80million readout channels about 50 percent of the total channels for the experiment All thecomponents are radiation hardened to cope with the extremely high radiation in the proximityof the interaction point

                    The next layer is the semi-conductor layer (SCT) similar in concept to the pixel detectorbut with long narrow strips instead of pixels to cover a larger area It has 4 double layers ofsilicon strips with 63 million readout channels

                    The outermost component of the inner detector is the Transition Radiation Tracker It is acombination of a straw tracker and a transition radiation detector The detecting elementsare drift tubes (straws) each 4mm in diameter and up to 144 cm long Each straw is filledwith gas which becomes ionised when the radiation passes through The straws are held at-1500V with a thin wire down the center This drives negatively ionised atoms to the centercausing a pulse of electricity in the wire This causes a pattern of hit straws allowing the pathof the particles to be determined Between the straws materials with widely varying indicesof refraction cause transition radiation when particles pass through leaving much strongersignals in some straws Transition radiation is strongest for highly relativistic particles (thelightest particles- electrons and positrons) allowing strong signals to be associated with theseparticles

                    23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 33

                    2312 Calorimeters

                    The inner detector is surrounded by a solenoidal magnet of 2 Tesla (there is also an outertoroidal magnet) but surrounding the inner magnet are the calorimeters to measure theenergy of particles by absorbing them There is an inner electromagnetic calorimeter whichmeasures the angle between the detectorrsquos beam axis and the particle (or more preciselypseudo-rapidity) and its angle in the perpendicular plane to a high degree of accuracy Theabsorbing material is lead and stainless steel and the sampling material liquid argon Thehadron calorimeter measures the energy of particles that interact via the strong force Theenergy absorbing material is scintillating steel

                    2313 Muon Specrometer

                    The muon spectrometer surrounds the calorimeters and has a set of 1200 chambers tomeasure the tracks of outgoing muons with high precision using a magnetic field andtriggering chambers Very few particles of other types pass through the calorimeters and aresubsequently measured

                    2314 Magnets

                    The magnets are in two layers the inner solenoidal and outer toroidal which together causecharged particles to curve in the magetic field allowing their momentum to be determinedThe outer toroidal magnet consists of 8 very large air core superconducting magnets whichsupply the field in the muon spectrometer

                    232 CMS Experiment

                    The CMS Detector is similar in design to the ATLAS Detector with a single solenoidalmagnet which is the largest of its type ever built (length 13m and diameter 7m) and allowsthe tracker and calorimeter detectors to be placed inside the coil in a compact way Thestrong magnetic field gives a high resolution for measuring muons electrons and photons ahigh quality central tracking system and a hermetic hadron calorimeter designed to preventhadrons from escaping

                    23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 34

                    Figure 211 CMS Experiment

                    Chapter 3

                    Fitting Models to the Observables

                    31 Simplified Models Considered

                    In order to extend the findings of previous studies [19] to include collider constraints wefirst performed Bayesian scans [7] over three simplified models real scalar dark matter φ Majorana fermion χ and real vector Xmicro The purpose was to obtain the most probable regionsof each model then to use the resulting posterior samples to investigate whether the LHCexperiments have any sensitivity to the preferred regions

                    The three models couple to the mediator with interactions shown in the following table

                    Table 31 Simplified Models

                    Hypothesis real scalar DM Majorana fermion DM real vector DM

                    DM mediator int Lφ sup microφ mφ φ 2S Lχ sup iλχ

                    2 χγ5χS LX sup microX mX2 X microXmicroS

                    The interactions between the mediator and the standard fermions is assumed to be

                    LS sup f f S (31)

                    and in line with minimal flavour violation we only consider third generation fermions- ief = b tτ

                    For the purposes of these scans we consider the following observables

                    32 Observables 36

                    32 Observables

                    321 Dark Matter Abundance

                    We have assumed a single dark matter candidate which freezes out in the early universe witha central value as determined by Planck [21] of

                    ΩDMh2 = 01199plusmn 0031 (32)

                    h is the reduced hubble constant

                    The experimental uncertainty of the background relic density [59] is dominated by atheoretical uncertainty of 50 of the value of the scanned Ωh2 [76] We have added thetheoretical and the experimental uncertainties in quadrature

                    SD =radic(05Ωh2)2 + 00312 (33)

                    This gives a log likelihood of

                    minus05lowast (Ωh2 minus 1199)2

                    SD2 minus log(radic

                    2πSD) (34)

                    322 Gamma Rays from the Galactic Center

                    Assuming that the gamma ray flux excess from the galactic centre observed by Fermi-LAT iscaused by self annihilation of dark matter particles the differential flux

                    d2Φ

                    dEdΩ=

                    lt σv gt8πmχ

                    2 J(ψ)sumf

                    B fdN f

                    γ

                    dE(35)

                    has been calculated using both Micromegas v425 with a Navarro-Frenk-White (NFW) darkmatter galactic distribution function

                    ρ(r) = ρ0(rrs)

                    minusγ

                    (1+ rrs)3minusγ (36)

                    with γ = 126 and an angle of 5 to the galactic centre [19]

                    32 Observables 37

                    Here Φ is the flux dE is the differential energy lt σv gt the velocity averaged annihi-lation cross section of the dark matter across the galactic centre mχ the mass of the darkmatter particle B f =lt σv gt f σv is the annihilation fraction into the f f final state anddN f

                    γ dE is the energy distribution of photons produced in the annihilation channel with thefinal state f f The J factor is described below

                    The angle ψ to the galactic centre determines the normalisation of the gamma ray fluxthrough the factor

                    J(ψ) =int

                    losρ(l2 + r⊙2 minus2lr⊙cosψ)dl (37)

                    where ρ(r) is the dark matter density at distance r from the centre of the galaxy r⊙=85 kpcis the solar distance from the Galactic center ρ0 is set to reproduce local dark matter densityof 03GeVcm3

                    The central values for the gamma ray spectrum were taken from Fig 5 of [77] and thestandard deviations of individual flux points from the error bars on the graph In commonwith [78] we ignore possible segment to segment correlations

                    For the Fermi-LAT log likelihood we assumed each point was distributed with indepen-dent Gaussian likelihood with the experimental error for each data point given by the errorbar presented in Fig5 of [76] for each point We also assumed an equal theoretical errorfor each point which when added to the experimental error in quadrature gave an effectivelog likelihood for each point of minus05lowast (giminusdi)

                    2

                    2lowastσ2i

                    where gi are the calculated values and di theexperimental values and σi the experimental errors

                    323 Direct Detection - LUX

                    The LUX experiment [16] is the most constraining direct search experiment to date for awide range of WIMP models We have used LUXCalc [79] (modified to handle momentumdependent nucleon cross sections) to calculate the likelihood of the expected number ofsignal events for the Majorana fermion model

                    The likelihood function is taken as the Poisson distribution

                    L(mχ σ | N) = P(N | mχ σ) =(b+micro)Neminus(b+micro)

                    N (38)

                    32 Observables 38

                    where b= expected background of 064 and micro = micro(mχσ ) is the expected number of signalevents calculated for a given model

                    micro = MTint

                    infin

                    0dEφ(E)

                    dRdE

                    (E) (39)

                    where MT is the detector mass times time exposure and φ(E) is a global efficiency factorincorporating trigger efficiencies energy resolution and analysis cuts

                    The differential recoil rate of dark matter on nucleii as a function of recoil energy E

                    dRdE

                    =ρX

                    mχmA

                    intdvv f (v)

                    dσASI

                    dER (310)

                    where mA is the nucleon mass f (v) is the dark matter velocity distribution and

                    dσSIA

                    dER= Gχ(q2)

                    4micro2A

                    Emaxπ[Z f χ

                    p +(AminusZ) f χn ]

                    2F2A (q) (311)

                    where Emax = 2micro2Av2mA Gχ(q2) = q2

                    4m2χ

                    [24] is an adjustment for a momentum dependantcross section which applies only to the Majorana model

                    f χ

                    N =λχ

                    2m2SgSNN assuming that the relic density is the central value of 1199 We have

                    implemented a scaling of the direct search cross section in our code by multiplying gSNN by( Ω

                    Ω0)05 as a convenient method of allowing for the fact that the actual relic density enters

                    into the calculation of the cross section as a square

                    FA(q) is the nucleus form factor and

                    microA =mχmA

                    (mχ +mA)(312)

                    is the reduced WIMP-nucleon mass

                    The strength of the dark matter nucleon interaction between dark matter particles andthird generation quarks is

                    gSNN =2

                    27mN fT G sum

                    f=bt

                    λ f

                    m f (313)

                    where fT G = 1minus f NTuminus f N

                    Tdminus fTs and f N

                    Tu= 02 f N

                    Td= 026 fTs = 043 [20]

                    33 Calculations 39

                    For the real scalar and real vector models where the spin independent cross sections arenot momentum dependent we based the likelihood on the LUX 95 CL from [16] using acomplementary error function Li(d|mχ I) = 05lowastCEr f (dminusx(mχ )

                    σ) where x is the LUX limit

                    and d the calculated SI cross section We assumed a LUX error σ of 01lowastd lowastradic

                    2

                    33 Calculations

                    We coded the Lagrangians for the three models in Feynrules [4] Calculation of the expectedobservables (dark matter relic density Fermi-LAT gamma ray spectrum and LUX nucleonscattering amplitudes) were performed using Micromegas v425

                    331 Mediator Decay

                    A number of searches have previously been performed at the LHC which show sensitivity tothe decay of scalar mediators in the form of Higgs bosons (in beyond the standard modeltheories) to bottom quarks top quarks and τ leptons (all third generation fermions) Two inparticular [43 44] provide limits on the cross-sections for the production of the scalar timesthe branching ratios to brsquos and τrsquos In order to check these limits against each of the modelswe ran parameter scans in Madgraph at leading order over the top 900 points from the nestedsampling output from MultiNest [19]

                    The two processes were

                    1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to apair of τ leptons in pp collisions [44]

                    bull generate p p gt b b S where S is the scalar mediator

                    The process measures the decay of a scalar in the presence of b-jets that decay to τ+τminus

                    leptons Fig31 shows the main diagrams contributing to the process The leftmost diagramis a loop process and the centre and right are tree level however we only got Madgraph togenerated the middle diagram since we are interested in τ leptons and b quarks in the finalstate

                    The limit given in [44] only applies in the case of narrow resonances where narrow istaken to mean that the width of the resonance is less than 10 of its mass To test the validity

                    33 Calculations 40

                    Figure 31 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof τ leptons

                    of applying these limits to our models we show in Figures 32 to 34 the percentage widthof the resonance vs mS[GeV ] (pole mass of the scalar mediator) λb and λt (the couplingconstants to the b and t quarks) The probabilities are the posterior probabilities obtainedfrom MultiNest scans of the Majorana model varying the coupling constants and scalarmediator and calculating the decay width of the mediator The results are represented as 2dposterior probability heat maps in the central panels and 1d marginalised distributions on thex and y axes of each graph

                    16 18 20 22 24 26 28 30

                    log10(mS[GeV])

                    001

                    002

                    003

                    004

                    005

                    Widthm

                    S

                    00

                    04

                    08

                    12

                    0 100 200

                    Posterior Probability

                    Figure 32 WidthmS vs mS

                    The WidthmS was lt 01 at almost all points except where the mass of the scalar wasgreater than 1000 GeV λb was greater than 1 or λτ was greater than 1

                    This can be seen from the graphs in Figs 323334

                    33 Calculations 41

                    4 3 2 1 0

                    λb

                    001

                    002

                    003

                    004

                    005

                    WidthmS

                    000

                    015

                    030

                    045

                    0 100 200

                    Posterior Probability

                    Figure 33 WidthmS vs λb

                    5 4 3 2 1 0

                    λτ

                    001

                    002

                    003

                    004

                    005

                    WidthmS

                    000

                    015

                    030

                    045

                    0 100 200

                    Posterior Probability

                    Figure 34 WidthmS vs λτ

                    The decay widths are calculated in Madgraph at leading order and branching fractionscan be calculated to b t τ and χ (DM) The formulae for the calculations can be found indecayspy which is a subroutine of Madgraph which we have documented in Appendix BB1 Using these we calculated the branching ratio to τ+τminus so that the cross section timesbranching ratio can be compared to the limits in [80] shown in Chapters 56 and 7 in Figs5564 and 74 for each model

                    2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                    This scan calculates the contributions of b-quark associated Higgs boson production inthe center and right diagrams of Fig 35 reproduced from [43] The scans did not includePythia and Delphes as only production cross sections were required and the producedparticles were in the final state and required no matching or merging

                    The Madgraph processes were

                    bull generate p p gt b S where S is the scalar mediator

                    bull add process p p gt b S j

                    bull add process p p gt b S

                    33 Calculations 42

                    Figure 35 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof b quarks in the presence of at least one b quark

                    bull add process p p gt b S j

                    The cross section times branching ratios to bb for mediator production in association withat least one b quark is shown for each model in Chapters 56 and 7 in Figs 5765 and 75

                    332 Collider Cuts Analyses

                    We generated two parameter scans with parameter combinations from the Multinest outputwhich were the basis of the plots of Fig 6 in [19] The plots in Fig 6 were marginalisedto two parameters each but the raw Multinest output contained 5 parameters plus the priorprobabilities of each combination normalised to total 10

                    The first scan was for χ produced with b quarks In order to force the mediator to decayto χ only we had χ in the final state and defined jets to include b quarks in Madgraph Thenwe generated the processes

                    bull generate p p gt χ χ j

                    bull add process p p gt χ χ j j

                    Jet matching was on

                    The second scan was for t quarks produced in the final state

                    bull generate p p gt χ χ tt

                    33 Calculations 43

                    No jet matching was required because we did not mix final state jet multiplicities at thematrix element level

                    The outputs from these two processes were normalised to 21 f bminus1 and combined

                    The results of these scans prior to collider cuts calculations are shown in Figs 5161and 71 These show the posterior probabilities of the various parameter combinations withdifferent constraints applied seperately and then together A more detailed commentary isgiven in each of the sections

                    We then sampled 900 points based on posterior probability from these scans and ran thesethrough Madgraph v425 then applied collider cuts with the C++ code using the limits fromthe following six ATLAS and CMS searches shown in Table32 and described below to findany exclusions

                    333 Description of Collider Cuts Analyses

                    In the following all masses and energies are in GeV and angles in radians unless specificallystated

                    3331 Lepstop0

                    Search for direct pair production of the top squark in all-hadronic final states in pro-tonndashproton collisions at

                    radics=8 TeV with the ATLAS detector[32]

                    This analysis searched for direct pair production of the supersymmetric partner to thetop quark using integrated luminosity of 201 f bminus1 The top squark was assumed to de-cay via t rarr t χ0

                    1 or t rarr bχ01 or t rarr bχ

                    plusmn1 rarr bW (lowast)χ1

                    0 where χ01 (χ

                    plusmn1 ) denotes the lightest

                    neutralino(chargino) in supersymmetric models The search targets a fully hadronic finalstate with four or more jets and large missing momentum The production of a mediator inassociation with top quarks followed by an invisible mediator decay to dark matter particleshas a similar final state to that considered in the ATLAS stop searches Thus we expect theATLAS reach to have sensitivity to the three dark matter models considered in this thesis

                    The collider cuts program mimicked the object reconstruction through jetlepton overlapremoval and selected signal electrons muons and τ particles with pT gt 10 GeV and

                    33 Calculations 44

                    Table 32 95 CL by Signal Region

                    Experiment Region Number

                    Lepstop0

                    SRA1 66SRA2 57SRA3 67SRA4 65SRC1 157SRC2 124SRC3 80

                    Lepstop1

                    SRtN1-1 857SRtN1-2 498SRtN1-3 389SRtN2 107SRtN3 85SRbC1 832SRbC2 195SRbC3 76

                    Lepstop2

                    L90 740L100 56L110 90L120 170

                    2bstop

                    SRA 857SRB 270SRA150 380SRA200 260SRA250 90SRA300 75SRA350 52

                    CMSTopDM1L SRA 1385

                    ATLASMonobjetSR1 1240SR2 790

                    33 Calculations 45

                    |η | lt 24724 and 247 respectively (for eminusmicrominusτ) Jets were selected with pT gt 20 GeVand |η |lt 45 b-jets were selected if they were tagged by random selection lt 70 and had|η |lt 25 and pT gt 20 GeV

                    These jets and leptons were further managed for overlap removal within ∆R lt 04 withnon b-jets removed if ∆R lt 02

                    The b-jets and non b-jets were further selected as signal jets if pT gt 35 GeV and |η |lt 28

                    These signal jets electrons and muons were finally binned according to the signal regionsin Tables 333435 below

                    Table 33 Selection criteria common to all signal regions

                    Trigger EmissT

                    Nlep 0b-tagged jets ⩾ 2

                    EmissT 150 GeV

                    |∆φ( jet pmissT )| gtπ5

                    mbminT gt175 GeV

                    Table 34 Selection criteria for signal regions A

                    SRA1 SRA2 SRA3 SRA4anti-kt R=04 jets ⩾ 6 pT gt 808035353535 GeV

                    m0b j j lt 225 GeV [50250] GeV

                    m1b j j lt 225 GeV [50400] GeV

                    min( jet i pmissT ) - gt50 GeV

                    τ veto yesEmiss

                    T gt150 GeV gt250 GeV gt300 GeV gt 350 GeV

                    Table 35 Selection criteria for signal regions C

                    SRC1 SRC2 SRC3anti-kt R=04 jets ⩾ 5 pT gt 8080353535 GeV

                    |∆φ(bb)| gt02 π

                    mbminT gt185 GeV gt200 GeV gt200 GeV

                    mbmaxT gt205 GeV gt290 GeV gt325 GeV

                    τ veto yesEmiss

                    T gt160 GeV gt160 GeV gt215 GeV

                    wherembmin

                    T =radic

                    2pbt Emiss

                    T [1minus cos∆φ(pbT pmiss

                    T )]gt 175 (314)

                    33 Calculations 46

                    and pbT is calculated for the b-tagged jet with the smallest ∆φ to the pmiss

                    T direction andmbmax

                    T is calculated with the pbT for the b-tagged jet with the largest ∆φ to the pmiss

                    T direction

                    m0b j j is obtained by first selecting the two jets with the highest b-tagging weight- from

                    the remaining jets the two jets closest in the η minusφ plane are combined to form a W bosoncandidate this candidate is then combined with the closest of the 2 b-tagged jets in the η minusφ

                    plane to form the first top candidate with mass m0b j j A second W boson candidate is formed

                    by repeating the procedure with the remaining jets this candidate is combined with thesecond of the selected b-tagged jets to form the second top candidate with mass m1

                    b j j Thecuts shown in Appendix1 Table A1 show the cumulative effect of the individual cuts appliedto the data the signal region cuts and the cumulative effects of individual cuts in SignalRegion C

                    3332 Lepstop1

                    Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

                    radics=8 TeV pp collisions using 21 f bminus1 of

                    ATLAS data[33]

                    The stop decay modes considered were those to a top quark and the lightest supersym-metric particle (LSP) as well as a bottom quark and the lightest chargino where the charginodecays to the LSP by emitting a W boson We should expect the three dark matter models inthis paper with couplings to t and b quarks to show some visibility in this search

                    The collider cuts program mimicked the object construction by selecting electrons andmuons with pT gt 10 GeV and |η |lt 247 and 24 respectively and jets with pT gt 20 GeVand |η |lt 10 b-jets with |η |lt 25 and pT gt 25 GeV were selected with random efficiency75 and mistag rate 2 (where non b-jets were mis-classified) We then removed any jetwithin ∆R = 02 of an electron and non overlapping jets with |η |lt 28 were used to removeany electrons and muons within ∆R = 04 of these while non-overlapping jets with |η |lt 25and pT gt 25 GeV were classified as signal jets Non overlapping electrons and muons withpT gt 25 GeV were classified as signal electrons and muons

                    Signal selection defined six regions to optimize sensitivity for different stop and LSPmasses as well as sensitivity to two considered stop decay scenarios Three signal regionslabelled SRbC 1-3 where bC is a moniker for b+Chargino and 3 regions labelled SRtN 1-3where tN is a moniker for t+Neutralino were optimised for the scenarios where t1 decaysto the respective particles indicated by the labels Increasing label numbers correspond toincreasingly strict selection criteria

                    33 Calculations 47

                    The dominant background in all signal regions arises from dileptonic tt events All threeSRtN selectons impose a requirement on the 3-jet mass m j j j of the hadronically decayingtop quark to specifically reject the tt background where both W bosons from the top quarkdecay leponically

                    For increasing stop mass and increasing difference between the stop and the LSP therequirents are tightened on Emiss

                    T on the ratio EmissT

                    radicHT where HT is the scalar sum of the

                    momenta of the four selected jets and also tightened on mT

                    To further reduce the dileptonic tt background for an event characterised by two one stepdecay chains a and b each producing a missing particle C use is made of the variable

                    mT 2 =min

                    pCTa + pC

                    T b = pmissT

                    [max(mTamtb)] (315)

                    where mTi is the transverse mass 1 of branch i for a given hypothetical allocation (pCTa pC

                    T b)

                    of the missing particle momenta Eqn 315 is extended for more general decay chains byamT 2 [81] which is an asymmetric form of mT 2 in which the missing particle is the W bosonin the branch for the lost lepton and the neutrino is the missing particle for the branch withthe observed charged lepton

                    ∆φ( jet12 pmissT ) the azimuthal angle between leading or subleading jets and missing mo-

                    mentum is used to suppress the background from mostly multijet events with mis-measuredEmiss

                    T

                    Events with four or more jets are selected where the jets must satisfy |η | lt 25 and pT gt80604025 GeV respectively

                    These signal particles were then classified according to the cutflows analysis in Fig36below and compared with the paper in Appendix1 Table A2

                    3333 Lepstop2

                    Search for direct top squark pair production in final states with two leptons in p pcollisions at

                    radics=8TeV with the ATLAS detector[34]

                    Three different analysis strategies are used to search for t1 pair production Two targett1 rarr b+ χ

                    plusmn1 decay and the three body t1 rarr bW χ0

                    1 decay via an off-shell top quark whilst

                    1The transverse mass is defined as m2T = 2plep

                    T EmissT (1minus cos(∆φ)) of branch i where ∆φ is the azimuthal

                    angle between the lepton and the missing transverse momentum

                    33 Calculations 48

                    Table 36 Signal Regions - Lepstop1

                    Requirement SRtN2 SRtN3 SRbC1 SRbC2 SRbC3∆φ( jet1 pmiss

                    t )gt - 08 08 08 08∆φ( jet2 pmiss

                    T )gt 08 08 08 08 08Emiss

                    T [GeV ]gt 200 275 150 160 160Emiss

                    T radic

                    HT [GeV12 ]gt 13 11 7 8 8

                    mT [GeV ]gt 140 200 120 120 120me f f [GeV ]gt - - - 550 700amT 2[GeV ]gt 170 175 - 175 200mT

                    T 2[GeV ]gt - 80 - - -m j j j Yes Yes - - -

                    one targets t1 rarr t + χ01 to an on shell quark decay mode The analysis is arranged to look

                    at complementary mass splittings between χplusmn1 and χ0

                    1 smaller and larger than the W bosonmass The smaller splitting provides sensitivity to t1 three body decay

                    Object reconstruction was carried out by selecting baseline electronsmuons and τrsquos withpT gt 10 GeV and |η |lt 24724 and 247 respectively Baseline jets and b-jets were selectedwith pT gt 20 GeV and |η |lt 25 and b-jets with 70 efficiency Baseline jets were removedif within ∆R lt 02 of an electron and electrons and muons were removed if within ∆R lt 04of surviving jets Signal electrons required pT gt 10 GeV and signal jets pT gt 20 GeV and|η |lt 25

                    The analysis made use of the variable mT 2 [82] the stransverse mass used to mea-sure the masses of pair-produced semi-invisibly decaying heavy particles It is definedas mT 2(pT1 pT2qT ) =

                    minqT1+qT2=qT

                    max[mT (pT1qT1)mT (pT2qT2)] where mT indicatesthe transverse mass pT1 and pT2 are vectors and qT = qT1 + qT2 The minimisation isperformed over all possible decompositions of qT

                    Events were selected if they passed either a single-electron a single-muon a double-electron a double-muon or an electron-muon trigger Events are required to have exactlytwo opposite sign leptons and at least one of the leptons or muons must have a momentumgreater than 25 GeV and the invariant mass (rest mass) of the two leptons is required to begreater than 20 GeV In order to reduce the number of background events containing twoleptons produced by the on-shell decay of the Z boson the invariant mas must be outside711-111 GeV range

                    Two additional selections are applied to reduce the number of background events withhigh mT 2 arising from events with large Emiss

                    T due to mismeasured jets ∆φb lt 15 and∆φ gt 10 The quantity ∆φb is the azimuthal angle between pll

                    T b = pmissT + pl1

                    T +Pl2T The

                    33 Calculations 49

                    vector pllT b is the opposite of the vector sum of all the hadronic activity in the event For WW

                    and tt backgrounds it measures the transverse boost of the WW system and for the signalthe transverse boost of the chargino-chargino system The ∆φ variable is the azimuthal anglebetween the pmiss

                    T vector and the direction of the closest jet

                    By requiring mT 2 gt 40 GeV the tt background is the dominant background in both thesame-flavoured (SF) and different-flavoured (DF) events

                    Four signal regions are then defined as shown in Table 37 SR M90 has the loosestselection requiring MT 2gt 90 GeV and no additional requirement on jets and providessensitivity to scenarios with a small difference between the masses of the top-squark andchargino SR M110 and M120 have a loose selection on jets requiring two jets with pT

                    gt 20 GeV and mT 2 to be greater than 110 GeV and 120 GeV respectively They providesensitivity to scenarios with small to moderate difference between the invariant masses of thestop quark and the charginos SR M100 has a tighter jet slection and provides sensitivity toscenarios with a large mass difference between the both stop quark and the chargino and thestop quark and the neutralino

                    The analysis cut regions are summarised in Fig37 below

                    Table 37 Signal Regions Lepstop2

                    SR M90 M100 M110 M120pT leading lepton gt 25 GeV

                    ∆φ(pmissT closest jet) gt10

                    ∆φ(pmissT pll

                    T b) lt15mT 2 gt90 GeV gt100 GeV gt110 GeV gt120 GeV

                    pT leading jet no selection gt100 GeV gt20 GeV gt20 GeVpT second jet no selection gt50 GeV gt20 GeV gt20 GeV

                    To validate the code we ran the cuts shown in Appendix 1 Table A3 with the givenMadgraph simulation and compared to [83] It was not clear what baseline cuts had beenemployed in the ATLAS paper so our results were normalised to the 2 SF electrons beforecomparison Some of the ATLAS cut numbers were less than 3 and we did not produce anyevents passing these cuts but clearly the margin of error at this level was greater than thenumber of ATLAS events

                    33 Calculations 50

                    3334 2bstop

                    Search for direct third-generation squark pair production in final states with miss-ing transverse momentum and two b-jets in

                    radics= 8 TeV pp collisions with the ATLAS

                    detector[31]

                    Two sets of scenarios are considered In the first the coloured sbottom is the onlyproduction process decaying via b1 rarr bχ0

                    1 In the second the stop is the only productionprocess decaying via t1 rarr t χplusmn

                    1 and the lightest chargino (χplusmn1 ) decays via a virtual W boson

                    into the final state χ01 f f prime Both scenarios are characterised by the presence of two jets

                    resulting from the hadronisation of b quarks and large missing transverse momentum Theproduction of a mediator in association with top quarks followed by an invisible mediatordecay to dark matter particles has a similar final state as that considered in the ATLAS stopsearches Thus we expect the ATLAS reach to have sensitivity to the three dark mattermodels considered in this thesis

                    The collider cuts program mimicked object reconstruction by selecting baseline electronswith pT gt 7 GeV and |η |lt 247 and muons with pT gt 6 GeV and |η |lt 24 Baseline jetswere selected with pT gt 20 GeV and |η |lt 28 We then removed any jets within ∆R lt 02of an electron and any electron within ∆R lt 04 of surviving jets Signal electrons andmuons were required to have pT gt 25 GeV b-jets were selected with 60 tagging efficiencyand |η |lt 25 and pT gt 20 GeV

                    Two sets of signal regions are defined to provide sensitivity to the kinematic topologiesassociated with the different mass splittings between the sbottom or the stop and the neu-tralino In all cases the presence of at least one primary vertex (with at least five tracks withpT gt 04 GeV) is required Events are selected with Emiss

                    T gt 150 GeV and no electrons ormuons identified in the final state For signal region selections jets with |η | lt 28 are orderedaccording to their pT and 2 jets are required to be b-tagged

                    The variables are defined as follows

                    bull ∆φmin is the minimal azimuthal angle ∆φ between any of the three leading jets andpmiss

                    T

                    bull me f f (k) = sumki=1(p jet

                    T )i +EmissT where the index refers to the pT ordered list of jets

                    bull HT3 is defined as the scalar sum of the pT of the n leading pT ordered jets withoutincluding the 3 leading jets HT3 = sum

                    ni=4(p jet

                    T )i

                    bull mbb is the invariant mass of the two b-tagged jets in the event

                    33 Calculations 51

                    bull mCT is the contransverse mass [82] and is a kinematical variable that can be usedto measure the masses of pair produced semi-invisibly decaying particles For twoidentical decays of heavy particles into two invisible particles v1 and v2 and twoinvisible particles mCT is defined as m2

                    CT (v1v2) = [ET (v1)+ET (v2)]2 minus [pT (v1)minus

                    pT (v2)]2 where ET =

                    radicp2

                    T +m2 The the contransverse mass is invariant under equaland opposite boosts of the parent particles in the transverse plane and is bounded fromabove by an analytical expression of the particle masses For production of sbottompairs the bound is given by mmax

                    CT =m2(b)minusm2(χ0

                    1 )

                    m(b) and for tt events the bound is 135

                    GeV A similar equation can be written for stop pairs in term of the stop and charginomasses

                    A definition of the signal regions is given in the Table38

                    Table 38 Signal Regions 2bstop

                    Description SRA SRBEvent cleaning All signal regions

                    Lepton veto No emicro after overlap removal with pT gt7(6) GeV for e(micro)Emiss

                    T gt 150 GeV gt 250 GeVLeading jet pT ( j1) gt 130 GeV gt 150 GeVSecond jet pT ( j2) gt 50 GeV gt 30 GeVThird jet pT ( j3) veto if gt 50 GeV gt 30 GeV

                    ∆φ(pmissT j1) - gt 25

                    b-tagging leading 2 jets(pT gt 50 GeV |η | lt 25)

                    2nd- and 3rd- leading jets(pT gt 30 GeV|η | lt 25)

                    ∆φmin gt 04 gt 04Emiss

                    T me f f (k) EmissT me f f (2) gt 025 Emiss

                    T me f f (3) gt 04mCT gt 150200250300350 GeV -HT3 - lt 50 GeVmbb gt200 GeV -

                    The analysis cuts are summarised in Table A4 of Appendix 1

                    3335 ATLASMonobjet

                    Search for dark matter in events with heavy quarks and missing transverse momentumin pp collisions with the ATLAS detector[30]

                    Events were selected with high missing transverse momentum when produced in associa-tion with one or more jets identified as containing b quarks Final states with top quarks wereselected by requiring a high jet multiplicity and in some cases a single lepton The research

                    33 Calculations 52

                    studied effective contact operators in the form of a scalar operator Oscalar = sumqmqMN

                    lowastqqχχ

                    where MNlowast is the effective mass of the coupling and the quark and DM fields are denoted by

                    q and χ respectively These are the equivalent of the simplified models in this paper with themediator integrated out

                    Object reconstruction selected baseline electrons muons jets and τrsquos with pT gt 20 GeVand |η |lt 2525 and 45 respectively Signal jets were selected by removing any jets within∆R lt 02 of an electron and electrons and muons were selected by removing them if theywere within ∆R lt 04 of surviving jets b-jets were tagged with an efficiency of 60

                    Only signal regions SR1 and SR2 were analysed

                    The analysis split events into signal regions SR1 and SR2 according to the followingCriteria

                    Table 39 Signal Region ATLASmonobjet

                    Cut SR1 SR2Jet multiplicity n j 1minus2 3minus4

                    bminusjet multiplicity nb gt 0 (60 eff) gt 0 (60 eff)Lepton multiplicity nl 0 0

                    EmissT gt300 GeV gt200 GeV

                    Jet kinematics pb1T gt100 GeV pb1

                    T gt100 GeV p j2T gt100 (60) GeV

                    ∆φ( j1EmissT ) gt 10 i = 12 gt 10 i = 14

                    Where p jiT (pbi

                    T ) represent the transverse momentum of the i-th jet b-tagged jet respec-tively If there is a second b-tagged jet the pT must be greater than 60 GeV

                    3336 CMSTop1L

                    Search for top-squark pair production in the single-lepton final state in pp collisionsat

                    radics=8 TeV[41]

                    This search was for pair production of top squarks in events with a single isolatedelectron or muon jets large missing transverse momentum and large transverse mass

                    (MT =radic

                    2EmissT pl

                    T (1minus cos(∆φ))) where plT is the leptonrsquos transverse momentum and ∆(φ)

                    is the difference between the azimuthal angles of the lepton and EmissT The 3 models

                    considered should have some visibility at certain parameter combinations in this regionThe search focuses on two decay modes of the top squark leading to the two processespp rarr t tlowast rarr tt χ0

                    1 χ01 rarr bbW+Wminusχ0

                    1 χ01 and pp rarr t tlowast rarr bbχ

                    +1 χ

                    minus1 rarr bbW+Wminusχ0

                    1 χ01 The

                    33 Calculations 53

                    lightest neutralino χ01 is considered the lightest supersymmetric particle and escapes without

                    detection The analysis is based on events where one of the W bosons decays leptonicallyand the other hadronically resulting in one isolated lepton and 4 jets two of which originatefrom b quarks The two neutralinos and the neutrino from the W decay can result in largemissing transverse momentom The kinematical end point of MT is lt the mass of the Wboson However for signal events the presence of LSPs in the final state allows MT to exceedthe mass of the W boson The search is therefore for events with large MT The dominantbackground with large MT arises from tt decays where W bosons decay leptonically withone of the leptons not being identified

                    To reduce the dominant tt background use was made of the MWT 2 variable defined as

                    the minimum mother particle mass compatible with the transverse momenta and mass shellconstraints [84]

                    Object reconstruction required electrons and muons to have pT gt 30 and 25 GeV respand |η |lt 25 and 21 respectively Baseline jets were required to have pT gt 30 and |η |lt 4and b-jets were tagged with an efficiency of 75 for |η |lt 24

                    Events were chosen if they contained one isolated electron or muon at least 4 jets and atleast one b-tagged jet and Emiss

                    T gt 100 GeV MT gt 160 GeV MWT 2 was required to be greater

                    than 200 GeV and the smallest azimuthal angle difference between baseline jets and EmissT

                    gt12

                    Chapter 4

                    Calculation Tools

                    41 Summary

                    Figure 41 Calculation Tools

                    The simplified models were implemented in Feynrules [4] and the model files importedinto MicrOmegas v425 which was used to calculate the log-likelihoods of the astrophysicalobservables MicrOmegas was modified to allow only s-channel decays in both relic densityand gamma ray calculations Direct detection constraints were easy to implement for thescalar and vector models by coding the analytical expression for the spin independent WIMP-nucleon scattering cross section but due to the momentum dependence of the WIMP-nucleon

                    42 FeynRules 55

                    scattering cross section of the Majorana fermion model LUXCalc v101 [79] was modifiedfor this model to implement momentum dependent cross sections [85] We used MicrOmegasv425 to calculate relic density values and to produce gamma ray spectra for comparisonwith the Fermi-LAT results Parameter scans were carried out using MultiNest v310 withthe parameter ranges given in Table 51 and using astrophysical constraints only Oncethese scans had converged a random sample of 900 points was chosen (weighted by theposterior probability) and then used to calculate collider constraints using the MadgraphMonte Carlo generator This included comparison with mediator production and decay limitsfrom the LHC experiments and the reproduction of searches for dark matter plus other finalstate objects which required dedicated C++ code that reproduces the cut and count analysesperformed by CMS and ATLAS

                    42 FeynRules

                    FeynRules is a Mathematicareg package that allows the calculation of Feynman rules inmomentum space for any QFT physics model The user needs to provide FeynRules withminimal information to describe the new model contained in the model-file This informationis then used to calculate a set of Fenman rules associated with the Lagrangian The Feynmanrules calculated by the code can then be used to implement new physics models into otherexisting tools such as Monte Carlo (MC) generators via a set of interfaces developed andmaintained by the MC authors

                    Feynrules can be run with a base model = SM (or MSSM) and user defined extensions tothe base model

                    43 LUXCalc

                    LUXCalc is a utility program for calculating and deriving WIMP-nucleon coupling limitsfrom the recent results of the LUX direct search dark matter experiment The program hasbeen used to derive limits on both spin-dependent and spin-independent WIMP-nucleoncouplings and it was found under standard astrophysical assumptions and common otherassumptions that LUX excludes the entire spin-dependent parameter space consistent witha dark matter interpretation of DAMArsquos anomalous signal and also found good agreementwith published LUX results [79]

                    We modified the program for the Majorana fermion model to implement momentumdependent cross -sections [85]

                    44 Multinest 56

                    44 Multinest

                    Multinest is an efficient Bayesian Inference Tool which uses a simultaneous ellipsoidalnested sampling method a Monte Carlo technique aimed at an efficient Bayesian Evidencecalculation with posterior inferences as a by product

                    Bayes theorem states that

                    Pr(θ |DH) =Pr(D|θ H)Pr(θ |H)

                    Pr(D|H) (41)

                    Here D is the data and H is the Hypothesis Pr(θ |DH) is the posterior probability ofthe parameters Pr(D|θ H) = L(θ) is the likelihood Pr(θ |H) is the prior and Pr(D|H) isthe evidence

                    The evidence Pr(D|H) =int

                    Pr(θ |DH)Pr(θ |H)d(θ) =int

                    L(θ)Pr(θ |H)d(θ) Thus ifwe define the prior volume dX = Pr(θ |H)d(θ) and

                    X(λ ) =int

                    L(θ)gtλ

                    Pr(θ |H)d(θ) (42)

                    where the integral extends over the region of parameter space within the iso likelihoodcontour L(θ) = λ then the evidence integral can be written

                    int 10 L (X)dX where L (X) the

                    inverse of equation 42 and is a monotonically decreasing function of X (the prior volumewhich is 1 for the total sample space and 0 for a single point)

                    Thus if one can evaluate likelihoods L (Xi) where 0 lt Xn lt lt X2 lt X1 lt X0 = 1the evidence can be estimated as sumLiwi where wi = 5(Ximinus1 minusXi+1) Ellipsoidal samplingapproximates the iso likelihood contour by a D dimensional covariance matrix of the currentset of active points New points are selected from the prior within this ellipsoidal bounduntil one is obtained that has a likelihood greater than the likelihood of the removed lowestlikelihood point This method is extended for multimodal sampling where there may bemore than one local maximum by clustering points that are well separated and constructingindividual ellipsoidal bounds for each cluster For more details consult [7]

                    The results of the analysis are expressed as a data file containing the sample probability-2loglikelihood followed by the parameters of the model for the particular sampling Theprobabilities are normalised to total 10

                    45 Madgraph 57

                    45 Madgraph

                    Madgraph is a general purpose matrix element calculator and Monte Carlo generator Basedon user-supplied Feynman rules (eg the output of Feynrules) the package calculates the tree-level matrix elements for any processes that the user specifies and generates samples of LHCevents including an accurate simulation of the proton collisions themselves (which requires aknowledge of the parton distribution functions that determine the relative proportion of eachparton inside the nucleus) Hadronisation of final state quarks is not possible in Madgraphand hence the code must be used in conjunction with Pythia which performs a parton shower

                    The Madgraph output was automatically showered by Pythia (Pythia describes partonradiation as successive parton emissions using Markov chain techniques based on Sudakovform factors and this description is only formally correct for the soft and collinear emissions)A Madgraph simulation is at the level of the fundamental particles (quarks and leptons)Madgraph generates a limited set of matrix element processes at LO representing hard (highenergy) emissions and passes these to Pythia for showering The final state partons in theMadgraph simulation must be matched to the Pythia final state partons to avoid doublecounting

                    The output from Pythia is automatically fed through Delphes where basic collider cutscan be performed (calculation of the numbers of particles and jets that would be observed atdifferent energies and locations within the collider) The output of the Delphes simulation is afile containing reconstructed events that can be analysed using the standard ROOT framework(the most popular data analysis package in particle physics) [86]

                    The parameters that gave the best fits to the direct [16] and indirect DM experiments [87]and relic density [59] are ranked by posterior probability from the Multinest [7] output

                    The parameters in the Madgraph simulations are stored in files in the cards sub directoryand in particular the run card param card and Delphes card record items relevent to thesimulation

                    In the run card the number of events the Beam 1 and 2 total energy in GeV (4000 each)and the type of jet matching were modified (to MLM type matching where the final statepartons in Madgraph are clustered according to the KT jet algorithm to find the equivalentparton shower history of the event there is a minimum cutoff scale for the momentum pT

                    given by the xqcut parameter) The maxjetflavour (=5 to include b quarks) and xqcut = 30GeV parameters were also modified

                    46 Collider Cuts C++ Code 58

                    The param card was modified to specify the masses of the new particles and the autodecay modes The Delphes card is overwritten with either the ATLAS or CMS defaultcards which record tracking efficiencies and isolation momentum and energy resolutionscalorimeter details jet finder and b and τ -tagging parameters However the isolation andtagging in this study were done by the collider cuts C++ code since multiple sets of cutswere applied to the same output from Pythia

                    When performing simulations of the three dark matter models it was necessary to definethe proton and jets to contain b quarks as this is not the default in Madgraph The processesgenerated included 1 and 2 jets

                    46 Collider Cuts C++ Code

                    Because of the volume of simulations required for the collider constraints we used C++code (originally developed by Martin White and Sky French and modified by me) that readsthe ROOT files and outputs the numbers of events that would have been observed under thechosen collider experiments had the model been true for the given parameters Thus the sameoutput files can be used for all of the collider cuts and hence no cuts were performed throughthe Delphes cards

                    In order to validate the C++ code we replicated the observed events with the papersdescribing the various collider searches (see Appendix)

                    Chapter 5

                    Majorana Model Results

                    51 Bayesian Scans

                    To build some intuition we first examined the constraining effect of each observable on itsown We ran separate scans for each constraint turned on and then for all the constraintstogether In Fig 51 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together The paramaters were scanned over the range in Table 51 below

                    Table 51 Scanned Ranges

                    Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                    Range 0minus103 0minus103 10minus5 minus10 10minus5 minus10 10minus5 minus10

                    In Fig51 we can see that the preferred DM mass is centered around 35 GeV primarily dueto the influence of the Fermi-LAT gamma ray constraint whilst the relic density constraintallows a wide range of possible mediator and DM masses Direct detection (LUX) constraintssplit the mass regions into a top left and bottom right region and together these split theoverall mass parameter regions into two regions by mediator mass but with DM mass around35 GeV Later studies using a lower excess gamma ray spectrum tend to increase the preferredDM mass up to 50 GeV [76] and [88]

                    The marginalised posterior probabilities for all combinations of parameters taken twoat a time are shown in Fig52 These plots have all constraints (Gamma LUX and RelicDensity) applied together The idea of these graphs is to study the posterior probabilities ofcombinations of two paramaters taken at a time (the off-diagonal graphs) The diagonal plotsshow marginalised histograms of a single parameter at a time- this gives an indication of thespread of each parameter The plots show that mS and λb show a high degree of correlationfor mS gt 100 GeV but there is also a range of values of mS around 35 GeV which are

                    51 Bayesian Scans 60

                    1 0 1 2 3 4log10(mχ)[GeV]

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                    s)[GeV

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                    1 0 1 2 3 4log10(mχ)[GeV]

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                    (d) All Constraints

                    Figure 51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether

                    51 Bayesian Scans 61

                    00

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                    τ)

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                    log10(λb)5 4 3 2 1 0 1

                    log10(λτ)

                    Figure 52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter

                    52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 62

                    possible for λb around 001 λt and λτ show a wide range of possible values for two possibleranges of mS lt 35 GeV and greater than 100 GeV mχ shows a narrow range of possiblevalues around 35 GeV for all values of the other parameters

                    52 Best fit Gamma Ray Spectrum for the Majorana FermionDM model

                    We can see from the Fig51(a) scan that the maximum log-likelihood Fermi-LAT gammaray spectrum on its own shows values that centre around 35 GeV This is not changed byadding a relic density constraint or a direct detection (LUX) constraint

                    The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraintsis shown in Fig53 The parameters for this point were

                    Table 52 Best Fit Parameters

                    Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                    Value 3332 49266 0322371 409990 0008106

                    10-1 100 101 102

                    E(GeV)

                    10

                    05

                    00

                    05

                    10

                    15

                    20

                    25

                    30

                    35

                    E2dφd

                    E(G

                    eVc

                    m2ss

                    r)

                    1e 6

                    Best fitData

                    Figure 53 Gamma Ray Spectrum

                    The relic density (Ωh2) at this point has a value of 0135 but the point is separated onlyslightly in posterior probability from points with a value of 01199 with mS lt 10 GeV (iethe posterior probability is almost the same differing by less than 1)

                    To better understand the interplay of the different constraints the three graphs in Fig54show the log likelihood surfaces and individual best fit points (best log likelihoods) for the γ

                    and Ω scans seperately and then combined The diagrams each choose a pair of couplings(fixing the remaining parameters) and display the 2d surfaces of minuslog(χ2) (proportional to

                    52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 63

                    the log likelihoods of the coupling combinations) obtained by varying the pair of couplingsThese graphs do not show marginalised posterior probabilities but rather are based on the-2log likelihood calculations directly from MicrOmegas for the given coupling combinationsWe have termed this value as χ2 for convenience and used the log of this value to give abetter range for the colours These log likelihood values are what drive the MultiNest searchof parameter space for the best parameter combinations

                    The graphs show a complex interplay of the different constraints resulting often in acompromise parameter combination The arrow on each graph points to the maximum point We have chosen a single representative set of couplings near the best overall fit point for allthe graphs

                    52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 64

                    00 05 10 15 20 25 30

                    log10(mχ)

                    00

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                    Both Maximum at mχ=8317 GeV mS=2884GeV The best fit point for these couplings maybe above below or between the individual bestfit points but will on average be between thepoints

                    Figure 54 Plots of log likelihoods by individual and combined constraints Masses in GeV

                    53 Collider Constraints 65

                    53 Collider Constraints

                    531 Mediator Decay

                    1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof τ leptons in pp collisions [44] Refer also to Chapter 3 for details

                    We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

                    The results of the calculations are shown in the graph Fig55 and can be compared withthe limits in the table from the paper [44] in Fig 56 The black line is the observed ATLASlimit

                    Figure 55 σ lowastBr(σ rarr ττ) versus Mass of Scalar

                    0 200 400 600 800

                    mS[GeV]

                    10

                    5

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                    log 1

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                    (bbS

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                    (Sgtττ

                    ))[pb]

                    Observed LimitLikely PointsExcluded Points

                    0

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                    0 5 10 15 20 25 30 35 40 45

                    We can see that only one point here exceeded the limit and the points showed a widespread over σ(bbS)lowastBr(σ rarr ττ) by mS space which is consistent with the spread of pointson the MultiNest scan on the mS by λτ plane causing a spread of branching ratios to the τ

                    quark It may seem odd that only one point was excluded here but this point was sampledwith a very low probability having high coupling constants of 597 to the τ quark and 278 tothe t quark and only just exceeds the limit As such it can be explained as a random outlier

                    53 Collider Constraints 66

                    Figure 56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar

                    2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                    This scan calculates the contributions of b-quark associated Higgs boson productionreproduced from [43] Refer to Chapter 3 for details

                    We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

                    The cross section times branching ratios to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig57 The black line is the CMSLimit

                    Figure 57 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar

                    0 200 400 600 800

                    mS[GeV]

                    15

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                    0

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                    53 Collider Constraints 67

                    The results of this scan were compared to the limits in [89] with the plot shown inFig58

                    Figure 58 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

                    We can see that there are a number of exclusions across the range of mS which occuredfor high values of the coupling constants gt 1 for λbλt λtau - these can be seen also on Fig59The excluded parameter combinations represent less than 2 of the total sampled number

                    532 Collider Cuts Analyses

                    We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude and the results are shown against theposterior probabilities of the parameter combinations in Fig 59

                    The circles in Figure 59 show which points were excluded by the collider cuts as well asthe limits on cross sections times branching ratios to each of τ+τminus and bb

                    All the excluded points under the heading Collider Cuts showed exclusion by the Lepstop0experiment (Search for direct pair production of the top quark in all hadronic final states

                    53 Collider Constraints 68

                    0 1 2 3

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                    σ lowastBr(σgt ττ)

                    (a) mχ by mS

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                    Figure 59 Excluded points from Collider Cuts and σBranching Ratio

                    53 Collider Constraints 69

                    [32]) Some of these same points were also excluded by Lepstop2 2bstop CMSTop andATLASMonobjet experiments There were expected events at other points but none weresignificant at the 95 exclusion level These significant exclusions occurred mainly wherethe mass of the mediator was over 100 GeV and higher values of the couplings to b and tquarks but there were a few points with mediator mass around 25 GeV The exclusions withmediator mass around 25 GeV were all accompanied by couplings to t quarks of order 10which is probably non-perturbative

                    Looking at the exclusions in Fig 59 we can see that these occur mainly in the regionof higher mediator mass gt 100 GeV and lt 400 GeV These exclusions come from theastrophysical constraints (Gamma) which show a strong correlation between λb and mS andhigh values of λt gt 1 However these mass regions do not overlap at the same couplingcombinations as the high likelihood region The green and white circles correspond to theexcluded points in Figs 57 and 58 respectively

                    The remaining regions of highest likelihood can be seen to be where 0001 λb lt 01accompanied by λt lt 0001 and mass of the mediator mS lt 10 GeV or 400 GeV lt mS lt1000GeV with mχ sim 35 GeV

                    The plot of λt by λb shows the significant regions do not overlap with the collider cutsand cross section constraints since the latter occur at high values of either coupling while thelikely region has low values of these couplings (also accompanied by low couplings to λτ )

                    Chapter 6

                    Real Scalar Model Results

                    61 Bayesian Scans

                    To build some intuition we first examined the constraining effect of each observable on itsown We ran seperate scans for each constraint turned on and then for all the constraintstogether In Figure 61 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together

                    In Figure 61 we can see that there are is a narrow region which dominates when allconstraints are taken into account This centers on mχ=9 GeV and a range of values for mS

                    from 35 GeV to 400 GeV The region is not the highest likelihood region for the Fermi-LATgamma ray spectrum on its own (as can be seen from the individual plot - there is a narrowband to the left of the highest likelihood band) nor for the LUX likelihood or relic densitybut represents a compromise between the three constraints

                    The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig62 The diagonal plots show marginalised histograms of a singleparameter at a time

                    61 Bayesian Scans 71

                    05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

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                    Figure 61 Real Scalar Dark Matter - By Individual Constraint and All Together

                    61 Bayesian Scans 72

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                    Figure 62 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter

                    62 Best fit Gamma Ray Spectrum for the Real Scalar DM model 73

                    62 Best fit Gamma Ray Spectrum for the Real Scalar DMmodel

                    We can see from Fig61 that the maximum log-likelihood Fermi-LAT Gamma ray spectrumon its own shows values of mχ that centre around 35 GeV However the LUX constraintin particular moves the best fit to a point around 9 GeV- the tight LUX constraint onlydisappears for low masses and at that point the proportion of τ+τminus annihilation needs toincrease to fit the constraints

                    The gamma ray spectrum at the points of best fit for Fermi-LAT including all constraintsis shown in Fig63 The parameters for these points were respectively and involve decays tomainly τ and b quarks respectively

                    Table 61 Best Fit Parameters

                    Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                    Value 932 3526 000049 0002561 000781

                    10-1 100 101 102

                    E(GeV)

                    10

                    05

                    00

                    05

                    10

                    15

                    20

                    25

                    30

                    35

                    E2dφdE

                    (GeVc

                    m2ss

                    r)

                    1e 6

                    Best fitData

                    Figure 63 Gamma Ray Spectrum

                    This is not a particularly good fit and it occcurs at a relic density Ωh2 of 335 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

                    63 Collider Constraints 74

                    63 Collider Constraints

                    631 Mediator Decay

                    1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44]

                    We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

                    The results of the scan are shown in the graph Fig64 and can be compared with thelimits in the table from the paper [44] in Fig 56The black line is the observed ATLAS limit

                    Figure 64 σ lowastBr(σ rarr ττ) versus Mass of Scalar

                    0 200 400 600 800

                    mS[GeV]

                    8

                    6

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                    log 1

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                    (bbS

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                    (Sgtττ

                    ))[pb]

                    Observed LimitLikely PointsExcluded Points

                    050

                    100150200250300350

                    0 10 20 30 40 50 60

                    We can see that there is a narrow band in the region of interest in the σ(bbS)lowastB(S gt ττ)

                    by mS plane due to the high degree of correlation between mS and λτ seen in the likelihoodscan Values for mS gt 300Gev are excluded

                    2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                    We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinations

                    63 Collider Constraints 75

                    randomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

                    The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig65 The black line is the CMSlimit

                    Figure 65 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

                    0 200 400 600 800

                    mS[GeV]

                    8

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                    Observed LimitLikely PointsExcluded Points

                    050

                    100150200250300350

                    0 10 20 30 40 50 60

                    The results of this scan were compared to the limits in [89] with the plot shown inFig58

                    We can see a similar shape for the region of interest to that in Fig64 due to the highcorrelation between mS and λb over the region preferred by astrophysical constraints resultingin branching ratios highly correlated to ms Only one point is excluded with mS gt 500GeV

                    632 Collider Cuts Analyses

                    We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expected

                    63 Collider Constraints 76

                    with 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 66

                    We can see that collider cuts show some exclusions only for masses mχ below the highlikelihood region but these exclusions do not occur at the same λb and λt combinations asseen from the plot of these two couplings except for a small region at the highest mS andcouplings

                    All but one of the collider cut exclusions were excluded by the Lepstop0 experiment (thiswas excluded by 2bstop with high coupling to b quarks of 73) A number of the excludedpoints were also excluded by Lepstop2 2bstop CMSTopDM1L and ATLASMonobJet-(around 30 of the total for each)

                    63 Collider Constraints 77

                    0 1 2 3

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                    Figure 66 Excluded points from Collider Cuts and σBranching Ratio

                    Chapter 7

                    Real Vector Dark Matter Results

                    71 Bayesian Scans

                    In the Figure 71 we can see that the preferred DM mass is centered around 35 GeV primarilydue to the influence of the Fermi-LAT gamma ray constraint with relic density having a widerange of possible mediator and DM masses The result of combining all the constraints is todefine a small region around mχ = 8 GeV and mS = 20 GeV which dominates the posteriorprobability

                    The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig72 The diagonal plots show marginalised histograms of a singleparameter at a time

                    71 Bayesian Scans 79

                    05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                    1

                    0

                    1

                    2

                    3

                    4

                    log 1

                    0(m

                    s)[GeV

                    ]

                    (a) Gamma Only

                    05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                    05

                    00

                    05

                    10

                    15

                    20

                    25

                    30

                    35

                    log 1

                    0(m

                    s)[GeV

                    ]

                    (b) Relic Density

                    1 0 1 2 3 4log10(mχ)[GeV]

                    05

                    00

                    05

                    10

                    15

                    20

                    25

                    30

                    35

                    log 1

                    0(m

                    s)[GeV

                    ]

                    (c) LUX

                    05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                    05

                    00

                    05

                    10

                    15

                    20

                    25

                    30

                    35

                    log 1

                    0(m

                    s)[GeV

                    ]

                    (d) All Constraints

                    Figure 71 Real Vector Dark Matter - By Individual Constraint and All Together

                    71 Bayesian Scans 80

                    00

                    05

                    10

                    15

                    20

                    25

                    30

                    log 1

                    0(m

                    χ)[GeV

                    ]

                    00

                    05

                    10

                    15

                    20

                    25

                    30

                    ms[Gev

                    ]

                    5

                    4

                    3

                    2

                    1

                    0

                    1

                    log 1

                    0(λ

                    t)

                    5

                    4

                    3

                    2

                    1

                    0

                    1

                    log 1

                    0(λ

                    b)

                    00 05 10 15 20 25 30

                    log10(mχ)[GeV]

                    5

                    4

                    3

                    2

                    1

                    0

                    1

                    log 1

                    0(λ

                    τ)

                    00 05 10 15 20 25 30

                    ms[Gev]5 4 3 2 1 0 1

                    log10(λt)5 4 3 2 1 0 1

                    log10(λb)5 4 3 2 1 0 1

                    log10(λτ)

                    Figure 72 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter

                    72 Best fit Gamma Ray Spectrum for the Real Vector DM model 81

                    72 Best fit Gamma Ray Spectrum for the Real Vector DMmodel

                    The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraints isshown in Fig73

                    Table 71 Best Fit Parameters

                    Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                    Value 8447 20685 0000022 0000746 0002439

                    10-1 100 101 102

                    E(GeV)

                    10

                    05

                    00

                    05

                    10

                    15

                    20

                    25

                    30

                    35

                    E2dφdE

                    (GeVc

                    m2s

                    sr)

                    1e 6

                    Best fitData

                    Figure 73 Gamma Ray Spectrum

                    This is not a particularly good fit and it occcurs at a relic density Ωh2 of 025 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

                    73 Collider Constraints

                    731 Mediator Decay

                    1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44] (See Chapter 3 for more details)

                    We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

                    73 Collider Constraints 82

                    The results of the scan are shown in the graph Fig74 and can be compared with thelimits in the table from the paper [44] in Fig 56

                    Figure 74 σ lowastBr(σ rarr ττ) versus Mass of Scalar

                    0 200 400 600 800

                    mS[GeV]

                    8

                    6

                    4

                    2

                    0

                    2

                    log 1

                    0(σ

                    (bbS

                    )lowastB

                    (Sgtττ

                    ))[pb]

                    Observed LimitLikely PointsExcluded Points

                    0100200300400500600700800

                    0 20 40 60 80 100120140

                    We can see that the region of highest likelihood is limited to mS lt 100 GeV and a narrowrange of couplings to λτ giving a narrow region of interest which falls well below the limit

                    2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                    We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

                    The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig75 The black line is the CMSlimit

                    The results of this scan were compared to the limits in [89] with the plot shown in Fig58

                    We can see that again the region of interest for mS is below 100 GeV and the points fallwell below the limit

                    73 Collider Constraints 83

                    Figure 75 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

                    0 200 400 600 800

                    mS[GeV]

                    8

                    6

                    4

                    2

                    0

                    2

                    4

                    log

                    10(σ

                    (bS

                    +X

                    )lowastB

                    (Sgt

                    bb))

                    [pb]

                    Observed LimitLikely PointsExcluded Points

                    0100200300400500600700800

                    0 20 40 60 80 100120140

                    732 Collider Cuts Analyses

                    We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 76

                    We can see from Figure 76 that like the real scalar model the region of highest likelihoodhas λb and λt lt 001 while the excluded points from collider cuts and cross sections areabove 01 Most of the excluded points have mχ gt than the values in the region of highestlikelihood and all points are widely separated in parameter space from this region

                    Once again 89 of excluded points were excluded by Lepstop0 with many also beingexcluded by the other experiments (50 by Lepstop2 and 56 by CMSTopDM1L 35 byLepstop2 and 20 by ATLASMonobjet)

                    73 Collider Constraints 84

                    0 1 2 3

                    log10(mχ)[GeV]

                    0

                    1

                    2

                    3

                    log 1

                    0(m

                    s)[GeV

                    ]Collider Cuts

                    σ lowastBr(σgt bS+X)

                    σ lowastBr(σgt ττ)

                    (a) mχ by mS

                    5 4 3 2 1 0 1

                    log10(λt)

                    0

                    1

                    2

                    3

                    log 1

                    0(m

                    s)[GeV

                    ](b) λt by mS

                    5 4 3 2 1 0 1

                    log10(λb)

                    5

                    4

                    3

                    2

                    1

                    0

                    1

                    log 1

                    0(λ

                    t)

                    (c) λb by λt

                    5 4 3 2 1 0 1

                    log10(λb)

                    5

                    4

                    3

                    2

                    1

                    0

                    1

                    log 1

                    0(λ

                    τ)

                    (d) λb by λτ

                    5 4 3 2 1 0 1

                    log10(λt)

                    5

                    4

                    3

                    2

                    1

                    0

                    1

                    log 1

                    0(λ

                    τ)

                    (e) λt by λτ

                    5 4 3 2 1 0 1

                    log10(λb)

                    0

                    1

                    2

                    3

                    log 1

                    0(m

                    s)[GeV

                    ]

                    (f) λb by mS

                    Figure 76 Excluded points from Collider Cuts and σBranching Ratio

                    Chapter 8

                    Conclusion

                    We first conducted Bayesian scans over mass and coupling parameters for three simplifiedmodels - Majorana fermion real scalar and real vector using direct detecion (LUX) indirectdetection (GCE γ rays) and relic density constraints to find the regions of highest likelihoodWe ran these scans one constraint at a time and then together to better understand the interplayof the different constraints on best fit and likely parameter regions

                    We also developed and tested C++ collider cut software specifically designed to mimickthe cuts applied in a number of ATLAS and CMS experiments that searched for Emiss

                    T inconnection with specific beyond the standard model (BSM) theories of Dark Matter

                    We then selected parameter combinations by randomly sampling the scanned resultsbased on the posterior probability distribution of the points and applied collider cuts andcross section times branching ratio calculations to these points to see if there would havebeen any sensitivity from the ATLAS and CMS experiments

                    We found that the Majorana fermion model showed the best overall fit to all the constraintswith relic density close to the experimental value of 01199 and a low χ2 error on the gammaspectrum fit The preferred mass mχ was around 35 GeV and mS ranging from lt10 GeV to500 GeV The smaller masses give a better fit for relic density and also are well separatedfrom current collider constraints mainly due the the fact that the exclusions occur mainlywith higher couplings to t quarks (greater than 1) which is consistent with the fact that manyof the experiments were targeting dark matter pair production associated with t quark decaysNevertheless there were some exclusions that did fall in the region sampled by randomselection from the posterior probability distribution

                    The preferred parameter regions were also consistent with the observation in [20] thatthe Majorana model showed the best overall fit with similar mass constraints We find thatthe couplings to the SM fermions should all be less that 001

                    86

                    The scalar and vector DM models give slightly less good fits especially to the currentrelic density estimate of 01199 but again the current collider constraints are well separatedfrom the best fit region and in fact none of the points randomly sampled from the posteriordistribution showed any exclusion The best fit parameter combination favours couplings toSM fermions of less than 001

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                    Appendix A

                    Validation of Calculation Tools

                    Lepstop0Search for direct pair production of the top squarkin all hadronic final states in proton=proton collisions atradic

                    s=8 TeV with the ATLAS detector [32]

                    Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1) mt1=600 GeV mn1=1 GeVWhere t1 is the top squark and n1 is the neutralino of the MSSM

                    94

                    Table A1 0 Leptons in the final state

                    Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 10000 460 460 100Emiss

                    T gt 130GeV 8574 395 451 88Leptonveto 6621 307 285 108Emiss

                    T gt 150GeV 6316 293 294 100Jet multiplicityand pT 1099 48 51 94|∆φ( jet pmiss

                    T )|gt π5 902 40 42 96gt= 2b jets 396 19 18 106τ veto 396 19 18 106mbmin

                    T gt 175GeV 270 13 13 105SRA1 79 4 4 104SRA2 70 3 3 93SRA3 67 3 3 97SRA4 52 2 2 91SRC exactly5 jets 1297 50 60 83SRC |∆φ( jet pmiss

                    T )|gt π5 1082 50 50 99SRC gt= 2b jets 430 20 20 97SRC τ veto 430 20 20 97SRC |∆φ(bb)|gt 02π 367 17 17 99SRC1 246 12 11 105SRC2 177 9 8 116SRC3 152 8 7 116

                    95

                    Lepstop1Search for direct top squark pair production infinal states with one isolated lepton jets and missing trans-verse momentum in

                    radics = 8 TeV pp collisions using 21 f bminus1

                    of ATLAS data[33]

                    Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1)(t1 gt b x+1 )(t1 gt b xminus1 )mt1=650 GeV mn1=1 GeV Where t1 is the top squark n1 is the neutralino xplusmn1 are thecharginos

                    96

                    Table A2 1 Lepton in the Final state

                    Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 100000 99990 99990 100Electron(= 1signal) 11558 11557 11785 984 jets(80604025) 5187 5186 6271 83gt= 1btag 4998 4997 5389 93Emiss

                    T gt 100GeV [allSRs] 4131 4131 4393 94Emiss

                    T radic

                    HT gt 5 [allSRs] 4026 4026 4286 94∆φ( jet2 pmiss

                    T )gt 08 [allSRs] 3565 3565 3817 93∆φ( jet1Pmiss

                    T )gt 08 [not SRtN2] 3446 3446 3723 93Emiss

                    T gt 200GeV (SRtN2) 2027 2027 2186 93Emiss

                    T radic

                    HT gt 13(SRtN2) 1156 1156 1172 99mT gt 140GeV (SRtN2) 839 839 974 86Emiss

                    T gt 275GeV (SRtN3) 948 948 965 98Emiss

                    T radic

                    HT gt 11(SRtN3) 932 932 937 99mT gt 200GeV (SRtN3) 456 456 566 81Emiss

                    T gt 150GeV (SRbC1minusSRbC3) 2803 2803 3080 91Emiss

                    T radic

                    HT gt 7(SRbC1minusSRbC3) 2742 2742 3017 91mT gt 12GeV (SRbC1minusSRbC3) 1917 1917 2347 82Emiss

                    T gt 160GeV (SRbC2SRbC3) 1836 1836 2257 81Emiss

                    T radic

                    HT gt 8(SRbC2SRbC3) 1788 1788 2175 82me f f gt 550GeV (SRbC2) 1741 1741 2042 85me f f gt 700GeV (SRbC3) 1075 1075 1398 77Muon(= 1signal) 11379 11378 11975 954 jets(80604025) 5264 5263 6046 87gt= 1btag 5101 5100 5711 89Emiss

                    T gt 100GeV [allSRs] 4199 4199 4770 88Emiss

                    T radic

                    HT gt 5 [all SRs] 4115 4115 4653 88∆φ( jet2 pmiss

                    T )gt 08 [all SRs] 3622 3622 4077 89∆φ( jet1 pmiss

                    T )gt 08 [not SRtN2] 3515 3515 3970 89Emiss

                    T gt 200GeV (SRtN2) 2136 2136 2534 84Emiss

                    T radic

                    HT gt 13(SRtN2) 1261 1261 1441 87mT gt 140GeV (SRtN2) 928 928 1033 90Emiss

                    T gt 275GeV (SRtN3) 1036 1036 1274 81Emiss

                    T radic

                    HT gt 11(SRtN3) 1018 1018 1235 82mT gt 200GeV (SRtN3) 497 497 612 81Emiss

                    T gt 150GeV (SRbC1minusSRbC3) 2891 2891 3380 86Emiss

                    T radic

                    HT gt 7(SRbC1minusSRbC3) 2821 2821 3290 86mT gt 120GeV (SRbC1minusSRbC3) 1977 1977 2290 86Emiss

                    T gt 160GeV (SRbC2SRbC3) 1913 1913 2220 86Emiss

                    T radic

                    HT gt 8(SRbC2SRbC3) 1863 1863 2152 87me f f gt 550GeV (SRbC2) 1826 1826 2116 86me f f gt 700GeV (SRbC3) 1131 1131 1452 78

                    97

                    Lepstop2Search for direct top squark pair production infinal states with two leptons in

                    radics =8 TeV pp collisions using

                    20 f bminus1 of ATLAS data[83][34]

                    Simulated in Madgraph with p p gt t1 t1(t1 gt b w+ n1)(t1 gt b wminusn1) mt1=180 GeV mn1=60 GeV Where t1 is the top squark n1 is the neutralino b1 is the bottom squark

                    Table A3 2 Leptons in the final state

                    Cut RawMe Scaled ATLAS MeATLAS PercentNoCuts 100000 324308 199000 1632BaselineCuts 3593 11652 7203 1622SF Signal Leptons 1699 5510 5510 1002OSSF Signal Leptons 1632 5293 5233 101mll gt 20Gev 1553 5037 5028 100leadinglepton pT 1479 4797 5048 95|mll minusmZ|gt 20Gev 1015 3292 3484 94∆φmin gt 10 850 2757 2160 128∆φb lt 15 680 2205 1757 126SRM90 [SF ] 13 42 60 70SRM120 [SF ] 0 0 19 0SRM100 +2 jets [SF ] 0 0 17 0SRM110 +2 jets [SF ] 0 0 1 02DF Signal Leptons 1737 5633 7164 792OSDF Signal Leptons 1670 5416 5396 100mll20Gev 1587 5147 5136 100leadinglepton pT 1501 4868 4931 99∆φmin gt 10 1247 4044 3122 130∆φb lt 15 1010 3276 2558 128SRM90 [SF ] 21 68 95 72SRM120 [SF ] 0 0 23 0SRM100 +2 jets [SF ] 0 0 19 0SRM110 +2 jets [SF ] 0 0 14 0

                    98

                    2bstopSearch for direct third-generation squark pair pro-duction in final states with missing transverse momentumand two b-jets[31]

                    Simulated in Madgraph with p p gt b1 b1(b1 gt b n1)(b1 gt b n1)

                    SRAmb1=500 GeV mn1=1 GeV SRBmb1=350 GeV mn1=320 GeVWhere b1 is the bottom squark and n1 is the neutralino

                    Table A4 2b jets in the final state

                    Cut RawMe Scaled ATLAS MeATLAS PercentSRA Nocuts 100000 173800 1738 100SRA MET gt 80 93407 162341 1606 101SRA Leptonveto 93302 162159 1505 108SRA MET gt 150 79449 138082 1323 104SRA Jet selection 10213 17750 119 149SRA mbb gt 200 8344 14502 96 151SRA mCT gt 150 6535 11358 82 139SRA mCT gt 200 5175 8994 67 134SRA mCT gt 250 3793 6592 51 129SRA mCT gt 300 2487 4322 35 123SRB NoCuts 100000 1624100 16241 100SRB leptonveto 11489 186593 4069 46SRB MET gt 250 2530 41090 757 54SRB Jet selection 63 1023 79 130SRB HT 3 lt 50 51 828 52 159

                    99

                    CMSTop1LSearch for the Production of Dark Matter in As-sociation with Top Quark Pairs in the Single-lepton FinalState in pp collisions[41]

                    Simulated in Madgraph with p p gt t t p1 p1

                    Where p1= Majorana fermion Table 3 shows Signal efficiencies and cross sections TheM scale was read from Figure 7 for 90 CL

                    Table A5 Signal Efficiencies 90 CL on σ limexp[ f b] on pp gt tt +χχ

                    Mχ[GeV ] My Signal Efficiency Quoted Efficiency My σ [ f b] Quoted σ [ f b]1 102 101 63 47

                    10 132 101 62 4650 143 120 50 38100 170 146 56 32200 185 173 69 27600 265 240 32 19

                    1000 320 276 41 17

                    Appendix B

                    Branching ratio calculations for narrowwidth approximation

                    B1 Code obtained from decayspy in Madgraph

                    Br(S rarr bb) = (minus24λ2b m2

                    b +6λ2b m2

                    s

                    radicminus4m2

                    bm2S +m4

                    S)16πm3S

                    Br(S rarr tt) = (6λ2t m2

                    S minus24λ2t m2

                    t

                    radicm4

                    S minus4ms2m2t )16πm3

                    S

                    Br(S rarr τ+

                    τminus) = (2λ

                    2τ m2

                    S minus8λ2τ m2

                    τ

                    radicm4

                    S minus4m2Sm2

                    τ)16πm3S

                    Br(S rarr χχ) = (2λ2χm2

                    S

                    radicm4

                    S minus4m2Sm2

                    χ)32πm3S

                    (B1)

                    Where

                    mS is the mass of the scalar mediator

                    mχ is the mass of the Dark Matter particle

                    mb is the mass of the b quark

                    mt is the mass of the t quark

                    mτ is the mass of the τ lepton

                    The coupling constants λ follow the same pattern

                    • TITLE Collider Constraints applied to Simplified Models of Dark Matter fitted to the Fermi-LAT gamma ray excess using Bayesian Techniques
                      • Dedication
                      • Declaration
                      • Acknowledgements
                      • Contents
                      • List of Figures
                      • List of Tables
                        • Chapter 1 Introduction
                        • Chapter 2 Review of Physics
                        • Chapter 3 Fitting Models to the Observables
                        • Chapter 4 Calculation Tools
                        • Chapter 5 Majorana Model Results
                        • Chapter 6 Real Scalar Model Results
                        • Chapter 7 Real Vector Dark Matter Results
                        • Chapter 8 Conclusion
                        • Bibliography
                        • Appendix A Validation of Calculation Tools
                        • Appendix B Branching ratio calculations for narrow width approximation

                      Chapter 1

                      Introduction

                      Dark matter (DM) was first postulated over 80 years ago when Swiss astronomer FritzZwicky observed a discrepancy between the amount of light emitted by a cluster of galaxiesand the total mass contained within the cluster inferred from the relative motion of thosegalaxies by a simple application of the theory of Newtonian gravitation The surprising resultof this observation was that the vast majority of the mass in the cluster did not emit lightwhich was contrary to the expectation that most of the mass would be carried by the starsSince that time further observations over a wide range of scales and experimental techniqueshave continued to point to the same result and refine it Some of these observations and otherevidence are discussed in section 22 We now know with certainty that in the entire Universeall of the matter we know about - stars planets gases and other cosmic objects such as blackholes can only account for less than 5 of the mass that we calculate to be there

                      A recent phenomenon that has received much attention is the significant deviation frombackground expectations of the Fermi Large Area Telescope(Fermi-LAT) gamma ray flux atthe galactic centre [1] A number of astrophysical explanations have been proposed includingmillisecond pulsars of supernova remnants [2] or burst-like continuous events at the galacticcentre but these are unresolved However it has also been noted that the observed Fermi-LATexcess is consistent with the annihilation of dark matter particles which would naturally beconcentrated at the Galactic centre in a manner consistent with the Navarro-Frenk-Whitedistribution of dark matter [3]

                      There are a number of other purely theoretical (particle physics) reasons to postulatethe existence of weakly interacting matter particles that could supply the missing mass andyet remain unobservable Weakly interacting massive particle (WIMPS) have been a majorfocus of Run I and ongoing Run II searches of the Large Hadron Collider (LHC) In spite

                      2

                      of substantial theoretical and experimental effort the microscopic properties of dark matterparticles are still unknown

                      One of the main problems with extracting these properties is the plethora of competingtheoretical models that provide dark matter candidates Another is the very weak interactionof these particles and the lack of experimental observables allowing one to discriminatebetween the competing models

                      A particularly fruitful approach is the simplified model approach which makes minimalassumptions about the DM candidates as extensions to the Standard Model interacting weaklywith ordinary matter through a mediator These can be coded as Lagrangian interaction termsusing tools such as FeynRules [4] and Calchep [5] and used to simulate Dark Matter relicdensities indirect detection expectations as well as direct detection expectations throughMicrOmegas [6] This in turn can be run under a Bayesian inference tool (see below)

                      Using a Bayesian inference tool such as MultiNest [7][8][9] one can scan the parameterspace of a particular model to find regions of highest probability for that model based onobservations of a range of phenomena including direct detection experiments such as ZEPLIN[10] XENON [11] DEAP [12] ARDM [13] DarkSide [14] PandaX [15] and LUX [16]indirect detection - eg Fermi-LAT gamma ray excess [17] Pamela-positron and antiprotonflux [18] and production at colliders MultiNest calculates the evidence value which is theaverage likelihood of the model weighted by prior probability and a posterior distributionwhich is the distribution of the parameter(s) after taking into account the observed databy scanning over a range of parameters based on prior probabilities This can be used tocompare models on a consistent basis The scans also provide information about the best fitparameters for the overall fit as well as for the seperate experimentsobservations

                      The simplified model approach has been taken in a number of papers to study the Fermi-LAT gamma ray excess (see literature review in section 12) and one in particular comparesthese models using a Bayesian inference tool which found that a Majorana fermion DMcandidate interacting via a scalar mediator with arbitrary couplings to third generationfermions showed the best fit [19] A subsequent paper [20] examined the most likelyparameter combinations that simultaneously fit the Planck relic density Fermi-LAT galacticcentre gamma ray AMS-02 positron flux and Pamela antiproton flux data

                      There has not as yet been a thorough study of the collider constraints on DM simplifiedmodels that fit the Fermi-LAT galactic centre excess In this thesis we reproduce the

                      11 Motivation 3

                      previous results of [20] using similar astrophysical constraints and then implement detailedsimulations of collider constraints on DM and mediator production

                      In Section 11 we review the motivation for this study and in Section 12 we considerrecent astronomical data which is consistent with the annihilation of DM particles andreview the literature and searches which attempt to explain it with a range of extensions tothe standard model

                      In Chapter 2 we review the Standard Model of particle physics (SM) the evidence fordark matter and the ATLAS and CMS experiments at the LHC

                      In Chapter 3 we describe the models considered in this paper the observables used inthe Bayesian scans and the collider experiments which provide limits against which theMagraph simulations of the three models can be compared

                      In Chapter 4 we review the calculation tools that have been used in this paper

                      In Chapters 5 6 and 7 we give the results of the Bayesian scans and collider cutsanalyses for each of the models and in Chapter 8 we summarise our conclusions

                      11 Motivation

                      The Fermi gamma ray space telescope has over the past few years detected a gamma raysignal from the inner few degrees around the galactic centre corresponding to a regionseveral hundred parsecs in radius The spectrum and angular distribution are compatible withthat from annihilating dark matter particles

                      A number of papers have recently attempted to explain the Gamma Ray excess in termsof extensions to the Standard Model taking into account constraints from ATLAS and CMScollider experiments direct detection of dark matter colliding off nucleii in well shieldeddetectors (eg LUX) as well as other indirect measurements (astrophysical observations)These papers take the approach of simplified models which describe the contact interactionsof a mediator beween DM and ordinary matter using an effective field theory approach(EFT) Until recently it has been difficult to discriminate between the results of these paperswhich look at different models and produce different statistical results A recent paper[19] Simplified Dark Matter Models Confront the Gamma Ray Excess used a Bayesianinference tool to compare a variety of simplified models using the Bayesian evidence values

                      12 Literature review 4

                      calculated for each model to find the most preferred explanation The motivation for thechosen models is discussed in section 121

                      A subsequent paper [20] Interpreting the Fermi-LAT gamma ray excess in the simplifiedframework studied the most likely model again using Bayesian inference to find the mostlikely parameters for the (most likely) model The data inputs (observables) to the scanwere the Planck measurement of the dark matter relic density [21] Fermi-LAT gamma rayflux data from the galactic center [17] cosmic positron flux data [22] cosmic anti-protonflux data [18] cosmic microwave background data and direct detection data from the LUXexperiment [16] The result of the scan was a posterior probability density function in theparameter space of the model

                      This paper extends the findings in [19] by adding collider constraints to the likelihoodestimations for the three models It is not a simple matter to add collider constraints becauseeach point in parameter space demands a significant amount of computing to calculate theexpected excess amount of observable particles corresponding to the particular experimentgiven the proposed simplified model A sampling method has been employed to calculatethe collider constraints using the probability of the parameters based on the astrophysicalobservables

                      12 Literature review

                      121 Simplified Models

                      A recent paper [23] summarised the requirements and properties that simplified models forLHC dark matter searches should have

                      The general principles are

                      bull Besides the Standard Model (SM) the Lagrangian should contain a DM candidate thatis absolutely stable or lives long enough to escape the LHC detector and a mediatorthat couples the two sectors

                      bull The Lagrangian should contain all terms that are renormalizable consistent withLorentz invariance SM symmetries and DM stability

                      bull Additional interactions should not violate exact and approximate global symmetries ofthe SM

                      12 Literature review 5

                      The examples of models that satisfy these requirements are

                      1 Scalar (or pseudo-scalar) s-channel mediator coupled to fermionic dark matter (Diracor Majorana)

                      2 Higgs portal DM where the DM is a scalar singlet or fermion singlet under the gaugesymmetries of the SM coupling to a scalar boson which mixes with the Higgs (Thefermion singlet is a specific realisation of 1 above)

                      3 DM is a mixture of an electroweak singlet and doublet as in the Minimal Supersym-metric SM (MSSM)

                      4 Vector (or Axial vector) s-channel mediator obtained by extending the SM gaugesymmetry by a new U(1)rsquo which is spontaneously broken such that the mediatoracquires mass

                      5 t-channel flavoured mediator (if the DM particle is a fermion χ the mediator can bea coloured scalar or vector particle φ -eg assuming a scalar mediator a couplingof the form φ χq requires either χ or φ to carry a flavor index to be consistent withminimal flavour violation (MFV) which postulates the flavour changing neutral currentstructure in the SM is preserved

                      Another recent paper [24] summarised the requirements and properties that simplifiedmodels for LHC dark matter searches should have The models studied in this paper arepossible candidates competing with Dirac fermion and complex scalar dark matter andmodels with vector and pseudo-scalar mediators Other models would have an elasticscattering cross section that will remain beyond the reach of direct detection experiments dueto the neutrino floor (the irreducible background produced by neutrino scattering) or arealready close to sensitivity to existing LUX and XENONIT experiments

                      A recent paper [25] Scalar Simplified Models for Dark Matter looks at Dirac fermiondark matter mediated by either a new scalar or pseudo-scalar to SM fermions This paperplaces bounds on the coupling combination gχgφ (coupling of DM to mediator times couplingof mediator to SM fermions) calculated from direct detection indirect detection and relicdensity by the mass of DM and mass of mediator Bounds are also calculated for colliderconstraints (mono-jet searches heavy-flavour searches and collider bounds on the 125 GeVHiggs are used to place limits on the monojet and heavy flavor channels which can betranslated into limits on the Higgs coupling to dark matter and experimental measurements

                      12 Literature review 6

                      of the Higgs width are used to constrain the addition of new channels to Higgs decay) Thepaper models the top loop in a full theory using the Monte Carlo generator MCFM [26]and extends the process to accommodate off-shell mediator production and decay to a darkmatter pair While this paper does not consider the Majorana fermion dark matter model theconstraints on the coupling product may have some relevence to the present study if only asan order of magnitude indicator

                      Constraining dark sectors with monojets and dijets [27] studies a simplified model withDirac fermion dark matter and an axial-vector mediator They use two classes of constraint-searches for DM production in events with large amounts of missing transverse energy inassociation with SM particles (eg monojet events) and direct constraints on the mediatorfrom monojet searches to show that for example where the choice of couplings of themediator to quarks and DM gA

                      q =gAχ=1 all mediator masses in the range 130 GeV ltMR lt3

                      TeV are excluded

                      The paper Constraining Dark Sectors at Colliders Beyond the Effective Theory Ap-proach [28] considers Higgs portal models with scalar pseudo-scalar vector and axial-vectormediators between the SM and the dark sector The paper takes account of loop processesand gluon fusion as well as quark-mediator interactions using MCFM Apart from mediatortype the models are characterised by free parameters of mediator width and mass darkmatter mass and the effective coupling combination gχgφ (coupling of DM to mediator timescoupling of mediator to SM fermions) The conclusion is that LHC searches can proberegions of the parameter space that are not able to be explored in direct detection experiments(since the latter measurements suffer from a velocity suppression of the WIMP-nucleoncross-section)

                      The paper Constraining the Fermi-LAT excess with multi-jet plus EmissT collider searches

                      [29] shows that models with a pseudo-scalar or scalar mediator are best constrained by multi-jet final states with Emiss

                      T The analysis was done using POWHEG-BOX-V2 at LO and usesM2

                      T variables together with multi-jet binning to exploit differences in signal and backgroundin the different regions of phase space to obtain a more stringent set of constraints on DMproduction This paper largely motivated the current work in that it showed that collidersearches that were not traditionally thought of as dark matter searches were neverthelessoften sensitive to dark matter produced in association with heavy quarks The combinationof searches used in this paper reflects this fact

                      12 Literature review 7

                      122 Collider Constraints

                      In order to constrain this model with collider constraints (direct detection limits) we considerthe following papers by the CMS and ATLAS collaborations which describe searches at theLHC

                      ATLAS Experiments

                      bull Search for dark matter in events with heavy quarks and missing transverse momentumin p p collisions with the ATLAS detector [30]

                      bull Search for direct third-generation squark pair production in final states with missingtransverse momentum and two b-jets in

                      radics= 8 TeV pp collisions with the ATLAS

                      detector[31]

                      bull Search for direct pair production of the top squark in all-hadronic final states inprotonndashproton collisions at

                      radics=8 TeV with the ATLAS detector [32]

                      bull Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

                      radic(s)=8TeV pp collisions using 21 f bminus1 of

                      ATLAS data [33]

                      bull Search for direct top squark pair production in final states with two leptons inradic

                      s = 8TeV pp collisions using 20 f bminus1 of ATLAS data [34]

                      bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

                      radics=8 TeV [35]

                      CMS Experiments

                      bull Searches for anomalous tt production in p p collisions atradic

                      s=8 TeV [36]

                      bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

                      radics=8 TeV [37]

                      bull Search for the production of dark matter in association with top-quark pairs in thesingle-lepton final state in proton-proton collisions at

                      radics = 8 TeV [38]

                      bull Search for new physics in monojet events in p p collisions atradic

                      s = 8 TeV(CMS) [39]

                      bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

                      s = 8 TeV [40]

                      12 Literature review 8

                      bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theSingle-lepton Final State in pp collisions at

                      radics=8 TeV [41]

                      bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

                      s=8 TeV [42]

                      bull Search for neutral MSSM Higgs bosons decaying into a pair of bottom quarks [43]

                      bull Search for neutral MSSM Higgs bosons decaying into a pair of τ leptons in p p collision[44]

                      bull Search for dark matter extra dimensions and unparticles in monojet events in proton-proton collisions at

                      radics=8 TeV [45]

                      In particular we have implemented Madgraph simulations of certain data analysesapplying to collider experiments described in papers [32][33][34][31][30][41] Theremaining models were not implemented in Madgraph because of inability to validate themodels against published results in the time available We developed C++ analysis code toimplement the collider cuts after showering with Pythia and basic analysis with Delpheswith output to root files The C++ code allowed multiple experimental cuts to be applied tothe same simulation output The C++ code was first checked against the results of the variousexperiments above and then the simulations were rerun using the three modelsrsquo Lagrangians(implemented in FeynRules) described in this paper

                      Chapter 2

                      Review of Physics

                      21 Standard Model

                      211 Introduction

                      The Standard Model of particle physics (SM) has sometimes been described as the theoryof Almost Everything The SM is a field theory which emerged in the second half of thetwentieth century from quantum mechanics and special relativity which themselves wereonly conceived after 1905 by Einsten Pauli Bohr Dirac and others In the first decades ofthe twenty first century the discovery of the Higgs boson [46][47] completed the pictureand confirmed the last previously unverified part of the SM The SM is a self consistenttheory that describes and classifies all known particles and their interactions (other thangravity) to an astonishing level of accuracy and detail However the theory is incomplete-it does not describe the complete theory of gravitation or account for DM (see below) ordark energy (the accelerating expansion of the universe) Related to these shortcomings aretheoretical deficiencies such as the asymmetry of matter and anti-matter the fact that the SMis inconsistent with general relativity (to the extent that one or both theories break down atspacetime singularities like the Big Bang and black hole event horizons)

                      212 Quantum Mechanics

                      Quantum Mechanics (QM) describes particles as probability waves which can be thoughtof as packets of energy which can be localised or spread through all space depending on how

                      21 Standard Model 10

                      accurately one measures the particlersquos momentum However QM is not itself consistent withspecial relativity It was realised too that because of relativity particles could be created anddestroyed and because of the uncertainty principle the theory should describe the probabilityof particles being created and destroyed on arbitrarily short timescales A quantum fieldtheory (QFT) is a way to describe this probability at every point of space and time

                      213 Field Theory

                      A field theory is described by a mathematical function of space and time called a LagrangianAs in classical mechanics where Lagrangians were introduced by Louis Lagrange at thebeginning of the nineteenth century to summarise dynamical systems the Lagrangian sum-marises the dynamics of space which is permeated by force fields The equations of motionarising from a Lagrangian are determined by finding the contours of the Lagrangian describedby stationary points (ie where the infinitesimal change in the Lagrangian is zero for a smallchange in field) Particles are generally described by singularities in the Lagrangian (wherethe values tend to infinity also called resonances)

                      214 Spin and Statistics

                      It was also realised that different types of particle needed to be represented by different fieldsin the Lagrangian Diracrsquos discovery that particles with spin 12 quantum numbers couldbe represented by a field obtained by effectively taking the square root of the relativisticequation (Klein Gordon equation) was probably the most significant advance in quantumfield theory It required the introduction of a matrix operators called the gamma matricesor Dirac matrices which helped explain the already known fact that spin half particles orfermions obey Fermi-Dirac statistics ie when two spin half particles are exchanged thesign of the wavefunction must change and hence they cannot occupy the same quantum stateunlike bosons with integer spin that can occupy the same quantum state

                      Other particles need to be represented by fields called vector pseudo-vector scalar (as inthe Klein Gordon equation) pseudo-scalar and complex fields A vector field is really fourfields one for each dimension of space plus time and is parameterised by the four gammamatrices γmicro which transform among themselves in a relativistically invariant way (Lorentzinvariance) A pseudo-vector field is a vector field which transforms in the same way but

                      21 Standard Model 11

                      with additional sign change under parity transformations (a simultaneous reversal of the threespatial directions) and is represented by γmicroγ5 A pseudo-scalar is a scalar which changes signunder parity transformations and a complex scalar field is represented by complex numbers

                      215 Feynman Diagrams

                      QFT took another giant leap forward when Richard Feynman realised that the propagationof particles described by the field could be described by expanding the so called S-matrixor scattering matrix derived from calculating the probability of initial states transitioning tofinal states The expansion is similar to a Taylor expansion in mathematics allowing for thespecial commuting properties of elements of the field The Feynman diagrams representingindividual terms in the expansion can be interpreted as interactions between the particlecomponents The S-Matrix (or more correctly the non trivial part called the T-Matrix) isderived using the interaction Hamiltonian of the field The Hamiltonian is an operator whoseeigenvalues or characteristic values give the energy quanta of the field The T-matrix canalso be described by a path integral which is an integral over all of quantum space weightedby the exponential of the Lagrangian This is completely equivalent to the Hamiltonianformulation and shows how the Lagrangian arises naturally out of the Hamiltonian (in factthe Lagrangian is the Legendre transformation of the Hamiltonian)

                      Figure 21 Feynman Diagram of electron interacting with a muon

                      γ

                      eminus

                      e+

                      micro+

                      microminus

                      The diagram Fig21 is a tree diagram (diagrams with closed loops indicate self in-teractions and are higher order terms in the expansion) and shows an electron (eminus) and ananti-muon (micro+) interacting by exchanging a photon

                      21 Standard Model 12

                      216 Gauge Symmetries and Quantum Electrodynamics (QED)

                      The Dirac equation with its gamma matrices ushered in the use of group theory to describethe transformation of fields (because the multiplication table of the gamma matrices isclosed and described by group theory) It also ushered in the idea of symmetries within theLagrangian- for example consider a Lagrangian

                      ψ(ipart minusm)ψ (21)

                      The Lagrangian is invariant under a global U(1) transformation which acts on the fieldsand hence the derivatives as follows

                      ψ rarr eiqαψ ψ rarr ψeminusiqα partmicroψ rarr eiqα

                      partmicroψ (22)

                      where qα is a global phase and α is a continuous parameter

                      A symmetry leads by Noetherrsquos theorem to a conserved current jmicro = qψγmicroψ and is thebasis of charge conservation since partmicro jmicro = 0 and q =

                      intd3x j0(x)

                      By promoting this transformation to a local one (ie α rarr α(x) where x = xmicro are thespace time co-ordinates) the transformations become

                      ψ rarr eiqα(x)ψ ψ rarr ψeminusiqα(x)partmicroψ rarr eiqα(x)(partmicroψ + iq(partmicroα(x))ψ) (23)

                      The derivative transformation introduces an extra term which would spoil the gaugeinvariance This can be solved by introducing a new field the photon field Amicro(x) whichinteracts with the fermion field ψ and transforms under a U(1) transformation

                      Amicro rarr Amicro minuspartmicroα(x) (24)

                      If we replace normal derivatives partmicro with covariant derivatives Dmicro = (partmicro + iqAmicro(x)) in theLagrangian then the extra term is cancelled and we can now see that the covariant derivativetransforms like ψ - under a local U(1) transform ie

                      Dmicro rarr Dmicroψprime = eiqα(x)Dmicroψ (25)

                      The additional term iqAmicro(x) can be interpreted in a geometric sense as a connectionwhich facilitates the parallel transport of vectors along a curve in a parallel consistent manner

                      21 Standard Model 13

                      We must add a gauge-invariant kinetic term to the Lagrangian to describe the dynamicsof the photon field This is given by the electromagnetic field strength tensor

                      Fmicroν = partmicroAν minuspartνAmicro (26)

                      The full Quantum Electrodynamics (QED) Lagrangian is then for electrons with mass mand charge -e

                      LQED = ψ(i Dminusm)ψ minus 14

                      Fmicroν(X)Fmicroν(x) (27)

                      This Lagrangian is Lorentz and U(1) guage invariant The interaction term is

                      Lint =+eψ Aψ = eψγmicro

                      ψAmicro = jmicro

                      EMAmicro (28)

                      where jmicro

                      EM is the electromagnetic four current

                      217 The Standard Electroweak Model

                      The Standard Electroweak (EW) Model was created by Glashow-Weinberg-Salam (also calledthe GWS model) and unifies the electomagnetic and weak interactions of the fundamentalparticles

                      The idea of gauge symmetry was extended to the idea of a non-abelian gauge symmetry(essentially the operators defining the symmetry do not commute and so are represented bymatrices- in this case the Pauli matrices for the SU(2) group) - the gauge group is the directproduct G = SU(2)

                      otimesU(1) It was known that weak interactions were mediated by Wplusmn

                      and Z0 particles where the superscripts are the signs of the electromagnetic charge Thesewere fairly massive but it was also known that through spontaneous symmetry breaking ofa continuous global non-abelian symmetry there would be one massive propagation modefor each spontaneously broken-symmetry generator and massless modes for the unbrokengenerators (the so-called Goldstone bosons)

                      This required an extra field to represent the gauge bosons called Bmicro (a vector fieldhaving the 4 spacetime degrees of freedom) With this identification the covariant derivativebecomes

                      Dmicro = partmicro minus igAmicro τ

                      2minus i

                      gprime

                      2Y Bmicro (29)

                      21 Standard Model 14

                      Here Y is the so called weak hypercharge of the field Bmicro which is a quantum numberdepending on particle (lepton or quark) g and gprime are two independent parameters in theGWS theory and the expression Amicro τ implies a sum over 3 components Aa

                      micro a=123 and thePauli matrices τa

                      This isnrsquot the full story because it had been known since 1956 when Chin Shuing Wuand her collaborators confirmed in an experiment prompted by Lee and Yang that nature isnot ambivalent to so-called parity It had been assumed that the laws of nature would notchange if all spatial coordinates were reversed (a parity transformation) The experimentinvolved the beta decay of cooled 60C atoms (more electrons were emitted in the backwardhemisphere with respect to spin direction of the atoms) This emission involved the weakinteraction and required that the Lagrangian for weak interactions contain terms which aremaximally parity violating- which turn out to be currents of the form

                      ψ(1minus γ5)γmicro

                      ψ (210)

                      The term

                      12(1minus γ

                      5) (211)

                      projects out a left-handed spinor and can be absorbed into the spinor to describe left-handedleptons (and as later discovered quarks) The right handed leptons are blind to the weakinteraction

                      The processes describing left-handed current interactions are shown in Fig 22

                      Analogously to isospin where the proton and neutron are considered two eigenstates ofthe nucleon we postulate that Bmicro is the gauge boson of an SU(2)transformation coupled toweak SU(2) doublets (

                      νe

                      eminus

                      )

                      (ud

                      ) (212)

                      We may now write the weak SU(2) currents as eg

                      jimicro = (ν e)Lγmicro

                      τ i

                      2

                      e

                      )L (213)

                      21 Standard Model 15

                      Figure 22 Weak Interaction Vertices [48]

                      where τ i are the Pauli-spin matrices and the subscript L denotes the left-handed componentof the spinor projected out by Eqn211 The third current corresponding to i=3 is called theneutral current as it doesnrsquot change the charge of the particle involved in the interaction

                      We could postulate that the electromagnetic current is a linear combination of the leftand right elecromagnetic fields eL = 1

                      2(1minus γ5)e and eR = 12(1+ γ5)e

                      jemmicro = eLγmicroQeL + eRγmicroQeR (214)

                      where Q is the electromagnetic operator and eR is a singlet However this is not invariantunder SU(2)L To construct an invariant current we need to use the SU(2)L doublet 212The current which is an SU(2)L invariant U(1) current is

                      jYmicro = (ν e)LγmicroYL

                      e

                      )L+ eRγmicroYReR (215)

                      where hypercharges YL and YR are associated with the U(1)Y symmetry jYmicro is a linearcombination of the electromagnetic current and the weak neutral current thus explaining whythe hypercharge differs for the left and right-handed components since the electromagneticcurrent has the same charge for both components whilst the weak current exists only forleft-handed particles j3

                      micro and the third component of weak isospin T 3 allows us to calculate

                      21 Standard Model 16

                      the relationship Y = 2(QminusT 3) which can be seen by writing jYmicro as a linear combination ofj3micro (the weak neutral current that doesnrsquot change the charge of the particle involved in the

                      interaction) and jemmicro (using the convention that the jYmicro is multiplied by 1

                      2 to match the samefactor implicit in j3

                      micro ) Substituting

                      τ3 =

                      (1 00 minus1

                      )(216)

                      into equation 213 for i=3 and subtracting equation 215 and equating to 12 times equation

                      214

                      we get

                      eLγmicroQeL + eRγmicroQeR minus (νLγmicro

                      12

                      νL minus eLγmicro

                      12

                      eL) =12

                      eRγmicroYReR +12(ν e)LγmicroYL

                      e

                      )L (217)

                      from which we can read out

                      YR = 2QYL = 2Q+1 (218)

                      and T3(eR) = 0 T3(νL) =12 and T3(eL) =

                      12 The latter three identities are implied by

                      the fraction 12 inserted into the definition of equation 213

                      The Lagrangian kinetic terms of the fermions can then be written

                      L =minus14

                      FmicroνFmicroν minus 14

                      GmicroνGmicroν

                      + sumgenerations

                      LL(i D)LL + lR(i D)lR + νR(i D)νR

                      + sumgenerations

                      QL(i D)QL +UR(i D)UR + DR(i D)DR

                      (219)

                      LL are the left-handed lepton doublets QL are the left-handed quark doublets lRνR theright handed lepton singlets URDR the right handed quark singlets

                      The field strength tensors are given by

                      Fmicroν = partmicroAν minuspartνAmicro minus ig[Amicro Aν ] (220)

                      21 Standard Model 17

                      andGmicroν = partmicroBν minuspartνBmicro (221)

                      Bmicro is an abelian U(1) gauge field while Amicro is non-abelian and Fmicroν contains the com-mutator of the field through the action of the covariant derivative [Dmicro Dν ] This equationcontains no mass terms terms for the field Amicro for good reason- this would violate gaugeinvariance We know that the gauge bosons of the weak force have substantial mass and theway that this problem has been solved is through the Higgs mechanism

                      218 Higgs Mechanism

                      To solve the problem of getting gauge invariant mass terms three independent groupspublished papers in 1964 The mechanism was named after Peter Higgs To introduce theconcept consider a complex scalar field φ that permeates all space Consider a potential forthe Higgs field of the form

                      minusmicro2φ

                      daggerφ +λ (φ dagger

                      φ2) (222)

                      which has the form of a mexican hat Fig 23 This potential is still U(1) gauge invariant andcan be added to our Lagrangian together with kinetic+field strength terms to give

                      L = (Dmicroφ)dagger(Dmicroφ)minusmicro2φ

                      daggerφ +λ (φ dagger

                      φ2)minus 1

                      4FmicroνFmicroν (223)

                      It is easily seen that this is invariant to the transformations

                      Amicro rarr Amicro minuspartmicroη(x) (224)

                      φ(x)rarr eieη(x)φ(x) (225)

                      The minimum of this field is not at 0 but in a circular ring around 0 with a vacuum

                      expectation value(vev)radic

                      micro2

                      2λequiv vradic

                      2

                      We can parameterise φ as v+h(x)radic2

                      ei π

                      Fπ where h and π are referred to as the Higgs Bosonand Goldstone Boson respectively and are real scalar fields with no vevs Fπ is a constant

                      21 Standard Model 18

                      Figure 23 Higgs Potential [49]

                      Substituting this back into the Lagrangian 223 we get

                      minus14

                      FmicroνFmicroν minusevAmicropartmicro

                      π+e2v2

                      2AmicroAmicro +

                      12(partmicrohpart

                      microhminus2micro2h2)+

                      12

                      partmicroπpartmicro

                      π+(hπinteractions)(226)

                      This Lagrangian now seems to describe a theory with a photon Amicro of mass = ev a Higgsboson h with mass =

                      radic2micro and a massless Goldstone π

                      However there is a term mixing Amicro and part microπ in this expression and this shows Amicro and π

                      are not independent normal coordinates and we can not conclude that the third term is a massterm This π minusAmicro mixing can be removed by making a gauge transformation in equations224 and 225 making

                      φrarrv+h(x)radic2

                      ei π

                      Fπminusieη(x) (227)

                      and setting πrarr π

                      Fπminus eη(x) = 0 This choosing of a point on the circle for the vev is called

                      spontaneous symmetry breaking much like a vertical pencil falling and pointing horizontallyin the plane This choice of gauge is called unitary gauge The Goldstone π will thencompletely disappear from the theory and one says that the Goldstone has been eaten to givethe photon mass

                      21 Standard Model 19

                      This can be confirmed by counting degrees of freedom before and after spontaneoussymmetry breaking The original Lagrangian had four degrees of freedom- two from thereal massless vector field Amicro (massless photons have only two independent polarisationstates) and two from the complex scalar field (one each from the h and π fields) The newLagrangian has one degree of freedom from the h field and 3 degrees of freedom from themassive real vector field Amicro and the π field has disappeared

                      The actual Higgs field is postulated to be a complex scalar doublet with four degrees offreedom

                      Φ =

                      (φ+

                      φ0

                      )(228)

                      which transforms under SU(2) in the fundamental representation and also transforms underU(1) hypercharge with quantum numbers

                      Table 21 Quantum numbers of the Higgs field

                      T 3 Q Yφ+

                      12 1 1

                      φ0 minus12 1 0

                      We can parameterise the Higgs field in terms of deviations from the vacuum

                      Φ(x) =(

                      η1(x)+ iη2(x)v+σ(x)+ iη3(x)

                      ) (229)

                      It can be shown that the three η fields can be transformed away by working in unitarygauge By analogy with the scalar Higgs field example above the classical action will beminimised for a constant value of the Higgs field Φ = Φ0 for which Φ

                      dagger0Φ0 = v2 This again

                      defines a range of minima and we can pick one arbitarily (this is equivalent to applying anSU(2)timesU(1) transform to the Higgs doublet transforming the top component to 0 and thebottom component to Φ0 = v) which corresponds to a vacuum of zero charge density andlt φ0 gt=+v

                      In this gauge we can write the Higgs doublet as

                      Φ =

                      (φ+

                      φ0

                      )rarr M

                      (0

                      v+ H(x)radic2

                      ) (230)

                      where H(x) is a real Higgs field v is the vacuum expectation and M is an SU(2)timesU(1)matrix The matrix Mdagger converts the isospinor into a down form (an SU(2) rotation) followedby a U(1) rotation to a real number

                      21 Standard Model 20

                      If we consider the Higgs part of the Lagrangian

                      minus14(Fmicroν)

                      2 minus 14(Bmicroν)

                      2 +(DmicroΦ)dagger(DmicroΦ)minusλ (Φdagger

                      Φminus v2)2 (231)

                      Substituting from equation 230 into this and noting that

                      DmicroΦ = partmicroΦminus igW amicro τ

                      aΦminus 1

                      2ig

                      primeBmicroΦ (232)

                      We can express as

                      DmicroΦ = (partmicro minus i2

                      (gA3

                      micro +gprimeBmicro g(A1micro minusA2

                      micro)

                      g(A1micro +A2

                      micro) minusgA3micro +gprimeBmicro

                      ))Φ equiv (partmicro minus i

                      2Amicro)Φ (233)

                      After some calculation the kinetic term is

                      (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

                      14(v+

                      Hradic2)2[A 2]22 (234)

                      where the 22 subscript is the index in the matrix

                      If we defineWplusmn

                      micro =1radic2(A1

                      micro∓iA2micro) (235)

                      then [A 2]22 is given by

                      [A 2]22 =

                      (gprimeBmicro +gA3

                      micro

                      radic2gW+

                      microradic2gWminus

                      micro gprimeBmicro minusgA3micro

                      ) (236)

                      We can now substitute this expression for [A 2]22 into equation 234 and get

                      (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

                      14(v+

                      Hradic2)2(2g2Wminus

                      micro W+micro +(gprimeBmicro minusgA3micro)

                      2) (237)

                      This expression contains mass terms for three massive gauge bosons Wplusmn and the combi-nation of vector fields gprimeBmicro minusgA3

                      micro where note

                      21 Standard Model 21

                      Table 22 Weak Quantum numbers of Lepton and Quarks

                      T 3 Q YνL

                      12 0 -1

                      lminusL minus12 -1 -1

                      νR 0 0 0lminusR 0 -1 -2UL

                      12

                      23

                      13

                      DL minus12 minus1

                      313

                      UR 0 23

                      43

                      DR 0 minus13 minus2

                      3

                      Wminusmicro = (W+

                      micro )dagger equivW 1micro minus iW 2

                      micro (238)

                      Then the mass terms can be written

                      12

                      v2g2|Wmicro |2 +14

                      v2(gprimeBmicro minusgA3micro)

                      2 (239)

                      W+micro and W microminus can be associated with the W boson and its anti-particle and (minusgprimeBmicro +

                      gA3micro) with the Z Boson (after normalisation by

                      radicg2 +(gprime

                      )2) The combination gprimeA3micro +gBmicro

                      is orthogonal to Z and remains massless This can be associated with the Photon (afternormalisation) These predict mass terms for the W and Z bosons mW = vgradic

                      2and mZ =

                      vradic2

                      radicg2 +(gprime

                      )2It is again instructive to count the degrees of freedom before and after the Higgs mech-

                      anism Before we had a complex doublet Φ with four degrees of freedom one masslessB with two degrees of freedom and three massless W a fields each with two each givinga total of twelve degrees of freedom After spontaneous symmetry breaking we have onereal scalar Higgs field with one degree of freedom three massive weak bosons with nineand one massless photon with two again giving twelve degrees of freedom One says thescalar degrees of freedom have been eaten to give the Wplusmn and Z bosons their longitudinalcomponents (ie mass)

                      Table 22 shows T 3 the weak isospin quantum number (which describes how a particletransforms under SU(2)) Y the weak hypercharge (which describes how the particle trans-

                      21 Standard Model 22

                      forms under U(1)) and Q the electric charge of leptons and quarks The former two enterdirectly into the Lagrangian while the latter can be derived as a function of the former two

                      Strictly speaking the right handed neutrino which has no charge may not exist and is notpart of the standard model although some theories postulate a mass for this (hypothetical)particle

                      Writing gauge invariant couplings between ELeRφ with the vacuum expectation ofφ = v (Yukawa matrices λ

                      i ju and λ

                      i jd respectively for the up and down quarks) we get mass

                      terms for the quarks (and similarly for the leptons)

                      Mass terms for quarks minussumi j[(λi jd Qi

                      Lφd jR)+λ

                      i ju εab(Qi

                      L)aφlowastb u j

                      R +hc]

                      Mass terms for leptonsminussumi j[(λi jl Li

                      Lφ l jR)+λ

                      i jν εab(Li

                      L)aφlowastb ν

                      jR +hc]

                      Here hc is the hermitian conjugate of the first terms L and Q are lepton and quarkdoublets l and ν are the lepton singlet and corresponding neutrino d and u are singlet downand up quarks εab is the Levi-Civita (anti-symmetric) tensor

                      If we write the quark generations in a basis that diagonalises the Higgs couplings weget a unitary transformations Uu and Ud for the up and down quarks respectively which iscompounded in the boson current (Udagger

                      u Ud) called the CKM matrix where off-diagonal termsallow mixing between quark generations (this is also called the mass basis as opposed tothe flavour basis) A similar argument can give masses to right handed sterile neutrinos butthese can never be measured However a lepton analogue of the CKM matrix the PNMSmatrix allows left-handed neutrinos to mix between various generations and oscillate betweenflavours as they propagate through space allowing a calculation of mass differences

                      219 Quantum Chromodynamics

                      The final plank in SM theory tackling the strong force Quantum Chromodynamics wasproposed by Gell-Mann and Zweig in 1963 Three generations of quarks were proposedwith charges of 23 for uc and t and -13 for ds and b Initially SU(2) isospin was consideredtransforming u and d states but later irreducible representations of SU(3) were found to fit theexperimental data The need for an additional quantum number was motivated by the fact thatthe lightest excited state of the nucleon consisted of sss (the ∆++) To reconcile the baryon

                      21 Standard Model 23

                      spectrum with spin statistics colour was proposed Quarks transform under the fundamentalor 3 representation of colour symmetry and anti-quarks under 3 The inner product of 3and 3 is an invariant under SU(3) as is a totally anti-symmetric combination ε i jkqiq jqk (andthe anti-quark equivalent) It was discovered though deep inelastic scattering that quarksshowed asymptotic freedom (the binding force grew as the seperation grew) Non-abeliangauge theories showed this same property and colour was identified with the gauge quantumnumbers of the quarks The quanta of the gauge field are called gluons Because the forcesshow asymptotic freedom - ie the interaction becomes weaker at high energies and shorterdistances perturbation theory can be used (ie Feynman diagrams) at these energies but forlonger distances and lower energies lattice simulations must be done

                      2110 Full SM Lagrangian

                      The full SM can be written

                      L =minus14

                      BmicroνBmicroν minus 18

                      tr(FmicroνFmicroν)minus 12

                      tr(GmicroνGmicroν)

                      + sumgenerations

                      (ν eL)σmicro iDmicro

                      (νL

                      eL

                      )+ eRσ

                      micro iDmicroeR + νRσmicro iDmicroνR +hc

                      + sumgenerations

                      (u dL)σmicro iDmicro

                      (uL

                      dL

                      )+ uRσ

                      micro iDmicrouR + dRσmicro iDmicrodR +hc

                      minussumi j[(λ

                      i jl Li

                      Lφ l jR)+λ

                      i jν ε

                      ab(LiL)aφ

                      lowastb ν

                      jR +hc]

                      minussumi j[(λ

                      i jd Qi

                      Lφd jR)+λ

                      i ju ε

                      ab(QiL)aφ

                      lowastb u j

                      R +hc]

                      + (Dmicroφ)dagger(Dmicroφ)minusmh[φφ minus v22]22v2

                      (240)

                      where σ micro are the extended Pauli matrices

                      (1 00 1

                      )

                      (0 11 0

                      )

                      (0 minusii 0

                      )

                      (1 00 minus1

                      )

                      The first line gives the gauge field strength terms the second the lepton dynamical termsthe third the quark dynamical terms the fourth the lepton (including neutrino) mass termsthe fifth the quark mass terms and the sixth the Higgs terms

                      The SU(3) gauge field is given by Gmicroν and which provides the colour quantum numbers

                      21 Standard Model 24

                      Figure 24 Standard Model Particles and Forces [50]

                      Note The covariant derivatives operating on the left-handed leptons contain only theU(1) and SU(2) gauge fields Bmicro Fmicro whereas those operating on left-handed quarks containU(1)SU(2) and SU(3) gauge fields Bmicro Fmicro Gmicro The covariant derivatives operating onright handed electrons and quarks are as above but do not contain the SU(2) gauge fieldsRight handed neutrinos have no gauge fields

                      The sums over i j above are over the different generations of leptons and quarks

                      The particles and forces that emerge from the SM are shown in Fig 24

                      22 Dark Matter 25

                      22 Dark Matter

                      221 Evidence for the existence of dark matter

                      2211 Bullet Cluster of galaxies

                      Most of the mass of the cluster as measured by a gravitational lensing map is concentratedaway from the centre of the object The visible matter (as measured by X-ray emission) ismostly present in the centre of the object The interpretation is that the two galaxy clustershave collided perpendicular to the line of sight with the mostly collisionless dark matterpassing straight through and the visible matter interacting and being trapped in the middleIt was estimated that the ratio of actual to visible matter was about 16 [51]

                      Figure 25 Bullet Cluster [52]

                      2212 Coma Cluster

                      The first evidence for dark matter came in the 1930s when Fritz Zwicky observed the Comacluster of galaxies over 1000 galaxies gravitationally bound to one another He observed thevelocities of individual galaxies in the cluster and estimated the amount of matter based onthe average galaxy He calculated that many of the galaxies should be able to escape fromthe cluster unless the mass was about 10 times higher than estimated

                      22 Dark Matter 26

                      2213 Rotation Curves [53]

                      Since those early observations many spiral galaxies have been observed with the rotationcurves calculated by doppler shift with the result that in all cases the outermost stars arerotating at rates fast enough to escape their host galaxies Indeed the rotation curves arealmost flat indicating large amounts of dark matter rotating with the galaxies and holdingthe galaxies together

                      Figure 26 Galaxy Rotation Curves [54]

                      2214 WIMPS MACHOS

                      The question is if the particles that make up most of the mass of the universe are not baryonswhat are they It is clear that the matter does not interact significantly with radiation so mustbe electrically neutral It has also not lost much of its kinetic energy sufficiently to relax intothe disks of galaxies like ordinary matter Massive particles may survive to the present ifthey carry a conserved additive or multiplicative quantum number This is so even if equalnumbers of particles and anti-particles can annihilate if the number densities become too lowto reduce the density further These particles have been dubbed Weakly interacting MassiveParticles (WIMPS) as they interact with ordinary matter only through a coupling strengthtypically encountered in the weak interaction They are predicted to have frozen out ofthe hot plasma and would give the correct relic density based solely on the weak interactioncross section (sometimes called the WIMP Miracle) Another possibility called MassiveCompact Halo Objects (MACHOS)- are brown dwarf stars Jupiter mass objects or remnantsfrom an earlier generation of star formation

                      22 Dark Matter 27

                      2215 MACHO Collaboration [55]

                      In order to estimate the mass of MACHOS a number of experiments have been conductedusing microlensing which makes use of the fact that if a massive object lies in the line ofsight of a much more distant star the light from the more distant star will be lensed and forma ring around the star It is extemely unlikely that the MACHO will pass directly through theline of sight but if there is a near miss the two images will be seperated by a small angleThe signature for microlensing is a time-symmetric brightening of a star as the MACHOpasses near the line of sight called a light curve One can obtain statistical information aboutthe MACHO masses from the shape and duration of the light curves and a large number ofstars (many million) must be monitored The MACHO Collaboration [55] have monitoredover 11 million stars and less than 20 events were detected The conclusion was that thiscould account for a small addition (perhaps up to 20 ) to the visible mass but not enoughto account for Dark Matter

                      2216 Big Bang Nucleosynthesis (BBN) [56]

                      Another source of evidence for DM comes from the theory of Nucleosynthesis in the fewseconds after the Big Bang In the first few minutes after the big bang the temperatures wereso high that all matter was dissociated but after about 3 minutes the temperature rapidlycooled to about 109 Kelvin Nucleosynthesis could begin when protons and neutrons couldstick together to form Deuterium (Heavy Hydrogen) and isotopes of Helium The ratio ofDeuterium to Hydrogen and Helium is very much a function of the baryon density and alsothe rate of expansion of the universe at the time of Nucleosynthesis There is a small windowwhen the density of Baryons is sufficient to account for the ratios of the various elementsThe rate of expansion of the universe is also a function of the total matter density The ratiosof these lighter elements have been calculated by spectral measurements with Helium some25 of the mass of the universe Deuterium 001 The theory of BBN accounts for theratios of all of the lighter elements if the ratio of Baryon mass to total mass is a few percentsuggesting that the missing matter is some exotic non-baryonic form

                      22 Dark Matter 28

                      2217 Cosmic Microwave Background [57]

                      The discovery of a uniform Microwave Background radiation by Penzias and Wilson in1965 was confirmation of the theory first proposed by Gamow and collaborators that theuniverse should be filled with black body radiation an afterglow of the big bang TheCOBE satelite launched in 1989 confirmed this black body radiation with temperature of2725+-002K with 95 confidence While extremely uniform it found tiny irregularitieswhich could be accounted for by irregularities in the density of the plasma at last scattering(gravitational redshifts or blueshifts called the Sachs Wolfe effect) and temperature andvelocity fluctuations in the plasma The anisotropy of the background has been furtheranalysed by WMAP (the Wilkinson Microwave Anisotropy Probe) launched in 2001 Thefluctuations in the density of the plasma are what account for the seeds of galaxy formationHowever detailed calculations have found that the fluctuations were too small to account forthe observed galaxy distribution without an electrically neutral gravitating matter in additionto ordinary matter that only became electrically neutral at the time of recombination [58]

                      In May 2009 the European Space Agency (ESA) launched the Planck sattelite a spacebased observatory to observe the universe at wavelengths between 03mm and 111 mmbroadly covering the far-infrared microwave and high frequency domains The missionrsquosmain goal was to study the cosmic microwave background across the whole sky at greatersensitivity and resolution than ever before Using the information gathered and the standardmodel of cosmology ESA was able to calculate cosmological parameters including the relicdensity of dark matter remaining after the Big Bang to a high degree of precision withinthe framework of the standard model of cosmology The standard model of cosmology canbe described by a relatively small number of parameters including the density of ordinarymatter dark matter and dark energy the speed of cosmic expansion at the present epoch (alsoknown as the Hubble constant) the geometry of the Universe and the relative amount ofthe primordial fluctuations embedded during inflation on different scales and their amplitude[59]

                      2218 LUX Experiment - Large Underground Xenon experiment [16]

                      The LUX experiment is conducted at the Sanford Underground research facility one mileunderground to minimise the background of cosmic rays The detector is a time projectionchamber allowing a 3D positioning of interactions occurring within its active volume LUX

                      22 Dark Matter 29

                      Figure 27 WMAP Cosmic Microwave Background Fluctuations [58]

                      Figure 28 Dark Matter Interactions [60]

                      uses as its target 368 kiliograms of liquefied ultra pure xenon which is a scintillator Interac-tions inside the xenon will create an amount of light proportional to the energy depositedThat light can be collected on arrays of light detectors sensitive to a single photon

                      Xenon is very pure and the amount of radiation originating within the target is limitedAlso being three times denser than water it can stop radiation originating from outside thedetector The 3D detector focuses on interactions near the center which is a quiet regionwhere rare dark matter interactions might be detected [16]

                      22 Dark Matter 30

                      222 Searches for dark matter

                      2221 Dark Matter Detection

                      Fig 28 illustrates the reactions studied under the different forms of experiment designed todetect DM

                      Direct detection experiments are mainly underground experiments with huge detectorsattempting to detect the recoil of dark matter off ordinary matter such as LUX [16] CDMS-II [61] XENON100 [11] COUPP [62] PICASSO [63] ZEPLIN [10] These involveinteractions between DM and SM particles with final states involving DM and SM particles

                      Indirect detection involves the search for the annihilation of DM particles in distantastrophysical sources where the DM content is expected to be large Example facilitiesinclude space based experiments (EGRET [64] PAMELA [18] FERMI-LAT [65]) balloonflights (ATIC [66] PEBS [67] PPB-BETS [68]) or atmospheric Cherenkov telescopes(HESS [69]MAGIC [70]) and neutrino telescopes (SuperKamiokande [71] IceCube [72]ANTARES [73] KM3NET [74]) Indirect Detection involves DM particles in the initial stateand SM particles in the final state

                      Collider detection involves SM particles in the initial state creating DM particles in thefinal state which can be detected by missing transverse energy and explained by variousmodels extending the SM

                      223 Possible signals of dark matter

                      224 Gamma Ray Excess at the Centre of the Galaxy [65]

                      The center of the galaxy is the most dense and active region and it is no surprise that itproduces intense gamma ray sources When astronomers subtract all known gamma raysources there remains an excess which extends out some 5000 light years from the centerHooper et al find that the annihilation of dark matter particles with a mass between 31 and40 Gev fits the excess to a remarkable degree based on Gamma Ray spectrum distributionand brightness

                      23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 31

                      Figure 29 Gamma Ray Excess from the Milky Way Center [75]

                      23 Background on ATLAS and CMS Experiments at theLarge Hadron collider (LHC)

                      The collider constraints in this paper are based on two of the experiments that are part of theLHC in CERN

                      Figure 210 ATLAS Experiment

                      The LHC accelerates proton beams in opposite directions around a 27 km ring of super-conducting magnets until they are made to collide at an interaction point inside one of the 4experiments placed in the ring An upgrade to the LHC has just been performed providing65 TeV per beam

                      23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 32

                      231 ATLAS Experiment

                      The ATLAS detector consists of concentric cylinders around the interaction point whichmeasure different aspects of the particles that are flung out of the collision point There are 4major parts the Inner Detector the Calorimeters the Muon spectrometer and the magnetsystems

                      2311 Inner Detector

                      The inner detector extends from a few centimeters to 12 meters from the interaction point andmeasures details of the charged particles and their momentum interacting with the materialIf particles seem to originate from a point other than the interaction point this may indicatethat the particle comes from the decay of a hadron with a bottom quark This is used to b-tagcertain particles

                      The inner detector consists of a pixel detector with three concentric layers and threeendcaps containing detecting silicon modules 250 microm thick The pixel detector has over 80million readout channels about 50 percent of the total channels for the experiment All thecomponents are radiation hardened to cope with the extremely high radiation in the proximityof the interaction point

                      The next layer is the semi-conductor layer (SCT) similar in concept to the pixel detectorbut with long narrow strips instead of pixels to cover a larger area It has 4 double layers ofsilicon strips with 63 million readout channels

                      The outermost component of the inner detector is the Transition Radiation Tracker It is acombination of a straw tracker and a transition radiation detector The detecting elementsare drift tubes (straws) each 4mm in diameter and up to 144 cm long Each straw is filledwith gas which becomes ionised when the radiation passes through The straws are held at-1500V with a thin wire down the center This drives negatively ionised atoms to the centercausing a pulse of electricity in the wire This causes a pattern of hit straws allowing the pathof the particles to be determined Between the straws materials with widely varying indicesof refraction cause transition radiation when particles pass through leaving much strongersignals in some straws Transition radiation is strongest for highly relativistic particles (thelightest particles- electrons and positrons) allowing strong signals to be associated with theseparticles

                      23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 33

                      2312 Calorimeters

                      The inner detector is surrounded by a solenoidal magnet of 2 Tesla (there is also an outertoroidal magnet) but surrounding the inner magnet are the calorimeters to measure theenergy of particles by absorbing them There is an inner electromagnetic calorimeter whichmeasures the angle between the detectorrsquos beam axis and the particle (or more preciselypseudo-rapidity) and its angle in the perpendicular plane to a high degree of accuracy Theabsorbing material is lead and stainless steel and the sampling material liquid argon Thehadron calorimeter measures the energy of particles that interact via the strong force Theenergy absorbing material is scintillating steel

                      2313 Muon Specrometer

                      The muon spectrometer surrounds the calorimeters and has a set of 1200 chambers tomeasure the tracks of outgoing muons with high precision using a magnetic field andtriggering chambers Very few particles of other types pass through the calorimeters and aresubsequently measured

                      2314 Magnets

                      The magnets are in two layers the inner solenoidal and outer toroidal which together causecharged particles to curve in the magetic field allowing their momentum to be determinedThe outer toroidal magnet consists of 8 very large air core superconducting magnets whichsupply the field in the muon spectrometer

                      232 CMS Experiment

                      The CMS Detector is similar in design to the ATLAS Detector with a single solenoidalmagnet which is the largest of its type ever built (length 13m and diameter 7m) and allowsthe tracker and calorimeter detectors to be placed inside the coil in a compact way Thestrong magnetic field gives a high resolution for measuring muons electrons and photons ahigh quality central tracking system and a hermetic hadron calorimeter designed to preventhadrons from escaping

                      23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 34

                      Figure 211 CMS Experiment

                      Chapter 3

                      Fitting Models to the Observables

                      31 Simplified Models Considered

                      In order to extend the findings of previous studies [19] to include collider constraints wefirst performed Bayesian scans [7] over three simplified models real scalar dark matter φ Majorana fermion χ and real vector Xmicro The purpose was to obtain the most probable regionsof each model then to use the resulting posterior samples to investigate whether the LHCexperiments have any sensitivity to the preferred regions

                      The three models couple to the mediator with interactions shown in the following table

                      Table 31 Simplified Models

                      Hypothesis real scalar DM Majorana fermion DM real vector DM

                      DM mediator int Lφ sup microφ mφ φ 2S Lχ sup iλχ

                      2 χγ5χS LX sup microX mX2 X microXmicroS

                      The interactions between the mediator and the standard fermions is assumed to be

                      LS sup f f S (31)

                      and in line with minimal flavour violation we only consider third generation fermions- ief = b tτ

                      For the purposes of these scans we consider the following observables

                      32 Observables 36

                      32 Observables

                      321 Dark Matter Abundance

                      We have assumed a single dark matter candidate which freezes out in the early universe witha central value as determined by Planck [21] of

                      ΩDMh2 = 01199plusmn 0031 (32)

                      h is the reduced hubble constant

                      The experimental uncertainty of the background relic density [59] is dominated by atheoretical uncertainty of 50 of the value of the scanned Ωh2 [76] We have added thetheoretical and the experimental uncertainties in quadrature

                      SD =radic(05Ωh2)2 + 00312 (33)

                      This gives a log likelihood of

                      minus05lowast (Ωh2 minus 1199)2

                      SD2 minus log(radic

                      2πSD) (34)

                      322 Gamma Rays from the Galactic Center

                      Assuming that the gamma ray flux excess from the galactic centre observed by Fermi-LAT iscaused by self annihilation of dark matter particles the differential flux

                      d2Φ

                      dEdΩ=

                      lt σv gt8πmχ

                      2 J(ψ)sumf

                      B fdN f

                      γ

                      dE(35)

                      has been calculated using both Micromegas v425 with a Navarro-Frenk-White (NFW) darkmatter galactic distribution function

                      ρ(r) = ρ0(rrs)

                      minusγ

                      (1+ rrs)3minusγ (36)

                      with γ = 126 and an angle of 5 to the galactic centre [19]

                      32 Observables 37

                      Here Φ is the flux dE is the differential energy lt σv gt the velocity averaged annihi-lation cross section of the dark matter across the galactic centre mχ the mass of the darkmatter particle B f =lt σv gt f σv is the annihilation fraction into the f f final state anddN f

                      γ dE is the energy distribution of photons produced in the annihilation channel with thefinal state f f The J factor is described below

                      The angle ψ to the galactic centre determines the normalisation of the gamma ray fluxthrough the factor

                      J(ψ) =int

                      losρ(l2 + r⊙2 minus2lr⊙cosψ)dl (37)

                      where ρ(r) is the dark matter density at distance r from the centre of the galaxy r⊙=85 kpcis the solar distance from the Galactic center ρ0 is set to reproduce local dark matter densityof 03GeVcm3

                      The central values for the gamma ray spectrum were taken from Fig 5 of [77] and thestandard deviations of individual flux points from the error bars on the graph In commonwith [78] we ignore possible segment to segment correlations

                      For the Fermi-LAT log likelihood we assumed each point was distributed with indepen-dent Gaussian likelihood with the experimental error for each data point given by the errorbar presented in Fig5 of [76] for each point We also assumed an equal theoretical errorfor each point which when added to the experimental error in quadrature gave an effectivelog likelihood for each point of minus05lowast (giminusdi)

                      2

                      2lowastσ2i

                      where gi are the calculated values and di theexperimental values and σi the experimental errors

                      323 Direct Detection - LUX

                      The LUX experiment [16] is the most constraining direct search experiment to date for awide range of WIMP models We have used LUXCalc [79] (modified to handle momentumdependent nucleon cross sections) to calculate the likelihood of the expected number ofsignal events for the Majorana fermion model

                      The likelihood function is taken as the Poisson distribution

                      L(mχ σ | N) = P(N | mχ σ) =(b+micro)Neminus(b+micro)

                      N (38)

                      32 Observables 38

                      where b= expected background of 064 and micro = micro(mχσ ) is the expected number of signalevents calculated for a given model

                      micro = MTint

                      infin

                      0dEφ(E)

                      dRdE

                      (E) (39)

                      where MT is the detector mass times time exposure and φ(E) is a global efficiency factorincorporating trigger efficiencies energy resolution and analysis cuts

                      The differential recoil rate of dark matter on nucleii as a function of recoil energy E

                      dRdE

                      =ρX

                      mχmA

                      intdvv f (v)

                      dσASI

                      dER (310)

                      where mA is the nucleon mass f (v) is the dark matter velocity distribution and

                      dσSIA

                      dER= Gχ(q2)

                      4micro2A

                      Emaxπ[Z f χ

                      p +(AminusZ) f χn ]

                      2F2A (q) (311)

                      where Emax = 2micro2Av2mA Gχ(q2) = q2

                      4m2χ

                      [24] is an adjustment for a momentum dependantcross section which applies only to the Majorana model

                      f χ

                      N =λχ

                      2m2SgSNN assuming that the relic density is the central value of 1199 We have

                      implemented a scaling of the direct search cross section in our code by multiplying gSNN by( Ω

                      Ω0)05 as a convenient method of allowing for the fact that the actual relic density enters

                      into the calculation of the cross section as a square

                      FA(q) is the nucleus form factor and

                      microA =mχmA

                      (mχ +mA)(312)

                      is the reduced WIMP-nucleon mass

                      The strength of the dark matter nucleon interaction between dark matter particles andthird generation quarks is

                      gSNN =2

                      27mN fT G sum

                      f=bt

                      λ f

                      m f (313)

                      where fT G = 1minus f NTuminus f N

                      Tdminus fTs and f N

                      Tu= 02 f N

                      Td= 026 fTs = 043 [20]

                      33 Calculations 39

                      For the real scalar and real vector models where the spin independent cross sections arenot momentum dependent we based the likelihood on the LUX 95 CL from [16] using acomplementary error function Li(d|mχ I) = 05lowastCEr f (dminusx(mχ )

                      σ) where x is the LUX limit

                      and d the calculated SI cross section We assumed a LUX error σ of 01lowastd lowastradic

                      2

                      33 Calculations

                      We coded the Lagrangians for the three models in Feynrules [4] Calculation of the expectedobservables (dark matter relic density Fermi-LAT gamma ray spectrum and LUX nucleonscattering amplitudes) were performed using Micromegas v425

                      331 Mediator Decay

                      A number of searches have previously been performed at the LHC which show sensitivity tothe decay of scalar mediators in the form of Higgs bosons (in beyond the standard modeltheories) to bottom quarks top quarks and τ leptons (all third generation fermions) Two inparticular [43 44] provide limits on the cross-sections for the production of the scalar timesthe branching ratios to brsquos and τrsquos In order to check these limits against each of the modelswe ran parameter scans in Madgraph at leading order over the top 900 points from the nestedsampling output from MultiNest [19]

                      The two processes were

                      1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to apair of τ leptons in pp collisions [44]

                      bull generate p p gt b b S where S is the scalar mediator

                      The process measures the decay of a scalar in the presence of b-jets that decay to τ+τminus

                      leptons Fig31 shows the main diagrams contributing to the process The leftmost diagramis a loop process and the centre and right are tree level however we only got Madgraph togenerated the middle diagram since we are interested in τ leptons and b quarks in the finalstate

                      The limit given in [44] only applies in the case of narrow resonances where narrow istaken to mean that the width of the resonance is less than 10 of its mass To test the validity

                      33 Calculations 40

                      Figure 31 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof τ leptons

                      of applying these limits to our models we show in Figures 32 to 34 the percentage widthof the resonance vs mS[GeV ] (pole mass of the scalar mediator) λb and λt (the couplingconstants to the b and t quarks) The probabilities are the posterior probabilities obtainedfrom MultiNest scans of the Majorana model varying the coupling constants and scalarmediator and calculating the decay width of the mediator The results are represented as 2dposterior probability heat maps in the central panels and 1d marginalised distributions on thex and y axes of each graph

                      16 18 20 22 24 26 28 30

                      log10(mS[GeV])

                      001

                      002

                      003

                      004

                      005

                      Widthm

                      S

                      00

                      04

                      08

                      12

                      0 100 200

                      Posterior Probability

                      Figure 32 WidthmS vs mS

                      The WidthmS was lt 01 at almost all points except where the mass of the scalar wasgreater than 1000 GeV λb was greater than 1 or λτ was greater than 1

                      This can be seen from the graphs in Figs 323334

                      33 Calculations 41

                      4 3 2 1 0

                      λb

                      001

                      002

                      003

                      004

                      005

                      WidthmS

                      000

                      015

                      030

                      045

                      0 100 200

                      Posterior Probability

                      Figure 33 WidthmS vs λb

                      5 4 3 2 1 0

                      λτ

                      001

                      002

                      003

                      004

                      005

                      WidthmS

                      000

                      015

                      030

                      045

                      0 100 200

                      Posterior Probability

                      Figure 34 WidthmS vs λτ

                      The decay widths are calculated in Madgraph at leading order and branching fractionscan be calculated to b t τ and χ (DM) The formulae for the calculations can be found indecayspy which is a subroutine of Madgraph which we have documented in Appendix BB1 Using these we calculated the branching ratio to τ+τminus so that the cross section timesbranching ratio can be compared to the limits in [80] shown in Chapters 56 and 7 in Figs5564 and 74 for each model

                      2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                      This scan calculates the contributions of b-quark associated Higgs boson production inthe center and right diagrams of Fig 35 reproduced from [43] The scans did not includePythia and Delphes as only production cross sections were required and the producedparticles were in the final state and required no matching or merging

                      The Madgraph processes were

                      bull generate p p gt b S where S is the scalar mediator

                      bull add process p p gt b S j

                      bull add process p p gt b S

                      33 Calculations 42

                      Figure 35 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof b quarks in the presence of at least one b quark

                      bull add process p p gt b S j

                      The cross section times branching ratios to bb for mediator production in association withat least one b quark is shown for each model in Chapters 56 and 7 in Figs 5765 and 75

                      332 Collider Cuts Analyses

                      We generated two parameter scans with parameter combinations from the Multinest outputwhich were the basis of the plots of Fig 6 in [19] The plots in Fig 6 were marginalisedto two parameters each but the raw Multinest output contained 5 parameters plus the priorprobabilities of each combination normalised to total 10

                      The first scan was for χ produced with b quarks In order to force the mediator to decayto χ only we had χ in the final state and defined jets to include b quarks in Madgraph Thenwe generated the processes

                      bull generate p p gt χ χ j

                      bull add process p p gt χ χ j j

                      Jet matching was on

                      The second scan was for t quarks produced in the final state

                      bull generate p p gt χ χ tt

                      33 Calculations 43

                      No jet matching was required because we did not mix final state jet multiplicities at thematrix element level

                      The outputs from these two processes were normalised to 21 f bminus1 and combined

                      The results of these scans prior to collider cuts calculations are shown in Figs 5161and 71 These show the posterior probabilities of the various parameter combinations withdifferent constraints applied seperately and then together A more detailed commentary isgiven in each of the sections

                      We then sampled 900 points based on posterior probability from these scans and ran thesethrough Madgraph v425 then applied collider cuts with the C++ code using the limits fromthe following six ATLAS and CMS searches shown in Table32 and described below to findany exclusions

                      333 Description of Collider Cuts Analyses

                      In the following all masses and energies are in GeV and angles in radians unless specificallystated

                      3331 Lepstop0

                      Search for direct pair production of the top squark in all-hadronic final states in pro-tonndashproton collisions at

                      radics=8 TeV with the ATLAS detector[32]

                      This analysis searched for direct pair production of the supersymmetric partner to thetop quark using integrated luminosity of 201 f bminus1 The top squark was assumed to de-cay via t rarr t χ0

                      1 or t rarr bχ01 or t rarr bχ

                      plusmn1 rarr bW (lowast)χ1

                      0 where χ01 (χ

                      plusmn1 ) denotes the lightest

                      neutralino(chargino) in supersymmetric models The search targets a fully hadronic finalstate with four or more jets and large missing momentum The production of a mediator inassociation with top quarks followed by an invisible mediator decay to dark matter particleshas a similar final state to that considered in the ATLAS stop searches Thus we expect theATLAS reach to have sensitivity to the three dark matter models considered in this thesis

                      The collider cuts program mimicked the object reconstruction through jetlepton overlapremoval and selected signal electrons muons and τ particles with pT gt 10 GeV and

                      33 Calculations 44

                      Table 32 95 CL by Signal Region

                      Experiment Region Number

                      Lepstop0

                      SRA1 66SRA2 57SRA3 67SRA4 65SRC1 157SRC2 124SRC3 80

                      Lepstop1

                      SRtN1-1 857SRtN1-2 498SRtN1-3 389SRtN2 107SRtN3 85SRbC1 832SRbC2 195SRbC3 76

                      Lepstop2

                      L90 740L100 56L110 90L120 170

                      2bstop

                      SRA 857SRB 270SRA150 380SRA200 260SRA250 90SRA300 75SRA350 52

                      CMSTopDM1L SRA 1385

                      ATLASMonobjetSR1 1240SR2 790

                      33 Calculations 45

                      |η | lt 24724 and 247 respectively (for eminusmicrominusτ) Jets were selected with pT gt 20 GeVand |η |lt 45 b-jets were selected if they were tagged by random selection lt 70 and had|η |lt 25 and pT gt 20 GeV

                      These jets and leptons were further managed for overlap removal within ∆R lt 04 withnon b-jets removed if ∆R lt 02

                      The b-jets and non b-jets were further selected as signal jets if pT gt 35 GeV and |η |lt 28

                      These signal jets electrons and muons were finally binned according to the signal regionsin Tables 333435 below

                      Table 33 Selection criteria common to all signal regions

                      Trigger EmissT

                      Nlep 0b-tagged jets ⩾ 2

                      EmissT 150 GeV

                      |∆φ( jet pmissT )| gtπ5

                      mbminT gt175 GeV

                      Table 34 Selection criteria for signal regions A

                      SRA1 SRA2 SRA3 SRA4anti-kt R=04 jets ⩾ 6 pT gt 808035353535 GeV

                      m0b j j lt 225 GeV [50250] GeV

                      m1b j j lt 225 GeV [50400] GeV

                      min( jet i pmissT ) - gt50 GeV

                      τ veto yesEmiss

                      T gt150 GeV gt250 GeV gt300 GeV gt 350 GeV

                      Table 35 Selection criteria for signal regions C

                      SRC1 SRC2 SRC3anti-kt R=04 jets ⩾ 5 pT gt 8080353535 GeV

                      |∆φ(bb)| gt02 π

                      mbminT gt185 GeV gt200 GeV gt200 GeV

                      mbmaxT gt205 GeV gt290 GeV gt325 GeV

                      τ veto yesEmiss

                      T gt160 GeV gt160 GeV gt215 GeV

                      wherembmin

                      T =radic

                      2pbt Emiss

                      T [1minus cos∆φ(pbT pmiss

                      T )]gt 175 (314)

                      33 Calculations 46

                      and pbT is calculated for the b-tagged jet with the smallest ∆φ to the pmiss

                      T direction andmbmax

                      T is calculated with the pbT for the b-tagged jet with the largest ∆φ to the pmiss

                      T direction

                      m0b j j is obtained by first selecting the two jets with the highest b-tagging weight- from

                      the remaining jets the two jets closest in the η minusφ plane are combined to form a W bosoncandidate this candidate is then combined with the closest of the 2 b-tagged jets in the η minusφ

                      plane to form the first top candidate with mass m0b j j A second W boson candidate is formed

                      by repeating the procedure with the remaining jets this candidate is combined with thesecond of the selected b-tagged jets to form the second top candidate with mass m1

                      b j j Thecuts shown in Appendix1 Table A1 show the cumulative effect of the individual cuts appliedto the data the signal region cuts and the cumulative effects of individual cuts in SignalRegion C

                      3332 Lepstop1

                      Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

                      radics=8 TeV pp collisions using 21 f bminus1 of

                      ATLAS data[33]

                      The stop decay modes considered were those to a top quark and the lightest supersym-metric particle (LSP) as well as a bottom quark and the lightest chargino where the charginodecays to the LSP by emitting a W boson We should expect the three dark matter models inthis paper with couplings to t and b quarks to show some visibility in this search

                      The collider cuts program mimicked the object construction by selecting electrons andmuons with pT gt 10 GeV and |η |lt 247 and 24 respectively and jets with pT gt 20 GeVand |η |lt 10 b-jets with |η |lt 25 and pT gt 25 GeV were selected with random efficiency75 and mistag rate 2 (where non b-jets were mis-classified) We then removed any jetwithin ∆R = 02 of an electron and non overlapping jets with |η |lt 28 were used to removeany electrons and muons within ∆R = 04 of these while non-overlapping jets with |η |lt 25and pT gt 25 GeV were classified as signal jets Non overlapping electrons and muons withpT gt 25 GeV were classified as signal electrons and muons

                      Signal selection defined six regions to optimize sensitivity for different stop and LSPmasses as well as sensitivity to two considered stop decay scenarios Three signal regionslabelled SRbC 1-3 where bC is a moniker for b+Chargino and 3 regions labelled SRtN 1-3where tN is a moniker for t+Neutralino were optimised for the scenarios where t1 decaysto the respective particles indicated by the labels Increasing label numbers correspond toincreasingly strict selection criteria

                      33 Calculations 47

                      The dominant background in all signal regions arises from dileptonic tt events All threeSRtN selectons impose a requirement on the 3-jet mass m j j j of the hadronically decayingtop quark to specifically reject the tt background where both W bosons from the top quarkdecay leponically

                      For increasing stop mass and increasing difference between the stop and the LSP therequirents are tightened on Emiss

                      T on the ratio EmissT

                      radicHT where HT is the scalar sum of the

                      momenta of the four selected jets and also tightened on mT

                      To further reduce the dileptonic tt background for an event characterised by two one stepdecay chains a and b each producing a missing particle C use is made of the variable

                      mT 2 =min

                      pCTa + pC

                      T b = pmissT

                      [max(mTamtb)] (315)

                      where mTi is the transverse mass 1 of branch i for a given hypothetical allocation (pCTa pC

                      T b)

                      of the missing particle momenta Eqn 315 is extended for more general decay chains byamT 2 [81] which is an asymmetric form of mT 2 in which the missing particle is the W bosonin the branch for the lost lepton and the neutrino is the missing particle for the branch withthe observed charged lepton

                      ∆φ( jet12 pmissT ) the azimuthal angle between leading or subleading jets and missing mo-

                      mentum is used to suppress the background from mostly multijet events with mis-measuredEmiss

                      T

                      Events with four or more jets are selected where the jets must satisfy |η | lt 25 and pT gt80604025 GeV respectively

                      These signal particles were then classified according to the cutflows analysis in Fig36below and compared with the paper in Appendix1 Table A2

                      3333 Lepstop2

                      Search for direct top squark pair production in final states with two leptons in p pcollisions at

                      radics=8TeV with the ATLAS detector[34]

                      Three different analysis strategies are used to search for t1 pair production Two targett1 rarr b+ χ

                      plusmn1 decay and the three body t1 rarr bW χ0

                      1 decay via an off-shell top quark whilst

                      1The transverse mass is defined as m2T = 2plep

                      T EmissT (1minus cos(∆φ)) of branch i where ∆φ is the azimuthal

                      angle between the lepton and the missing transverse momentum

                      33 Calculations 48

                      Table 36 Signal Regions - Lepstop1

                      Requirement SRtN2 SRtN3 SRbC1 SRbC2 SRbC3∆φ( jet1 pmiss

                      t )gt - 08 08 08 08∆φ( jet2 pmiss

                      T )gt 08 08 08 08 08Emiss

                      T [GeV ]gt 200 275 150 160 160Emiss

                      T radic

                      HT [GeV12 ]gt 13 11 7 8 8

                      mT [GeV ]gt 140 200 120 120 120me f f [GeV ]gt - - - 550 700amT 2[GeV ]gt 170 175 - 175 200mT

                      T 2[GeV ]gt - 80 - - -m j j j Yes Yes - - -

                      one targets t1 rarr t + χ01 to an on shell quark decay mode The analysis is arranged to look

                      at complementary mass splittings between χplusmn1 and χ0

                      1 smaller and larger than the W bosonmass The smaller splitting provides sensitivity to t1 three body decay

                      Object reconstruction was carried out by selecting baseline electronsmuons and τrsquos withpT gt 10 GeV and |η |lt 24724 and 247 respectively Baseline jets and b-jets were selectedwith pT gt 20 GeV and |η |lt 25 and b-jets with 70 efficiency Baseline jets were removedif within ∆R lt 02 of an electron and electrons and muons were removed if within ∆R lt 04of surviving jets Signal electrons required pT gt 10 GeV and signal jets pT gt 20 GeV and|η |lt 25

                      The analysis made use of the variable mT 2 [82] the stransverse mass used to mea-sure the masses of pair-produced semi-invisibly decaying heavy particles It is definedas mT 2(pT1 pT2qT ) =

                      minqT1+qT2=qT

                      max[mT (pT1qT1)mT (pT2qT2)] where mT indicatesthe transverse mass pT1 and pT2 are vectors and qT = qT1 + qT2 The minimisation isperformed over all possible decompositions of qT

                      Events were selected if they passed either a single-electron a single-muon a double-electron a double-muon or an electron-muon trigger Events are required to have exactlytwo opposite sign leptons and at least one of the leptons or muons must have a momentumgreater than 25 GeV and the invariant mass (rest mass) of the two leptons is required to begreater than 20 GeV In order to reduce the number of background events containing twoleptons produced by the on-shell decay of the Z boson the invariant mas must be outside711-111 GeV range

                      Two additional selections are applied to reduce the number of background events withhigh mT 2 arising from events with large Emiss

                      T due to mismeasured jets ∆φb lt 15 and∆φ gt 10 The quantity ∆φb is the azimuthal angle between pll

                      T b = pmissT + pl1

                      T +Pl2T The

                      33 Calculations 49

                      vector pllT b is the opposite of the vector sum of all the hadronic activity in the event For WW

                      and tt backgrounds it measures the transverse boost of the WW system and for the signalthe transverse boost of the chargino-chargino system The ∆φ variable is the azimuthal anglebetween the pmiss

                      T vector and the direction of the closest jet

                      By requiring mT 2 gt 40 GeV the tt background is the dominant background in both thesame-flavoured (SF) and different-flavoured (DF) events

                      Four signal regions are then defined as shown in Table 37 SR M90 has the loosestselection requiring MT 2gt 90 GeV and no additional requirement on jets and providessensitivity to scenarios with a small difference between the masses of the top-squark andchargino SR M110 and M120 have a loose selection on jets requiring two jets with pT

                      gt 20 GeV and mT 2 to be greater than 110 GeV and 120 GeV respectively They providesensitivity to scenarios with small to moderate difference between the invariant masses of thestop quark and the charginos SR M100 has a tighter jet slection and provides sensitivity toscenarios with a large mass difference between the both stop quark and the chargino and thestop quark and the neutralino

                      The analysis cut regions are summarised in Fig37 below

                      Table 37 Signal Regions Lepstop2

                      SR M90 M100 M110 M120pT leading lepton gt 25 GeV

                      ∆φ(pmissT closest jet) gt10

                      ∆φ(pmissT pll

                      T b) lt15mT 2 gt90 GeV gt100 GeV gt110 GeV gt120 GeV

                      pT leading jet no selection gt100 GeV gt20 GeV gt20 GeVpT second jet no selection gt50 GeV gt20 GeV gt20 GeV

                      To validate the code we ran the cuts shown in Appendix 1 Table A3 with the givenMadgraph simulation and compared to [83] It was not clear what baseline cuts had beenemployed in the ATLAS paper so our results were normalised to the 2 SF electrons beforecomparison Some of the ATLAS cut numbers were less than 3 and we did not produce anyevents passing these cuts but clearly the margin of error at this level was greater than thenumber of ATLAS events

                      33 Calculations 50

                      3334 2bstop

                      Search for direct third-generation squark pair production in final states with miss-ing transverse momentum and two b-jets in

                      radics= 8 TeV pp collisions with the ATLAS

                      detector[31]

                      Two sets of scenarios are considered In the first the coloured sbottom is the onlyproduction process decaying via b1 rarr bχ0

                      1 In the second the stop is the only productionprocess decaying via t1 rarr t χplusmn

                      1 and the lightest chargino (χplusmn1 ) decays via a virtual W boson

                      into the final state χ01 f f prime Both scenarios are characterised by the presence of two jets

                      resulting from the hadronisation of b quarks and large missing transverse momentum Theproduction of a mediator in association with top quarks followed by an invisible mediatordecay to dark matter particles has a similar final state as that considered in the ATLAS stopsearches Thus we expect the ATLAS reach to have sensitivity to the three dark mattermodels considered in this thesis

                      The collider cuts program mimicked object reconstruction by selecting baseline electronswith pT gt 7 GeV and |η |lt 247 and muons with pT gt 6 GeV and |η |lt 24 Baseline jetswere selected with pT gt 20 GeV and |η |lt 28 We then removed any jets within ∆R lt 02of an electron and any electron within ∆R lt 04 of surviving jets Signal electrons andmuons were required to have pT gt 25 GeV b-jets were selected with 60 tagging efficiencyand |η |lt 25 and pT gt 20 GeV

                      Two sets of signal regions are defined to provide sensitivity to the kinematic topologiesassociated with the different mass splittings between the sbottom or the stop and the neu-tralino In all cases the presence of at least one primary vertex (with at least five tracks withpT gt 04 GeV) is required Events are selected with Emiss

                      T gt 150 GeV and no electrons ormuons identified in the final state For signal region selections jets with |η | lt 28 are orderedaccording to their pT and 2 jets are required to be b-tagged

                      The variables are defined as follows

                      bull ∆φmin is the minimal azimuthal angle ∆φ between any of the three leading jets andpmiss

                      T

                      bull me f f (k) = sumki=1(p jet

                      T )i +EmissT where the index refers to the pT ordered list of jets

                      bull HT3 is defined as the scalar sum of the pT of the n leading pT ordered jets withoutincluding the 3 leading jets HT3 = sum

                      ni=4(p jet

                      T )i

                      bull mbb is the invariant mass of the two b-tagged jets in the event

                      33 Calculations 51

                      bull mCT is the contransverse mass [82] and is a kinematical variable that can be usedto measure the masses of pair produced semi-invisibly decaying particles For twoidentical decays of heavy particles into two invisible particles v1 and v2 and twoinvisible particles mCT is defined as m2

                      CT (v1v2) = [ET (v1)+ET (v2)]2 minus [pT (v1)minus

                      pT (v2)]2 where ET =

                      radicp2

                      T +m2 The the contransverse mass is invariant under equaland opposite boosts of the parent particles in the transverse plane and is bounded fromabove by an analytical expression of the particle masses For production of sbottompairs the bound is given by mmax

                      CT =m2(b)minusm2(χ0

                      1 )

                      m(b) and for tt events the bound is 135

                      GeV A similar equation can be written for stop pairs in term of the stop and charginomasses

                      A definition of the signal regions is given in the Table38

                      Table 38 Signal Regions 2bstop

                      Description SRA SRBEvent cleaning All signal regions

                      Lepton veto No emicro after overlap removal with pT gt7(6) GeV for e(micro)Emiss

                      T gt 150 GeV gt 250 GeVLeading jet pT ( j1) gt 130 GeV gt 150 GeVSecond jet pT ( j2) gt 50 GeV gt 30 GeVThird jet pT ( j3) veto if gt 50 GeV gt 30 GeV

                      ∆φ(pmissT j1) - gt 25

                      b-tagging leading 2 jets(pT gt 50 GeV |η | lt 25)

                      2nd- and 3rd- leading jets(pT gt 30 GeV|η | lt 25)

                      ∆φmin gt 04 gt 04Emiss

                      T me f f (k) EmissT me f f (2) gt 025 Emiss

                      T me f f (3) gt 04mCT gt 150200250300350 GeV -HT3 - lt 50 GeVmbb gt200 GeV -

                      The analysis cuts are summarised in Table A4 of Appendix 1

                      3335 ATLASMonobjet

                      Search for dark matter in events with heavy quarks and missing transverse momentumin pp collisions with the ATLAS detector[30]

                      Events were selected with high missing transverse momentum when produced in associa-tion with one or more jets identified as containing b quarks Final states with top quarks wereselected by requiring a high jet multiplicity and in some cases a single lepton The research

                      33 Calculations 52

                      studied effective contact operators in the form of a scalar operator Oscalar = sumqmqMN

                      lowastqqχχ

                      where MNlowast is the effective mass of the coupling and the quark and DM fields are denoted by

                      q and χ respectively These are the equivalent of the simplified models in this paper with themediator integrated out

                      Object reconstruction selected baseline electrons muons jets and τrsquos with pT gt 20 GeVand |η |lt 2525 and 45 respectively Signal jets were selected by removing any jets within∆R lt 02 of an electron and electrons and muons were selected by removing them if theywere within ∆R lt 04 of surviving jets b-jets were tagged with an efficiency of 60

                      Only signal regions SR1 and SR2 were analysed

                      The analysis split events into signal regions SR1 and SR2 according to the followingCriteria

                      Table 39 Signal Region ATLASmonobjet

                      Cut SR1 SR2Jet multiplicity n j 1minus2 3minus4

                      bminusjet multiplicity nb gt 0 (60 eff) gt 0 (60 eff)Lepton multiplicity nl 0 0

                      EmissT gt300 GeV gt200 GeV

                      Jet kinematics pb1T gt100 GeV pb1

                      T gt100 GeV p j2T gt100 (60) GeV

                      ∆φ( j1EmissT ) gt 10 i = 12 gt 10 i = 14

                      Where p jiT (pbi

                      T ) represent the transverse momentum of the i-th jet b-tagged jet respec-tively If there is a second b-tagged jet the pT must be greater than 60 GeV

                      3336 CMSTop1L

                      Search for top-squark pair production in the single-lepton final state in pp collisionsat

                      radics=8 TeV[41]

                      This search was for pair production of top squarks in events with a single isolatedelectron or muon jets large missing transverse momentum and large transverse mass

                      (MT =radic

                      2EmissT pl

                      T (1minus cos(∆φ))) where plT is the leptonrsquos transverse momentum and ∆(φ)

                      is the difference between the azimuthal angles of the lepton and EmissT The 3 models

                      considered should have some visibility at certain parameter combinations in this regionThe search focuses on two decay modes of the top squark leading to the two processespp rarr t tlowast rarr tt χ0

                      1 χ01 rarr bbW+Wminusχ0

                      1 χ01 and pp rarr t tlowast rarr bbχ

                      +1 χ

                      minus1 rarr bbW+Wminusχ0

                      1 χ01 The

                      33 Calculations 53

                      lightest neutralino χ01 is considered the lightest supersymmetric particle and escapes without

                      detection The analysis is based on events where one of the W bosons decays leptonicallyand the other hadronically resulting in one isolated lepton and 4 jets two of which originatefrom b quarks The two neutralinos and the neutrino from the W decay can result in largemissing transverse momentom The kinematical end point of MT is lt the mass of the Wboson However for signal events the presence of LSPs in the final state allows MT to exceedthe mass of the W boson The search is therefore for events with large MT The dominantbackground with large MT arises from tt decays where W bosons decay leptonically withone of the leptons not being identified

                      To reduce the dominant tt background use was made of the MWT 2 variable defined as

                      the minimum mother particle mass compatible with the transverse momenta and mass shellconstraints [84]

                      Object reconstruction required electrons and muons to have pT gt 30 and 25 GeV respand |η |lt 25 and 21 respectively Baseline jets were required to have pT gt 30 and |η |lt 4and b-jets were tagged with an efficiency of 75 for |η |lt 24

                      Events were chosen if they contained one isolated electron or muon at least 4 jets and atleast one b-tagged jet and Emiss

                      T gt 100 GeV MT gt 160 GeV MWT 2 was required to be greater

                      than 200 GeV and the smallest azimuthal angle difference between baseline jets and EmissT

                      gt12

                      Chapter 4

                      Calculation Tools

                      41 Summary

                      Figure 41 Calculation Tools

                      The simplified models were implemented in Feynrules [4] and the model files importedinto MicrOmegas v425 which was used to calculate the log-likelihoods of the astrophysicalobservables MicrOmegas was modified to allow only s-channel decays in both relic densityand gamma ray calculations Direct detection constraints were easy to implement for thescalar and vector models by coding the analytical expression for the spin independent WIMP-nucleon scattering cross section but due to the momentum dependence of the WIMP-nucleon

                      42 FeynRules 55

                      scattering cross section of the Majorana fermion model LUXCalc v101 [79] was modifiedfor this model to implement momentum dependent cross sections [85] We used MicrOmegasv425 to calculate relic density values and to produce gamma ray spectra for comparisonwith the Fermi-LAT results Parameter scans were carried out using MultiNest v310 withthe parameter ranges given in Table 51 and using astrophysical constraints only Oncethese scans had converged a random sample of 900 points was chosen (weighted by theposterior probability) and then used to calculate collider constraints using the MadgraphMonte Carlo generator This included comparison with mediator production and decay limitsfrom the LHC experiments and the reproduction of searches for dark matter plus other finalstate objects which required dedicated C++ code that reproduces the cut and count analysesperformed by CMS and ATLAS

                      42 FeynRules

                      FeynRules is a Mathematicareg package that allows the calculation of Feynman rules inmomentum space for any QFT physics model The user needs to provide FeynRules withminimal information to describe the new model contained in the model-file This informationis then used to calculate a set of Fenman rules associated with the Lagrangian The Feynmanrules calculated by the code can then be used to implement new physics models into otherexisting tools such as Monte Carlo (MC) generators via a set of interfaces developed andmaintained by the MC authors

                      Feynrules can be run with a base model = SM (or MSSM) and user defined extensions tothe base model

                      43 LUXCalc

                      LUXCalc is a utility program for calculating and deriving WIMP-nucleon coupling limitsfrom the recent results of the LUX direct search dark matter experiment The program hasbeen used to derive limits on both spin-dependent and spin-independent WIMP-nucleoncouplings and it was found under standard astrophysical assumptions and common otherassumptions that LUX excludes the entire spin-dependent parameter space consistent witha dark matter interpretation of DAMArsquos anomalous signal and also found good agreementwith published LUX results [79]

                      We modified the program for the Majorana fermion model to implement momentumdependent cross -sections [85]

                      44 Multinest 56

                      44 Multinest

                      Multinest is an efficient Bayesian Inference Tool which uses a simultaneous ellipsoidalnested sampling method a Monte Carlo technique aimed at an efficient Bayesian Evidencecalculation with posterior inferences as a by product

                      Bayes theorem states that

                      Pr(θ |DH) =Pr(D|θ H)Pr(θ |H)

                      Pr(D|H) (41)

                      Here D is the data and H is the Hypothesis Pr(θ |DH) is the posterior probability ofthe parameters Pr(D|θ H) = L(θ) is the likelihood Pr(θ |H) is the prior and Pr(D|H) isthe evidence

                      The evidence Pr(D|H) =int

                      Pr(θ |DH)Pr(θ |H)d(θ) =int

                      L(θ)Pr(θ |H)d(θ) Thus ifwe define the prior volume dX = Pr(θ |H)d(θ) and

                      X(λ ) =int

                      L(θ)gtλ

                      Pr(θ |H)d(θ) (42)

                      where the integral extends over the region of parameter space within the iso likelihoodcontour L(θ) = λ then the evidence integral can be written

                      int 10 L (X)dX where L (X) the

                      inverse of equation 42 and is a monotonically decreasing function of X (the prior volumewhich is 1 for the total sample space and 0 for a single point)

                      Thus if one can evaluate likelihoods L (Xi) where 0 lt Xn lt lt X2 lt X1 lt X0 = 1the evidence can be estimated as sumLiwi where wi = 5(Ximinus1 minusXi+1) Ellipsoidal samplingapproximates the iso likelihood contour by a D dimensional covariance matrix of the currentset of active points New points are selected from the prior within this ellipsoidal bounduntil one is obtained that has a likelihood greater than the likelihood of the removed lowestlikelihood point This method is extended for multimodal sampling where there may bemore than one local maximum by clustering points that are well separated and constructingindividual ellipsoidal bounds for each cluster For more details consult [7]

                      The results of the analysis are expressed as a data file containing the sample probability-2loglikelihood followed by the parameters of the model for the particular sampling Theprobabilities are normalised to total 10

                      45 Madgraph 57

                      45 Madgraph

                      Madgraph is a general purpose matrix element calculator and Monte Carlo generator Basedon user-supplied Feynman rules (eg the output of Feynrules) the package calculates the tree-level matrix elements for any processes that the user specifies and generates samples of LHCevents including an accurate simulation of the proton collisions themselves (which requires aknowledge of the parton distribution functions that determine the relative proportion of eachparton inside the nucleus) Hadronisation of final state quarks is not possible in Madgraphand hence the code must be used in conjunction with Pythia which performs a parton shower

                      The Madgraph output was automatically showered by Pythia (Pythia describes partonradiation as successive parton emissions using Markov chain techniques based on Sudakovform factors and this description is only formally correct for the soft and collinear emissions)A Madgraph simulation is at the level of the fundamental particles (quarks and leptons)Madgraph generates a limited set of matrix element processes at LO representing hard (highenergy) emissions and passes these to Pythia for showering The final state partons in theMadgraph simulation must be matched to the Pythia final state partons to avoid doublecounting

                      The output from Pythia is automatically fed through Delphes where basic collider cutscan be performed (calculation of the numbers of particles and jets that would be observed atdifferent energies and locations within the collider) The output of the Delphes simulation is afile containing reconstructed events that can be analysed using the standard ROOT framework(the most popular data analysis package in particle physics) [86]

                      The parameters that gave the best fits to the direct [16] and indirect DM experiments [87]and relic density [59] are ranked by posterior probability from the Multinest [7] output

                      The parameters in the Madgraph simulations are stored in files in the cards sub directoryand in particular the run card param card and Delphes card record items relevent to thesimulation

                      In the run card the number of events the Beam 1 and 2 total energy in GeV (4000 each)and the type of jet matching were modified (to MLM type matching where the final statepartons in Madgraph are clustered according to the KT jet algorithm to find the equivalentparton shower history of the event there is a minimum cutoff scale for the momentum pT

                      given by the xqcut parameter) The maxjetflavour (=5 to include b quarks) and xqcut = 30GeV parameters were also modified

                      46 Collider Cuts C++ Code 58

                      The param card was modified to specify the masses of the new particles and the autodecay modes The Delphes card is overwritten with either the ATLAS or CMS defaultcards which record tracking efficiencies and isolation momentum and energy resolutionscalorimeter details jet finder and b and τ -tagging parameters However the isolation andtagging in this study were done by the collider cuts C++ code since multiple sets of cutswere applied to the same output from Pythia

                      When performing simulations of the three dark matter models it was necessary to definethe proton and jets to contain b quarks as this is not the default in Madgraph The processesgenerated included 1 and 2 jets

                      46 Collider Cuts C++ Code

                      Because of the volume of simulations required for the collider constraints we used C++code (originally developed by Martin White and Sky French and modified by me) that readsthe ROOT files and outputs the numbers of events that would have been observed under thechosen collider experiments had the model been true for the given parameters Thus the sameoutput files can be used for all of the collider cuts and hence no cuts were performed throughthe Delphes cards

                      In order to validate the C++ code we replicated the observed events with the papersdescribing the various collider searches (see Appendix)

                      Chapter 5

                      Majorana Model Results

                      51 Bayesian Scans

                      To build some intuition we first examined the constraining effect of each observable on itsown We ran separate scans for each constraint turned on and then for all the constraintstogether In Fig 51 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together The paramaters were scanned over the range in Table 51 below

                      Table 51 Scanned Ranges

                      Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                      Range 0minus103 0minus103 10minus5 minus10 10minus5 minus10 10minus5 minus10

                      In Fig51 we can see that the preferred DM mass is centered around 35 GeV primarily dueto the influence of the Fermi-LAT gamma ray constraint whilst the relic density constraintallows a wide range of possible mediator and DM masses Direct detection (LUX) constraintssplit the mass regions into a top left and bottom right region and together these split theoverall mass parameter regions into two regions by mediator mass but with DM mass around35 GeV Later studies using a lower excess gamma ray spectrum tend to increase the preferredDM mass up to 50 GeV [76] and [88]

                      The marginalised posterior probabilities for all combinations of parameters taken twoat a time are shown in Fig52 These plots have all constraints (Gamma LUX and RelicDensity) applied together The idea of these graphs is to study the posterior probabilities ofcombinations of two paramaters taken at a time (the off-diagonal graphs) The diagonal plotsshow marginalised histograms of a single parameter at a time- this gives an indication of thespread of each parameter The plots show that mS and λb show a high degree of correlationfor mS gt 100 GeV but there is also a range of values of mS around 35 GeV which are

                      51 Bayesian Scans 60

                      1 0 1 2 3 4log10(mχ)[GeV]

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                      1 0 1 2 3 4log10(mχ)[GeV]

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                      (d) All Constraints

                      Figure 51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether

                      51 Bayesian Scans 61

                      00

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                      τ)

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                      ms[Gev]5 4 3 2 1 0 1

                      log10(λt)5 4 3 2 1 0 1

                      log10(λb)5 4 3 2 1 0 1

                      log10(λτ)

                      Figure 52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter

                      52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 62

                      possible for λb around 001 λt and λτ show a wide range of possible values for two possibleranges of mS lt 35 GeV and greater than 100 GeV mχ shows a narrow range of possiblevalues around 35 GeV for all values of the other parameters

                      52 Best fit Gamma Ray Spectrum for the Majorana FermionDM model

                      We can see from the Fig51(a) scan that the maximum log-likelihood Fermi-LAT gammaray spectrum on its own shows values that centre around 35 GeV This is not changed byadding a relic density constraint or a direct detection (LUX) constraint

                      The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraintsis shown in Fig53 The parameters for this point were

                      Table 52 Best Fit Parameters

                      Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                      Value 3332 49266 0322371 409990 0008106

                      10-1 100 101 102

                      E(GeV)

                      10

                      05

                      00

                      05

                      10

                      15

                      20

                      25

                      30

                      35

                      E2dφd

                      E(G

                      eVc

                      m2ss

                      r)

                      1e 6

                      Best fitData

                      Figure 53 Gamma Ray Spectrum

                      The relic density (Ωh2) at this point has a value of 0135 but the point is separated onlyslightly in posterior probability from points with a value of 01199 with mS lt 10 GeV (iethe posterior probability is almost the same differing by less than 1)

                      To better understand the interplay of the different constraints the three graphs in Fig54show the log likelihood surfaces and individual best fit points (best log likelihoods) for the γ

                      and Ω scans seperately and then combined The diagrams each choose a pair of couplings(fixing the remaining parameters) and display the 2d surfaces of minuslog(χ2) (proportional to

                      52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 63

                      the log likelihoods of the coupling combinations) obtained by varying the pair of couplingsThese graphs do not show marginalised posterior probabilities but rather are based on the-2log likelihood calculations directly from MicrOmegas for the given coupling combinationsWe have termed this value as χ2 for convenience and used the log of this value to give abetter range for the colours These log likelihood values are what drive the MultiNest searchof parameter space for the best parameter combinations

                      The graphs show a complex interplay of the different constraints resulting often in acompromise parameter combination The arrow on each graph points to the maximum point We have chosen a single representative set of couplings near the best overall fit point for allthe graphs

                      52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 64

                      00 05 10 15 20 25 30

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                      00

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                      Both Maximum at mχ=8317 GeV mS=2884GeV The best fit point for these couplings maybe above below or between the individual bestfit points but will on average be between thepoints

                      Figure 54 Plots of log likelihoods by individual and combined constraints Masses in GeV

                      53 Collider Constraints 65

                      53 Collider Constraints

                      531 Mediator Decay

                      1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof τ leptons in pp collisions [44] Refer also to Chapter 3 for details

                      We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

                      The results of the calculations are shown in the graph Fig55 and can be compared withthe limits in the table from the paper [44] in Fig 56 The black line is the observed ATLASlimit

                      Figure 55 σ lowastBr(σ rarr ττ) versus Mass of Scalar

                      0 200 400 600 800

                      mS[GeV]

                      10

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                      (bbS

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                      (Sgtττ

                      ))[pb]

                      Observed LimitLikely PointsExcluded Points

                      0

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                      0 5 10 15 20 25 30 35 40 45

                      We can see that only one point here exceeded the limit and the points showed a widespread over σ(bbS)lowastBr(σ rarr ττ) by mS space which is consistent with the spread of pointson the MultiNest scan on the mS by λτ plane causing a spread of branching ratios to the τ

                      quark It may seem odd that only one point was excluded here but this point was sampledwith a very low probability having high coupling constants of 597 to the τ quark and 278 tothe t quark and only just exceeds the limit As such it can be explained as a random outlier

                      53 Collider Constraints 66

                      Figure 56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar

                      2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                      This scan calculates the contributions of b-quark associated Higgs boson productionreproduced from [43] Refer to Chapter 3 for details

                      We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

                      The cross section times branching ratios to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig57 The black line is the CMSLimit

                      Figure 57 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar

                      0 200 400 600 800

                      mS[GeV]

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                      53 Collider Constraints 67

                      The results of this scan were compared to the limits in [89] with the plot shown inFig58

                      Figure 58 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

                      We can see that there are a number of exclusions across the range of mS which occuredfor high values of the coupling constants gt 1 for λbλt λtau - these can be seen also on Fig59The excluded parameter combinations represent less than 2 of the total sampled number

                      532 Collider Cuts Analyses

                      We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude and the results are shown against theposterior probabilities of the parameter combinations in Fig 59

                      The circles in Figure 59 show which points were excluded by the collider cuts as well asthe limits on cross sections times branching ratios to each of τ+τminus and bb

                      All the excluded points under the heading Collider Cuts showed exclusion by the Lepstop0experiment (Search for direct pair production of the top quark in all hadronic final states

                      53 Collider Constraints 68

                      0 1 2 3

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                      σ lowastBr(σgt ττ)

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                      Figure 59 Excluded points from Collider Cuts and σBranching Ratio

                      53 Collider Constraints 69

                      [32]) Some of these same points were also excluded by Lepstop2 2bstop CMSTop andATLASMonobjet experiments There were expected events at other points but none weresignificant at the 95 exclusion level These significant exclusions occurred mainly wherethe mass of the mediator was over 100 GeV and higher values of the couplings to b and tquarks but there were a few points with mediator mass around 25 GeV The exclusions withmediator mass around 25 GeV were all accompanied by couplings to t quarks of order 10which is probably non-perturbative

                      Looking at the exclusions in Fig 59 we can see that these occur mainly in the regionof higher mediator mass gt 100 GeV and lt 400 GeV These exclusions come from theastrophysical constraints (Gamma) which show a strong correlation between λb and mS andhigh values of λt gt 1 However these mass regions do not overlap at the same couplingcombinations as the high likelihood region The green and white circles correspond to theexcluded points in Figs 57 and 58 respectively

                      The remaining regions of highest likelihood can be seen to be where 0001 λb lt 01accompanied by λt lt 0001 and mass of the mediator mS lt 10 GeV or 400 GeV lt mS lt1000GeV with mχ sim 35 GeV

                      The plot of λt by λb shows the significant regions do not overlap with the collider cutsand cross section constraints since the latter occur at high values of either coupling while thelikely region has low values of these couplings (also accompanied by low couplings to λτ )

                      Chapter 6

                      Real Scalar Model Results

                      61 Bayesian Scans

                      To build some intuition we first examined the constraining effect of each observable on itsown We ran seperate scans for each constraint turned on and then for all the constraintstogether In Figure 61 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together

                      In Figure 61 we can see that there are is a narrow region which dominates when allconstraints are taken into account This centers on mχ=9 GeV and a range of values for mS

                      from 35 GeV to 400 GeV The region is not the highest likelihood region for the Fermi-LATgamma ray spectrum on its own (as can be seen from the individual plot - there is a narrowband to the left of the highest likelihood band) nor for the LUX likelihood or relic densitybut represents a compromise between the three constraints

                      The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig62 The diagonal plots show marginalised histograms of a singleparameter at a time

                      61 Bayesian Scans 71

                      05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

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                      Figure 61 Real Scalar Dark Matter - By Individual Constraint and All Together

                      61 Bayesian Scans 72

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                      Figure 62 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter

                      62 Best fit Gamma Ray Spectrum for the Real Scalar DM model 73

                      62 Best fit Gamma Ray Spectrum for the Real Scalar DMmodel

                      We can see from Fig61 that the maximum log-likelihood Fermi-LAT Gamma ray spectrumon its own shows values of mχ that centre around 35 GeV However the LUX constraintin particular moves the best fit to a point around 9 GeV- the tight LUX constraint onlydisappears for low masses and at that point the proportion of τ+τminus annihilation needs toincrease to fit the constraints

                      The gamma ray spectrum at the points of best fit for Fermi-LAT including all constraintsis shown in Fig63 The parameters for these points were respectively and involve decays tomainly τ and b quarks respectively

                      Table 61 Best Fit Parameters

                      Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                      Value 932 3526 000049 0002561 000781

                      10-1 100 101 102

                      E(GeV)

                      10

                      05

                      00

                      05

                      10

                      15

                      20

                      25

                      30

                      35

                      E2dφdE

                      (GeVc

                      m2ss

                      r)

                      1e 6

                      Best fitData

                      Figure 63 Gamma Ray Spectrum

                      This is not a particularly good fit and it occcurs at a relic density Ωh2 of 335 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

                      63 Collider Constraints 74

                      63 Collider Constraints

                      631 Mediator Decay

                      1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44]

                      We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

                      The results of the scan are shown in the graph Fig64 and can be compared with thelimits in the table from the paper [44] in Fig 56The black line is the observed ATLAS limit

                      Figure 64 σ lowastBr(σ rarr ττ) versus Mass of Scalar

                      0 200 400 600 800

                      mS[GeV]

                      8

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                      Observed LimitLikely PointsExcluded Points

                      050

                      100150200250300350

                      0 10 20 30 40 50 60

                      We can see that there is a narrow band in the region of interest in the σ(bbS)lowastB(S gt ττ)

                      by mS plane due to the high degree of correlation between mS and λτ seen in the likelihoodscan Values for mS gt 300Gev are excluded

                      2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                      We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinations

                      63 Collider Constraints 75

                      randomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

                      The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig65 The black line is the CMSlimit

                      Figure 65 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

                      0 200 400 600 800

                      mS[GeV]

                      8

                      6

                      4

                      2

                      0

                      2

                      4

                      log

                      10(σ

                      (bS

                      +X

                      )lowastB

                      (Sgt

                      bb))

                      [pb]

                      Observed LimitLikely PointsExcluded Points

                      050

                      100150200250300350

                      0 10 20 30 40 50 60

                      The results of this scan were compared to the limits in [89] with the plot shown inFig58

                      We can see a similar shape for the region of interest to that in Fig64 due to the highcorrelation between mS and λb over the region preferred by astrophysical constraints resultingin branching ratios highly correlated to ms Only one point is excluded with mS gt 500GeV

                      632 Collider Cuts Analyses

                      We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expected

                      63 Collider Constraints 76

                      with 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 66

                      We can see that collider cuts show some exclusions only for masses mχ below the highlikelihood region but these exclusions do not occur at the same λb and λt combinations asseen from the plot of these two couplings except for a small region at the highest mS andcouplings

                      All but one of the collider cut exclusions were excluded by the Lepstop0 experiment (thiswas excluded by 2bstop with high coupling to b quarks of 73) A number of the excludedpoints were also excluded by Lepstop2 2bstop CMSTopDM1L and ATLASMonobJet-(around 30 of the total for each)

                      63 Collider Constraints 77

                      0 1 2 3

                      log10(mχ)[GeV]

                      0

                      1

                      2

                      3

                      log 1

                      0(m

                      s)[GeV

                      ]Collider Cuts

                      σ lowastBr(σgt bS+X)

                      σ lowastBr(σgt ττ)

                      (a) mχ by mS

                      5 4 3 2 1 0 1

                      log10(λt)

                      0

                      1

                      2

                      3

                      log 1

                      0(m

                      s)[GeV

                      ](b) λt by mS

                      5 4 3 2 1 0 1

                      log10(λb)

                      5

                      4

                      3

                      2

                      1

                      0

                      1

                      log 1

                      0(λ

                      t)

                      (c) λb by λt

                      5 4 3 2 1 0 1

                      log10(λb)

                      6

                      5

                      4

                      3

                      2

                      1

                      0

                      1

                      2

                      log 1

                      0(λ

                      τ)

                      (d) λb by λτ

                      5 4 3 2 1 0 1

                      log10(λt)

                      6

                      5

                      4

                      3

                      2

                      1

                      0

                      1

                      2

                      log 1

                      0(λ

                      τ)

                      (e) λt by λτ

                      5 4 3 2 1 0 1

                      log10(λb)

                      0

                      1

                      2

                      3

                      log 1

                      0(m

                      s)[GeV

                      ]

                      (f) λb by mS

                      Figure 66 Excluded points from Collider Cuts and σBranching Ratio

                      Chapter 7

                      Real Vector Dark Matter Results

                      71 Bayesian Scans

                      In the Figure 71 we can see that the preferred DM mass is centered around 35 GeV primarilydue to the influence of the Fermi-LAT gamma ray constraint with relic density having a widerange of possible mediator and DM masses The result of combining all the constraints is todefine a small region around mχ = 8 GeV and mS = 20 GeV which dominates the posteriorprobability

                      The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig72 The diagonal plots show marginalised histograms of a singleparameter at a time

                      71 Bayesian Scans 79

                      05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                      1

                      0

                      1

                      2

                      3

                      4

                      log 1

                      0(m

                      s)[GeV

                      ]

                      (a) Gamma Only

                      05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                      05

                      00

                      05

                      10

                      15

                      20

                      25

                      30

                      35

                      log 1

                      0(m

                      s)[GeV

                      ]

                      (b) Relic Density

                      1 0 1 2 3 4log10(mχ)[GeV]

                      05

                      00

                      05

                      10

                      15

                      20

                      25

                      30

                      35

                      log 1

                      0(m

                      s)[GeV

                      ]

                      (c) LUX

                      05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                      05

                      00

                      05

                      10

                      15

                      20

                      25

                      30

                      35

                      log 1

                      0(m

                      s)[GeV

                      ]

                      (d) All Constraints

                      Figure 71 Real Vector Dark Matter - By Individual Constraint and All Together

                      71 Bayesian Scans 80

                      00

                      05

                      10

                      15

                      20

                      25

                      30

                      log 1

                      0(m

                      χ)[GeV

                      ]

                      00

                      05

                      10

                      15

                      20

                      25

                      30

                      ms[Gev

                      ]

                      5

                      4

                      3

                      2

                      1

                      0

                      1

                      log 1

                      0(λ

                      t)

                      5

                      4

                      3

                      2

                      1

                      0

                      1

                      log 1

                      0(λ

                      b)

                      00 05 10 15 20 25 30

                      log10(mχ)[GeV]

                      5

                      4

                      3

                      2

                      1

                      0

                      1

                      log 1

                      0(λ

                      τ)

                      00 05 10 15 20 25 30

                      ms[Gev]5 4 3 2 1 0 1

                      log10(λt)5 4 3 2 1 0 1

                      log10(λb)5 4 3 2 1 0 1

                      log10(λτ)

                      Figure 72 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter

                      72 Best fit Gamma Ray Spectrum for the Real Vector DM model 81

                      72 Best fit Gamma Ray Spectrum for the Real Vector DMmodel

                      The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraints isshown in Fig73

                      Table 71 Best Fit Parameters

                      Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                      Value 8447 20685 0000022 0000746 0002439

                      10-1 100 101 102

                      E(GeV)

                      10

                      05

                      00

                      05

                      10

                      15

                      20

                      25

                      30

                      35

                      E2dφdE

                      (GeVc

                      m2s

                      sr)

                      1e 6

                      Best fitData

                      Figure 73 Gamma Ray Spectrum

                      This is not a particularly good fit and it occcurs at a relic density Ωh2 of 025 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

                      73 Collider Constraints

                      731 Mediator Decay

                      1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44] (See Chapter 3 for more details)

                      We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

                      73 Collider Constraints 82

                      The results of the scan are shown in the graph Fig74 and can be compared with thelimits in the table from the paper [44] in Fig 56

                      Figure 74 σ lowastBr(σ rarr ττ) versus Mass of Scalar

                      0 200 400 600 800

                      mS[GeV]

                      8

                      6

                      4

                      2

                      0

                      2

                      log 1

                      0(σ

                      (bbS

                      )lowastB

                      (Sgtττ

                      ))[pb]

                      Observed LimitLikely PointsExcluded Points

                      0100200300400500600700800

                      0 20 40 60 80 100120140

                      We can see that the region of highest likelihood is limited to mS lt 100 GeV and a narrowrange of couplings to λτ giving a narrow region of interest which falls well below the limit

                      2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                      We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

                      The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig75 The black line is the CMSlimit

                      The results of this scan were compared to the limits in [89] with the plot shown in Fig58

                      We can see that again the region of interest for mS is below 100 GeV and the points fallwell below the limit

                      73 Collider Constraints 83

                      Figure 75 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

                      0 200 400 600 800

                      mS[GeV]

                      8

                      6

                      4

                      2

                      0

                      2

                      4

                      log

                      10(σ

                      (bS

                      +X

                      )lowastB

                      (Sgt

                      bb))

                      [pb]

                      Observed LimitLikely PointsExcluded Points

                      0100200300400500600700800

                      0 20 40 60 80 100120140

                      732 Collider Cuts Analyses

                      We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 76

                      We can see from Figure 76 that like the real scalar model the region of highest likelihoodhas λb and λt lt 001 while the excluded points from collider cuts and cross sections areabove 01 Most of the excluded points have mχ gt than the values in the region of highestlikelihood and all points are widely separated in parameter space from this region

                      Once again 89 of excluded points were excluded by Lepstop0 with many also beingexcluded by the other experiments (50 by Lepstop2 and 56 by CMSTopDM1L 35 byLepstop2 and 20 by ATLASMonobjet)

                      73 Collider Constraints 84

                      0 1 2 3

                      log10(mχ)[GeV]

                      0

                      1

                      2

                      3

                      log 1

                      0(m

                      s)[GeV

                      ]Collider Cuts

                      σ lowastBr(σgt bS+X)

                      σ lowastBr(σgt ττ)

                      (a) mχ by mS

                      5 4 3 2 1 0 1

                      log10(λt)

                      0

                      1

                      2

                      3

                      log 1

                      0(m

                      s)[GeV

                      ](b) λt by mS

                      5 4 3 2 1 0 1

                      log10(λb)

                      5

                      4

                      3

                      2

                      1

                      0

                      1

                      log 1

                      0(λ

                      t)

                      (c) λb by λt

                      5 4 3 2 1 0 1

                      log10(λb)

                      5

                      4

                      3

                      2

                      1

                      0

                      1

                      log 1

                      0(λ

                      τ)

                      (d) λb by λτ

                      5 4 3 2 1 0 1

                      log10(λt)

                      5

                      4

                      3

                      2

                      1

                      0

                      1

                      log 1

                      0(λ

                      τ)

                      (e) λt by λτ

                      5 4 3 2 1 0 1

                      log10(λb)

                      0

                      1

                      2

                      3

                      log 1

                      0(m

                      s)[GeV

                      ]

                      (f) λb by mS

                      Figure 76 Excluded points from Collider Cuts and σBranching Ratio

                      Chapter 8

                      Conclusion

                      We first conducted Bayesian scans over mass and coupling parameters for three simplifiedmodels - Majorana fermion real scalar and real vector using direct detecion (LUX) indirectdetection (GCE γ rays) and relic density constraints to find the regions of highest likelihoodWe ran these scans one constraint at a time and then together to better understand the interplayof the different constraints on best fit and likely parameter regions

                      We also developed and tested C++ collider cut software specifically designed to mimickthe cuts applied in a number of ATLAS and CMS experiments that searched for Emiss

                      T inconnection with specific beyond the standard model (BSM) theories of Dark Matter

                      We then selected parameter combinations by randomly sampling the scanned resultsbased on the posterior probability distribution of the points and applied collider cuts andcross section times branching ratio calculations to these points to see if there would havebeen any sensitivity from the ATLAS and CMS experiments

                      We found that the Majorana fermion model showed the best overall fit to all the constraintswith relic density close to the experimental value of 01199 and a low χ2 error on the gammaspectrum fit The preferred mass mχ was around 35 GeV and mS ranging from lt10 GeV to500 GeV The smaller masses give a better fit for relic density and also are well separatedfrom current collider constraints mainly due the the fact that the exclusions occur mainlywith higher couplings to t quarks (greater than 1) which is consistent with the fact that manyof the experiments were targeting dark matter pair production associated with t quark decaysNevertheless there were some exclusions that did fall in the region sampled by randomselection from the posterior probability distribution

                      The preferred parameter regions were also consistent with the observation in [20] thatthe Majorana model showed the best overall fit with similar mass constraints We find thatthe couplings to the SM fermions should all be less that 001

                      86

                      The scalar and vector DM models give slightly less good fits especially to the currentrelic density estimate of 01199 but again the current collider constraints are well separatedfrom the best fit region and in fact none of the points randomly sampled from the posteriordistribution showed any exclusion The best fit parameter combination favours couplings toSM fermions of less than 001

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                      Appendix A

                      Validation of Calculation Tools

                      Lepstop0Search for direct pair production of the top squarkin all hadronic final states in proton=proton collisions atradic

                      s=8 TeV with the ATLAS detector [32]

                      Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1) mt1=600 GeV mn1=1 GeVWhere t1 is the top squark and n1 is the neutralino of the MSSM

                      94

                      Table A1 0 Leptons in the final state

                      Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 10000 460 460 100Emiss

                      T gt 130GeV 8574 395 451 88Leptonveto 6621 307 285 108Emiss

                      T gt 150GeV 6316 293 294 100Jet multiplicityand pT 1099 48 51 94|∆φ( jet pmiss

                      T )|gt π5 902 40 42 96gt= 2b jets 396 19 18 106τ veto 396 19 18 106mbmin

                      T gt 175GeV 270 13 13 105SRA1 79 4 4 104SRA2 70 3 3 93SRA3 67 3 3 97SRA4 52 2 2 91SRC exactly5 jets 1297 50 60 83SRC |∆φ( jet pmiss

                      T )|gt π5 1082 50 50 99SRC gt= 2b jets 430 20 20 97SRC τ veto 430 20 20 97SRC |∆φ(bb)|gt 02π 367 17 17 99SRC1 246 12 11 105SRC2 177 9 8 116SRC3 152 8 7 116

                      95

                      Lepstop1Search for direct top squark pair production infinal states with one isolated lepton jets and missing trans-verse momentum in

                      radics = 8 TeV pp collisions using 21 f bminus1

                      of ATLAS data[33]

                      Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1)(t1 gt b x+1 )(t1 gt b xminus1 )mt1=650 GeV mn1=1 GeV Where t1 is the top squark n1 is the neutralino xplusmn1 are thecharginos

                      96

                      Table A2 1 Lepton in the Final state

                      Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 100000 99990 99990 100Electron(= 1signal) 11558 11557 11785 984 jets(80604025) 5187 5186 6271 83gt= 1btag 4998 4997 5389 93Emiss

                      T gt 100GeV [allSRs] 4131 4131 4393 94Emiss

                      T radic

                      HT gt 5 [allSRs] 4026 4026 4286 94∆φ( jet2 pmiss

                      T )gt 08 [allSRs] 3565 3565 3817 93∆φ( jet1Pmiss

                      T )gt 08 [not SRtN2] 3446 3446 3723 93Emiss

                      T gt 200GeV (SRtN2) 2027 2027 2186 93Emiss

                      T radic

                      HT gt 13(SRtN2) 1156 1156 1172 99mT gt 140GeV (SRtN2) 839 839 974 86Emiss

                      T gt 275GeV (SRtN3) 948 948 965 98Emiss

                      T radic

                      HT gt 11(SRtN3) 932 932 937 99mT gt 200GeV (SRtN3) 456 456 566 81Emiss

                      T gt 150GeV (SRbC1minusSRbC3) 2803 2803 3080 91Emiss

                      T radic

                      HT gt 7(SRbC1minusSRbC3) 2742 2742 3017 91mT gt 12GeV (SRbC1minusSRbC3) 1917 1917 2347 82Emiss

                      T gt 160GeV (SRbC2SRbC3) 1836 1836 2257 81Emiss

                      T radic

                      HT gt 8(SRbC2SRbC3) 1788 1788 2175 82me f f gt 550GeV (SRbC2) 1741 1741 2042 85me f f gt 700GeV (SRbC3) 1075 1075 1398 77Muon(= 1signal) 11379 11378 11975 954 jets(80604025) 5264 5263 6046 87gt= 1btag 5101 5100 5711 89Emiss

                      T gt 100GeV [allSRs] 4199 4199 4770 88Emiss

                      T radic

                      HT gt 5 [all SRs] 4115 4115 4653 88∆φ( jet2 pmiss

                      T )gt 08 [all SRs] 3622 3622 4077 89∆φ( jet1 pmiss

                      T )gt 08 [not SRtN2] 3515 3515 3970 89Emiss

                      T gt 200GeV (SRtN2) 2136 2136 2534 84Emiss

                      T radic

                      HT gt 13(SRtN2) 1261 1261 1441 87mT gt 140GeV (SRtN2) 928 928 1033 90Emiss

                      T gt 275GeV (SRtN3) 1036 1036 1274 81Emiss

                      T radic

                      HT gt 11(SRtN3) 1018 1018 1235 82mT gt 200GeV (SRtN3) 497 497 612 81Emiss

                      T gt 150GeV (SRbC1minusSRbC3) 2891 2891 3380 86Emiss

                      T radic

                      HT gt 7(SRbC1minusSRbC3) 2821 2821 3290 86mT gt 120GeV (SRbC1minusSRbC3) 1977 1977 2290 86Emiss

                      T gt 160GeV (SRbC2SRbC3) 1913 1913 2220 86Emiss

                      T radic

                      HT gt 8(SRbC2SRbC3) 1863 1863 2152 87me f f gt 550GeV (SRbC2) 1826 1826 2116 86me f f gt 700GeV (SRbC3) 1131 1131 1452 78

                      97

                      Lepstop2Search for direct top squark pair production infinal states with two leptons in

                      radics =8 TeV pp collisions using

                      20 f bminus1 of ATLAS data[83][34]

                      Simulated in Madgraph with p p gt t1 t1(t1 gt b w+ n1)(t1 gt b wminusn1) mt1=180 GeV mn1=60 GeV Where t1 is the top squark n1 is the neutralino b1 is the bottom squark

                      Table A3 2 Leptons in the final state

                      Cut RawMe Scaled ATLAS MeATLAS PercentNoCuts 100000 324308 199000 1632BaselineCuts 3593 11652 7203 1622SF Signal Leptons 1699 5510 5510 1002OSSF Signal Leptons 1632 5293 5233 101mll gt 20Gev 1553 5037 5028 100leadinglepton pT 1479 4797 5048 95|mll minusmZ|gt 20Gev 1015 3292 3484 94∆φmin gt 10 850 2757 2160 128∆φb lt 15 680 2205 1757 126SRM90 [SF ] 13 42 60 70SRM120 [SF ] 0 0 19 0SRM100 +2 jets [SF ] 0 0 17 0SRM110 +2 jets [SF ] 0 0 1 02DF Signal Leptons 1737 5633 7164 792OSDF Signal Leptons 1670 5416 5396 100mll20Gev 1587 5147 5136 100leadinglepton pT 1501 4868 4931 99∆φmin gt 10 1247 4044 3122 130∆φb lt 15 1010 3276 2558 128SRM90 [SF ] 21 68 95 72SRM120 [SF ] 0 0 23 0SRM100 +2 jets [SF ] 0 0 19 0SRM110 +2 jets [SF ] 0 0 14 0

                      98

                      2bstopSearch for direct third-generation squark pair pro-duction in final states with missing transverse momentumand two b-jets[31]

                      Simulated in Madgraph with p p gt b1 b1(b1 gt b n1)(b1 gt b n1)

                      SRAmb1=500 GeV mn1=1 GeV SRBmb1=350 GeV mn1=320 GeVWhere b1 is the bottom squark and n1 is the neutralino

                      Table A4 2b jets in the final state

                      Cut RawMe Scaled ATLAS MeATLAS PercentSRA Nocuts 100000 173800 1738 100SRA MET gt 80 93407 162341 1606 101SRA Leptonveto 93302 162159 1505 108SRA MET gt 150 79449 138082 1323 104SRA Jet selection 10213 17750 119 149SRA mbb gt 200 8344 14502 96 151SRA mCT gt 150 6535 11358 82 139SRA mCT gt 200 5175 8994 67 134SRA mCT gt 250 3793 6592 51 129SRA mCT gt 300 2487 4322 35 123SRB NoCuts 100000 1624100 16241 100SRB leptonveto 11489 186593 4069 46SRB MET gt 250 2530 41090 757 54SRB Jet selection 63 1023 79 130SRB HT 3 lt 50 51 828 52 159

                      99

                      CMSTop1LSearch for the Production of Dark Matter in As-sociation with Top Quark Pairs in the Single-lepton FinalState in pp collisions[41]

                      Simulated in Madgraph with p p gt t t p1 p1

                      Where p1= Majorana fermion Table 3 shows Signal efficiencies and cross sections TheM scale was read from Figure 7 for 90 CL

                      Table A5 Signal Efficiencies 90 CL on σ limexp[ f b] on pp gt tt +χχ

                      Mχ[GeV ] My Signal Efficiency Quoted Efficiency My σ [ f b] Quoted σ [ f b]1 102 101 63 47

                      10 132 101 62 4650 143 120 50 38100 170 146 56 32200 185 173 69 27600 265 240 32 19

                      1000 320 276 41 17

                      Appendix B

                      Branching ratio calculations for narrowwidth approximation

                      B1 Code obtained from decayspy in Madgraph

                      Br(S rarr bb) = (minus24λ2b m2

                      b +6λ2b m2

                      s

                      radicminus4m2

                      bm2S +m4

                      S)16πm3S

                      Br(S rarr tt) = (6λ2t m2

                      S minus24λ2t m2

                      t

                      radicm4

                      S minus4ms2m2t )16πm3

                      S

                      Br(S rarr τ+

                      τminus) = (2λ

                      2τ m2

                      S minus8λ2τ m2

                      τ

                      radicm4

                      S minus4m2Sm2

                      τ)16πm3S

                      Br(S rarr χχ) = (2λ2χm2

                      S

                      radicm4

                      S minus4m2Sm2

                      χ)32πm3S

                      (B1)

                      Where

                      mS is the mass of the scalar mediator

                      mχ is the mass of the Dark Matter particle

                      mb is the mass of the b quark

                      mt is the mass of the t quark

                      mτ is the mass of the τ lepton

                      The coupling constants λ follow the same pattern

                      • TITLE Collider Constraints applied to Simplified Models of Dark Matter fitted to the Fermi-LAT gamma ray excess using Bayesian Techniques
                        • Dedication
                        • Declaration
                        • Acknowledgements
                        • Contents
                        • List of Figures
                        • List of Tables
                          • Chapter 1 Introduction
                          • Chapter 2 Review of Physics
                          • Chapter 3 Fitting Models to the Observables
                          • Chapter 4 Calculation Tools
                          • Chapter 5 Majorana Model Results
                          • Chapter 6 Real Scalar Model Results
                          • Chapter 7 Real Vector Dark Matter Results
                          • Chapter 8 Conclusion
                          • Bibliography
                          • Appendix A Validation of Calculation Tools
                          • Appendix B Branching ratio calculations for narrow width approximation

                        2

                        of substantial theoretical and experimental effort the microscopic properties of dark matterparticles are still unknown

                        One of the main problems with extracting these properties is the plethora of competingtheoretical models that provide dark matter candidates Another is the very weak interactionof these particles and the lack of experimental observables allowing one to discriminatebetween the competing models

                        A particularly fruitful approach is the simplified model approach which makes minimalassumptions about the DM candidates as extensions to the Standard Model interacting weaklywith ordinary matter through a mediator These can be coded as Lagrangian interaction termsusing tools such as FeynRules [4] and Calchep [5] and used to simulate Dark Matter relicdensities indirect detection expectations as well as direct detection expectations throughMicrOmegas [6] This in turn can be run under a Bayesian inference tool (see below)

                        Using a Bayesian inference tool such as MultiNest [7][8][9] one can scan the parameterspace of a particular model to find regions of highest probability for that model based onobservations of a range of phenomena including direct detection experiments such as ZEPLIN[10] XENON [11] DEAP [12] ARDM [13] DarkSide [14] PandaX [15] and LUX [16]indirect detection - eg Fermi-LAT gamma ray excess [17] Pamela-positron and antiprotonflux [18] and production at colliders MultiNest calculates the evidence value which is theaverage likelihood of the model weighted by prior probability and a posterior distributionwhich is the distribution of the parameter(s) after taking into account the observed databy scanning over a range of parameters based on prior probabilities This can be used tocompare models on a consistent basis The scans also provide information about the best fitparameters for the overall fit as well as for the seperate experimentsobservations

                        The simplified model approach has been taken in a number of papers to study the Fermi-LAT gamma ray excess (see literature review in section 12) and one in particular comparesthese models using a Bayesian inference tool which found that a Majorana fermion DMcandidate interacting via a scalar mediator with arbitrary couplings to third generationfermions showed the best fit [19] A subsequent paper [20] examined the most likelyparameter combinations that simultaneously fit the Planck relic density Fermi-LAT galacticcentre gamma ray AMS-02 positron flux and Pamela antiproton flux data

                        There has not as yet been a thorough study of the collider constraints on DM simplifiedmodels that fit the Fermi-LAT galactic centre excess In this thesis we reproduce the

                        11 Motivation 3

                        previous results of [20] using similar astrophysical constraints and then implement detailedsimulations of collider constraints on DM and mediator production

                        In Section 11 we review the motivation for this study and in Section 12 we considerrecent astronomical data which is consistent with the annihilation of DM particles andreview the literature and searches which attempt to explain it with a range of extensions tothe standard model

                        In Chapter 2 we review the Standard Model of particle physics (SM) the evidence fordark matter and the ATLAS and CMS experiments at the LHC

                        In Chapter 3 we describe the models considered in this paper the observables used inthe Bayesian scans and the collider experiments which provide limits against which theMagraph simulations of the three models can be compared

                        In Chapter 4 we review the calculation tools that have been used in this paper

                        In Chapters 5 6 and 7 we give the results of the Bayesian scans and collider cutsanalyses for each of the models and in Chapter 8 we summarise our conclusions

                        11 Motivation

                        The Fermi gamma ray space telescope has over the past few years detected a gamma raysignal from the inner few degrees around the galactic centre corresponding to a regionseveral hundred parsecs in radius The spectrum and angular distribution are compatible withthat from annihilating dark matter particles

                        A number of papers have recently attempted to explain the Gamma Ray excess in termsof extensions to the Standard Model taking into account constraints from ATLAS and CMScollider experiments direct detection of dark matter colliding off nucleii in well shieldeddetectors (eg LUX) as well as other indirect measurements (astrophysical observations)These papers take the approach of simplified models which describe the contact interactionsof a mediator beween DM and ordinary matter using an effective field theory approach(EFT) Until recently it has been difficult to discriminate between the results of these paperswhich look at different models and produce different statistical results A recent paper[19] Simplified Dark Matter Models Confront the Gamma Ray Excess used a Bayesianinference tool to compare a variety of simplified models using the Bayesian evidence values

                        12 Literature review 4

                        calculated for each model to find the most preferred explanation The motivation for thechosen models is discussed in section 121

                        A subsequent paper [20] Interpreting the Fermi-LAT gamma ray excess in the simplifiedframework studied the most likely model again using Bayesian inference to find the mostlikely parameters for the (most likely) model The data inputs (observables) to the scanwere the Planck measurement of the dark matter relic density [21] Fermi-LAT gamma rayflux data from the galactic center [17] cosmic positron flux data [22] cosmic anti-protonflux data [18] cosmic microwave background data and direct detection data from the LUXexperiment [16] The result of the scan was a posterior probability density function in theparameter space of the model

                        This paper extends the findings in [19] by adding collider constraints to the likelihoodestimations for the three models It is not a simple matter to add collider constraints becauseeach point in parameter space demands a significant amount of computing to calculate theexpected excess amount of observable particles corresponding to the particular experimentgiven the proposed simplified model A sampling method has been employed to calculatethe collider constraints using the probability of the parameters based on the astrophysicalobservables

                        12 Literature review

                        121 Simplified Models

                        A recent paper [23] summarised the requirements and properties that simplified models forLHC dark matter searches should have

                        The general principles are

                        bull Besides the Standard Model (SM) the Lagrangian should contain a DM candidate thatis absolutely stable or lives long enough to escape the LHC detector and a mediatorthat couples the two sectors

                        bull The Lagrangian should contain all terms that are renormalizable consistent withLorentz invariance SM symmetries and DM stability

                        bull Additional interactions should not violate exact and approximate global symmetries ofthe SM

                        12 Literature review 5

                        The examples of models that satisfy these requirements are

                        1 Scalar (or pseudo-scalar) s-channel mediator coupled to fermionic dark matter (Diracor Majorana)

                        2 Higgs portal DM where the DM is a scalar singlet or fermion singlet under the gaugesymmetries of the SM coupling to a scalar boson which mixes with the Higgs (Thefermion singlet is a specific realisation of 1 above)

                        3 DM is a mixture of an electroweak singlet and doublet as in the Minimal Supersym-metric SM (MSSM)

                        4 Vector (or Axial vector) s-channel mediator obtained by extending the SM gaugesymmetry by a new U(1)rsquo which is spontaneously broken such that the mediatoracquires mass

                        5 t-channel flavoured mediator (if the DM particle is a fermion χ the mediator can bea coloured scalar or vector particle φ -eg assuming a scalar mediator a couplingof the form φ χq requires either χ or φ to carry a flavor index to be consistent withminimal flavour violation (MFV) which postulates the flavour changing neutral currentstructure in the SM is preserved

                        Another recent paper [24] summarised the requirements and properties that simplifiedmodels for LHC dark matter searches should have The models studied in this paper arepossible candidates competing with Dirac fermion and complex scalar dark matter andmodels with vector and pseudo-scalar mediators Other models would have an elasticscattering cross section that will remain beyond the reach of direct detection experiments dueto the neutrino floor (the irreducible background produced by neutrino scattering) or arealready close to sensitivity to existing LUX and XENONIT experiments

                        A recent paper [25] Scalar Simplified Models for Dark Matter looks at Dirac fermiondark matter mediated by either a new scalar or pseudo-scalar to SM fermions This paperplaces bounds on the coupling combination gχgφ (coupling of DM to mediator times couplingof mediator to SM fermions) calculated from direct detection indirect detection and relicdensity by the mass of DM and mass of mediator Bounds are also calculated for colliderconstraints (mono-jet searches heavy-flavour searches and collider bounds on the 125 GeVHiggs are used to place limits on the monojet and heavy flavor channels which can betranslated into limits on the Higgs coupling to dark matter and experimental measurements

                        12 Literature review 6

                        of the Higgs width are used to constrain the addition of new channels to Higgs decay) Thepaper models the top loop in a full theory using the Monte Carlo generator MCFM [26]and extends the process to accommodate off-shell mediator production and decay to a darkmatter pair While this paper does not consider the Majorana fermion dark matter model theconstraints on the coupling product may have some relevence to the present study if only asan order of magnitude indicator

                        Constraining dark sectors with monojets and dijets [27] studies a simplified model withDirac fermion dark matter and an axial-vector mediator They use two classes of constraint-searches for DM production in events with large amounts of missing transverse energy inassociation with SM particles (eg monojet events) and direct constraints on the mediatorfrom monojet searches to show that for example where the choice of couplings of themediator to quarks and DM gA

                        q =gAχ=1 all mediator masses in the range 130 GeV ltMR lt3

                        TeV are excluded

                        The paper Constraining Dark Sectors at Colliders Beyond the Effective Theory Ap-proach [28] considers Higgs portal models with scalar pseudo-scalar vector and axial-vectormediators between the SM and the dark sector The paper takes account of loop processesand gluon fusion as well as quark-mediator interactions using MCFM Apart from mediatortype the models are characterised by free parameters of mediator width and mass darkmatter mass and the effective coupling combination gχgφ (coupling of DM to mediator timescoupling of mediator to SM fermions) The conclusion is that LHC searches can proberegions of the parameter space that are not able to be explored in direct detection experiments(since the latter measurements suffer from a velocity suppression of the WIMP-nucleoncross-section)

                        The paper Constraining the Fermi-LAT excess with multi-jet plus EmissT collider searches

                        [29] shows that models with a pseudo-scalar or scalar mediator are best constrained by multi-jet final states with Emiss

                        T The analysis was done using POWHEG-BOX-V2 at LO and usesM2

                        T variables together with multi-jet binning to exploit differences in signal and backgroundin the different regions of phase space to obtain a more stringent set of constraints on DMproduction This paper largely motivated the current work in that it showed that collidersearches that were not traditionally thought of as dark matter searches were neverthelessoften sensitive to dark matter produced in association with heavy quarks The combinationof searches used in this paper reflects this fact

                        12 Literature review 7

                        122 Collider Constraints

                        In order to constrain this model with collider constraints (direct detection limits) we considerthe following papers by the CMS and ATLAS collaborations which describe searches at theLHC

                        ATLAS Experiments

                        bull Search for dark matter in events with heavy quarks and missing transverse momentumin p p collisions with the ATLAS detector [30]

                        bull Search for direct third-generation squark pair production in final states with missingtransverse momentum and two b-jets in

                        radics= 8 TeV pp collisions with the ATLAS

                        detector[31]

                        bull Search for direct pair production of the top squark in all-hadronic final states inprotonndashproton collisions at

                        radics=8 TeV with the ATLAS detector [32]

                        bull Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

                        radic(s)=8TeV pp collisions using 21 f bminus1 of

                        ATLAS data [33]

                        bull Search for direct top squark pair production in final states with two leptons inradic

                        s = 8TeV pp collisions using 20 f bminus1 of ATLAS data [34]

                        bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

                        radics=8 TeV [35]

                        CMS Experiments

                        bull Searches for anomalous tt production in p p collisions atradic

                        s=8 TeV [36]

                        bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theDi-lepton Final State in pp collisions at

                        radics=8 TeV [37]

                        bull Search for the production of dark matter in association with top-quark pairs in thesingle-lepton final state in proton-proton collisions at

                        radics = 8 TeV [38]

                        bull Search for new physics in monojet events in p p collisions atradic

                        s = 8 TeV(CMS) [39]

                        bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

                        s = 8 TeV [40]

                        12 Literature review 8

                        bull Search for the Production of Dark Matter in Association with Top Quark Pairs in theSingle-lepton Final State in pp collisions at

                        radics=8 TeV [41]

                        bull Search for new phenomena in monophoton final states in proton-proton collisions atradic

                        s=8 TeV [42]

                        bull Search for neutral MSSM Higgs bosons decaying into a pair of bottom quarks [43]

                        bull Search for neutral MSSM Higgs bosons decaying into a pair of τ leptons in p p collision[44]

                        bull Search for dark matter extra dimensions and unparticles in monojet events in proton-proton collisions at

                        radics=8 TeV [45]

                        In particular we have implemented Madgraph simulations of certain data analysesapplying to collider experiments described in papers [32][33][34][31][30][41] Theremaining models were not implemented in Madgraph because of inability to validate themodels against published results in the time available We developed C++ analysis code toimplement the collider cuts after showering with Pythia and basic analysis with Delpheswith output to root files The C++ code allowed multiple experimental cuts to be applied tothe same simulation output The C++ code was first checked against the results of the variousexperiments above and then the simulations were rerun using the three modelsrsquo Lagrangians(implemented in FeynRules) described in this paper

                        Chapter 2

                        Review of Physics

                        21 Standard Model

                        211 Introduction

                        The Standard Model of particle physics (SM) has sometimes been described as the theoryof Almost Everything The SM is a field theory which emerged in the second half of thetwentieth century from quantum mechanics and special relativity which themselves wereonly conceived after 1905 by Einsten Pauli Bohr Dirac and others In the first decades ofthe twenty first century the discovery of the Higgs boson [46][47] completed the pictureand confirmed the last previously unverified part of the SM The SM is a self consistenttheory that describes and classifies all known particles and their interactions (other thangravity) to an astonishing level of accuracy and detail However the theory is incomplete-it does not describe the complete theory of gravitation or account for DM (see below) ordark energy (the accelerating expansion of the universe) Related to these shortcomings aretheoretical deficiencies such as the asymmetry of matter and anti-matter the fact that the SMis inconsistent with general relativity (to the extent that one or both theories break down atspacetime singularities like the Big Bang and black hole event horizons)

                        212 Quantum Mechanics

                        Quantum Mechanics (QM) describes particles as probability waves which can be thoughtof as packets of energy which can be localised or spread through all space depending on how

                        21 Standard Model 10

                        accurately one measures the particlersquos momentum However QM is not itself consistent withspecial relativity It was realised too that because of relativity particles could be created anddestroyed and because of the uncertainty principle the theory should describe the probabilityof particles being created and destroyed on arbitrarily short timescales A quantum fieldtheory (QFT) is a way to describe this probability at every point of space and time

                        213 Field Theory

                        A field theory is described by a mathematical function of space and time called a LagrangianAs in classical mechanics where Lagrangians were introduced by Louis Lagrange at thebeginning of the nineteenth century to summarise dynamical systems the Lagrangian sum-marises the dynamics of space which is permeated by force fields The equations of motionarising from a Lagrangian are determined by finding the contours of the Lagrangian describedby stationary points (ie where the infinitesimal change in the Lagrangian is zero for a smallchange in field) Particles are generally described by singularities in the Lagrangian (wherethe values tend to infinity also called resonances)

                        214 Spin and Statistics

                        It was also realised that different types of particle needed to be represented by different fieldsin the Lagrangian Diracrsquos discovery that particles with spin 12 quantum numbers couldbe represented by a field obtained by effectively taking the square root of the relativisticequation (Klein Gordon equation) was probably the most significant advance in quantumfield theory It required the introduction of a matrix operators called the gamma matricesor Dirac matrices which helped explain the already known fact that spin half particles orfermions obey Fermi-Dirac statistics ie when two spin half particles are exchanged thesign of the wavefunction must change and hence they cannot occupy the same quantum stateunlike bosons with integer spin that can occupy the same quantum state

                        Other particles need to be represented by fields called vector pseudo-vector scalar (as inthe Klein Gordon equation) pseudo-scalar and complex fields A vector field is really fourfields one for each dimension of space plus time and is parameterised by the four gammamatrices γmicro which transform among themselves in a relativistically invariant way (Lorentzinvariance) A pseudo-vector field is a vector field which transforms in the same way but

                        21 Standard Model 11

                        with additional sign change under parity transformations (a simultaneous reversal of the threespatial directions) and is represented by γmicroγ5 A pseudo-scalar is a scalar which changes signunder parity transformations and a complex scalar field is represented by complex numbers

                        215 Feynman Diagrams

                        QFT took another giant leap forward when Richard Feynman realised that the propagationof particles described by the field could be described by expanding the so called S-matrixor scattering matrix derived from calculating the probability of initial states transitioning tofinal states The expansion is similar to a Taylor expansion in mathematics allowing for thespecial commuting properties of elements of the field The Feynman diagrams representingindividual terms in the expansion can be interpreted as interactions between the particlecomponents The S-Matrix (or more correctly the non trivial part called the T-Matrix) isderived using the interaction Hamiltonian of the field The Hamiltonian is an operator whoseeigenvalues or characteristic values give the energy quanta of the field The T-matrix canalso be described by a path integral which is an integral over all of quantum space weightedby the exponential of the Lagrangian This is completely equivalent to the Hamiltonianformulation and shows how the Lagrangian arises naturally out of the Hamiltonian (in factthe Lagrangian is the Legendre transformation of the Hamiltonian)

                        Figure 21 Feynman Diagram of electron interacting with a muon

                        γ

                        eminus

                        e+

                        micro+

                        microminus

                        The diagram Fig21 is a tree diagram (diagrams with closed loops indicate self in-teractions and are higher order terms in the expansion) and shows an electron (eminus) and ananti-muon (micro+) interacting by exchanging a photon

                        21 Standard Model 12

                        216 Gauge Symmetries and Quantum Electrodynamics (QED)

                        The Dirac equation with its gamma matrices ushered in the use of group theory to describethe transformation of fields (because the multiplication table of the gamma matrices isclosed and described by group theory) It also ushered in the idea of symmetries within theLagrangian- for example consider a Lagrangian

                        ψ(ipart minusm)ψ (21)

                        The Lagrangian is invariant under a global U(1) transformation which acts on the fieldsand hence the derivatives as follows

                        ψ rarr eiqαψ ψ rarr ψeminusiqα partmicroψ rarr eiqα

                        partmicroψ (22)

                        where qα is a global phase and α is a continuous parameter

                        A symmetry leads by Noetherrsquos theorem to a conserved current jmicro = qψγmicroψ and is thebasis of charge conservation since partmicro jmicro = 0 and q =

                        intd3x j0(x)

                        By promoting this transformation to a local one (ie α rarr α(x) where x = xmicro are thespace time co-ordinates) the transformations become

                        ψ rarr eiqα(x)ψ ψ rarr ψeminusiqα(x)partmicroψ rarr eiqα(x)(partmicroψ + iq(partmicroα(x))ψ) (23)

                        The derivative transformation introduces an extra term which would spoil the gaugeinvariance This can be solved by introducing a new field the photon field Amicro(x) whichinteracts with the fermion field ψ and transforms under a U(1) transformation

                        Amicro rarr Amicro minuspartmicroα(x) (24)

                        If we replace normal derivatives partmicro with covariant derivatives Dmicro = (partmicro + iqAmicro(x)) in theLagrangian then the extra term is cancelled and we can now see that the covariant derivativetransforms like ψ - under a local U(1) transform ie

                        Dmicro rarr Dmicroψprime = eiqα(x)Dmicroψ (25)

                        The additional term iqAmicro(x) can be interpreted in a geometric sense as a connectionwhich facilitates the parallel transport of vectors along a curve in a parallel consistent manner

                        21 Standard Model 13

                        We must add a gauge-invariant kinetic term to the Lagrangian to describe the dynamicsof the photon field This is given by the electromagnetic field strength tensor

                        Fmicroν = partmicroAν minuspartνAmicro (26)

                        The full Quantum Electrodynamics (QED) Lagrangian is then for electrons with mass mand charge -e

                        LQED = ψ(i Dminusm)ψ minus 14

                        Fmicroν(X)Fmicroν(x) (27)

                        This Lagrangian is Lorentz and U(1) guage invariant The interaction term is

                        Lint =+eψ Aψ = eψγmicro

                        ψAmicro = jmicro

                        EMAmicro (28)

                        where jmicro

                        EM is the electromagnetic four current

                        217 The Standard Electroweak Model

                        The Standard Electroweak (EW) Model was created by Glashow-Weinberg-Salam (also calledthe GWS model) and unifies the electomagnetic and weak interactions of the fundamentalparticles

                        The idea of gauge symmetry was extended to the idea of a non-abelian gauge symmetry(essentially the operators defining the symmetry do not commute and so are represented bymatrices- in this case the Pauli matrices for the SU(2) group) - the gauge group is the directproduct G = SU(2)

                        otimesU(1) It was known that weak interactions were mediated by Wplusmn

                        and Z0 particles where the superscripts are the signs of the electromagnetic charge Thesewere fairly massive but it was also known that through spontaneous symmetry breaking ofa continuous global non-abelian symmetry there would be one massive propagation modefor each spontaneously broken-symmetry generator and massless modes for the unbrokengenerators (the so-called Goldstone bosons)

                        This required an extra field to represent the gauge bosons called Bmicro (a vector fieldhaving the 4 spacetime degrees of freedom) With this identification the covariant derivativebecomes

                        Dmicro = partmicro minus igAmicro τ

                        2minus i

                        gprime

                        2Y Bmicro (29)

                        21 Standard Model 14

                        Here Y is the so called weak hypercharge of the field Bmicro which is a quantum numberdepending on particle (lepton or quark) g and gprime are two independent parameters in theGWS theory and the expression Amicro τ implies a sum over 3 components Aa

                        micro a=123 and thePauli matrices τa

                        This isnrsquot the full story because it had been known since 1956 when Chin Shuing Wuand her collaborators confirmed in an experiment prompted by Lee and Yang that nature isnot ambivalent to so-called parity It had been assumed that the laws of nature would notchange if all spatial coordinates were reversed (a parity transformation) The experimentinvolved the beta decay of cooled 60C atoms (more electrons were emitted in the backwardhemisphere with respect to spin direction of the atoms) This emission involved the weakinteraction and required that the Lagrangian for weak interactions contain terms which aremaximally parity violating- which turn out to be currents of the form

                        ψ(1minus γ5)γmicro

                        ψ (210)

                        The term

                        12(1minus γ

                        5) (211)

                        projects out a left-handed spinor and can be absorbed into the spinor to describe left-handedleptons (and as later discovered quarks) The right handed leptons are blind to the weakinteraction

                        The processes describing left-handed current interactions are shown in Fig 22

                        Analogously to isospin where the proton and neutron are considered two eigenstates ofthe nucleon we postulate that Bmicro is the gauge boson of an SU(2)transformation coupled toweak SU(2) doublets (

                        νe

                        eminus

                        )

                        (ud

                        ) (212)

                        We may now write the weak SU(2) currents as eg

                        jimicro = (ν e)Lγmicro

                        τ i

                        2

                        e

                        )L (213)

                        21 Standard Model 15

                        Figure 22 Weak Interaction Vertices [48]

                        where τ i are the Pauli-spin matrices and the subscript L denotes the left-handed componentof the spinor projected out by Eqn211 The third current corresponding to i=3 is called theneutral current as it doesnrsquot change the charge of the particle involved in the interaction

                        We could postulate that the electromagnetic current is a linear combination of the leftand right elecromagnetic fields eL = 1

                        2(1minus γ5)e and eR = 12(1+ γ5)e

                        jemmicro = eLγmicroQeL + eRγmicroQeR (214)

                        where Q is the electromagnetic operator and eR is a singlet However this is not invariantunder SU(2)L To construct an invariant current we need to use the SU(2)L doublet 212The current which is an SU(2)L invariant U(1) current is

                        jYmicro = (ν e)LγmicroYL

                        e

                        )L+ eRγmicroYReR (215)

                        where hypercharges YL and YR are associated with the U(1)Y symmetry jYmicro is a linearcombination of the electromagnetic current and the weak neutral current thus explaining whythe hypercharge differs for the left and right-handed components since the electromagneticcurrent has the same charge for both components whilst the weak current exists only forleft-handed particles j3

                        micro and the third component of weak isospin T 3 allows us to calculate

                        21 Standard Model 16

                        the relationship Y = 2(QminusT 3) which can be seen by writing jYmicro as a linear combination ofj3micro (the weak neutral current that doesnrsquot change the charge of the particle involved in the

                        interaction) and jemmicro (using the convention that the jYmicro is multiplied by 1

                        2 to match the samefactor implicit in j3

                        micro ) Substituting

                        τ3 =

                        (1 00 minus1

                        )(216)

                        into equation 213 for i=3 and subtracting equation 215 and equating to 12 times equation

                        214

                        we get

                        eLγmicroQeL + eRγmicroQeR minus (νLγmicro

                        12

                        νL minus eLγmicro

                        12

                        eL) =12

                        eRγmicroYReR +12(ν e)LγmicroYL

                        e

                        )L (217)

                        from which we can read out

                        YR = 2QYL = 2Q+1 (218)

                        and T3(eR) = 0 T3(νL) =12 and T3(eL) =

                        12 The latter three identities are implied by

                        the fraction 12 inserted into the definition of equation 213

                        The Lagrangian kinetic terms of the fermions can then be written

                        L =minus14

                        FmicroνFmicroν minus 14

                        GmicroνGmicroν

                        + sumgenerations

                        LL(i D)LL + lR(i D)lR + νR(i D)νR

                        + sumgenerations

                        QL(i D)QL +UR(i D)UR + DR(i D)DR

                        (219)

                        LL are the left-handed lepton doublets QL are the left-handed quark doublets lRνR theright handed lepton singlets URDR the right handed quark singlets

                        The field strength tensors are given by

                        Fmicroν = partmicroAν minuspartνAmicro minus ig[Amicro Aν ] (220)

                        21 Standard Model 17

                        andGmicroν = partmicroBν minuspartνBmicro (221)

                        Bmicro is an abelian U(1) gauge field while Amicro is non-abelian and Fmicroν contains the com-mutator of the field through the action of the covariant derivative [Dmicro Dν ] This equationcontains no mass terms terms for the field Amicro for good reason- this would violate gaugeinvariance We know that the gauge bosons of the weak force have substantial mass and theway that this problem has been solved is through the Higgs mechanism

                        218 Higgs Mechanism

                        To solve the problem of getting gauge invariant mass terms three independent groupspublished papers in 1964 The mechanism was named after Peter Higgs To introduce theconcept consider a complex scalar field φ that permeates all space Consider a potential forthe Higgs field of the form

                        minusmicro2φ

                        daggerφ +λ (φ dagger

                        φ2) (222)

                        which has the form of a mexican hat Fig 23 This potential is still U(1) gauge invariant andcan be added to our Lagrangian together with kinetic+field strength terms to give

                        L = (Dmicroφ)dagger(Dmicroφ)minusmicro2φ

                        daggerφ +λ (φ dagger

                        φ2)minus 1

                        4FmicroνFmicroν (223)

                        It is easily seen that this is invariant to the transformations

                        Amicro rarr Amicro minuspartmicroη(x) (224)

                        φ(x)rarr eieη(x)φ(x) (225)

                        The minimum of this field is not at 0 but in a circular ring around 0 with a vacuum

                        expectation value(vev)radic

                        micro2

                        2λequiv vradic

                        2

                        We can parameterise φ as v+h(x)radic2

                        ei π

                        Fπ where h and π are referred to as the Higgs Bosonand Goldstone Boson respectively and are real scalar fields with no vevs Fπ is a constant

                        21 Standard Model 18

                        Figure 23 Higgs Potential [49]

                        Substituting this back into the Lagrangian 223 we get

                        minus14

                        FmicroνFmicroν minusevAmicropartmicro

                        π+e2v2

                        2AmicroAmicro +

                        12(partmicrohpart

                        microhminus2micro2h2)+

                        12

                        partmicroπpartmicro

                        π+(hπinteractions)(226)

                        This Lagrangian now seems to describe a theory with a photon Amicro of mass = ev a Higgsboson h with mass =

                        radic2micro and a massless Goldstone π

                        However there is a term mixing Amicro and part microπ in this expression and this shows Amicro and π

                        are not independent normal coordinates and we can not conclude that the third term is a massterm This π minusAmicro mixing can be removed by making a gauge transformation in equations224 and 225 making

                        φrarrv+h(x)radic2

                        ei π

                        Fπminusieη(x) (227)

                        and setting πrarr π

                        Fπminus eη(x) = 0 This choosing of a point on the circle for the vev is called

                        spontaneous symmetry breaking much like a vertical pencil falling and pointing horizontallyin the plane This choice of gauge is called unitary gauge The Goldstone π will thencompletely disappear from the theory and one says that the Goldstone has been eaten to givethe photon mass

                        21 Standard Model 19

                        This can be confirmed by counting degrees of freedom before and after spontaneoussymmetry breaking The original Lagrangian had four degrees of freedom- two from thereal massless vector field Amicro (massless photons have only two independent polarisationstates) and two from the complex scalar field (one each from the h and π fields) The newLagrangian has one degree of freedom from the h field and 3 degrees of freedom from themassive real vector field Amicro and the π field has disappeared

                        The actual Higgs field is postulated to be a complex scalar doublet with four degrees offreedom

                        Φ =

                        (φ+

                        φ0

                        )(228)

                        which transforms under SU(2) in the fundamental representation and also transforms underU(1) hypercharge with quantum numbers

                        Table 21 Quantum numbers of the Higgs field

                        T 3 Q Yφ+

                        12 1 1

                        φ0 minus12 1 0

                        We can parameterise the Higgs field in terms of deviations from the vacuum

                        Φ(x) =(

                        η1(x)+ iη2(x)v+σ(x)+ iη3(x)

                        ) (229)

                        It can be shown that the three η fields can be transformed away by working in unitarygauge By analogy with the scalar Higgs field example above the classical action will beminimised for a constant value of the Higgs field Φ = Φ0 for which Φ

                        dagger0Φ0 = v2 This again

                        defines a range of minima and we can pick one arbitarily (this is equivalent to applying anSU(2)timesU(1) transform to the Higgs doublet transforming the top component to 0 and thebottom component to Φ0 = v) which corresponds to a vacuum of zero charge density andlt φ0 gt=+v

                        In this gauge we can write the Higgs doublet as

                        Φ =

                        (φ+

                        φ0

                        )rarr M

                        (0

                        v+ H(x)radic2

                        ) (230)

                        where H(x) is a real Higgs field v is the vacuum expectation and M is an SU(2)timesU(1)matrix The matrix Mdagger converts the isospinor into a down form (an SU(2) rotation) followedby a U(1) rotation to a real number

                        21 Standard Model 20

                        If we consider the Higgs part of the Lagrangian

                        minus14(Fmicroν)

                        2 minus 14(Bmicroν)

                        2 +(DmicroΦ)dagger(DmicroΦ)minusλ (Φdagger

                        Φminus v2)2 (231)

                        Substituting from equation 230 into this and noting that

                        DmicroΦ = partmicroΦminus igW amicro τ

                        aΦminus 1

                        2ig

                        primeBmicroΦ (232)

                        We can express as

                        DmicroΦ = (partmicro minus i2

                        (gA3

                        micro +gprimeBmicro g(A1micro minusA2

                        micro)

                        g(A1micro +A2

                        micro) minusgA3micro +gprimeBmicro

                        ))Φ equiv (partmicro minus i

                        2Amicro)Φ (233)

                        After some calculation the kinetic term is

                        (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

                        14(v+

                        Hradic2)2[A 2]22 (234)

                        where the 22 subscript is the index in the matrix

                        If we defineWplusmn

                        micro =1radic2(A1

                        micro∓iA2micro) (235)

                        then [A 2]22 is given by

                        [A 2]22 =

                        (gprimeBmicro +gA3

                        micro

                        radic2gW+

                        microradic2gWminus

                        micro gprimeBmicro minusgA3micro

                        ) (236)

                        We can now substitute this expression for [A 2]22 into equation 234 and get

                        (DmicroΦ)dagger(DmicroΦ) =12(partH)2 +

                        14(v+

                        Hradic2)2(2g2Wminus

                        micro W+micro +(gprimeBmicro minusgA3micro)

                        2) (237)

                        This expression contains mass terms for three massive gauge bosons Wplusmn and the combi-nation of vector fields gprimeBmicro minusgA3

                        micro where note

                        21 Standard Model 21

                        Table 22 Weak Quantum numbers of Lepton and Quarks

                        T 3 Q YνL

                        12 0 -1

                        lminusL minus12 -1 -1

                        νR 0 0 0lminusR 0 -1 -2UL

                        12

                        23

                        13

                        DL minus12 minus1

                        313

                        UR 0 23

                        43

                        DR 0 minus13 minus2

                        3

                        Wminusmicro = (W+

                        micro )dagger equivW 1micro minus iW 2

                        micro (238)

                        Then the mass terms can be written

                        12

                        v2g2|Wmicro |2 +14

                        v2(gprimeBmicro minusgA3micro)

                        2 (239)

                        W+micro and W microminus can be associated with the W boson and its anti-particle and (minusgprimeBmicro +

                        gA3micro) with the Z Boson (after normalisation by

                        radicg2 +(gprime

                        )2) The combination gprimeA3micro +gBmicro

                        is orthogonal to Z and remains massless This can be associated with the Photon (afternormalisation) These predict mass terms for the W and Z bosons mW = vgradic

                        2and mZ =

                        vradic2

                        radicg2 +(gprime

                        )2It is again instructive to count the degrees of freedom before and after the Higgs mech-

                        anism Before we had a complex doublet Φ with four degrees of freedom one masslessB with two degrees of freedom and three massless W a fields each with two each givinga total of twelve degrees of freedom After spontaneous symmetry breaking we have onereal scalar Higgs field with one degree of freedom three massive weak bosons with nineand one massless photon with two again giving twelve degrees of freedom One says thescalar degrees of freedom have been eaten to give the Wplusmn and Z bosons their longitudinalcomponents (ie mass)

                        Table 22 shows T 3 the weak isospin quantum number (which describes how a particletransforms under SU(2)) Y the weak hypercharge (which describes how the particle trans-

                        21 Standard Model 22

                        forms under U(1)) and Q the electric charge of leptons and quarks The former two enterdirectly into the Lagrangian while the latter can be derived as a function of the former two

                        Strictly speaking the right handed neutrino which has no charge may not exist and is notpart of the standard model although some theories postulate a mass for this (hypothetical)particle

                        Writing gauge invariant couplings between ELeRφ with the vacuum expectation ofφ = v (Yukawa matrices λ

                        i ju and λ

                        i jd respectively for the up and down quarks) we get mass

                        terms for the quarks (and similarly for the leptons)

                        Mass terms for quarks minussumi j[(λi jd Qi

                        Lφd jR)+λ

                        i ju εab(Qi

                        L)aφlowastb u j

                        R +hc]

                        Mass terms for leptonsminussumi j[(λi jl Li

                        Lφ l jR)+λ

                        i jν εab(Li

                        L)aφlowastb ν

                        jR +hc]

                        Here hc is the hermitian conjugate of the first terms L and Q are lepton and quarkdoublets l and ν are the lepton singlet and corresponding neutrino d and u are singlet downand up quarks εab is the Levi-Civita (anti-symmetric) tensor

                        If we write the quark generations in a basis that diagonalises the Higgs couplings weget a unitary transformations Uu and Ud for the up and down quarks respectively which iscompounded in the boson current (Udagger

                        u Ud) called the CKM matrix where off-diagonal termsallow mixing between quark generations (this is also called the mass basis as opposed tothe flavour basis) A similar argument can give masses to right handed sterile neutrinos butthese can never be measured However a lepton analogue of the CKM matrix the PNMSmatrix allows left-handed neutrinos to mix between various generations and oscillate betweenflavours as they propagate through space allowing a calculation of mass differences

                        219 Quantum Chromodynamics

                        The final plank in SM theory tackling the strong force Quantum Chromodynamics wasproposed by Gell-Mann and Zweig in 1963 Three generations of quarks were proposedwith charges of 23 for uc and t and -13 for ds and b Initially SU(2) isospin was consideredtransforming u and d states but later irreducible representations of SU(3) were found to fit theexperimental data The need for an additional quantum number was motivated by the fact thatthe lightest excited state of the nucleon consisted of sss (the ∆++) To reconcile the baryon

                        21 Standard Model 23

                        spectrum with spin statistics colour was proposed Quarks transform under the fundamentalor 3 representation of colour symmetry and anti-quarks under 3 The inner product of 3and 3 is an invariant under SU(3) as is a totally anti-symmetric combination ε i jkqiq jqk (andthe anti-quark equivalent) It was discovered though deep inelastic scattering that quarksshowed asymptotic freedom (the binding force grew as the seperation grew) Non-abeliangauge theories showed this same property and colour was identified with the gauge quantumnumbers of the quarks The quanta of the gauge field are called gluons Because the forcesshow asymptotic freedom - ie the interaction becomes weaker at high energies and shorterdistances perturbation theory can be used (ie Feynman diagrams) at these energies but forlonger distances and lower energies lattice simulations must be done

                        2110 Full SM Lagrangian

                        The full SM can be written

                        L =minus14

                        BmicroνBmicroν minus 18

                        tr(FmicroνFmicroν)minus 12

                        tr(GmicroνGmicroν)

                        + sumgenerations

                        (ν eL)σmicro iDmicro

                        (νL

                        eL

                        )+ eRσ

                        micro iDmicroeR + νRσmicro iDmicroνR +hc

                        + sumgenerations

                        (u dL)σmicro iDmicro

                        (uL

                        dL

                        )+ uRσ

                        micro iDmicrouR + dRσmicro iDmicrodR +hc

                        minussumi j[(λ

                        i jl Li

                        Lφ l jR)+λ

                        i jν ε

                        ab(LiL)aφ

                        lowastb ν

                        jR +hc]

                        minussumi j[(λ

                        i jd Qi

                        Lφd jR)+λ

                        i ju ε

                        ab(QiL)aφ

                        lowastb u j

                        R +hc]

                        + (Dmicroφ)dagger(Dmicroφ)minusmh[φφ minus v22]22v2

                        (240)

                        where σ micro are the extended Pauli matrices

                        (1 00 1

                        )

                        (0 11 0

                        )

                        (0 minusii 0

                        )

                        (1 00 minus1

                        )

                        The first line gives the gauge field strength terms the second the lepton dynamical termsthe third the quark dynamical terms the fourth the lepton (including neutrino) mass termsthe fifth the quark mass terms and the sixth the Higgs terms

                        The SU(3) gauge field is given by Gmicroν and which provides the colour quantum numbers

                        21 Standard Model 24

                        Figure 24 Standard Model Particles and Forces [50]

                        Note The covariant derivatives operating on the left-handed leptons contain only theU(1) and SU(2) gauge fields Bmicro Fmicro whereas those operating on left-handed quarks containU(1)SU(2) and SU(3) gauge fields Bmicro Fmicro Gmicro The covariant derivatives operating onright handed electrons and quarks are as above but do not contain the SU(2) gauge fieldsRight handed neutrinos have no gauge fields

                        The sums over i j above are over the different generations of leptons and quarks

                        The particles and forces that emerge from the SM are shown in Fig 24

                        22 Dark Matter 25

                        22 Dark Matter

                        221 Evidence for the existence of dark matter

                        2211 Bullet Cluster of galaxies

                        Most of the mass of the cluster as measured by a gravitational lensing map is concentratedaway from the centre of the object The visible matter (as measured by X-ray emission) ismostly present in the centre of the object The interpretation is that the two galaxy clustershave collided perpendicular to the line of sight with the mostly collisionless dark matterpassing straight through and the visible matter interacting and being trapped in the middleIt was estimated that the ratio of actual to visible matter was about 16 [51]

                        Figure 25 Bullet Cluster [52]

                        2212 Coma Cluster

                        The first evidence for dark matter came in the 1930s when Fritz Zwicky observed the Comacluster of galaxies over 1000 galaxies gravitationally bound to one another He observed thevelocities of individual galaxies in the cluster and estimated the amount of matter based onthe average galaxy He calculated that many of the galaxies should be able to escape fromthe cluster unless the mass was about 10 times higher than estimated

                        22 Dark Matter 26

                        2213 Rotation Curves [53]

                        Since those early observations many spiral galaxies have been observed with the rotationcurves calculated by doppler shift with the result that in all cases the outermost stars arerotating at rates fast enough to escape their host galaxies Indeed the rotation curves arealmost flat indicating large amounts of dark matter rotating with the galaxies and holdingthe galaxies together

                        Figure 26 Galaxy Rotation Curves [54]

                        2214 WIMPS MACHOS

                        The question is if the particles that make up most of the mass of the universe are not baryonswhat are they It is clear that the matter does not interact significantly with radiation so mustbe electrically neutral It has also not lost much of its kinetic energy sufficiently to relax intothe disks of galaxies like ordinary matter Massive particles may survive to the present ifthey carry a conserved additive or multiplicative quantum number This is so even if equalnumbers of particles and anti-particles can annihilate if the number densities become too lowto reduce the density further These particles have been dubbed Weakly interacting MassiveParticles (WIMPS) as they interact with ordinary matter only through a coupling strengthtypically encountered in the weak interaction They are predicted to have frozen out ofthe hot plasma and would give the correct relic density based solely on the weak interactioncross section (sometimes called the WIMP Miracle) Another possibility called MassiveCompact Halo Objects (MACHOS)- are brown dwarf stars Jupiter mass objects or remnantsfrom an earlier generation of star formation

                        22 Dark Matter 27

                        2215 MACHO Collaboration [55]

                        In order to estimate the mass of MACHOS a number of experiments have been conductedusing microlensing which makes use of the fact that if a massive object lies in the line ofsight of a much more distant star the light from the more distant star will be lensed and forma ring around the star It is extemely unlikely that the MACHO will pass directly through theline of sight but if there is a near miss the two images will be seperated by a small angleThe signature for microlensing is a time-symmetric brightening of a star as the MACHOpasses near the line of sight called a light curve One can obtain statistical information aboutthe MACHO masses from the shape and duration of the light curves and a large number ofstars (many million) must be monitored The MACHO Collaboration [55] have monitoredover 11 million stars and less than 20 events were detected The conclusion was that thiscould account for a small addition (perhaps up to 20 ) to the visible mass but not enoughto account for Dark Matter

                        2216 Big Bang Nucleosynthesis (BBN) [56]

                        Another source of evidence for DM comes from the theory of Nucleosynthesis in the fewseconds after the Big Bang In the first few minutes after the big bang the temperatures wereso high that all matter was dissociated but after about 3 minutes the temperature rapidlycooled to about 109 Kelvin Nucleosynthesis could begin when protons and neutrons couldstick together to form Deuterium (Heavy Hydrogen) and isotopes of Helium The ratio ofDeuterium to Hydrogen and Helium is very much a function of the baryon density and alsothe rate of expansion of the universe at the time of Nucleosynthesis There is a small windowwhen the density of Baryons is sufficient to account for the ratios of the various elementsThe rate of expansion of the universe is also a function of the total matter density The ratiosof these lighter elements have been calculated by spectral measurements with Helium some25 of the mass of the universe Deuterium 001 The theory of BBN accounts for theratios of all of the lighter elements if the ratio of Baryon mass to total mass is a few percentsuggesting that the missing matter is some exotic non-baryonic form

                        22 Dark Matter 28

                        2217 Cosmic Microwave Background [57]

                        The discovery of a uniform Microwave Background radiation by Penzias and Wilson in1965 was confirmation of the theory first proposed by Gamow and collaborators that theuniverse should be filled with black body radiation an afterglow of the big bang TheCOBE satelite launched in 1989 confirmed this black body radiation with temperature of2725+-002K with 95 confidence While extremely uniform it found tiny irregularitieswhich could be accounted for by irregularities in the density of the plasma at last scattering(gravitational redshifts or blueshifts called the Sachs Wolfe effect) and temperature andvelocity fluctuations in the plasma The anisotropy of the background has been furtheranalysed by WMAP (the Wilkinson Microwave Anisotropy Probe) launched in 2001 Thefluctuations in the density of the plasma are what account for the seeds of galaxy formationHowever detailed calculations have found that the fluctuations were too small to account forthe observed galaxy distribution without an electrically neutral gravitating matter in additionto ordinary matter that only became electrically neutral at the time of recombination [58]

                        In May 2009 the European Space Agency (ESA) launched the Planck sattelite a spacebased observatory to observe the universe at wavelengths between 03mm and 111 mmbroadly covering the far-infrared microwave and high frequency domains The missionrsquosmain goal was to study the cosmic microwave background across the whole sky at greatersensitivity and resolution than ever before Using the information gathered and the standardmodel of cosmology ESA was able to calculate cosmological parameters including the relicdensity of dark matter remaining after the Big Bang to a high degree of precision withinthe framework of the standard model of cosmology The standard model of cosmology canbe described by a relatively small number of parameters including the density of ordinarymatter dark matter and dark energy the speed of cosmic expansion at the present epoch (alsoknown as the Hubble constant) the geometry of the Universe and the relative amount ofthe primordial fluctuations embedded during inflation on different scales and their amplitude[59]

                        2218 LUX Experiment - Large Underground Xenon experiment [16]

                        The LUX experiment is conducted at the Sanford Underground research facility one mileunderground to minimise the background of cosmic rays The detector is a time projectionchamber allowing a 3D positioning of interactions occurring within its active volume LUX

                        22 Dark Matter 29

                        Figure 27 WMAP Cosmic Microwave Background Fluctuations [58]

                        Figure 28 Dark Matter Interactions [60]

                        uses as its target 368 kiliograms of liquefied ultra pure xenon which is a scintillator Interac-tions inside the xenon will create an amount of light proportional to the energy depositedThat light can be collected on arrays of light detectors sensitive to a single photon

                        Xenon is very pure and the amount of radiation originating within the target is limitedAlso being three times denser than water it can stop radiation originating from outside thedetector The 3D detector focuses on interactions near the center which is a quiet regionwhere rare dark matter interactions might be detected [16]

                        22 Dark Matter 30

                        222 Searches for dark matter

                        2221 Dark Matter Detection

                        Fig 28 illustrates the reactions studied under the different forms of experiment designed todetect DM

                        Direct detection experiments are mainly underground experiments with huge detectorsattempting to detect the recoil of dark matter off ordinary matter such as LUX [16] CDMS-II [61] XENON100 [11] COUPP [62] PICASSO [63] ZEPLIN [10] These involveinteractions between DM and SM particles with final states involving DM and SM particles

                        Indirect detection involves the search for the annihilation of DM particles in distantastrophysical sources where the DM content is expected to be large Example facilitiesinclude space based experiments (EGRET [64] PAMELA [18] FERMI-LAT [65]) balloonflights (ATIC [66] PEBS [67] PPB-BETS [68]) or atmospheric Cherenkov telescopes(HESS [69]MAGIC [70]) and neutrino telescopes (SuperKamiokande [71] IceCube [72]ANTARES [73] KM3NET [74]) Indirect Detection involves DM particles in the initial stateand SM particles in the final state

                        Collider detection involves SM particles in the initial state creating DM particles in thefinal state which can be detected by missing transverse energy and explained by variousmodels extending the SM

                        223 Possible signals of dark matter

                        224 Gamma Ray Excess at the Centre of the Galaxy [65]

                        The center of the galaxy is the most dense and active region and it is no surprise that itproduces intense gamma ray sources When astronomers subtract all known gamma raysources there remains an excess which extends out some 5000 light years from the centerHooper et al find that the annihilation of dark matter particles with a mass between 31 and40 Gev fits the excess to a remarkable degree based on Gamma Ray spectrum distributionand brightness

                        23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 31

                        Figure 29 Gamma Ray Excess from the Milky Way Center [75]

                        23 Background on ATLAS and CMS Experiments at theLarge Hadron collider (LHC)

                        The collider constraints in this paper are based on two of the experiments that are part of theLHC in CERN

                        Figure 210 ATLAS Experiment

                        The LHC accelerates proton beams in opposite directions around a 27 km ring of super-conducting magnets until they are made to collide at an interaction point inside one of the 4experiments placed in the ring An upgrade to the LHC has just been performed providing65 TeV per beam

                        23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 32

                        231 ATLAS Experiment

                        The ATLAS detector consists of concentric cylinders around the interaction point whichmeasure different aspects of the particles that are flung out of the collision point There are 4major parts the Inner Detector the Calorimeters the Muon spectrometer and the magnetsystems

                        2311 Inner Detector

                        The inner detector extends from a few centimeters to 12 meters from the interaction point andmeasures details of the charged particles and their momentum interacting with the materialIf particles seem to originate from a point other than the interaction point this may indicatethat the particle comes from the decay of a hadron with a bottom quark This is used to b-tagcertain particles

                        The inner detector consists of a pixel detector with three concentric layers and threeendcaps containing detecting silicon modules 250 microm thick The pixel detector has over 80million readout channels about 50 percent of the total channels for the experiment All thecomponents are radiation hardened to cope with the extremely high radiation in the proximityof the interaction point

                        The next layer is the semi-conductor layer (SCT) similar in concept to the pixel detectorbut with long narrow strips instead of pixels to cover a larger area It has 4 double layers ofsilicon strips with 63 million readout channels

                        The outermost component of the inner detector is the Transition Radiation Tracker It is acombination of a straw tracker and a transition radiation detector The detecting elementsare drift tubes (straws) each 4mm in diameter and up to 144 cm long Each straw is filledwith gas which becomes ionised when the radiation passes through The straws are held at-1500V with a thin wire down the center This drives negatively ionised atoms to the centercausing a pulse of electricity in the wire This causes a pattern of hit straws allowing the pathof the particles to be determined Between the straws materials with widely varying indicesof refraction cause transition radiation when particles pass through leaving much strongersignals in some straws Transition radiation is strongest for highly relativistic particles (thelightest particles- electrons and positrons) allowing strong signals to be associated with theseparticles

                        23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 33

                        2312 Calorimeters

                        The inner detector is surrounded by a solenoidal magnet of 2 Tesla (there is also an outertoroidal magnet) but surrounding the inner magnet are the calorimeters to measure theenergy of particles by absorbing them There is an inner electromagnetic calorimeter whichmeasures the angle between the detectorrsquos beam axis and the particle (or more preciselypseudo-rapidity) and its angle in the perpendicular plane to a high degree of accuracy Theabsorbing material is lead and stainless steel and the sampling material liquid argon Thehadron calorimeter measures the energy of particles that interact via the strong force Theenergy absorbing material is scintillating steel

                        2313 Muon Specrometer

                        The muon spectrometer surrounds the calorimeters and has a set of 1200 chambers tomeasure the tracks of outgoing muons with high precision using a magnetic field andtriggering chambers Very few particles of other types pass through the calorimeters and aresubsequently measured

                        2314 Magnets

                        The magnets are in two layers the inner solenoidal and outer toroidal which together causecharged particles to curve in the magetic field allowing their momentum to be determinedThe outer toroidal magnet consists of 8 very large air core superconducting magnets whichsupply the field in the muon spectrometer

                        232 CMS Experiment

                        The CMS Detector is similar in design to the ATLAS Detector with a single solenoidalmagnet which is the largest of its type ever built (length 13m and diameter 7m) and allowsthe tracker and calorimeter detectors to be placed inside the coil in a compact way Thestrong magnetic field gives a high resolution for measuring muons electrons and photons ahigh quality central tracking system and a hermetic hadron calorimeter designed to preventhadrons from escaping

                        23 Background on ATLAS and CMS Experiments at the Large Hadron collider (LHC) 34

                        Figure 211 CMS Experiment

                        Chapter 3

                        Fitting Models to the Observables

                        31 Simplified Models Considered

                        In order to extend the findings of previous studies [19] to include collider constraints wefirst performed Bayesian scans [7] over three simplified models real scalar dark matter φ Majorana fermion χ and real vector Xmicro The purpose was to obtain the most probable regionsof each model then to use the resulting posterior samples to investigate whether the LHCexperiments have any sensitivity to the preferred regions

                        The three models couple to the mediator with interactions shown in the following table

                        Table 31 Simplified Models

                        Hypothesis real scalar DM Majorana fermion DM real vector DM

                        DM mediator int Lφ sup microφ mφ φ 2S Lχ sup iλχ

                        2 χγ5χS LX sup microX mX2 X microXmicroS

                        The interactions between the mediator and the standard fermions is assumed to be

                        LS sup f f S (31)

                        and in line with minimal flavour violation we only consider third generation fermions- ief = b tτ

                        For the purposes of these scans we consider the following observables

                        32 Observables 36

                        32 Observables

                        321 Dark Matter Abundance

                        We have assumed a single dark matter candidate which freezes out in the early universe witha central value as determined by Planck [21] of

                        ΩDMh2 = 01199plusmn 0031 (32)

                        h is the reduced hubble constant

                        The experimental uncertainty of the background relic density [59] is dominated by atheoretical uncertainty of 50 of the value of the scanned Ωh2 [76] We have added thetheoretical and the experimental uncertainties in quadrature

                        SD =radic(05Ωh2)2 + 00312 (33)

                        This gives a log likelihood of

                        minus05lowast (Ωh2 minus 1199)2

                        SD2 minus log(radic

                        2πSD) (34)

                        322 Gamma Rays from the Galactic Center

                        Assuming that the gamma ray flux excess from the galactic centre observed by Fermi-LAT iscaused by self annihilation of dark matter particles the differential flux

                        d2Φ

                        dEdΩ=

                        lt σv gt8πmχ

                        2 J(ψ)sumf

                        B fdN f

                        γ

                        dE(35)

                        has been calculated using both Micromegas v425 with a Navarro-Frenk-White (NFW) darkmatter galactic distribution function

                        ρ(r) = ρ0(rrs)

                        minusγ

                        (1+ rrs)3minusγ (36)

                        with γ = 126 and an angle of 5 to the galactic centre [19]

                        32 Observables 37

                        Here Φ is the flux dE is the differential energy lt σv gt the velocity averaged annihi-lation cross section of the dark matter across the galactic centre mχ the mass of the darkmatter particle B f =lt σv gt f σv is the annihilation fraction into the f f final state anddN f

                        γ dE is the energy distribution of photons produced in the annihilation channel with thefinal state f f The J factor is described below

                        The angle ψ to the galactic centre determines the normalisation of the gamma ray fluxthrough the factor

                        J(ψ) =int

                        losρ(l2 + r⊙2 minus2lr⊙cosψ)dl (37)

                        where ρ(r) is the dark matter density at distance r from the centre of the galaxy r⊙=85 kpcis the solar distance from the Galactic center ρ0 is set to reproduce local dark matter densityof 03GeVcm3

                        The central values for the gamma ray spectrum were taken from Fig 5 of [77] and thestandard deviations of individual flux points from the error bars on the graph In commonwith [78] we ignore possible segment to segment correlations

                        For the Fermi-LAT log likelihood we assumed each point was distributed with indepen-dent Gaussian likelihood with the experimental error for each data point given by the errorbar presented in Fig5 of [76] for each point We also assumed an equal theoretical errorfor each point which when added to the experimental error in quadrature gave an effectivelog likelihood for each point of minus05lowast (giminusdi)

                        2

                        2lowastσ2i

                        where gi are the calculated values and di theexperimental values and σi the experimental errors

                        323 Direct Detection - LUX

                        The LUX experiment [16] is the most constraining direct search experiment to date for awide range of WIMP models We have used LUXCalc [79] (modified to handle momentumdependent nucleon cross sections) to calculate the likelihood of the expected number ofsignal events for the Majorana fermion model

                        The likelihood function is taken as the Poisson distribution

                        L(mχ σ | N) = P(N | mχ σ) =(b+micro)Neminus(b+micro)

                        N (38)

                        32 Observables 38

                        where b= expected background of 064 and micro = micro(mχσ ) is the expected number of signalevents calculated for a given model

                        micro = MTint

                        infin

                        0dEφ(E)

                        dRdE

                        (E) (39)

                        where MT is the detector mass times time exposure and φ(E) is a global efficiency factorincorporating trigger efficiencies energy resolution and analysis cuts

                        The differential recoil rate of dark matter on nucleii as a function of recoil energy E

                        dRdE

                        =ρX

                        mχmA

                        intdvv f (v)

                        dσASI

                        dER (310)

                        where mA is the nucleon mass f (v) is the dark matter velocity distribution and

                        dσSIA

                        dER= Gχ(q2)

                        4micro2A

                        Emaxπ[Z f χ

                        p +(AminusZ) f χn ]

                        2F2A (q) (311)

                        where Emax = 2micro2Av2mA Gχ(q2) = q2

                        4m2χ

                        [24] is an adjustment for a momentum dependantcross section which applies only to the Majorana model

                        f χ

                        N =λχ

                        2m2SgSNN assuming that the relic density is the central value of 1199 We have

                        implemented a scaling of the direct search cross section in our code by multiplying gSNN by( Ω

                        Ω0)05 as a convenient method of allowing for the fact that the actual relic density enters

                        into the calculation of the cross section as a square

                        FA(q) is the nucleus form factor and

                        microA =mχmA

                        (mχ +mA)(312)

                        is the reduced WIMP-nucleon mass

                        The strength of the dark matter nucleon interaction between dark matter particles andthird generation quarks is

                        gSNN =2

                        27mN fT G sum

                        f=bt

                        λ f

                        m f (313)

                        where fT G = 1minus f NTuminus f N

                        Tdminus fTs and f N

                        Tu= 02 f N

                        Td= 026 fTs = 043 [20]

                        33 Calculations 39

                        For the real scalar and real vector models where the spin independent cross sections arenot momentum dependent we based the likelihood on the LUX 95 CL from [16] using acomplementary error function Li(d|mχ I) = 05lowastCEr f (dminusx(mχ )

                        σ) where x is the LUX limit

                        and d the calculated SI cross section We assumed a LUX error σ of 01lowastd lowastradic

                        2

                        33 Calculations

                        We coded the Lagrangians for the three models in Feynrules [4] Calculation of the expectedobservables (dark matter relic density Fermi-LAT gamma ray spectrum and LUX nucleonscattering amplitudes) were performed using Micromegas v425

                        331 Mediator Decay

                        A number of searches have previously been performed at the LHC which show sensitivity tothe decay of scalar mediators in the form of Higgs bosons (in beyond the standard modeltheories) to bottom quarks top quarks and τ leptons (all third generation fermions) Two inparticular [43 44] provide limits on the cross-sections for the production of the scalar timesthe branching ratios to brsquos and τrsquos In order to check these limits against each of the modelswe ran parameter scans in Madgraph at leading order over the top 900 points from the nestedsampling output from MultiNest [19]

                        The two processes were

                        1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to apair of τ leptons in pp collisions [44]

                        bull generate p p gt b b S where S is the scalar mediator

                        The process measures the decay of a scalar in the presence of b-jets that decay to τ+τminus

                        leptons Fig31 shows the main diagrams contributing to the process The leftmost diagramis a loop process and the centre and right are tree level however we only got Madgraph togenerated the middle diagram since we are interested in τ leptons and b quarks in the finalstate

                        The limit given in [44] only applies in the case of narrow resonances where narrow istaken to mean that the width of the resonance is less than 10 of its mass To test the validity

                        33 Calculations 40

                        Figure 31 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof τ leptons

                        of applying these limits to our models we show in Figures 32 to 34 the percentage widthof the resonance vs mS[GeV ] (pole mass of the scalar mediator) λb and λt (the couplingconstants to the b and t quarks) The probabilities are the posterior probabilities obtainedfrom MultiNest scans of the Majorana model varying the coupling constants and scalarmediator and calculating the decay width of the mediator The results are represented as 2dposterior probability heat maps in the central panels and 1d marginalised distributions on thex and y axes of each graph

                        16 18 20 22 24 26 28 30

                        log10(mS[GeV])

                        001

                        002

                        003

                        004

                        005

                        Widthm

                        S

                        00

                        04

                        08

                        12

                        0 100 200

                        Posterior Probability

                        Figure 32 WidthmS vs mS

                        The WidthmS was lt 01 at almost all points except where the mass of the scalar wasgreater than 1000 GeV λb was greater than 1 or λτ was greater than 1

                        This can be seen from the graphs in Figs 323334

                        33 Calculations 41

                        4 3 2 1 0

                        λb

                        001

                        002

                        003

                        004

                        005

                        WidthmS

                        000

                        015

                        030

                        045

                        0 100 200

                        Posterior Probability

                        Figure 33 WidthmS vs λb

                        5 4 3 2 1 0

                        λτ

                        001

                        002

                        003

                        004

                        005

                        WidthmS

                        000

                        015

                        030

                        045

                        0 100 200

                        Posterior Probability

                        Figure 34 WidthmS vs λτ

                        The decay widths are calculated in Madgraph at leading order and branching fractionscan be calculated to b t τ and χ (DM) The formulae for the calculations can be found indecayspy which is a subroutine of Madgraph which we have documented in Appendix BB1 Using these we calculated the branching ratio to τ+τminus so that the cross section timesbranching ratio can be compared to the limits in [80] shown in Chapters 56 and 7 in Figs5564 and 74 for each model

                        2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                        This scan calculates the contributions of b-quark associated Higgs boson production inthe center and right diagrams of Fig 35 reproduced from [43] The scans did not includePythia and Delphes as only production cross sections were required and the producedparticles were in the final state and required no matching or merging

                        The Madgraph processes were

                        bull generate p p gt b S where S is the scalar mediator

                        bull add process p p gt b S j

                        bull add process p p gt b S

                        33 Calculations 42

                        Figure 35 Main Feyman diagrams leading to the cross section for scalar decaying to a pairof b quarks in the presence of at least one b quark

                        bull add process p p gt b S j

                        The cross section times branching ratios to bb for mediator production in association withat least one b quark is shown for each model in Chapters 56 and 7 in Figs 5765 and 75

                        332 Collider Cuts Analyses

                        We generated two parameter scans with parameter combinations from the Multinest outputwhich were the basis of the plots of Fig 6 in [19] The plots in Fig 6 were marginalisedto two parameters each but the raw Multinest output contained 5 parameters plus the priorprobabilities of each combination normalised to total 10

                        The first scan was for χ produced with b quarks In order to force the mediator to decayto χ only we had χ in the final state and defined jets to include b quarks in Madgraph Thenwe generated the processes

                        bull generate p p gt χ χ j

                        bull add process p p gt χ χ j j

                        Jet matching was on

                        The second scan was for t quarks produced in the final state

                        bull generate p p gt χ χ tt

                        33 Calculations 43

                        No jet matching was required because we did not mix final state jet multiplicities at thematrix element level

                        The outputs from these two processes were normalised to 21 f bminus1 and combined

                        The results of these scans prior to collider cuts calculations are shown in Figs 5161and 71 These show the posterior probabilities of the various parameter combinations withdifferent constraints applied seperately and then together A more detailed commentary isgiven in each of the sections

                        We then sampled 900 points based on posterior probability from these scans and ran thesethrough Madgraph v425 then applied collider cuts with the C++ code using the limits fromthe following six ATLAS and CMS searches shown in Table32 and described below to findany exclusions

                        333 Description of Collider Cuts Analyses

                        In the following all masses and energies are in GeV and angles in radians unless specificallystated

                        3331 Lepstop0

                        Search for direct pair production of the top squark in all-hadronic final states in pro-tonndashproton collisions at

                        radics=8 TeV with the ATLAS detector[32]

                        This analysis searched for direct pair production of the supersymmetric partner to thetop quark using integrated luminosity of 201 f bminus1 The top squark was assumed to de-cay via t rarr t χ0

                        1 or t rarr bχ01 or t rarr bχ

                        plusmn1 rarr bW (lowast)χ1

                        0 where χ01 (χ

                        plusmn1 ) denotes the lightest

                        neutralino(chargino) in supersymmetric models The search targets a fully hadronic finalstate with four or more jets and large missing momentum The production of a mediator inassociation with top quarks followed by an invisible mediator decay to dark matter particleshas a similar final state to that considered in the ATLAS stop searches Thus we expect theATLAS reach to have sensitivity to the three dark matter models considered in this thesis

                        The collider cuts program mimicked the object reconstruction through jetlepton overlapremoval and selected signal electrons muons and τ particles with pT gt 10 GeV and

                        33 Calculations 44

                        Table 32 95 CL by Signal Region

                        Experiment Region Number

                        Lepstop0

                        SRA1 66SRA2 57SRA3 67SRA4 65SRC1 157SRC2 124SRC3 80

                        Lepstop1

                        SRtN1-1 857SRtN1-2 498SRtN1-3 389SRtN2 107SRtN3 85SRbC1 832SRbC2 195SRbC3 76

                        Lepstop2

                        L90 740L100 56L110 90L120 170

                        2bstop

                        SRA 857SRB 270SRA150 380SRA200 260SRA250 90SRA300 75SRA350 52

                        CMSTopDM1L SRA 1385

                        ATLASMonobjetSR1 1240SR2 790

                        33 Calculations 45

                        |η | lt 24724 and 247 respectively (for eminusmicrominusτ) Jets were selected with pT gt 20 GeVand |η |lt 45 b-jets were selected if they were tagged by random selection lt 70 and had|η |lt 25 and pT gt 20 GeV

                        These jets and leptons were further managed for overlap removal within ∆R lt 04 withnon b-jets removed if ∆R lt 02

                        The b-jets and non b-jets were further selected as signal jets if pT gt 35 GeV and |η |lt 28

                        These signal jets electrons and muons were finally binned according to the signal regionsin Tables 333435 below

                        Table 33 Selection criteria common to all signal regions

                        Trigger EmissT

                        Nlep 0b-tagged jets ⩾ 2

                        EmissT 150 GeV

                        |∆φ( jet pmissT )| gtπ5

                        mbminT gt175 GeV

                        Table 34 Selection criteria for signal regions A

                        SRA1 SRA2 SRA3 SRA4anti-kt R=04 jets ⩾ 6 pT gt 808035353535 GeV

                        m0b j j lt 225 GeV [50250] GeV

                        m1b j j lt 225 GeV [50400] GeV

                        min( jet i pmissT ) - gt50 GeV

                        τ veto yesEmiss

                        T gt150 GeV gt250 GeV gt300 GeV gt 350 GeV

                        Table 35 Selection criteria for signal regions C

                        SRC1 SRC2 SRC3anti-kt R=04 jets ⩾ 5 pT gt 8080353535 GeV

                        |∆φ(bb)| gt02 π

                        mbminT gt185 GeV gt200 GeV gt200 GeV

                        mbmaxT gt205 GeV gt290 GeV gt325 GeV

                        τ veto yesEmiss

                        T gt160 GeV gt160 GeV gt215 GeV

                        wherembmin

                        T =radic

                        2pbt Emiss

                        T [1minus cos∆φ(pbT pmiss

                        T )]gt 175 (314)

                        33 Calculations 46

                        and pbT is calculated for the b-tagged jet with the smallest ∆φ to the pmiss

                        T direction andmbmax

                        T is calculated with the pbT for the b-tagged jet with the largest ∆φ to the pmiss

                        T direction

                        m0b j j is obtained by first selecting the two jets with the highest b-tagging weight- from

                        the remaining jets the two jets closest in the η minusφ plane are combined to form a W bosoncandidate this candidate is then combined with the closest of the 2 b-tagged jets in the η minusφ

                        plane to form the first top candidate with mass m0b j j A second W boson candidate is formed

                        by repeating the procedure with the remaining jets this candidate is combined with thesecond of the selected b-tagged jets to form the second top candidate with mass m1

                        b j j Thecuts shown in Appendix1 Table A1 show the cumulative effect of the individual cuts appliedto the data the signal region cuts and the cumulative effects of individual cuts in SignalRegion C

                        3332 Lepstop1

                        Search for direct top squark pair production in final states with one isolated leptonjets and missing transverse momentum in

                        radics=8 TeV pp collisions using 21 f bminus1 of

                        ATLAS data[33]

                        The stop decay modes considered were those to a top quark and the lightest supersym-metric particle (LSP) as well as a bottom quark and the lightest chargino where the charginodecays to the LSP by emitting a W boson We should expect the three dark matter models inthis paper with couplings to t and b quarks to show some visibility in this search

                        The collider cuts program mimicked the object construction by selecting electrons andmuons with pT gt 10 GeV and |η |lt 247 and 24 respectively and jets with pT gt 20 GeVand |η |lt 10 b-jets with |η |lt 25 and pT gt 25 GeV were selected with random efficiency75 and mistag rate 2 (where non b-jets were mis-classified) We then removed any jetwithin ∆R = 02 of an electron and non overlapping jets with |η |lt 28 were used to removeany electrons and muons within ∆R = 04 of these while non-overlapping jets with |η |lt 25and pT gt 25 GeV were classified as signal jets Non overlapping electrons and muons withpT gt 25 GeV were classified as signal electrons and muons

                        Signal selection defined six regions to optimize sensitivity for different stop and LSPmasses as well as sensitivity to two considered stop decay scenarios Three signal regionslabelled SRbC 1-3 where bC is a moniker for b+Chargino and 3 regions labelled SRtN 1-3where tN is a moniker for t+Neutralino were optimised for the scenarios where t1 decaysto the respective particles indicated by the labels Increasing label numbers correspond toincreasingly strict selection criteria

                        33 Calculations 47

                        The dominant background in all signal regions arises from dileptonic tt events All threeSRtN selectons impose a requirement on the 3-jet mass m j j j of the hadronically decayingtop quark to specifically reject the tt background where both W bosons from the top quarkdecay leponically

                        For increasing stop mass and increasing difference between the stop and the LSP therequirents are tightened on Emiss

                        T on the ratio EmissT

                        radicHT where HT is the scalar sum of the

                        momenta of the four selected jets and also tightened on mT

                        To further reduce the dileptonic tt background for an event characterised by two one stepdecay chains a and b each producing a missing particle C use is made of the variable

                        mT 2 =min

                        pCTa + pC

                        T b = pmissT

                        [max(mTamtb)] (315)

                        where mTi is the transverse mass 1 of branch i for a given hypothetical allocation (pCTa pC

                        T b)

                        of the missing particle momenta Eqn 315 is extended for more general decay chains byamT 2 [81] which is an asymmetric form of mT 2 in which the missing particle is the W bosonin the branch for the lost lepton and the neutrino is the missing particle for the branch withthe observed charged lepton

                        ∆φ( jet12 pmissT ) the azimuthal angle between leading or subleading jets and missing mo-

                        mentum is used to suppress the background from mostly multijet events with mis-measuredEmiss

                        T

                        Events with four or more jets are selected where the jets must satisfy |η | lt 25 and pT gt80604025 GeV respectively

                        These signal particles were then classified according to the cutflows analysis in Fig36below and compared with the paper in Appendix1 Table A2

                        3333 Lepstop2

                        Search for direct top squark pair production in final states with two leptons in p pcollisions at

                        radics=8TeV with the ATLAS detector[34]

                        Three different analysis strategies are used to search for t1 pair production Two targett1 rarr b+ χ

                        plusmn1 decay and the three body t1 rarr bW χ0

                        1 decay via an off-shell top quark whilst

                        1The transverse mass is defined as m2T = 2plep

                        T EmissT (1minus cos(∆φ)) of branch i where ∆φ is the azimuthal

                        angle between the lepton and the missing transverse momentum

                        33 Calculations 48

                        Table 36 Signal Regions - Lepstop1

                        Requirement SRtN2 SRtN3 SRbC1 SRbC2 SRbC3∆φ( jet1 pmiss

                        t )gt - 08 08 08 08∆φ( jet2 pmiss

                        T )gt 08 08 08 08 08Emiss

                        T [GeV ]gt 200 275 150 160 160Emiss

                        T radic

                        HT [GeV12 ]gt 13 11 7 8 8

                        mT [GeV ]gt 140 200 120 120 120me f f [GeV ]gt - - - 550 700amT 2[GeV ]gt 170 175 - 175 200mT

                        T 2[GeV ]gt - 80 - - -m j j j Yes Yes - - -

                        one targets t1 rarr t + χ01 to an on shell quark decay mode The analysis is arranged to look

                        at complementary mass splittings between χplusmn1 and χ0

                        1 smaller and larger than the W bosonmass The smaller splitting provides sensitivity to t1 three body decay

                        Object reconstruction was carried out by selecting baseline electronsmuons and τrsquos withpT gt 10 GeV and |η |lt 24724 and 247 respectively Baseline jets and b-jets were selectedwith pT gt 20 GeV and |η |lt 25 and b-jets with 70 efficiency Baseline jets were removedif within ∆R lt 02 of an electron and electrons and muons were removed if within ∆R lt 04of surviving jets Signal electrons required pT gt 10 GeV and signal jets pT gt 20 GeV and|η |lt 25

                        The analysis made use of the variable mT 2 [82] the stransverse mass used to mea-sure the masses of pair-produced semi-invisibly decaying heavy particles It is definedas mT 2(pT1 pT2qT ) =

                        minqT1+qT2=qT

                        max[mT (pT1qT1)mT (pT2qT2)] where mT indicatesthe transverse mass pT1 and pT2 are vectors and qT = qT1 + qT2 The minimisation isperformed over all possible decompositions of qT

                        Events were selected if they passed either a single-electron a single-muon a double-electron a double-muon or an electron-muon trigger Events are required to have exactlytwo opposite sign leptons and at least one of the leptons or muons must have a momentumgreater than 25 GeV and the invariant mass (rest mass) of the two leptons is required to begreater than 20 GeV In order to reduce the number of background events containing twoleptons produced by the on-shell decay of the Z boson the invariant mas must be outside711-111 GeV range

                        Two additional selections are applied to reduce the number of background events withhigh mT 2 arising from events with large Emiss

                        T due to mismeasured jets ∆φb lt 15 and∆φ gt 10 The quantity ∆φb is the azimuthal angle between pll

                        T b = pmissT + pl1

                        T +Pl2T The

                        33 Calculations 49

                        vector pllT b is the opposite of the vector sum of all the hadronic activity in the event For WW

                        and tt backgrounds it measures the transverse boost of the WW system and for the signalthe transverse boost of the chargino-chargino system The ∆φ variable is the azimuthal anglebetween the pmiss

                        T vector and the direction of the closest jet

                        By requiring mT 2 gt 40 GeV the tt background is the dominant background in both thesame-flavoured (SF) and different-flavoured (DF) events

                        Four signal regions are then defined as shown in Table 37 SR M90 has the loosestselection requiring MT 2gt 90 GeV and no additional requirement on jets and providessensitivity to scenarios with a small difference between the masses of the top-squark andchargino SR M110 and M120 have a loose selection on jets requiring two jets with pT

                        gt 20 GeV and mT 2 to be greater than 110 GeV and 120 GeV respectively They providesensitivity to scenarios with small to moderate difference between the invariant masses of thestop quark and the charginos SR M100 has a tighter jet slection and provides sensitivity toscenarios with a large mass difference between the both stop quark and the chargino and thestop quark and the neutralino

                        The analysis cut regions are summarised in Fig37 below

                        Table 37 Signal Regions Lepstop2

                        SR M90 M100 M110 M120pT leading lepton gt 25 GeV

                        ∆φ(pmissT closest jet) gt10

                        ∆φ(pmissT pll

                        T b) lt15mT 2 gt90 GeV gt100 GeV gt110 GeV gt120 GeV

                        pT leading jet no selection gt100 GeV gt20 GeV gt20 GeVpT second jet no selection gt50 GeV gt20 GeV gt20 GeV

                        To validate the code we ran the cuts shown in Appendix 1 Table A3 with the givenMadgraph simulation and compared to [83] It was not clear what baseline cuts had beenemployed in the ATLAS paper so our results were normalised to the 2 SF electrons beforecomparison Some of the ATLAS cut numbers were less than 3 and we did not produce anyevents passing these cuts but clearly the margin of error at this level was greater than thenumber of ATLAS events

                        33 Calculations 50

                        3334 2bstop

                        Search for direct third-generation squark pair production in final states with miss-ing transverse momentum and two b-jets in

                        radics= 8 TeV pp collisions with the ATLAS

                        detector[31]

                        Two sets of scenarios are considered In the first the coloured sbottom is the onlyproduction process decaying via b1 rarr bχ0

                        1 In the second the stop is the only productionprocess decaying via t1 rarr t χplusmn

                        1 and the lightest chargino (χplusmn1 ) decays via a virtual W boson

                        into the final state χ01 f f prime Both scenarios are characterised by the presence of two jets

                        resulting from the hadronisation of b quarks and large missing transverse momentum Theproduction of a mediator in association with top quarks followed by an invisible mediatordecay to dark matter particles has a similar final state as that considered in the ATLAS stopsearches Thus we expect the ATLAS reach to have sensitivity to the three dark mattermodels considered in this thesis

                        The collider cuts program mimicked object reconstruction by selecting baseline electronswith pT gt 7 GeV and |η |lt 247 and muons with pT gt 6 GeV and |η |lt 24 Baseline jetswere selected with pT gt 20 GeV and |η |lt 28 We then removed any jets within ∆R lt 02of an electron and any electron within ∆R lt 04 of surviving jets Signal electrons andmuons were required to have pT gt 25 GeV b-jets were selected with 60 tagging efficiencyand |η |lt 25 and pT gt 20 GeV

                        Two sets of signal regions are defined to provide sensitivity to the kinematic topologiesassociated with the different mass splittings between the sbottom or the stop and the neu-tralino In all cases the presence of at least one primary vertex (with at least five tracks withpT gt 04 GeV) is required Events are selected with Emiss

                        T gt 150 GeV and no electrons ormuons identified in the final state For signal region selections jets with |η | lt 28 are orderedaccording to their pT and 2 jets are required to be b-tagged

                        The variables are defined as follows

                        bull ∆φmin is the minimal azimuthal angle ∆φ between any of the three leading jets andpmiss

                        T

                        bull me f f (k) = sumki=1(p jet

                        T )i +EmissT where the index refers to the pT ordered list of jets

                        bull HT3 is defined as the scalar sum of the pT of the n leading pT ordered jets withoutincluding the 3 leading jets HT3 = sum

                        ni=4(p jet

                        T )i

                        bull mbb is the invariant mass of the two b-tagged jets in the event

                        33 Calculations 51

                        bull mCT is the contransverse mass [82] and is a kinematical variable that can be usedto measure the masses of pair produced semi-invisibly decaying particles For twoidentical decays of heavy particles into two invisible particles v1 and v2 and twoinvisible particles mCT is defined as m2

                        CT (v1v2) = [ET (v1)+ET (v2)]2 minus [pT (v1)minus

                        pT (v2)]2 where ET =

                        radicp2

                        T +m2 The the contransverse mass is invariant under equaland opposite boosts of the parent particles in the transverse plane and is bounded fromabove by an analytical expression of the particle masses For production of sbottompairs the bound is given by mmax

                        CT =m2(b)minusm2(χ0

                        1 )

                        m(b) and for tt events the bound is 135

                        GeV A similar equation can be written for stop pairs in term of the stop and charginomasses

                        A definition of the signal regions is given in the Table38

                        Table 38 Signal Regions 2bstop

                        Description SRA SRBEvent cleaning All signal regions

                        Lepton veto No emicro after overlap removal with pT gt7(6) GeV for e(micro)Emiss

                        T gt 150 GeV gt 250 GeVLeading jet pT ( j1) gt 130 GeV gt 150 GeVSecond jet pT ( j2) gt 50 GeV gt 30 GeVThird jet pT ( j3) veto if gt 50 GeV gt 30 GeV

                        ∆φ(pmissT j1) - gt 25

                        b-tagging leading 2 jets(pT gt 50 GeV |η | lt 25)

                        2nd- and 3rd- leading jets(pT gt 30 GeV|η | lt 25)

                        ∆φmin gt 04 gt 04Emiss

                        T me f f (k) EmissT me f f (2) gt 025 Emiss

                        T me f f (3) gt 04mCT gt 150200250300350 GeV -HT3 - lt 50 GeVmbb gt200 GeV -

                        The analysis cuts are summarised in Table A4 of Appendix 1

                        3335 ATLASMonobjet

                        Search for dark matter in events with heavy quarks and missing transverse momentumin pp collisions with the ATLAS detector[30]

                        Events were selected with high missing transverse momentum when produced in associa-tion with one or more jets identified as containing b quarks Final states with top quarks wereselected by requiring a high jet multiplicity and in some cases a single lepton The research

                        33 Calculations 52

                        studied effective contact operators in the form of a scalar operator Oscalar = sumqmqMN

                        lowastqqχχ

                        where MNlowast is the effective mass of the coupling and the quark and DM fields are denoted by

                        q and χ respectively These are the equivalent of the simplified models in this paper with themediator integrated out

                        Object reconstruction selected baseline electrons muons jets and τrsquos with pT gt 20 GeVand |η |lt 2525 and 45 respectively Signal jets were selected by removing any jets within∆R lt 02 of an electron and electrons and muons were selected by removing them if theywere within ∆R lt 04 of surviving jets b-jets were tagged with an efficiency of 60

                        Only signal regions SR1 and SR2 were analysed

                        The analysis split events into signal regions SR1 and SR2 according to the followingCriteria

                        Table 39 Signal Region ATLASmonobjet

                        Cut SR1 SR2Jet multiplicity n j 1minus2 3minus4

                        bminusjet multiplicity nb gt 0 (60 eff) gt 0 (60 eff)Lepton multiplicity nl 0 0

                        EmissT gt300 GeV gt200 GeV

                        Jet kinematics pb1T gt100 GeV pb1

                        T gt100 GeV p j2T gt100 (60) GeV

                        ∆φ( j1EmissT ) gt 10 i = 12 gt 10 i = 14

                        Where p jiT (pbi

                        T ) represent the transverse momentum of the i-th jet b-tagged jet respec-tively If there is a second b-tagged jet the pT must be greater than 60 GeV

                        3336 CMSTop1L

                        Search for top-squark pair production in the single-lepton final state in pp collisionsat

                        radics=8 TeV[41]

                        This search was for pair production of top squarks in events with a single isolatedelectron or muon jets large missing transverse momentum and large transverse mass

                        (MT =radic

                        2EmissT pl

                        T (1minus cos(∆φ))) where plT is the leptonrsquos transverse momentum and ∆(φ)

                        is the difference between the azimuthal angles of the lepton and EmissT The 3 models

                        considered should have some visibility at certain parameter combinations in this regionThe search focuses on two decay modes of the top squark leading to the two processespp rarr t tlowast rarr tt χ0

                        1 χ01 rarr bbW+Wminusχ0

                        1 χ01 and pp rarr t tlowast rarr bbχ

                        +1 χ

                        minus1 rarr bbW+Wminusχ0

                        1 χ01 The

                        33 Calculations 53

                        lightest neutralino χ01 is considered the lightest supersymmetric particle and escapes without

                        detection The analysis is based on events where one of the W bosons decays leptonicallyand the other hadronically resulting in one isolated lepton and 4 jets two of which originatefrom b quarks The two neutralinos and the neutrino from the W decay can result in largemissing transverse momentom The kinematical end point of MT is lt the mass of the Wboson However for signal events the presence of LSPs in the final state allows MT to exceedthe mass of the W boson The search is therefore for events with large MT The dominantbackground with large MT arises from tt decays where W bosons decay leptonically withone of the leptons not being identified

                        To reduce the dominant tt background use was made of the MWT 2 variable defined as

                        the minimum mother particle mass compatible with the transverse momenta and mass shellconstraints [84]

                        Object reconstruction required electrons and muons to have pT gt 30 and 25 GeV respand |η |lt 25 and 21 respectively Baseline jets were required to have pT gt 30 and |η |lt 4and b-jets were tagged with an efficiency of 75 for |η |lt 24

                        Events were chosen if they contained one isolated electron or muon at least 4 jets and atleast one b-tagged jet and Emiss

                        T gt 100 GeV MT gt 160 GeV MWT 2 was required to be greater

                        than 200 GeV and the smallest azimuthal angle difference between baseline jets and EmissT

                        gt12

                        Chapter 4

                        Calculation Tools

                        41 Summary

                        Figure 41 Calculation Tools

                        The simplified models were implemented in Feynrules [4] and the model files importedinto MicrOmegas v425 which was used to calculate the log-likelihoods of the astrophysicalobservables MicrOmegas was modified to allow only s-channel decays in both relic densityand gamma ray calculations Direct detection constraints were easy to implement for thescalar and vector models by coding the analytical expression for the spin independent WIMP-nucleon scattering cross section but due to the momentum dependence of the WIMP-nucleon

                        42 FeynRules 55

                        scattering cross section of the Majorana fermion model LUXCalc v101 [79] was modifiedfor this model to implement momentum dependent cross sections [85] We used MicrOmegasv425 to calculate relic density values and to produce gamma ray spectra for comparisonwith the Fermi-LAT results Parameter scans were carried out using MultiNest v310 withthe parameter ranges given in Table 51 and using astrophysical constraints only Oncethese scans had converged a random sample of 900 points was chosen (weighted by theposterior probability) and then used to calculate collider constraints using the MadgraphMonte Carlo generator This included comparison with mediator production and decay limitsfrom the LHC experiments and the reproduction of searches for dark matter plus other finalstate objects which required dedicated C++ code that reproduces the cut and count analysesperformed by CMS and ATLAS

                        42 FeynRules

                        FeynRules is a Mathematicareg package that allows the calculation of Feynman rules inmomentum space for any QFT physics model The user needs to provide FeynRules withminimal information to describe the new model contained in the model-file This informationis then used to calculate a set of Fenman rules associated with the Lagrangian The Feynmanrules calculated by the code can then be used to implement new physics models into otherexisting tools such as Monte Carlo (MC) generators via a set of interfaces developed andmaintained by the MC authors

                        Feynrules can be run with a base model = SM (or MSSM) and user defined extensions tothe base model

                        43 LUXCalc

                        LUXCalc is a utility program for calculating and deriving WIMP-nucleon coupling limitsfrom the recent results of the LUX direct search dark matter experiment The program hasbeen used to derive limits on both spin-dependent and spin-independent WIMP-nucleoncouplings and it was found under standard astrophysical assumptions and common otherassumptions that LUX excludes the entire spin-dependent parameter space consistent witha dark matter interpretation of DAMArsquos anomalous signal and also found good agreementwith published LUX results [79]

                        We modified the program for the Majorana fermion model to implement momentumdependent cross -sections [85]

                        44 Multinest 56

                        44 Multinest

                        Multinest is an efficient Bayesian Inference Tool which uses a simultaneous ellipsoidalnested sampling method a Monte Carlo technique aimed at an efficient Bayesian Evidencecalculation with posterior inferences as a by product

                        Bayes theorem states that

                        Pr(θ |DH) =Pr(D|θ H)Pr(θ |H)

                        Pr(D|H) (41)

                        Here D is the data and H is the Hypothesis Pr(θ |DH) is the posterior probability ofthe parameters Pr(D|θ H) = L(θ) is the likelihood Pr(θ |H) is the prior and Pr(D|H) isthe evidence

                        The evidence Pr(D|H) =int

                        Pr(θ |DH)Pr(θ |H)d(θ) =int

                        L(θ)Pr(θ |H)d(θ) Thus ifwe define the prior volume dX = Pr(θ |H)d(θ) and

                        X(λ ) =int

                        L(θ)gtλ

                        Pr(θ |H)d(θ) (42)

                        where the integral extends over the region of parameter space within the iso likelihoodcontour L(θ) = λ then the evidence integral can be written

                        int 10 L (X)dX where L (X) the

                        inverse of equation 42 and is a monotonically decreasing function of X (the prior volumewhich is 1 for the total sample space and 0 for a single point)

                        Thus if one can evaluate likelihoods L (Xi) where 0 lt Xn lt lt X2 lt X1 lt X0 = 1the evidence can be estimated as sumLiwi where wi = 5(Ximinus1 minusXi+1) Ellipsoidal samplingapproximates the iso likelihood contour by a D dimensional covariance matrix of the currentset of active points New points are selected from the prior within this ellipsoidal bounduntil one is obtained that has a likelihood greater than the likelihood of the removed lowestlikelihood point This method is extended for multimodal sampling where there may bemore than one local maximum by clustering points that are well separated and constructingindividual ellipsoidal bounds for each cluster For more details consult [7]

                        The results of the analysis are expressed as a data file containing the sample probability-2loglikelihood followed by the parameters of the model for the particular sampling Theprobabilities are normalised to total 10

                        45 Madgraph 57

                        45 Madgraph

                        Madgraph is a general purpose matrix element calculator and Monte Carlo generator Basedon user-supplied Feynman rules (eg the output of Feynrules) the package calculates the tree-level matrix elements for any processes that the user specifies and generates samples of LHCevents including an accurate simulation of the proton collisions themselves (which requires aknowledge of the parton distribution functions that determine the relative proportion of eachparton inside the nucleus) Hadronisation of final state quarks is not possible in Madgraphand hence the code must be used in conjunction with Pythia which performs a parton shower

                        The Madgraph output was automatically showered by Pythia (Pythia describes partonradiation as successive parton emissions using Markov chain techniques based on Sudakovform factors and this description is only formally correct for the soft and collinear emissions)A Madgraph simulation is at the level of the fundamental particles (quarks and leptons)Madgraph generates a limited set of matrix element processes at LO representing hard (highenergy) emissions and passes these to Pythia for showering The final state partons in theMadgraph simulation must be matched to the Pythia final state partons to avoid doublecounting

                        The output from Pythia is automatically fed through Delphes where basic collider cutscan be performed (calculation of the numbers of particles and jets that would be observed atdifferent energies and locations within the collider) The output of the Delphes simulation is afile containing reconstructed events that can be analysed using the standard ROOT framework(the most popular data analysis package in particle physics) [86]

                        The parameters that gave the best fits to the direct [16] and indirect DM experiments [87]and relic density [59] are ranked by posterior probability from the Multinest [7] output

                        The parameters in the Madgraph simulations are stored in files in the cards sub directoryand in particular the run card param card and Delphes card record items relevent to thesimulation

                        In the run card the number of events the Beam 1 and 2 total energy in GeV (4000 each)and the type of jet matching were modified (to MLM type matching where the final statepartons in Madgraph are clustered according to the KT jet algorithm to find the equivalentparton shower history of the event there is a minimum cutoff scale for the momentum pT

                        given by the xqcut parameter) The maxjetflavour (=5 to include b quarks) and xqcut = 30GeV parameters were also modified

                        46 Collider Cuts C++ Code 58

                        The param card was modified to specify the masses of the new particles and the autodecay modes The Delphes card is overwritten with either the ATLAS or CMS defaultcards which record tracking efficiencies and isolation momentum and energy resolutionscalorimeter details jet finder and b and τ -tagging parameters However the isolation andtagging in this study were done by the collider cuts C++ code since multiple sets of cutswere applied to the same output from Pythia

                        When performing simulations of the three dark matter models it was necessary to definethe proton and jets to contain b quarks as this is not the default in Madgraph The processesgenerated included 1 and 2 jets

                        46 Collider Cuts C++ Code

                        Because of the volume of simulations required for the collider constraints we used C++code (originally developed by Martin White and Sky French and modified by me) that readsthe ROOT files and outputs the numbers of events that would have been observed under thechosen collider experiments had the model been true for the given parameters Thus the sameoutput files can be used for all of the collider cuts and hence no cuts were performed throughthe Delphes cards

                        In order to validate the C++ code we replicated the observed events with the papersdescribing the various collider searches (see Appendix)

                        Chapter 5

                        Majorana Model Results

                        51 Bayesian Scans

                        To build some intuition we first examined the constraining effect of each observable on itsown We ran separate scans for each constraint turned on and then for all the constraintstogether In Fig 51 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together The paramaters were scanned over the range in Table 51 below

                        Table 51 Scanned Ranges

                        Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                        Range 0minus103 0minus103 10minus5 minus10 10minus5 minus10 10minus5 minus10

                        In Fig51 we can see that the preferred DM mass is centered around 35 GeV primarily dueto the influence of the Fermi-LAT gamma ray constraint whilst the relic density constraintallows a wide range of possible mediator and DM masses Direct detection (LUX) constraintssplit the mass regions into a top left and bottom right region and together these split theoverall mass parameter regions into two regions by mediator mass but with DM mass around35 GeV Later studies using a lower excess gamma ray spectrum tend to increase the preferredDM mass up to 50 GeV [76] and [88]

                        The marginalised posterior probabilities for all combinations of parameters taken twoat a time are shown in Fig52 These plots have all constraints (Gamma LUX and RelicDensity) applied together The idea of these graphs is to study the posterior probabilities ofcombinations of two paramaters taken at a time (the off-diagonal graphs) The diagonal plotsshow marginalised histograms of a single parameter at a time- this gives an indication of thespread of each parameter The plots show that mS and λb show a high degree of correlationfor mS gt 100 GeV but there is also a range of values of mS around 35 GeV which are

                        51 Bayesian Scans 60

                        1 0 1 2 3 4log10(mχ)[GeV]

                        1

                        0

                        1

                        2

                        3

                        4

                        log 1

                        0(m

                        s)[GeV

                        ]

                        (a) Gamma Only

                        1 0 1 2 3 4log10(mχ)[GeV]

                        1

                        0

                        1

                        2

                        3

                        4

                        log 1

                        0(m

                        s)[GeV

                        ]

                        (b) Relic Density

                        1 0 1 2 3 4log10(mχ)[GeV]

                        1

                        0

                        1

                        2

                        3

                        4

                        log 1

                        0(m

                        s)[GeV

                        ]

                        (c) LUX

                        05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                        05

                        00

                        05

                        10

                        15

                        20

                        25

                        30

                        35

                        log 1

                        0(m

                        s)[GeV

                        ]

                        (d) All Constraints

                        Figure 51 Majorana Dark Matter - Posterior probability by individual constraint and alltogether

                        51 Bayesian Scans 61

                        00

                        05

                        10

                        15

                        20

                        25

                        30

                        log 1

                        0(m

                        χ)[GeV

                        ]

                        00

                        05

                        10

                        15

                        20

                        25

                        30

                        ms[Gev

                        ]

                        5

                        4

                        3

                        2

                        1

                        0

                        1

                        log 1

                        0(λ

                        t)

                        5

                        4

                        3

                        2

                        1

                        0

                        1

                        log 1

                        0(λ

                        b)

                        00 05 10 15 20 25 30

                        log10(mχ)[GeV]

                        5

                        4

                        3

                        2

                        1

                        0

                        1

                        log 1

                        0(λ

                        τ)

                        00 05 10 15 20 25 30

                        ms[Gev]5 4 3 2 1 0 1

                        log10(λt)5 4 3 2 1 0 1

                        log10(λb)5 4 3 2 1 0 1

                        log10(λτ)

                        Figure 52 Majorana Dark Matter - Marginalised Posterior Probabilities by Parameter

                        52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 62

                        possible for λb around 001 λt and λτ show a wide range of possible values for two possibleranges of mS lt 35 GeV and greater than 100 GeV mχ shows a narrow range of possiblevalues around 35 GeV for all values of the other parameters

                        52 Best fit Gamma Ray Spectrum for the Majorana FermionDM model

                        We can see from the Fig51(a) scan that the maximum log-likelihood Fermi-LAT gammaray spectrum on its own shows values that centre around 35 GeV This is not changed byadding a relic density constraint or a direct detection (LUX) constraint

                        The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraintsis shown in Fig53 The parameters for this point were

                        Table 52 Best Fit Parameters

                        Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                        Value 3332 49266 0322371 409990 0008106

                        10-1 100 101 102

                        E(GeV)

                        10

                        05

                        00

                        05

                        10

                        15

                        20

                        25

                        30

                        35

                        E2dφd

                        E(G

                        eVc

                        m2ss

                        r)

                        1e 6

                        Best fitData

                        Figure 53 Gamma Ray Spectrum

                        The relic density (Ωh2) at this point has a value of 0135 but the point is separated onlyslightly in posterior probability from points with a value of 01199 with mS lt 10 GeV (iethe posterior probability is almost the same differing by less than 1)

                        To better understand the interplay of the different constraints the three graphs in Fig54show the log likelihood surfaces and individual best fit points (best log likelihoods) for the γ

                        and Ω scans seperately and then combined The diagrams each choose a pair of couplings(fixing the remaining parameters) and display the 2d surfaces of minuslog(χ2) (proportional to

                        52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 63

                        the log likelihoods of the coupling combinations) obtained by varying the pair of couplingsThese graphs do not show marginalised posterior probabilities but rather are based on the-2log likelihood calculations directly from MicrOmegas for the given coupling combinationsWe have termed this value as χ2 for convenience and used the log of this value to give abetter range for the colours These log likelihood values are what drive the MultiNest searchof parameter space for the best parameter combinations

                        The graphs show a complex interplay of the different constraints resulting often in acompromise parameter combination The arrow on each graph points to the maximum point We have chosen a single representative set of couplings near the best overall fit point for allthe graphs

                        52 Best fit Gamma Ray Spectrum for the Majorana Fermion DM model 64

                        00 05 10 15 20 25 30

                        log10(mχ)

                        00

                        05

                        10

                        15

                        20

                        25

                        30

                        log

                        10(m

                        S)

                        Max

                        minuslog10(χ2(Γ)) λt = 487 λτ = 024 λb = 0344

                        16

                        14

                        12

                        10

                        8

                        6

                        4

                        2

                        0

                        γ Maximum at mχ=416 GeV mS=2188 GeV

                        00 05 10 15 20 25 30

                        log10(mχ)

                        00

                        05

                        10

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                        20

                        25

                        30

                        log

                        10(m

                        S)

                        Max

                        minuslog10(χ2(Omega)) λt = 487 λτ = 024 λb = 0344

                        28

                        24

                        20

                        16

                        12

                        08

                        04

                        00

                        04

                        Ω Maximum at mχ=363 GeV mS=1659 GeV

                        00 05 10 15 20 25 30

                        log10(mχ)

                        00

                        05

                        10

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                        20

                        25

                        30

                        log

                        10(m

                        S)

                        Max

                        minuslog10(χ2(Both)) λt = 487 λτ = 024 λb = 0344

                        16

                        14

                        12

                        10

                        8

                        6

                        4

                        2

                        0

                        Both Maximum at mχ=8317 GeV mS=2884GeV The best fit point for these couplings maybe above below or between the individual bestfit points but will on average be between thepoints

                        Figure 54 Plots of log likelihoods by individual and combined constraints Masses in GeV

                        53 Collider Constraints 65

                        53 Collider Constraints

                        531 Mediator Decay

                        1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof τ leptons in pp collisions [44] Refer also to Chapter 3 for details

                        We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

                        The results of the calculations are shown in the graph Fig55 and can be compared withthe limits in the table from the paper [44] in Fig 56 The black line is the observed ATLASlimit

                        Figure 55 σ lowastBr(σ rarr ττ) versus Mass of Scalar

                        0 200 400 600 800

                        mS[GeV]

                        10

                        5

                        0

                        log 1

                        0(σ

                        (bbS

                        )lowastB

                        (Sgtττ

                        ))[pb]

                        Observed LimitLikely PointsExcluded Points

                        0

                        20

                        40

                        60

                        80

                        100

                        120

                        0 5 10 15 20 25 30 35 40 45

                        We can see that only one point here exceeded the limit and the points showed a widespread over σ(bbS)lowastBr(σ rarr ττ) by mS space which is consistent with the spread of pointson the MultiNest scan on the mS by λτ plane causing a spread of branching ratios to the τ

                        quark It may seem odd that only one point was excluded here but this point was sampledwith a very low probability having high coupling constants of 597 to the τ quark and 278 tothe t quark and only just exceeds the limit As such it can be explained as a random outlier

                        53 Collider Constraints 66

                        Figure 56 σ lowastBr(σ rarr τ+τminus) versus Mass of Scalar

                        2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                        This scan calculates the contributions of b-quark associated Higgs boson productionreproduced from [43] Refer to Chapter 3 for details

                        We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

                        The cross section times branching ratios to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig57 The black line is the CMSLimit

                        Figure 57 σ lowastBr(σ where rarr bS+X) versus Mass of Scalar

                        0 200 400 600 800

                        mS[GeV]

                        15

                        10

                        5

                        0

                        5

                        log

                        10(σ

                        (bS

                        +X

                        )lowastB

                        (Sgt

                        bb))

                        [pb]

                        Observed LimitLikely PointsExcluded Points

                        0

                        20

                        40

                        60

                        80

                        100

                        120

                        0 50 100 150 200 250

                        53 Collider Constraints 67

                        The results of this scan were compared to the limits in [89] with the plot shown inFig58

                        Figure 58 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

                        We can see that there are a number of exclusions across the range of mS which occuredfor high values of the coupling constants gt 1 for λbλt λtau - these can be seen also on Fig59The excluded parameter combinations represent less than 2 of the total sampled number

                        532 Collider Cuts Analyses

                        We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude and the results are shown against theposterior probabilities of the parameter combinations in Fig 59

                        The circles in Figure 59 show which points were excluded by the collider cuts as well asthe limits on cross sections times branching ratios to each of τ+τminus and bb

                        All the excluded points under the heading Collider Cuts showed exclusion by the Lepstop0experiment (Search for direct pair production of the top quark in all hadronic final states

                        53 Collider Constraints 68

                        0 1 2 3

                        log10(mχ)[GeV]

                        0

                        1

                        2

                        3

                        log 1

                        0(m

                        s)[GeV

                        ]Collider Cuts

                        σ lowastBr(σgt bS+X)

                        σ lowastBr(σgt ττ)

                        (a) mχ by mS

                        6 5 4 3 2 1 0 1 2

                        log10(λt)

                        0

                        1

                        2

                        3

                        log 1

                        0(m

                        s)[GeV

                        ](b) λt by mS

                        5 4 3 2 1 0 1

                        log10(λb)

                        6

                        5

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                        0(λ

                        t)

                        (c) λb by λt

                        5 4 3 2 1 0 1

                        log10(λb)

                        6

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                        τ)

                        (d) λb by λτ

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                        log10(λt)

                        6

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                        τ)

                        (e) λt by λτ

                        5 4 3 2 1 0 1

                        log10(λb)

                        0

                        1

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                        3

                        log 1

                        0(m

                        s)[GeV

                        ]

                        (f) λb by mS

                        Figure 59 Excluded points from Collider Cuts and σBranching Ratio

                        53 Collider Constraints 69

                        [32]) Some of these same points were also excluded by Lepstop2 2bstop CMSTop andATLASMonobjet experiments There were expected events at other points but none weresignificant at the 95 exclusion level These significant exclusions occurred mainly wherethe mass of the mediator was over 100 GeV and higher values of the couplings to b and tquarks but there were a few points with mediator mass around 25 GeV The exclusions withmediator mass around 25 GeV were all accompanied by couplings to t quarks of order 10which is probably non-perturbative

                        Looking at the exclusions in Fig 59 we can see that these occur mainly in the regionof higher mediator mass gt 100 GeV and lt 400 GeV These exclusions come from theastrophysical constraints (Gamma) which show a strong correlation between λb and mS andhigh values of λt gt 1 However these mass regions do not overlap at the same couplingcombinations as the high likelihood region The green and white circles correspond to theexcluded points in Figs 57 and 58 respectively

                        The remaining regions of highest likelihood can be seen to be where 0001 λb lt 01accompanied by λt lt 0001 and mass of the mediator mS lt 10 GeV or 400 GeV lt mS lt1000GeV with mχ sim 35 GeV

                        The plot of λt by λb shows the significant regions do not overlap with the collider cutsand cross section constraints since the latter occur at high values of either coupling while thelikely region has low values of these couplings (also accompanied by low couplings to λτ )

                        Chapter 6

                        Real Scalar Model Results

                        61 Bayesian Scans

                        To build some intuition we first examined the constraining effect of each observable on itsown We ran seperate scans for each constraint turned on and then for all the constraintstogether In Figure 61 we show the marginalised posterior distributions for each of the relicdensity (Ω) direct detection (LUX) and indirect detection (γ) scans and finally for the threeconstraints taken together

                        In Figure 61 we can see that there are is a narrow region which dominates when allconstraints are taken into account This centers on mχ=9 GeV and a range of values for mS

                        from 35 GeV to 400 GeV The region is not the highest likelihood region for the Fermi-LATgamma ray spectrum on its own (as can be seen from the individual plot - there is a narrowband to the left of the highest likelihood band) nor for the LUX likelihood or relic densitybut represents a compromise between the three constraints

                        The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig62 The diagonal plots show marginalised histograms of a singleparameter at a time

                        61 Bayesian Scans 71

                        05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                        05

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                        15

                        20

                        25

                        30

                        35

                        log 1

                        0(m

                        s)[GeV

                        ]

                        (d) All Constraints

                        Figure 61 Real Scalar Dark Matter - By Individual Constraint and All Together

                        61 Bayesian Scans 72

                        00

                        05

                        10

                        15

                        20

                        25

                        30

                        log 1

                        0(m

                        χ)[GeV

                        ]

                        00

                        05

                        10

                        15

                        20

                        25

                        30

                        ms[Gev

                        ]

                        5

                        4

                        3

                        2

                        1

                        0

                        1

                        log 1

                        0(λ

                        t)

                        5

                        4

                        3

                        2

                        1

                        0

                        1

                        log 1

                        0(λ

                        b)

                        00 05 10 15 20 25 30

                        log10(mχ)[GeV]

                        5

                        4

                        3

                        2

                        1

                        0

                        1

                        log 1

                        0(λ

                        τ)

                        00 05 10 15 20 25 30

                        ms[Gev]5 4 3 2 1 0 1

                        log10(λt)5 4 3 2 1 0 1

                        log10(λb)5 4 3 2 1 0 1

                        log10(λτ)

                        Figure 62 Real Scalar Matter - Marginalised Posterior Probabilities by Parameter

                        62 Best fit Gamma Ray Spectrum for the Real Scalar DM model 73

                        62 Best fit Gamma Ray Spectrum for the Real Scalar DMmodel

                        We can see from Fig61 that the maximum log-likelihood Fermi-LAT Gamma ray spectrumon its own shows values of mχ that centre around 35 GeV However the LUX constraintin particular moves the best fit to a point around 9 GeV- the tight LUX constraint onlydisappears for low masses and at that point the proportion of τ+τminus annihilation needs toincrease to fit the constraints

                        The gamma ray spectrum at the points of best fit for Fermi-LAT including all constraintsis shown in Fig63 The parameters for these points were respectively and involve decays tomainly τ and b quarks respectively

                        Table 61 Best Fit Parameters

                        Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                        Value 932 3526 000049 0002561 000781

                        10-1 100 101 102

                        E(GeV)

                        10

                        05

                        00

                        05

                        10

                        15

                        20

                        25

                        30

                        35

                        E2dφdE

                        (GeVc

                        m2ss

                        r)

                        1e 6

                        Best fitData

                        Figure 63 Gamma Ray Spectrum

                        This is not a particularly good fit and it occcurs at a relic density Ωh2 of 335 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

                        63 Collider Constraints 74

                        63 Collider Constraints

                        631 Mediator Decay

                        1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44]

                        We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

                        The results of the scan are shown in the graph Fig64 and can be compared with thelimits in the table from the paper [44] in Fig 56The black line is the observed ATLAS limit

                        Figure 64 σ lowastBr(σ rarr ττ) versus Mass of Scalar

                        0 200 400 600 800

                        mS[GeV]

                        8

                        6

                        4

                        2

                        0

                        2

                        4

                        log 1

                        0(σ

                        (bbS

                        )lowastB

                        (Sgtττ

                        ))[pb]

                        Observed LimitLikely PointsExcluded Points

                        050

                        100150200250300350

                        0 10 20 30 40 50 60

                        We can see that there is a narrow band in the region of interest in the σ(bbS)lowastB(S gt ττ)

                        by mS plane due to the high degree of correlation between mS and λτ seen in the likelihoodscan Values for mS gt 300Gev are excluded

                        2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                        We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinations

                        63 Collider Constraints 75

                        randomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

                        The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig65 The black line is the CMSlimit

                        Figure 65 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

                        0 200 400 600 800

                        mS[GeV]

                        8

                        6

                        4

                        2

                        0

                        2

                        4

                        log

                        10(σ

                        (bS

                        +X

                        )lowastB

                        (Sgt

                        bb))

                        [pb]

                        Observed LimitLikely PointsExcluded Points

                        050

                        100150200250300350

                        0 10 20 30 40 50 60

                        The results of this scan were compared to the limits in [89] with the plot shown inFig58

                        We can see a similar shape for the region of interest to that in Fig64 due to the highcorrelation between mS and λb over the region preferred by astrophysical constraints resultingin branching ratios highly correlated to ms Only one point is excluded with mS gt 500GeV

                        632 Collider Cuts Analyses

                        We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expected

                        63 Collider Constraints 76

                        with 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 66

                        We can see that collider cuts show some exclusions only for masses mχ below the highlikelihood region but these exclusions do not occur at the same λb and λt combinations asseen from the plot of these two couplings except for a small region at the highest mS andcouplings

                        All but one of the collider cut exclusions were excluded by the Lepstop0 experiment (thiswas excluded by 2bstop with high coupling to b quarks of 73) A number of the excludedpoints were also excluded by Lepstop2 2bstop CMSTopDM1L and ATLASMonobJet-(around 30 of the total for each)

                        63 Collider Constraints 77

                        0 1 2 3

                        log10(mχ)[GeV]

                        0

                        1

                        2

                        3

                        log 1

                        0(m

                        s)[GeV

                        ]Collider Cuts

                        σ lowastBr(σgt bS+X)

                        σ lowastBr(σgt ττ)

                        (a) mχ by mS

                        5 4 3 2 1 0 1

                        log10(λt)

                        0

                        1

                        2

                        3

                        log 1

                        0(m

                        s)[GeV

                        ](b) λt by mS

                        5 4 3 2 1 0 1

                        log10(λb)

                        5

                        4

                        3

                        2

                        1

                        0

                        1

                        log 1

                        0(λ

                        t)

                        (c) λb by λt

                        5 4 3 2 1 0 1

                        log10(λb)

                        6

                        5

                        4

                        3

                        2

                        1

                        0

                        1

                        2

                        log 1

                        0(λ

                        τ)

                        (d) λb by λτ

                        5 4 3 2 1 0 1

                        log10(λt)

                        6

                        5

                        4

                        3

                        2

                        1

                        0

                        1

                        2

                        log 1

                        0(λ

                        τ)

                        (e) λt by λτ

                        5 4 3 2 1 0 1

                        log10(λb)

                        0

                        1

                        2

                        3

                        log 1

                        0(m

                        s)[GeV

                        ]

                        (f) λb by mS

                        Figure 66 Excluded points from Collider Cuts and σBranching Ratio

                        Chapter 7

                        Real Vector Dark Matter Results

                        71 Bayesian Scans

                        In the Figure 71 we can see that the preferred DM mass is centered around 35 GeV primarilydue to the influence of the Fermi-LAT gamma ray constraint with relic density having a widerange of possible mediator and DM masses The result of combining all the constraints is todefine a small region around mχ = 8 GeV and mS = 20 GeV which dominates the posteriorprobability

                        The marginalised posterior probabilities for all combinations of parameters taken two ata time are shown in Fig72 The diagonal plots show marginalised histograms of a singleparameter at a time

                        71 Bayesian Scans 79

                        05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                        1

                        0

                        1

                        2

                        3

                        4

                        log 1

                        0(m

                        s)[GeV

                        ]

                        (a) Gamma Only

                        05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                        05

                        00

                        05

                        10

                        15

                        20

                        25

                        30

                        35

                        log 1

                        0(m

                        s)[GeV

                        ]

                        (b) Relic Density

                        1 0 1 2 3 4log10(mχ)[GeV]

                        05

                        00

                        05

                        10

                        15

                        20

                        25

                        30

                        35

                        log 1

                        0(m

                        s)[GeV

                        ]

                        (c) LUX

                        05 00 05 10 15 20 25 30 35log10(mχ)[GeV]

                        05

                        00

                        05

                        10

                        15

                        20

                        25

                        30

                        35

                        log 1

                        0(m

                        s)[GeV

                        ]

                        (d) All Constraints

                        Figure 71 Real Vector Dark Matter - By Individual Constraint and All Together

                        71 Bayesian Scans 80

                        00

                        05

                        10

                        15

                        20

                        25

                        30

                        log 1

                        0(m

                        χ)[GeV

                        ]

                        00

                        05

                        10

                        15

                        20

                        25

                        30

                        ms[Gev

                        ]

                        5

                        4

                        3

                        2

                        1

                        0

                        1

                        log 1

                        0(λ

                        t)

                        5

                        4

                        3

                        2

                        1

                        0

                        1

                        log 1

                        0(λ

                        b)

                        00 05 10 15 20 25 30

                        log10(mχ)[GeV]

                        5

                        4

                        3

                        2

                        1

                        0

                        1

                        log 1

                        0(λ

                        τ)

                        00 05 10 15 20 25 30

                        ms[Gev]5 4 3 2 1 0 1

                        log10(λt)5 4 3 2 1 0 1

                        log10(λb)5 4 3 2 1 0 1

                        log10(λτ)

                        Figure 72 Real Vector Dark Matter - Marginalised Posterior Probabilities by Parameter

                        72 Best fit Gamma Ray Spectrum for the Real Vector DM model 81

                        72 Best fit Gamma Ray Spectrum for the Real Vector DMmodel

                        The gamma ray spectrum at the point of best fit for Fermi-LAT including all constraints isshown in Fig73

                        Table 71 Best Fit Parameters

                        Parameter mχ [GeV ] mS[GeV ] λt λb λτ

                        Value 8447 20685 0000022 0000746 0002439

                        10-1 100 101 102

                        E(GeV)

                        10

                        05

                        00

                        05

                        10

                        15

                        20

                        25

                        30

                        35

                        E2dφdE

                        (GeVc

                        m2s

                        sr)

                        1e 6

                        Best fitData

                        Figure 73 Gamma Ray Spectrum

                        This is not a particularly good fit and it occcurs at a relic density Ωh2 of 025 This isconsistent with the findings of [19] which found that the Majorana Model showed the bestoverall fit to the constraints

                        73 Collider Constraints

                        731 Mediator Decay

                        1) To compare with the limits in Search for neutral MSSM Higgs bosons decaying to a pairof tau leptons in pp collisions [44] (See Chapter 3 for more details)

                        We generated proton proton collisions with b b and the scalar mediator S in the finalstate through Madgraphv223 randomly picking 900 of the parameter combinations withthe posterior probabilities generated by MultiNest We also calculated the branching ratiosof the decay of the scalar mediator to ττ using B1 and the product of the cross section forthe process and this branching ratio

                        73 Collider Constraints 82

                        The results of the scan are shown in the graph Fig74 and can be compared with thelimits in the table from the paper [44] in Fig 56

                        Figure 74 σ lowastBr(σ rarr ττ) versus Mass of Scalar

                        0 200 400 600 800

                        mS[GeV]

                        8

                        6

                        4

                        2

                        0

                        2

                        log 1

                        0(σ

                        (bbS

                        )lowastB

                        (Sgtττ

                        ))[pb]

                        Observed LimitLikely PointsExcluded Points

                        0100200300400500600700800

                        0 20 40 60 80 100120140

                        We can see that the region of highest likelihood is limited to mS lt 100 GeV and a narrowrange of couplings to λτ giving a narrow region of interest which falls well below the limit

                        2) To compare with the limits in Search for neutral MSSM Higgs bosons decaying intoa pair of bottom quarks [43]

                        We generated proton proton collisions in Madgraphv223 with b(b) quarks and the scalarmediator b(b) quarks and the scalar mediator and one jet picking 900 parameter combinationsrandomly with posterior probabilities generated by MultiNest We also calculated thebranching ratios of the decay of the scalar mediator to bb using B1 and the product of thecross section for the process and this branching ratio

                        The cross section times branching ratio to bb in association with the production ofmediator and at least one b quark are shown in the plot Fig75 The black line is the CMSlimit

                        The results of this scan were compared to the limits in [89] with the plot shown in Fig58

                        We can see that again the region of interest for mS is below 100 GeV and the points fallwell below the limit

                        73 Collider Constraints 83

                        Figure 75 σ lowastBr(σ rarr bS+X) versus Mass of Scalar

                        0 200 400 600 800

                        mS[GeV]

                        8

                        6

                        4

                        2

                        0

                        2

                        4

                        log

                        10(σ

                        (bS

                        +X

                        )lowastB

                        (Sgt

                        bb))

                        [pb]

                        Observed LimitLikely PointsExcluded Points

                        0100200300400500600700800

                        0 20 40 60 80 100120140

                        732 Collider Cuts Analyses

                        We generated proton proton collisions in Madgraphv223 with χχ and one and two b-jets inthe final state as well as χχ and tt in the final state picking parameter combinations from asample of 900 points based on posterior probabilities generated by MultiNest scans thenapplied the collider cuts program described in Chapter 4 on the outputs from these Thenumbers of events generated in each run were more than 10 times the number expected for21 f bminus1 to ensure that we had significant results before scaling by the number expectedwith 21 f bminus1 The scaled numbers of events generated were compared with 95 CLs shownin Chapter 3 to determine which points to exclude No points showed any exclusion as aresult of these cuts This was then extended to a sample of 1200 points taken at randomfrom the full set of scanned points with no reference to posterior probability to see where theexclusions lay in relation to the most likely region The results are shown in Figure 76

                        We can see from Figure 76 that like the real scalar model the region of highest likelihoodhas λb and λt lt 001 while the excluded points from collider cuts and cross sections areabove 01 Most of the excluded points have mχ gt than the values in the region of highestlikelihood and all points are widely separated in parameter space from this region

                        Once again 89 of excluded points were excluded by Lepstop0 with many also beingexcluded by the other experiments (50 by Lepstop2 and 56 by CMSTopDM1L 35 byLepstop2 and 20 by ATLASMonobjet)

                        73 Collider Constraints 84

                        0 1 2 3

                        log10(mχ)[GeV]

                        0

                        1

                        2

                        3

                        log 1

                        0(m

                        s)[GeV

                        ]Collider Cuts

                        σ lowastBr(σgt bS+X)

                        σ lowastBr(σgt ττ)

                        (a) mχ by mS

                        5 4 3 2 1 0 1

                        log10(λt)

                        0

                        1

                        2

                        3

                        log 1

                        0(m

                        s)[GeV

                        ](b) λt by mS

                        5 4 3 2 1 0 1

                        log10(λb)

                        5

                        4

                        3

                        2

                        1

                        0

                        1

                        log 1

                        0(λ

                        t)

                        (c) λb by λt

                        5 4 3 2 1 0 1

                        log10(λb)

                        5

                        4

                        3

                        2

                        1

                        0

                        1

                        log 1

                        0(λ

                        τ)

                        (d) λb by λτ

                        5 4 3 2 1 0 1

                        log10(λt)

                        5

                        4

                        3

                        2

                        1

                        0

                        1

                        log 1

                        0(λ

                        τ)

                        (e) λt by λτ

                        5 4 3 2 1 0 1

                        log10(λb)

                        0

                        1

                        2

                        3

                        log 1

                        0(m

                        s)[GeV

                        ]

                        (f) λb by mS

                        Figure 76 Excluded points from Collider Cuts and σBranching Ratio

                        Chapter 8

                        Conclusion

                        We first conducted Bayesian scans over mass and coupling parameters for three simplifiedmodels - Majorana fermion real scalar and real vector using direct detecion (LUX) indirectdetection (GCE γ rays) and relic density constraints to find the regions of highest likelihoodWe ran these scans one constraint at a time and then together to better understand the interplayof the different constraints on best fit and likely parameter regions

                        We also developed and tested C++ collider cut software specifically designed to mimickthe cuts applied in a number of ATLAS and CMS experiments that searched for Emiss

                        T inconnection with specific beyond the standard model (BSM) theories of Dark Matter

                        We then selected parameter combinations by randomly sampling the scanned resultsbased on the posterior probability distribution of the points and applied collider cuts andcross section times branching ratio calculations to these points to see if there would havebeen any sensitivity from the ATLAS and CMS experiments

                        We found that the Majorana fermion model showed the best overall fit to all the constraintswith relic density close to the experimental value of 01199 and a low χ2 error on the gammaspectrum fit The preferred mass mχ was around 35 GeV and mS ranging from lt10 GeV to500 GeV The smaller masses give a better fit for relic density and also are well separatedfrom current collider constraints mainly due the the fact that the exclusions occur mainlywith higher couplings to t quarks (greater than 1) which is consistent with the fact that manyof the experiments were targeting dark matter pair production associated with t quark decaysNevertheless there were some exclusions that did fall in the region sampled by randomselection from the posterior probability distribution

                        The preferred parameter regions were also consistent with the observation in [20] thatthe Majorana model showed the best overall fit with similar mass constraints We find thatthe couplings to the SM fermions should all be less that 001

                        86

                        The scalar and vector DM models give slightly less good fits especially to the currentrelic density estimate of 01199 but again the current collider constraints are well separatedfrom the best fit region and in fact none of the points randomly sampled from the posteriordistribution showed any exclusion The best fit parameter combination favours couplings toSM fermions of less than 001

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                        Appendix A

                        Validation of Calculation Tools

                        Lepstop0Search for direct pair production of the top squarkin all hadronic final states in proton=proton collisions atradic

                        s=8 TeV with the ATLAS detector [32]

                        Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1) mt1=600 GeV mn1=1 GeVWhere t1 is the top squark and n1 is the neutralino of the MSSM

                        94

                        Table A1 0 Leptons in the final state

                        Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 10000 460 460 100Emiss

                        T gt 130GeV 8574 395 451 88Leptonveto 6621 307 285 108Emiss

                        T gt 150GeV 6316 293 294 100Jet multiplicityand pT 1099 48 51 94|∆φ( jet pmiss

                        T )|gt π5 902 40 42 96gt= 2b jets 396 19 18 106τ veto 396 19 18 106mbmin

                        T gt 175GeV 270 13 13 105SRA1 79 4 4 104SRA2 70 3 3 93SRA3 67 3 3 97SRA4 52 2 2 91SRC exactly5 jets 1297 50 60 83SRC |∆φ( jet pmiss

                        T )|gt π5 1082 50 50 99SRC gt= 2b jets 430 20 20 97SRC τ veto 430 20 20 97SRC |∆φ(bb)|gt 02π 367 17 17 99SRC1 246 12 11 105SRC2 177 9 8 116SRC3 152 8 7 116

                        95

                        Lepstop1Search for direct top squark pair production infinal states with one isolated lepton jets and missing trans-verse momentum in

                        radics = 8 TeV pp collisions using 21 f bminus1

                        of ATLAS data[33]

                        Simulated in Madgraph with p p gt t1 t1(t1 gt t n1)(t1 gt t n1)(t1 gt b x+1 )(t1 gt b xminus1 )mt1=650 GeV mn1=1 GeV Where t1 is the top squark n1 is the neutralino xplusmn1 are thecharginos

                        96

                        Table A2 1 Lepton in the Final state

                        Cut RawMe Scaled ATLAS MeATLAS PercentNocuts 100000 99990 99990 100Electron(= 1signal) 11558 11557 11785 984 jets(80604025) 5187 5186 6271 83gt= 1btag 4998 4997 5389 93Emiss

                        T gt 100GeV [allSRs] 4131 4131 4393 94Emiss

                        T radic

                        HT gt 5 [allSRs] 4026 4026 4286 94∆φ( jet2 pmiss

                        T )gt 08 [allSRs] 3565 3565 3817 93∆φ( jet1Pmiss

                        T )gt 08 [not SRtN2] 3446 3446 3723 93Emiss

                        T gt 200GeV (SRtN2) 2027 2027 2186 93Emiss

                        T radic

                        HT gt 13(SRtN2) 1156 1156 1172 99mT gt 140GeV (SRtN2) 839 839 974 86Emiss

                        T gt 275GeV (SRtN3) 948 948 965 98Emiss

                        T radic

                        HT gt 11(SRtN3) 932 932 937 99mT gt 200GeV (SRtN3) 456 456 566 81Emiss

                        T gt 150GeV (SRbC1minusSRbC3) 2803 2803 3080 91Emiss

                        T radic

                        HT gt 7(SRbC1minusSRbC3) 2742 2742 3017 91mT gt 12GeV (SRbC1minusSRbC3) 1917 1917 2347 82Emiss

                        T gt 160GeV (SRbC2SRbC3) 1836 1836 2257 81Emiss

                        T radic

                        HT gt 8(SRbC2SRbC3) 1788 1788 2175 82me f f gt 550GeV (SRbC2) 1741 1741 2042 85me f f gt 700GeV (SRbC3) 1075 1075 1398 77Muon(= 1signal) 11379 11378 11975 954 jets(80604025) 5264 5263 6046 87gt= 1btag 5101 5100 5711 89Emiss

                        T gt 100GeV [allSRs] 4199 4199 4770 88Emiss

                        T radic

                        HT gt 5 [all SRs] 4115 4115 4653 88∆φ( jet2 pmiss

                        T )gt 08 [all SRs] 3622 3622 4077 89∆φ( jet1 pmiss

                        T )gt 08 [not SRtN2] 3515 3515 3970 89Emiss

                        T gt 200GeV (SRtN2) 2136 2136 2534 84Emiss

                        T radic

                        HT gt 13(SRtN2) 1261 1261 1441 87mT gt 140GeV (SRtN2) 928 928 1033 90Emiss

                        T gt 275GeV (SRtN3) 1036 1036 1274 81Emiss

                        T radic

                        HT gt 11(SRtN3) 1018 1018 1235 82mT gt 200GeV (SRtN3) 497 497 612 81Emiss

                        T gt 150GeV (SRbC1minusSRbC3) 2891 2891 3380 86Emiss

                        T radic

                        HT gt 7(SRbC1minusSRbC3) 2821 2821 3290 86mT gt 120GeV (SRbC1minusSRbC3) 1977 1977 2290 86Emiss

                        T gt 160GeV (SRbC2SRbC3) 1913 1913 2220 86Emiss

                        T radic

                        HT gt 8(SRbC2SRbC3) 1863 1863 2152 87me f f gt 550GeV (SRbC2) 1826 1826 2116 86me f f gt 700GeV (SRbC3) 1131 1131 1452 78

                        97

                        Lepstop2Search for direct top squark pair production infinal states with two leptons in

                        radics =8 TeV pp collisions using

                        20 f bminus1 of ATLAS data[83][34]

                        Simulated in Madgraph with p p gt t1 t1(t1 gt b w+ n1)(t1 gt b wminusn1) mt1=180 GeV mn1=60 GeV Where t1 is the top squark n1 is the neutralino b1 is the bottom squark

                        Table A3 2 Leptons in the final state

                        Cut RawMe Scaled ATLAS MeATLAS PercentNoCuts 100000 324308 199000 1632BaselineCuts 3593 11652 7203 1622SF Signal Leptons 1699 5510 5510 1002OSSF Signal Leptons 1632 5293 5233 101mll gt 20Gev 1553 5037 5028 100leadinglepton pT 1479 4797 5048 95|mll minusmZ|gt 20Gev 1015 3292 3484 94∆φmin gt 10 850 2757 2160 128∆φb lt 15 680 2205 1757 126SRM90 [SF ] 13 42 60 70SRM120 [SF ] 0 0 19 0SRM100 +2 jets [SF ] 0 0 17 0SRM110 +2 jets [SF ] 0 0 1 02DF Signal Leptons 1737 5633 7164 792OSDF Signal Leptons 1670 5416 5396 100mll20Gev 1587 5147 5136 100leadinglepton pT 1501 4868 4931 99∆φmin gt 10 1247 4044 3122 130∆φb lt 15 1010 3276 2558 128SRM90 [SF ] 21 68 95 72SRM120 [SF ] 0 0 23 0SRM100 +2 jets [SF ] 0 0 19 0SRM110 +2 jets [SF ] 0 0 14 0

                        98

                        2bstopSearch for direct third-generation squark pair pro-duction in final states with missing transverse momentumand two b-jets[31]

                        Simulated in Madgraph with p p gt b1 b1(b1 gt b n1)(b1 gt b n1)

                        SRAmb1=500 GeV mn1=1 GeV SRBmb1=350 GeV mn1=320 GeVWhere b1 is the bottom squark and n1 is the neutralino

                        Table A4 2b jets in the final state

                        Cut RawMe Scaled ATLAS MeATLAS PercentSRA Nocuts 100000 173800 1738 100SRA MET gt 80 93407 162341 1606 101SRA Leptonveto 93302 162159 1505 108SRA MET gt 150 79449 138082 1323 104SRA Jet selection 10213 17750 119 149SRA mbb gt 200 8344 14502 96 151SRA mCT gt 150 6535 11358 82 139SRA mCT gt 200 5175 8994 67 134SRA mCT gt 250 3793 6592 51 129SRA mCT gt 300 2487 4322 35 123SRB NoCuts 100000 1624100 16241 100SRB leptonveto 11489 186593 4069 46SRB MET gt 250 2530 41090 757 54SRB Jet selection 63 1023 79 130SRB HT 3 lt 50 51 828 52 159

                        99

                        CMSTop1LSearch for the Production of Dark Matter in As-sociation with Top Quark Pairs in the Single-lepton FinalState in pp collisions[41]

                        Simulated in Madgraph with p p gt t t p1 p1

                        Where p1= Majorana fermion Table 3 shows Signal efficiencies and cross sections TheM scale was read from Figure 7 for 90 CL

                        Table A5 Signal Efficiencies 90 CL on σ limexp[ f b] on pp gt tt +χχ

                        Mχ[GeV ] My Signal Efficiency Quoted Efficiency My σ [ f b] Quoted σ [ f b]1 102 101 63 47

                        10 132 101 62 4650 143 120 50 38100 170 146 56 32200 185 173 69 27600 265 240 32 19

                        1000 320 276 41 17

                        Appendix B

                        Branching ratio calculations for narrowwidth approximation

                        B1 Code obtained from decayspy in Madgraph

                        Br(S rarr bb) = (minus24λ2b m2

                        b +6λ2b m2

                        s

                        radicminus4m2

                        bm2S +m4

                        S)16πm3S

                        Br(S rarr tt) = (6λ2t m2

                        S minus24λ2t m2

                        t

                        radicm4

                        S minus4ms2m2t )16πm3

                        S

                        Br(S rarr τ+

                        τminus) = (2λ

                        2τ m2

                        S minus8λ2τ m2

                        τ

                        radicm4

                        S minus4m2Sm2

                        τ)16πm3S

                        Br(S rarr χχ) = (2λ2χm2

                        S

                        radicm4

                        S minus4m2Sm2

                        χ)32πm3S

                        (B1)

                        Where

                        mS is the mass of the scalar mediator

                        mχ is the mass of the Dark Matter particle

                        mb is the mass of the b quark

                        mt is the mass of the t quark

                        mτ is the mass of the τ lepton

                        The coupling constants λ follow the same pattern

                        • TITLE Collider Constraints applied to Simplified Models of Dark Matter fitted to the Fermi-LAT gamma ray excess using Bayesian Techniques
                          • Dedication
                          • Declaration
                          • Acknowledgements
                          • Contents
                          • List of Figures
                          • List of Tables
                            • Chapter 1 Introduction
                            • Chapter 2 Review of Physics
                            • Chapter 3 Fitting Models to the Observables
                            • Chapter 4 Calculation Tools
                            • Chapter 5 Majorana Model Results
                            • Chapter 6 Real Scalar Model Results
                            • Chapter 7 Real Vector Dark Matter Results
                            • Chapter 8 Conclusion
                            • Bibliography
                            • Appendix A Validation of Calculation Tools
                            • Appendix B Branching ratio calculations for narrow width approximation

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