8D Likelihood Effective Higgs Couplings Extraction Framework in h → 4

LPT-Orsay-13-100, CALT-68-2874, FERMILAB-PUB-13-555-T, nuhep-th/13-06

8D Likelihood Effective Higgs Couplings Extraction Framework in h→ 4`

Yi Chen,∗ Emanuele Di Marco,† and Maria Spiropulu‡

Lauritsen Laboratory of Physics, California Institute of Technology,1200 E. California Blvd, Pasadena, CA, 92115, USA

Joe Lykken§

Theoretical Physics Department, Fermilab, P.O. Box 500, Batavia, IL 60510, USA

Roberto Vega-Morales¶

Laboratoire de Physique Theorique d’Orsay, UMR8627-CNRS,Universite Paris-Sud 11, F-91405 Orsay Cedex, France

Theoretical Physics Department, Fermilab, P.O. Box 500, Batavia, IL 60510, USA andDepartment of Physics and Astronomy, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA

Si Xie∗∗

Lauritsen Laboratory of Physics, California Institute of Technology, Pasadena, CA, 92115

We present an overview of a comprehensive analysis framework aimed at performing direct ex-traction of all possible effective Higgs couplings to neutral electroweak gauge bosons in the decay toelectrons and muons, the so called ‘golden channel’. Our framework is based primarily on a maxi-mum likelihood method constructed from analytic expressions of the fully differential cross sectionsfor h→ 4` and for the dominant irreducible qq → 4` background, where 4` = 2e2µ, 4e, 4µ. Detectoreffects are included by an explicit convolution of these analytic expressions with the appropriatetransfer function over all center of mass variables. Utilizing the full set of observables, we constructan un-binned detector-level likelihood which is continuous in the effective couplings. We considerpossible ZZ, Zγ, and γγ couplings simultaneously, allowing for general CP odd/even admixtures. Abroad overview is given of how the convolution is performed and we discuss the principles and the-oretical basis of the framework. This framework can be used in a variety of ways to study Higgscouplings in the golden channel using data obtained at the LHC and other future colliders.

I. INTRODUCTION

The recent discovery of the Higgs boson at the LHC [1,2] with properties resembling those predicted by theStandard Model, shifts our attention to the determi-nation of its precise nature and to establish whetheror not the Higgs boson possesses any anomalous cou-plings to Standard Model particles. In this study we focuson couplings to neutral electroweak gauge bosons. Sincethese ‘anomalous effects’ are expected to be small if atall present, constraining or measuring of these couplingsshould preferably be done through direct parameter ex-traction with minimal theoretical assumptions. The vastliterature [3–37] on Higgs decays to four charged leptons(electrons and muons) through neutral electroweak gaugebosons, suggests that the so called ‘golden channel’, canbe a powerful means towards accomplishing this goal.

A number of frameworks have been established uti-lizing the Matrix Element Method to study the goldenchannel aiming to determine these potentially anomalous

∗ [email protected]† [email protected]‡ [email protected]§ [email protected]¶ [email protected]∗∗ [email protected]

couplings. These primarily rely on Monte Carlo genera-tors such as the JHU generator [13, 17, 32] or on Mad-graph implementations [22, 31]. They have the advantageof flexibility to include various Higgs production and de-cay channels and are especially useful for constructingkinematic discriminators to distinguish between compet-ing hypotheses.

Focusing on the golden channel only1, we propose anovel analysis framework largely based on an analytic im-plementation. It is designed to maximize the informationcontained in each event with the aim of direct extractionof the various effective Higgs couplings. It is generallyacknowledged in the literature that analytic methods areoptimal for performing this direct multi-parameter ex-traction within practical and reasonable computationalprocessing resources [13, 17, 32]. In this work, we alsodemonstrate that within an analytic framework one canreadily include the relevant detector effects and obtain adetector-level likelihood function in terms of the full set ofobservables available in the four lepton final state. Thisis accomplished by the explicit convolution of analyticexpressions for the ‘truth level’ fully differential crosssections with a transfer function which parametrizes thedetector resolution and acceptance effects.

1 Though we will not discuss it explicitly here, we are also able toextend our framework to the h→ γγ and h→ 2`γ channels.

2

This analysis framework has already proved useful inconstraining effective Higgs couplings as demonstratedin a recent CMS analysis [38, 39]. It was shown thatfor simplified cases of constraining one or two param-eters, our framework gives comparable performance toother established analysis methods for the golden chan-nel [13, 17, 22, 31, 32, 38, 39]. In this work we present anoverview of the framework and discuss the principles andtheoretical basis. In particular we sketch how the vari-ous components of the detector level likelihood are con-structed with emphasis on how the convolution integralis performed as well as various validations. How the like-lihood can then be used to perform multi-parameter ex-traction of effective Higgs couplings is also discussed. Wehope that the additional features of our framework arealso found useful in the next phase of the LHC and fu-ture colliders. Much more information on the frameworkincluding technical details can be found in [19, 35, 37–41].

II. OVERVIEW OF FRAMEWORK

Though ‘truth’ level (or generator) studies of h → 4`(4` = 2e2µ, 4e, 4µ) give a good approximate estimate ofthe expected sensitivity to the Higgs ZZ, Zγ, and γγcouplings [37], when analyzing data obtained at the LHC(or future colliders) a detector level likelihood which ac-counts for the various detector effects is necessary. Sincegenerally detector level likelihoods are obtained via theuse of Monte Carlo methods, it becomes difficult to ob-tain the full multi-dimensional likelihood for the 4` finalstate. Typically one needs to fill large multi-dimensionaltemplates that require an impractical amount of com-puting time. There are also potential collateral binningand ‘smoothing’ side-effects often associated with thesemethods. In the case of the golden channel this necessi-tates the use of kinematic discriminants which ‘collapse’the fully multi-dimensional likelihood into two or per-haps three detector level observables [32]. This approachis normally taken to facilitate the inclusion of detectoreffects, but is not optimal when fitting to a large numberof parameters simultaneously [17, 42]. This is unfortu-nate in the case of the golden channel where in principlethere are twelve observables which can be used to ex-tract a large number of parameters at once, includingtheir correlations. It would be satisfying and useful tohave a framework which is free of these issues and capa-ble of utilizing all available information in the four leptonfinal state at detector level.

A. From ‘Truth’ to ‘Detector’ Level

This is accomplished in our framework by performingan explicit convolution of the generator (‘truth’) levelprobability density, formed out of the signal and back-ground differential cross sections, with a transfer functionwhich encapsulates the relevant detector effects. This can

be represented schematically as follows,

P ( ~XR| ~A) =

∫P ( ~XG| ~A)T ( ~XR| ~XG)d ~XG. (1)

Here we take ~X to represent the full set of center of massvariables, of which there are twelve in the golden channel,

to be discussed more below, and ~A represents some set of

lagrangian parameters. The transfer function T ( ~XR| ~XG)takes us from generator (G) level to reconstructed (R)level observables and represents the probability of recon-

structing the observables ~XR given the generator level

observable ~XG. It is treated as a function of ~XR whichtakes ~XG as input. As will be described more in Sec. V A,once the integration in Eq.(1) is performed we must thennormalize over all twelve reconstructed level observablesto obtain the detector level pdf.

The integral in Eq.(1) is the defining feature of ourframework and has been obtained for both the h → 4`signal as well as the dominant qq → 4` background,which have been computed analytically in accompany-ing studies [19, 35, 43]. We emphasize that the integralhas not been obtained via Monte Carlo methods. Insteadwe have explicitly performed the integration by utiliz-ing various analytic and well-established numerical meth-ods [41, 44] (for studies that perform similar convolu-tions using Monte Carlo methods see [31, 45–47]). Thisensures that (arbitrarily) high precision is maintainedat each step, producing what is effectively an ‘analyticfunction’ in terms of detector level variables once theconvolution has been performed. After performing this12-dimensional integration and normalizing, we are leftwith a probability density function (pdf ) from which weconstruct an un-binned twelve-dimensional detector levellikelihood which is a continuous function of the effectivecouplings (or Lagrangian parameters) and takes as itsinput, up to twelve reconstructed (detector-level) cen-ter of mass observables. In the current implementationwe will average over the four production variables toreduce the systematic uncertainties, thus obtaining aneight-dimensional likelihood in terms of just decay ob-servables. However, this step is in principle not necessary.

We also emphasize that the convolution integral islargely independent of detector transfer function and gen-erator level differential cross section and in particularhow accurate the descriptions of the ‘truth’ level pdfsor the detector properties are. This means the frameworkcan in principle be adapted to any detector which studiesh→ 4` (or any X → 4`) and, in addition, as theoreticalcalculations of the generator level differential cross sec-tions improve they can easily be incorporated into theconvolution integral. Thus, there is ample for room opti-mization in our framework as time goes on. The gener-ality of the convolution also allows for other beyond theStandard Model physics such as exotic Higgs decays [48]to be easily be incorporated into the h→ 4` framework.

3

B. Analytic Parameterizations

As we discuss below, it is essential to first have an-alytic parameterizations of the ‘truth’ level differentialcross sections in order to perform the convolution inte-gral in Eq.(1). These can be obtained in essentially twodifferent ways. The first is to simply analytically com-pute the differential cross section starting from Feynmandiagrams, which of course is not always possible. The sec-ond is to obtain an ‘analytic’ parameterization by fittingto a large Monte Carlo sample with some appropriatelyparametrized function. This becomes quite difficult whenthe function is multi-dimensional as is the case in thegolden channel with twelve center of mass observablesand requires large samples and an accurate interpola-tion procedure. In this framework we have implementeda hybrid of these two approaches with the primary com-ponent coming from analytic expressions of the leadingorder h → 4` and qq → 4` fully differential cross sec-tions [19, 35, 43].

Since NLO effects in the golden channel are gener-ally small [49–51], these leading order differential crosssections represent the dominant contributions to the 4`‘truth’ level likelihood. There are however, a number ofsub-dominant effects which appear at higher order andshould be accounted for. These include production andadditional background effects. In these cases, the secondmethod of parametric fits to simulated data is typicallythe optimal route. To do this we follow a similar proce-dure as found in [32] while further details on the imple-mentation into our framework can be found in [41]. Weemphasize however that the convolution integral is in-dependent of these matters allowing for easy implemen-tation of more precise ‘truth’ level likelihoods as theybecome available over time.

Of course when considering the detector level likeli-hood there are additional, but again sub-dominant, ef-fects not present at ‘truth’ level which should be ac-counted for, such as detector momentum resolution andacceptance effects. These can be parametrized via trans-fer functions which can be optimized for a particular de-tector as done recently in [38, 39] which incorporates aparameterization of the CMS detector into our frame-work. Since these typically would be supplied by the ex-perimentalist we do not discuss their construction in de-tail here, but note that as knowledge of the detectorsimproves and parameterizations of the transfer functionsbecome more accurate, they can easily be incorporatedinto the convolution integral in Eq.(1), but again the inte-gration is independent of these matters. More details onthe construction and implementation, as well as the val-idation, of the transfer functions is found in [38, 39, 41].

C. Fast Parameter Extraction

The convolution integral in Eq.(1) allows us to (ef-fectively) obtain an analytic function in both detector

level observables and lagrangian parameters. This is be-cause, via the explicit 12-dimensional integration over allcenter of mass variables, we are able to obtain a 1-to-1 mapping from the ‘truth’ level likelihood to the ‘de-tector’ level likelihood. This allows us, during parame-ter extraction, to effectively work directly with the la-grangian parameters, but at detector level which givesus the ability to easily perform multi-parameter extrac-tion with the same speed and flexibility as was done atgenerator level [35, 37]. Being able to fit to multiple pa-rameters simultaneously is important since it allows forstrong tests of models which often predict correlationsbetween the various parameters. We point out that ourframework allows us to do this while avoiding relying onhypothesis testing or on the construction of kinematicdiscriminants which is less optimal when extracting mul-tiple parameters than maximizing the full likelihood [52]where all observables are used. Furthermore, the ana-lytic nature of our framework allows for a great deal offlexibility in performing a variety of types of parameterextractions and re-parameterizations.

D. Comments on Assumptionsand Approximations

In performing the convolution integral in Eq.(1) wehave relied on two key assumptions. The first is thatangular resolution effects due to detector smearing canbe neglected, which is an excellent approximation forthe LHC detectors [53–55]. Second, we have assumedin the transfer function that each lepton is independentof the others which again is a very good approximationsince leptons are clean and well-measured objects in theCMS and ATLAS detectors once standard lepton selec-tion criteria are imposed [53–55]. With these simplifyingassumptions the convolution integral can then be per-formed as will be described below and in much more de-tail in [40] and [41].

Even after the convolution is performed however, wemust still normalize the detector level differential crosssection. Since this can not be done analytically one mustresort to Monte Carlo techniques. Thus, strictly speakingthe final pdf is not analytic. However, as we will discussmore in Sec. V A, due to the manner in which the analyticexpressions are organized, a high precision on the normal-ization can be obtained in a short amount of computingtime. Furthermore, by fitting to ratios of couplings, wecan circumvent the need for the absolute normalizationwhich greatly simplifies the computational procedure andallows us to achieve a high precision [40, 41] leading inthe end to a detector level pdf which is effectively analyticin reconstructed observables and lagrangian parameters.

Of course there are components of both the ‘truth’and detector level likelihoods which can not be includedin the convolution integral of Eq.(1). These correspondto any components for which a sufficiently accurate an-alytic parameterization can not be obtained. These may

4

include potential higher order contributions to both sig-nal and background differential cross sections as well asadditional fake backgrounds such as Z + X. For theseone must resort to more conventional Monte Carlo tech-niques and the construction of large (binned) ‘look-up’tables. The effects of binning can me mitigated througha linear multi-dimensional interpolation technique whichis described in more detail in [41]. Fortunately these com-ponents are sub dominant in the golden channel and canbe assigned systematics to study their effects [38, 39, 41].

There are also various other systematics associatedwith both detector and theoretical uncertainties whichshould properly be accounted for. Since these compo-nents do not have a large effect on the final sensitivity(especially once sizable data sets are accumulated) andare not directly related to the convolution, we will dis-cuss them only briefly below, but see [38, 39, 41] for moredetails on how they are implemented into the frameworkin a real experimental analysis.

In constructing the detector level likelihood we haveovercome many of the technical challenges which in thepast have made it impossible to use the fully multi-dimensional likelihood during parameter fitting. Belowwe sketch in more detail how these various challengeshave been overcome, but many of the details are tech-nically beyond the scope of this paper so we refer thereader to [19, 35, 37–41] for more details.

III. THE ‘TRUTH LEVEL’ PDF

Before obtaining the detector level likelihood one mustof course first construct the ‘truth’ level (or generatorlevel) likelihood. As we discuss, the generator level likeli-hood is composed of a ‘decay’ and ‘production’ differen-tial spectrum. In our framework, the primary componentis constructed out of analytic expressions for the h→ 4`signal and the dominant qq → 4` background differen-tial cross sections. Analytic expressions have been shownto be useful in likelihood methods where the full kine-matics of an event can be exploited. This is especiallytrue for the golden channel as has been demonstrated innumerous studies [13–15, 17, 18, 29, 32, 35, 37]. For adetailed description of the analytic calculations for thesignal and background fully differential cross sections aswell as their validation we refer the reader to accompa-nying studies [19, 35].

Below we give an overview of how the ‘truth’ level like-lihood is constructed and define the twelve center of massvariables in the four lepton final state. We briefly dis-cuss our parameterization of the Higgs couplings to elec-troweak gauge bosons and how the analytic expressionsare combined with the appropriate production spectrato form the full truth level differential cross section. Wealso discuss in this section how the production spectrumis obtained and comment on the additional backgroundspresent in the golden channel.

Θ

FIG. 1. Definition of angles in the four lepton center of massframe X.

A. Center of Mass Observables

Here we describe the various center of mass variableswhich will be used as our set of observables when con-structing the likelihood. The kinematics of four leptonevents are illustrated in Fig. 1. The invariant masses aredefined as the following:

•√s ≡ M4` ≡ mh – The invariant mass of the four

lepton system or the Higgs mass in case of signal.

• M1 – The invariant mass of the lepton pair systemwhich reconstructs closest to the Z mass.

• M2 – The invariant mass of the other lepton pairsystem and interpreted asM2 < M1. This conditionholds as long as

√s . 2mZ .

These invariant masses are all independent subject tothe constraint (M1 + M2) ≤

√s and serve as the

most strongly discriminating observables between differ-ent signal hypothesis as well as between signal and back-ground. Note also that the 4e/4µ final state can be recon-structed in two different ways due to the identical finalstate interference. This is a quantum mechanical effectthat occurs at the amplitude level and thus both recon-structions are valid. The definitions M1 and M2 remainunchanged however.

The angular variables are defined as:

• Θ – The production angle between the momentumvectors of the lepton pair which reconstructs to M1

and the total 4` system momentum.

• θ1,2 – Polar angle of the momentum vectors ofe−, µ− in the lepton pair rest frame.

• Φ1 – The angle between the plane formed by theM1 lepton pair and the ‘production plane’ formed

5

out of the momenta of the incoming partons andthe momenta of the two lepton pair systems.

• Φ – The angle between the decay planes of the finalstate lepton pairs in the rest frame of the 4` system.

We group the angular variables as follows ~Ω =(Θ, cos θ1, cos θ2,Φ1,Φ). These angular variables are use-ful in aiding to distinguish different signal hypothesis andin particular between those with different CP proper-ties, as well as in discriminating signal from background.There are also additional production variables associatedwith the initial partonic state four momentum:

• ~pT – The momentum in the transverse direction.

• Y – Defined as the motion along the longitudinaldirection.

• φ – Defines a global rotation of the event in the 4`rest frame and in general does not aid greatly indiscriminating power.

Including Y and ~pT as observables in the likelihood in-creases the discriminating power of the golden chan-nel. However, including these production variables canintroduce large uncertainties since their spectra includes

parton distribution functions as well as NLO contribu-tions which should be included. Thus, they are oftenintegrated out of the final likelihood [38, 39] and notused during parameter extraction. This reduces the po-tential discriminating power, but this is compensatedby the smaller systematic uncertainties one obtains bynot including them in the likelihood (or averaging overthem). We will discuss more below how in our frameworkone can easily either include them in the final likelihoodor average over them to mitigate the effects of the uncer-tainties associated with these variables. However as withφ, it is crucial to include them when performing the con-volution with the transfer function in order to obtain theproper detector level likelihood. These variables exhaustthe twelve possible center of mass observables availablein the golden channel.

B. Parameterization of Scalar-Tensor Couplings

Assuming only Lorentz invariance, the general cou-plings of a spin-0 particle to two spin-1 vector bosonscan be parametrized in terms of effective couplings bythe following tensor structure,

Γµνi =i

v

(Ai1m2

Zgµν +Ai2(kν1k

µ2 − k1 · k2g

µν) +Ai3εµναβk1αk2β +(Ai4(

k21 + k2

2

m2Z

) +Ai5(s

m2h

))m2Zg

µν

), (2)

where in the golden channel i = ZZ,Zγ, γγ. The vari-ables k1 and k2 represent the four momentum of the in-termediate vector bosons with v the Higgs vacuum ex-pectation value (vev) which we have chosen as our over-all normalization. The Ain are dimensionless and in prin-ciple arbitrary complex form factors with possible mo-mentum dependence (or more precisely a s, k2

1, k22 depen-

dence) making Eq.(2) completely general. Note that thetensor structure forAi5 is only distinguishable fromAi1 for

off-shell Higgs decays as discussed in [36]. For a purelyStandard Model Higgs we have AZZ1 = 2 at tree levelwhile all other effective couplings are generated at higherloop order and at most O(. 10−2).

Of course it is often possible to expand the Ain in apower series of momenta keeping only the leading (con-stant) terms. By keeping the leading terms in this expan-sion there is a one-to-one mapping from this vertex ontothe effective Lagrangian2,

L ⊃ 1

4v

(2AZZ1o m

2ZhZ

µZµ +AZZ2o hZµνZµν +AZZ3o hZ

µνZµν − 4AZZ4o hZµZµ − 2AZZ5o (

mZ

mh)2hZµZ

µ

+ 2AZγ2o hFµνZµν + 2AZγ3o hF

µνZµν +Aγγ2o hFµνFµν +Aγγ3o hF

µν Fµν

), (3)

where the Aino represent the leading, momentum inde-pendent coefficients and electromagnetic gauge invari-

ance requires AZγ,γγ1o = AZγ,γγ4o = AZγ,γγ5o = 0. We have

2 This lagrangian has been implemented [56] into the Feyn-Rules/Madgraph [57, 58] framework for validation purposes.

defined Zµ and Aµ as the Z and photon fields respec-tively while Vµν = ∂µVν − ∂νVµ are the usual bosonicfield strengths and the dual field strengths are defined as

Vµν = 12εµνρσV

ρσ. The effective lagrangian in Eq.(3) iscomposed of the leading terms in a derivative expansion(up to two derivatives) and is useful for parametrizingpotentially large new physics effects generated by loops

6

of heavy particles and a convenient framework for assess-ing the potential sensitivity to the leading operators [37]involving photons and Z bosons.

Thus, although Eq.(2) is a redundant parameterizationof the tensor structure, it is a convenient, yet more gen-eral, parametrization for fitting to effective Lagrangianparameters that might be generated in various modelsat dimension five or less as in Eq.(3). The parameteriza-tion in Eq.(2) can of course be mapped, with appropri-ate translation of the parameters, onto Lagrangians withdimension greater than five or to an underlying dimen-sion six lagrangian in a theory of electroweak symmetrybreaking such as in the Standard Model. We will workexplicitly with the vertex in Eq.(2) which has been usedto calculate the fully differential cross section for h→ 4`and when performing parameter extraction, but otherparameterizations can be easily accommodated.

This flexibility of parameterization also allows for othernew physics, such as exotic Higgs decays involving exoticfermions or vector bosons [48], to be easily included inthe framework. Furthermore, by using this parameteriza-tion, explicit computations of either Standard Model ornew physics loop effects which would generate these mo-mentum dependent form factors can easily be includedinto the framework. This allows for the ability to in prin-ciple extract the parameters from whichever underlyingtheory is responsible for generating them. We leave amore detailed investigation of these loop effects to on-going work [59].

C. Signal and Background FullyDifferential Cross Sections

In the case of signal we have computed analytically thefully differential cross section in the observables describedin Sec.III A for the process h→ ZZ+Zγ+γγ → 4` usingthe parameterization in Eq.(2). We have included all pos-sible interference effects between tensor structures as wellas identical final states in the case of 4e/4µ. For the ir-reducible background we have computed analytically theprocess qq → ZZ + Zγ + γγ → 4` which includes thes-channel (resonant) 4` process as well as the t-channel(diboson production) 4` process and again have includedall possible interference effects. All vector bosons are al-lowed to be on or off-shell and we do not distinguishbetween them in what follows. The details of these cal-culations can be found in [19, 35, 37, 43] along with thevalidation procedures and studies of the distributions aswell as the various interference effects. We have combinedthese analytic expressions with functions parametrizingthe production spectra and implemented them into ouranalysis framework.

We note that it is important to include all possibleHiggs couplings including the Zγ and γγ contributionsin the signal differential cross section since the Higgs ap-pears to be mostly Standard Model-like [60] and we areprimarily searching for small anomalous deviations from

the Standard Model prediction. Thus when attemptingto extract specific couplings we must be sure that onesmall effect is not being mistaken for another. This isparticularly relevant since many of the couplings may becorrelated with one another.

Furthermore, it has been shown recently [37] that for‘true’ points near the Standard Model, the greatest sen-sitivity to the anomalous couplings (non AZZ1o ) is for theZγ and especially γγ operators (see Eq.(3)). Including allpossible couplings and doing a simultaneous fit ensuresthat we minimize the possibility of misinterpretation orof introducing a bias when attempting to extract thesecouplings. Searching for these small effects is also why itis important to include the interference effects betweenthe identical final state leptons as well as the relevantdetector effects and background.

We also comment that in principal there are NLO con-tributions to the h → 4` decay processes, but these areexpected to be small at ∼ 125 GeV [49, 50] and notrelevant until higher precision is obtained once largerdata sets are gathered. Eventually however, these effectsshould be included and their implementation into ourframework is currently ongoing.

D. Combining Production and Decay

To be able to perform a fit for the effective Higgs cou-plings, we must first construct the fully differential crosssection for the observables as a function of the unde-termined parameters ( ~A). This differential cross sectionconsists of two components which we assume to be factor-ized: the parton level (‘decay’) differential cross sectionas discussed in Section III C, and the ‘production’ spec-trum. The full production plus decay fully differentialcross section can be expressed as the following,

P (~pT , Y, φ, s,M1,M2, ~Ω| ~A) = (4)

dWprod(s, ~pT , Y, φ)

dsdY d~pT dφ× dσ4`(s,M1,M2, ~Ω| ~A)

dM21 dM

22 d~Ω

,

where, since the Higgs is a spin-0 particle, we can ex-plicitly assume that the decay process can be factorizedfrom the production mechanism. For the background thisexplicit factorization does not occur, but still turns outto be an adequate approximation [41] especially if the~pT and Y variables are averaged over once the convo-lution is performed. The parton level fully differentialcross section (σ4`) is treated as being at fixed s whereone obtains the input s value from the production spec-trum (Wprod). The production spectrum for the signaland background depend on the parton distribution func-tions and can not be computed analytically. For the sig-nal which we assume decays on-shell, the s spectrum istaken to be a delta function centered at m2

h, which fora Standard Model Higgs at 125 GeV is an excellent ap-proximation. Note however that this assumption can berelaxed in our framework to consider more general s spec-

7

tra as would be found for example in the case of a newheavy scalar with a large width.

E. Comments Production Spectra

Here we discuss how the Wprod(s, ~pT , Y, φ) productionspectrum in Eq.(4) is obtained. This function involveshigher order effects as well as parton distribution func-tions and thus can not be computed analytically. To in-clude them in the total differential cross sections thereare various options. One can in principal generate enoughMonte Carlo events to accurately fill the full spectrum inthe (s, Y, ~pT , φ) variables. As this is computationally in-tensive we take an approximate approach in which weinterpolate analytic functions for the signal and back-ground from one or two dimensional projections gener-ated from the Madgraph [61] and POWHEG [62] MonteCarlo generators following a similar procedure as foundin [32]. Having an analytic parameterization for thesefunctions also allows for faster integration when imple-menting them into the convolution procedure describedabove. This procedure of interpolating the one or two di-mensional projections neglects correlations between theproduction variables. However, since in the signal casethere is an explicit factorization between production anddecay, the effects of this approximation on parameter ex-traction in our analysis are small.

In addition, to mitigate these effects further, one canalways average over Y and ~pT as well as fit to ratiosof couplings while taking the Higgs mass and overallnormalization as input from the total rate (so called‘geolocating’ [36]) as was done in a recent implementa-tion of our framework into a CMS experimental analy-sis [38, 39]. Note however, the overall normalization andHiggs mass can in principle be extracted in our frame-work, but as this requires extra careful treatment of theproduction spectra and additional backgrounds we defera discussion of this to future work.

F. Comments on Additional Backgrounds

For the background there are also the higher ordercontributions such as the gg → 4` and Z + X pro-cesses. These make up the parts of the likelihood whichcan not currently be included in the convolution inte-gral since a sufficiently accurate analytic parameteri-zation has yet to be obtained. Thus for these compo-nents we must resort to constructing large ‘look-up’ ta-bles via Monte Carlo generation. Again, the effects ofthe necessary binning can be mitigated through a lin-ear multi-dimensional interpolation technique [41]. Ad-ditionally, there will be systematic uncertainties associ-ated with these components. Fortunately, the gg → 4`component only makes up ∼ 3− 5% relative to qq → 4`around 125 GeV [51]. The Z + X background on theother hand does make up a sizable contribution of the

total background which is comparable to, but smallerthan, the largest qq component (see Table 2 in [38] orTable 3 in [39]). This component however can in princi-ple be reduced further in the future by requiring morestringent lepton acceptance criteria once more data iscollected. For now the use of the linearly interpolated‘look-up’ templates and associated systematics is foundto be sufficient. Once the templates are built, includingthese components in the final likelihood is straightfor-ward as we briefly sketch in Sec. V B and discussed inmore detail in [38, 39, 41].

IV. THE ‘DETECTOR LEVEL’ PDF

The convolution integral in Eq.(1) is conceptuallystraightforward, but in practice is challenging to perform,both for computational and algorithmic reasons. The keyassumption which makes it possible is that the directionof lepton momenta are measured with infinite precisionwhich at CMS and ATLAS is a very good approxima-tion. This allows us, through a change of variables, to re-duce the 12-dimensional integral into a more manageable4-dimensional integral over the four energies of the lep-tons which are altered by detector resolution effects. Typ-ically this 4-dimensional integral is done using MonteCarlo techniques [31], thus losing the advantage of hav-ing analytic control over the likelihood or assuming thatthe resolution effects can also be neglecting making theintegral trivial. As discussed in Sec. II we instead performthis integration explicitly using a combination of numer-ical and analytic methods which allow us to maintainarbitrarily high precision at each step involved. Thereare a number of technical details involved in this proce-dure which are beyond the scope of this ‘overview’ of theframework, but the details can be found in [40, 41]. Weinstead briefly sketch an overview of the convolution in-tegral and show its validation.

A. Transforming from CM Basisto Lepton Smearing Basis

Beginning from Eq.(1) we first discuss the construc-tion of the background detector level pdf. The construc-tion of the signal will be discussed separately as thereis a subtle, but important, difference in performing theconvolution. Since there are no undetermined parametersin the background the generator and detector-level (un-normalized) differential cross sections are given simply by

PB( ~XG) and PB( ~XR) respectively and the convolutionintegral can be written schematically as,

PB( ~XR) =

∫PB( ~XG)T ( ~XR| ~XG)d ~XG. (5)

The set of variables ~X ≡ (~pT , Y, φ, s,M1,M2, ~Ω) exhauststhe twelve degrees of freedom (note that ~pT has 2 com-

ponents and ~Ω contains 5 angles) available to the four

8

(massless) final state leptons. The differential volume el-

ement is given by d ~X = dsdM21 dM

22 d~Ω · d~pT dY dφ.

To perform this convolution with the transfer functionwe must first transform to the basis in which the detectorsmearing of the lepton momenta is parameterized. Thisrequires transforming from the basis of the twelve cen-ter of mass variables defined in Sec. III A to the threemomentum basis for the four final state leptons. In thisbasis the lepton three momenta ~pi can be decomposed interms of the component of the lepton momentum parallelto the direction (pi||) of motion and the two components

perpendicular to the direction of motion (~pi⊥) (which arezero at generator level). We then make the assumptionthat detector smearing will only affect parallel compo-nents pi|| while the perpendicular components ~pi⊥ areleft invariant. Note that this assumption is equivalentto assuming angular resolution effects due to detectorsmearing can be neglected, which is an excellent approx-imation for the LHC detectors [53–55]. In the (pi||, ~pi⊥)basis only the transfer function associated with pi|| isnon-trivial while the one associated with the perpendic-ular components can be represented simply as a deltafunction for each perpendicular direction, thus allowingfor trivial integration over the eight ~pi⊥ variables.

With these assumptions the integral in Eq.(5) can thenbe represented as follows,

PB( ~XR) =

∫PB( ~XG)T (~c |~PG)

× |JB |dc1dc3dMG1

2dMG

2

2, (6)

where we have defined the lepton momenta ‘smearing fac-tors’ ci = pi

R|| /pi

G|| and,

~c = (c1, c2, c3, c4), ~PG = (~p1G, ~p2

G, ~p3G, ~p4

G). (7)

We have also defined |JB | which is the 12 × 12 Jaco-bian which parametrizes the (non-linear) transformationthat takes us from the center of mass basis to the lep-ton smearing basis. The construction of this Jacobian ishighly non trivial and requires a combination of analyticand numerical techniques which are beyond the scopeof this overview, but the relevant details can be foundin [40, 41].

We thus see in Eq.(6) that what started out as atwelve dimensional integral has been reduced to a muchmore manageable integration over four variables. The de-tails and validation of this four dimensional integration,which is done using a recursive numerical integrationtechnique [44] can also be found in [40, 41].

To construct the detector level signal differential crosssection (again un-normalized), which is now a function of

the effective couplings ~A, we follow the same procedureas for the background starting from,

PS( ~XR| ~A) =

∫PS( ~XG| ~A)T ( ~XR| ~XG)d ~XG. (8)

We again use the assumptions which allow us to performthe trivial integration over the eight ~pi⊥ variables, but

instead transform to the following integration basis

PS( ~XR| ~A) =

∫PS( ~XG| ~A)T (~c |~PG)

× |JS |dc1dsGdM21

GdM2

2G

(9)

We now also use the fact that, as mentioned below Eq.(4),the s spectrum for the signal is ∝ δ(sG−m2

h) (where mh

is the generated Higgs mass), enabling us to perform theintegration over dsG as well. Thus, we have for the finalsignal detector level differential cross section,

PS( ~XR| ~A) =

∫PS( ~XG| ~A)T (~c |~PG) (10)

× |JS |dc1dM21

GdM2

2G∣∣∣sG=m2

h

,

where again |JS | represent the 12 × 12 Jacobian (whichis different from |JB |) taking us from the CM basis tothe lepton smearing basis. By using a delta function tomodel the width of the resonance, there is one less di-mension to integrate over as compared to the backgroundcase. While this makes it easier computationally in onerespect, an additional complication arises since we haveto integrate along a trajectory in which sG is kept con-stant. This places an additional constraint when perform-

ing the MG1

2, MG

22

integration which further complicatesmatters and must be properly taken into account. Ex-plicit details of this integration and its validation alongwith the derivation of the signal Jacobian |JS | in Eq.(10)are given in [40, 41].

B. Comments on Transfer Function

Detector response effects including effects from selec-tion inefficiency may be parameterized into transfer func-tions in the following way,

T (ci|~piG) = δ(~piR⊥ − ~pi

G⊥)

S(piR|| ; pi

G|| )× ε(~pi

R⊥, pi

R|| ), (11)

The response function S, parameterizes the probabilityfor a lepton with actual momentum ~p G

i to be recon-structed with momentum ~p R

i , while δ is the Dirac deltafunction in the perpendicular components, and ε is theselection efficiency. With typical lepton selection criteriaemployed by the LHC experiments [54, 55], it is a goodapproximation that each lepton is independent. Thus, thefull transfer function for the event may be written as:

T (~c |~PG) =

4∏i=1

T (ci|~piG). (12)

We treat T (~c |~PG) as a function of ~c which takes the

generator level momenta ~PG as input. The only effectof imperfect momentum measurement on the productionspectra is to provide a small smearing of the ~pT spectrumfor the four lepton system. We can mitigate the effects

9

Detector Level Background Projections

1M40 50 60 70 80 90 100110120

1d

Mσd

σ1

0

0.05

0.1

0.15

0.2

0.25

0.3

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

|T

p|0 50 100 150 200 250

| Tpd

|σd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.22

HM115 120 125 130 135

Hd

Mσd

σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

Y-2 -1 0 1 2

d Yσ

d σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1M40 50 60 70 80 90 100110120

1d

Mσd

σ1

0

0.05

0.1

0.15

0.2

0.25

0.3

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

|T

p|0 50 100 150 200 250

| Tpd

|σd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.22

HM115 120 125 130 135

Hd

Mσd

σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

Y-2 -1 0 1 2

d Yσ

d σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1M40 50 60 70 80 90 100110120

1d

Mσd

σ1

0

0.05

0.1

0.15

0.2

0.25

0.3

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

|T

p|0 50 100 150 200 250

| Tpd

|σd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.22

s115 120 125 130 135

sd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

Y-2 -1 0 1 2

d Yσ

d σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1M40 50 60 70 80 90 100110120

1d

Mσd

σ1

0

0.05

0.1

0.15

0.2

0.25

0.3

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

HTP

0 50100150200250300350400450500

H Td

Pσd

σ1

00.05

0.10.15

0.20.25

0.30.35

0.4

HM115 120 125 130 135

Hd

Mσd

σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

HY-2 -1 0 1 2

Hd

Yσd

σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

FIG. 2. Projections of the Y , |~pT |,√s ≡ M4` and cos Θ

(see Sec. III A for definitions) spectra showing validation ofthe convolution described in Eq.(5) for the background. Inblue we show the ‘boosted’ Madgraph sample with acceptancecuts and detector smearing applied while in red we show pro-jections from our differential cross section after the convolu-tion integration.

of the smearing by averaging over the production spectrawhen performing parameter extractions. Further detailson the construction and implementation of the transferfunction can be found in [38, 39, 41].

C. Validation of Convolution Integral

As validation of the convolution integral we first showin Fig. 2-Fig. 5 projections for signal and background. Wecompare in these plots the distributions for a Madgraphsample which has had detector smearing and acceptanceeffects applied to it versus projections generated from ourdetector level differential cross sections obtained after theconvolution described above.

We have obtained the signal and background pro-duction spectrum for the (s, ~pT , Y, φ) variables fromPOWHEG and boosted the Madgraph events and thosefrom our projections accordingly. We have used the in-terpolation procedure described in Sec. III E to build theproduction spectra for the signal and background differ-ential cross sections and combined them with the analyticexpressions for the h→ 4` and qq → 4` processes. For thesignal we show the tree level Standard Model point whereAZZ1 = 2 and all other couplings are set to zero. For bothsignal and background we show only the 2e2µ final state,but results for 4e (or 4µ) are found in [38, 39, 41].

A further validation beyond these projections howeveris to look at the likelihoods (the differential cross sectionevaluated for a set of observables) for both the signal and

Detector Level Background Projections

1M40 50 60 70 80 90 100110120

1d

Mσd

σ1

0

0.05

0.1

0.15

0.2

0.25

0.3

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

HTP

0 50100150200250300350400450500

H Td

Pσd

σ1

00.05

0.10.15

0.20.25

0.30.35

0.4

HM115 120 125 130 135

Hd

Mσd

σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

HY-2 -1 0 1 2

Hd

Yσd

σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1M40 50 60 70 80 90 100110120

1d

Mσd

σ1

0

0.05

0.1

0.15

0.2

0.25

0.3

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

HTP

0 50100150200250300350400450500

H Td

Pσd

σ1

00.05

0.10.15

0.20.25

0.30.35

0.4

HM115 120 125 130 135

Hd

Mσd

σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

HY-2 -1 0 1 2

Hd

Yσd

σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1M40 50 60 70 80 90 100110120

1d

Mσd

σ1

0

0.05

0.1

0.15

0.2

0.25

0.3

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

HTP

0 50100150200250300350400450500

H Td

Pσd

σ1

00.05

0.10.15

0.20.25

0.30.35

0.4

HM115 120 125 130 135

Hd

Mσd

σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

HY-2 -1 0 1 2

Hd

Yσd

σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1M40 50 60 70 80 90 100110120

1d

Mσd

σ1

0

0.05

0.1

0.15

0.2

0.25

0.3

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

HTP

0 50100150200250300350400450500

H Td

Pσd

σ1

00.05

0.10.15

0.20.25

0.30.35

0.4

HM115 120 125 130 135

Hd

Mσd

σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

HY-2 -1 0 1 2

Hd

Yσd

σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1M40 50 60 70 80 90 100110120

1d

Mσd

σ1

0

0.05

0.1

0.15

0.2

0.25

0.3

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

HTP

0 50100150200250300350400450500

H Td

Pσd

σ1

00.05

0.10.15

0.20.25

0.30.35

0.4

HM115 120 125 130 135

Hd

Mσd

σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

HY-2 -1 0 1 2

Hd

Yσd

σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1M40 50 60 70 80 90 100110120

1d

Mσd

σ1

0

0.05

0.1

0.15

0.2

0.25

0.3

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

HTP

0 50100150200250300350400450500

H Td

Pσd

σ1

00.05

0.10.15

0.20.25

0.30.35

0.4

HM115 120 125 130 135

Hd

Mσd

σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

HY-2 -1 0 1 2

Hd

Yσd

σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

FIG. 3. Projections of the M1, M2, cos θ1, cos θ2, Φ andΦ1 (see Sec. III A for definitions) spectra showing validationof the convolution described in Eq.(5) for the background. Inblue we show the ‘boosted’ Madgraph sample with acceptancecuts and detector smearing applied while in red we show pro-jections from our differential cross section after the convolu-tion integration.

background which contain the full correlations betweenthe different variables. We show these in Fig. 6 for a CMS-like phase space and a very large number of events. Toobtain these likelihoods we have evaluated our detector-level differential cross section with the Madgraph samplewhich has had detector smearing and acceptance effectsapplied and plotted it on top of the result of evaluatingour detector-level differential cross section with eventsgenerated from the expression itself. We find the agree-ment between the two results to be very good. Furtherdetails are found in the accompanying documents [40, 41].

These plots should not be taken as validation of thecomplete detector-level differential cross sections whichmust be validated with full simulation and data. They aremeant only to show the validation of the convolution pro-cedure as well as the construction of the generator-leveldifferential cross sections including the analytic compu-tations. Complete validations of the full detector levellikelihoods including the various production and back-ground effects can be found in [38, 39, 41].

10

Detector Level Signal Projections

1M40 50 60 70 80 90 100 110

1d

Mσd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

|T

p|0 50 100 150 200 250

| Tpd

|σd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

s115 120 125 130 135

sσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

Y-2 -1 0 1 2

d Yσ

d σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

1M40 50 60 70 80 90 100 110

1d

Mσd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

|T

p|0 50 100 150 200 250

| Tpd

|σd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

s115 120 125 130 135

sσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

Y-2 -1 0 1 2

d Yσ

d σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

1M40 50 60 70 80 90 100 110

1d

Mσd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

|T

p|0 50 100 150 200 250

| Tpd

|σd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

s115 120 125 130 135

sσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

Y-2 -1 0 1 2

d Yσ

d σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

1M40 50 60 70 80 90 100 110

1d

Mσd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

|T

p|0 50 100 150 200 250

| Tpd

|σd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

s115 120 125 130 135

sσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

Y-2 -1 0 1 2

d Yσ

d σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

FIG. 4. Projections of the Y , |~pT |,√s ≡ M4` and cos Θ

(see Sec. III A for definitions) spectra showing validation ofthe convolution described in Eq.(8) for the tree level SM sig-nal. In blue we show the ‘boosted’ Madgraph sample withacceptance cuts and detector smearing applied while in redwe show projections from our differential cross section afterthe convolution integration.

V. CONSTRUCTION OF LIKELIHOODSAND PARAMETER EXTRACTION

With the detector level differential cross sections ob-tained in Eq.(6) and Eq.(10) in hand we can then go on toconstruct the full likelihood for a particular dataset. Be-fore doing so, we must properly normalize the back-ground and signal differential cross sections by perform-

ing the full integration over all twelve reconstructed ~Xvariables where from now on we drop the superscriptR since we only deal with detector level observablesin what follows. In this section we present a schematicoverview of the normalization procedure. We also at thisstage briefly discuss averaging over the production vari-ables (Y, ~pT , φ) and the implementation of systematic un-certainties through the use of nuisance parameters inthe likelihood functions. Further details can be foundin [40, 41].

A. Normalization of Background and Signal

One can reduce the effects of production uncertain-ties by averaging over the detector level production vari-ables (Y, ~pT , φ). This is straightforwardly done for thebackground differential cross sections by the following 4-dimensional integration,

PB(s,M1,M2, ~Ω) =

∫PB( ~X)dY d~pT dφ. (13)

Detector Level Signal Projections

1M40 50 60 70 80 90 100 110

1d

Mσd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

|T

p|0 50 100 150 200 250

| Tpd

|σd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

s115 120 125 130 135

sσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

Y-2 -1 0 1 2

d Yσ

d σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

1M40 50 60 70 80 90 100 110

1d

Mσd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

|T

p|0 50 100 150 200 250

| Tpd

|σd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

s115 120 125 130 135

sσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

Y-2 -1 0 1 2

d Yσ

d σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

1M40 50 60 70 80 90 100 110

1d

Mσd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

|T

p|0 50 100 150 200 250

| Tpd

|σd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

s115 120 125 130 135

sσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

Y-2 -1 0 1 2

d Yσ

d σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

1M40 50 60 70 80 90 100 110

1d

Mσd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

|T

p|0 50 100 150 200 250

| Tpd

|σd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

s115 120 125 130 135

sσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

Y-2 -1 0 1 2

d Yσ

d σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

1M40 50 60 70 80 90 100 110

1d

Mσd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

|T

p|0 50 100 150 200 250

| Tpd

|σd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

s115 120 125 130 135

sσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

Y-2 -1 0 1 2

d Yσ

d σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

1M40 50 60 70 80 90 100 110

1d

Mσd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

2M10 20 30 40 50 60 70

2d

Mσd

σ1

0

0.02

0.04

0.06

0.08

0.1

|T

p|0 50 100 150 200 250

| Tpd

|σd

σ1

00.020.040.060.08

0.10.120.140.160.18

0.20.220.24

s115 120 125 130 135

sσd

σ1

0

0.02

0.04

0.06

0.08

0.1

0.12

Y-2 -1 0 1 2

d Yσ

d σ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Θcos -1 -0.5 0 0.5 1

Θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

2θcos -1 -0.5 0 0.5 1

2θd

cos σ

d σ1

0

0.01

0.02

0.03

0.04

0.05

1θcos -1 -0.5 0 0.5 1

1θd

cos σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

Φ0 1 2 3 4 5 6

Φd σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.04

1Φ0 1 2 3 4 5 6

1Φ

d σ

d σ1

00.005

0.010.015

0.020.025

0.030.035

0.040.045

FIG. 5. Projections of the M1, M2, cos θ1, cos θ2, Φ andΦ1 (see Sec. III A for definitions) spectra showing validationof the convolution described in Eq.(8) for the tree level SMsignal. In blue we show the ‘boosted’ Madgraph sample withacceptance cuts and detector smearing applied while in redwe show projections from our differential cross section afterthe convolution integration.

An overall volume factor is not shown because for thepurpose of likelihood maximization this constant factoris not relevant. What matters is that the relative nor-malization between all components in the likelihood isdone consistently. With this differential cross sections interms of the eight center of mass decay observables wecan obtain the overall normalization via a Monte Carlointegration procedure described in [40, 41],

NB =

∫PB(s,M1,M2, ~Ω)

× dsdM21 dM

22 d~Ω, (14)

which gives our final normalized background pdf as,

PB(s,M1,M2, ~Ω) = N−1B × PB(s,M1,M2, ~Ω). (15)

We have calculated the qq → 4` expression as a sum ofthe separate individual contributions [19, 35] making itpossible to easily perform the integration on each smallerpiece to obtain each normalization and then simply sumover them to obtain the overall normalization.

11

Detector Level Background Likelihood

log(PDF)-48 -46 -44 -42 -40 -38 -36 -34

a.u.

-510

-410

-310

-210

Detector Level Signal Likelihood

log(PDF)-68 -66 -64 -62 -60 -58 -56 -54 -52

a.u.

-510

-410

-310

-210

FIG. 6. Validation of the convolution integrals describedin Eq.(5) and Eq.(8). In blue we show the ‘boosted’ Madgraphsample with acceptance cuts and detector smearing appliedwhile in red we show projections from our differential crosssections after the convolution integration for the tree level SMsignal and background likelihood.

Similarly for the signal we have for the averaging over(Y, ~pT , φ) variables,

PS(s,M1,M2, ~Ω| ~A) =

∫PS( ~X| ~A)dY d~pT . (16)

To obtain the overall normalization in the signal case wefirst note that it is a function of the underlying param-

eters ~A defined in Eq.(2)). However, from the calcula-tion of the parton level differential cross section presentedin [19, 35] or from considering Eq.(2) it is clear (assum-

ing constant effective couplings) that PS(s,M1,M2, ~Ω| ~A)

is a sum over terms each of which is proportional toAinAj∗m . Thus we can write,

PS(s,M1,M2, ~Ω| ~A) =∑ij

∑nm

AinAj∗m × PS(s,M1,M2, ~Ω)ijnm, (17)

where PS(s,M1,M2, ~Ω)ijnm represents the individual dif-ferential cross sections with the couplings factoredout. The separate normalizations for each term can noweasily be obtained via,

N ijnm =

∫PS(s,M1,M2, ~Ω)ijnm

× dsdM21 dM

22 d~Ω, (18)

from which we can now obtain the total overall normal-ization for the signal pdf as,

NS( ~A) =∑ij

∑nm

AinAj∗m ×N ijnm. (19)

This gives finally for the normalized signal pdf,

PS(s,M1,M2, ~Ω| ~A) =

N−1S ( ~A)× PS(s,M1,M2, ~Ω| ~A). (20)

Since each N ijnm is computed, one does not need to com-

pute the normalization each time a new hypothesis for ~Ais constructed. The procedure outlined here also works on

more general polynomial functions of the parameters ~Awhich one finds after expanding potentially momentum-dependent form factors in powers of momenta. See [40]for this more general discussion.

Note also that if we take the Higgs mass as a fixed inputand only fit for ratios of parameters and not their overallnormalization, we do not need the absolute normalizationof the differential cross sections. It thus suffices to havethe relative normalization between the different compo-nents correct when performing the maximization. Thisfact greatly reduces the computational complexity. In-stead of propagating the full normalization and aligningunits correctly so that when one integrates over all 8 di-mensions unity is obtained, it is sufficient to do a MonteCarlo integration using a fixed sample size in a consis-tent and sufficiently large range. The meaning of the loglikelihood difference remains unchanged with this con-struction. Further details of the normalization procedurefor both signal and background are found in [40, 41].

B. Signal Plus Background pdfand Final Likelihood

With Eq.(15) and Eq.(20) in hand we can now buildthe signal plus background pdf from which the total like-lihood will be constructed. The signal plus background

12

pdf can be written as,

PS+B(O|F iB , ~A) =∑i

F iB × PiB(s,M1,M2, ~Ω)

+ (1−∑i

F iB)× PS(s,M1,M2, ~Ω| ~A). (21)

where O ≡ (s,M1,M2, ~Ω) is our final set of observablesto be used in the construction of the likelihood and F iBis the background fraction for a particular component,each of which must also be extracted.

The sum over background components is given by,∑i

F iB × PiB(O) = F qqB P qqB (O)

+ F ggB P ggB (O) + FZ+XB PZ+X

B (O), (22)

where P qqB (O) is the dominant qq → 4` component andis obtained via the convolution integral in Eq.(1). Thesub-dominant gg → 4` and Z + X components, givenby P ggB (O) and PZ+X

B (O) respectively, must be obtainedvia the linearly interpolated ‘look-up’ tables from largeMonte Carlo samples as discussed in Sec. III F.

We can now write the likelihood of obtaining a partic-ular dataset containing N events as,

L(F iB , ~A) =

N∏OPS+B(O|F iB , ~A). (23)

This likelihood can also be combined with an appropri-ate poisson weighting factor to account for the probabil-ity of observing a given number of events [38, 39, 41]. Inthe case of multiple final states (for example 4e, 4µ and2e2µ), we build the likelihood function and implementthe appropriate systematic uncertainties for each one sep-arately. We now briefly discuss the implementation of thesystematic uncertainties.

C. Including Systematic Uncertainties

Systematic uncertainties must be accounted for givenour imperfect knowledge of various aspects of the analy-sis procedure. The lepton momentum resolution, the sizeof the backgrounds, and the exact production spectra aresome important examples. For each of these systematicuncertainties we can associate an undetermined parame-ter which parametrizes our ignorance of the correspond-ing effect. Since we are not directly interested in theseparameters, but only use them to estimate our system-atic uncertainties, they are deemed nuisance parametersand are subsequently profiled over [38, 39].

This is done by generating alternative pdfs using dif-ferent values for the nuisance parameter of interest. Togive one important example, we generate pdfs with nar-rower or wider lepton response functions to parameterizeour knowledge of the lepton momentum resolution. If wedefine the nominal pdf to be P0(O) and the alternativeas P1(O), one can parameterize the dependence of the

likelihood on a nuisance parameter n by interpolatingbetween the nominal and the alternative pdfs as follows:

P(O|n) = (1− n)P0(O) + nP1(O)

= P0(O) + n [P1(O)− P0(O)] . (24)

It is instructive to observe that, for all values of n, thenormalization of the total pdf stays the same. Given theasymmetric nature of many systematic uncertainties, it ismore appropriate to generate many “check-points” alongthe axis of n and to do piece-wise interpolation with-out the need of worrying about the normalization. Non-central values of n are a priori disfavored, therefore onecan impose a prior on top of the interpolated likelihood:

P(O|n) = P(O|n)G(n), (25)

where G(n) is typically a Gaussian centered at the centralvalue of n. In the case of multiple systematic uncertain-ties, one can replace n by a vector of nuisance parameters~n, and the prior G(n) by G(~n). In general G(~n) is a mul-tivariate Gaussian-like function with primary axes whichare some combination of different nuisance parameter di-rections. However one can carefully define the nuisanceparameters such that correlations between them are neg-ligible. In this limit G(~n) can be written as the productof many Gaussian-like functions. This procedure for in-cluding systematic uncertainties has been implementedin a recent CMS analysis utilizing our framework [38, 39]and further details can be found in [41].

D. Comments on Parameter Extraction

As discussed in [13, 17, 32] the advantage of analyticapproaches is that the likelihood can be maximized for alarge set of parameters in the most optimal way with-out losing information. Our framework allows for the‘analytic’ nature of these approaches to be maintainedat detector level giving us the ability to perform fastand accurate multi-parameter fits for lagrangian param-eters directly from the data. This is possible once theconvolution in Eq.(1) is performed and after normaliza-tion of the signal and background pdfs allowing us to

obtain the full detector level likelihood L( ~A) for a par-ticular dataset. With the likelihood in hand a maximiza-tion procedure to find the global maximum can be per-formed to obtain the value of the parameters for whichthe likelihood is maximized. For this task we have incor-porated the well established MINUIT [63] function mini-mization/maximization code into our framework. We findexcellent rates of convergence and a high degree of stabil-ity in locating the global maximum of the likelihood aswell as accurate extraction of the parameters as demon-strated in [38–41] where more details can be found.

One important feature of the procedure is that thecomputationally intensive component of evaluating thelikelihood only needs to be done for the events in the finaldataset used in the fit for a given experiment. Therefore

13

the computationally expensive pieces can be calculatedon the computing grid prior to the analysis of the data,and the fit for parameter extraction itself is then com-pleted within a few seconds. This allows for a great dealof flexibility, including testing alternative parameteriza-tions, when fitting the undetermined parameters. Manyexamples of the types of parameter extractions which canbe done within our framework, both at generator and atdetector level, can be found in [35, 37–39, 41].

VI. SUMMARY AND CONCLUSIONS

In this study we build upon an earlier study [35] to con-struct a comprehensive analysis framework aimed at ex-tracting as much information as possible from the Higgsgolden channel. Our framework is based on a maximumlikelihood method constructed from analytic expressionsof the fully differential cross sections for the h→ 4` decayas well as the dominant irreducible qq → 4` backgroundwhich were computed in [19, 35]. As our main result, wehave constructed the full 12-dimensional detector levellikelihood utilizing all observables available in the goldenchannel. This allows us to perform parameter extractionof the various possible Higgs couplings, including generalCP odd/even admixtures and any possible phases.

The detector-level likelihood is obtained by the explicitconvolution of a transfer function, encapsulating the rele-vant detector effects, with the generator-level probabilitydensity formed out of the signal and background differen-tial cross sections. After performing this 12-dimensionalconvolution integral and its normalization we obtain aprobability density function from which we construct anun-binned detector-level likelihood which is a continuousfunction of the effective couplings.

In summary we have given broad overview of a frame-work optimized for extracting Higgs couplings in thegolden channel. We have sketched how the convolution isperformed and shown various validations as well as dis-cussed the principles and theoretical basis of the frame-work. Many of the technical details as well as results us-ing our framework can be found in [19, 35, 37, 38, 40,41]. This framework has already proved useful in a recentCMS analysis [38, 39] and can be used in the future ina variety of ways to study Higgs couplings in the goldenchannel using data obtained at the LHC and other futurecolliders.

Acknowledgments: The authors are grateful to Ar-tur Apresyan, Michalis Bachtis, Adam Falkowski, PatrickFox, Andrei Gritsan, Roni Harnik, Alex Mott, PavelNadolsky, Daniel Stolarski, Reisaburo Tanaka, NhanTran, Roberto Vega, and Felix Yu for helpful discus-sions as well as the ATLAS and CMS collaborations fortheir encouragement and interest in this work. R.V.M. issupported by the Fermilab Graduate Student Fellow-ship in Theoretical Physics and the ERC AdvancedGrant Higgs@LHC. Fermilab is operated by Fermi Re-search Alliance, LLC, under Contract No. DE-AC02-07CH11359 with the United States Department of En-ergy. Y.C. is supported by the Weston Havens Founda-tion and DOE grant DE-FG02-92-ER-40701. This work isalso sponsored in part by the DOE grant No. DE-FG02-91ER40684. R.V.M is also grateful to the CalTech physicsdepartment for their hospitality during which much ofthis work was done. This work used the Extreme Sci-ence and Engineering Discovery Environment (XSEDE),which is supported by National Science Foundation grantnumber OCI-1053575.

[1] G. Aad et al. (ATLAS Collaboration), Phys.Lett. B716,1 (2012), 1207.7214.

[2] S. Chatrchyan et al. (CMS Collaboration), Phys.Lett.B716, 30 (2012), 1207.7235.

[3] C. A. Nelson, Phys.Rev. D37, 1220 (1988).[4] A. Soni and R. Xu, Phys.Rev. D48, 5259 (1993), hep-

ph/9301225.[5] D. Chang, W.-Y. Keung, and I. Phillips, Phys.Rev. D48,

3225 (1993), hep-ph/9303226.[6] V. D. Barger, K.-m. Cheung, A. Djouadi, B. A.

Kniehl, and P. Zerwas, Phys.Rev. D49, 79 (1994), hep-ph/9306270.

[7] T. Arens and L. Sehgal, Z.Phys. C66, 89 (1995), hep-ph/9409396.

[8] S. Choi, . Miller, D.J., M. Muhlleitner, and P. Zerwas,Phys.Lett. B553, 61 (2003), hep-ph/0210077.

[9] C. Buszello, I. Fleck, P. Marquard, and J. van der Bij,Eur.Phys.J. C32, 209 (2004), hep-ph/0212396.

[10] R. M. Godbole, . Miller, D.J., and M. M. Muhlleitner,JHEP 0712, 031 (2007), 0708.0458.

[11] V. Kovalchuk, J.Exp.Theor.Phys. 107, 774 (2008).

[12] Q.-H. Cao, C. Jackson, W.-Y. Keung, I. Low, and J. Shu,Phys.Rev. D81, 015010 (2010), 0911.3398.

[13] Y. Gao, A. V. Gritsan, Z. Guo, K. Melnikov, M. Schulze,et al., Phys.Rev. D81, 075022 (2010), 1001.3396.

[14] A. De Rujula, J. Lykken, M. Pierini, C. Rogan, andM. Spiropulu, Phys.Rev. D82, 013003 (2010), 1001.5300.

[15] J. S. Gainer, K. Kumar, I. Low, and R. Vega-Morales,JHEP 1111, 027 (2011), 1108.2274.

[16] B. Coleppa, K. Kumar, and H. E. Logan (2012),1208.2692.

[17] S. Bolognesi, Y. Gao, A. V. Gritsan, K. Melnikov,M. Schulze, et al. (2012), 1208.4018.

[18] D. Stolarski and R. Vega-Morales, Phys.Rev. D86,117504 (2012), 1208.4840.

[19] Y. Chen, N. Tran, and R. Vega-Morales, JHEP 1301,182 (2013), 1211.1959.

[20] R. Boughezal, T. J. LeCompte, and F. Petriello (2012),1208.4311.

[21] A. Belyaev, N. D. Christensen, and A. Pukhov (2012),1207.6082.

[22] P. Avery, D. Bourilkov, M. Chen, T. Cheng, A. Drozdet-

14

skiy, et al. (2012), 1210.0896.[23] J. M. Campbell, W. T. Giele, and C. Williams (2012),

1205.3434.[24] J. M. Campbell, W. T. Giele, and C. Williams (2012),

1204.4424.[25] S. Chatrchyan et al. (CMS Collaboration), JHEP 1204,

036 (2012), 1202.1416.[26] S. Chatrchyan et al. (CMS Collaboration), Phys.Lett.

B716, 30 (2012), 1207.7235.[27] S. Chatrchyan et al. (CMS Collaboration),

Phys.Rev.Lett. 110, 081803 (2013), 1212.6639.[28] T. Modak, D. Sahoo, R. Sinha, and H.-Y. Cheng (2013),

1301.5404.[29] Y. Sun, X.-F. Wang, and D.-N. Gao (2013), 1309.4171.[30] J. S. Gainer, J. Lykken, K. T. Matchev, S. Mrenna, and

M. Park, Phys.Rev.Lett. 111, 041801 (2013), 1304.4936.[31] P. Artoisenet, P. de Aquino, F. Demartin, R. Frederix,

S. Frixione, et al. (2013), 1306.6464.[32] I. Anderson, S. Bolognesi, F. Caola, Y. Gao, A. V. Grit-

san, et al. (2013), 1309.4819.[33] M. Chen, T. Cheng, J. S. Gainer, A. Korytov, K. T.

Matchev, et al. (2013), 1310.1397.[34] G. Buchalla, O. Cata, and G. D’Ambrosio (2013),

1310.2574.[35] Y. Chen and R. Vega-Morales, JHEP 1404, 057 (2014),

1310.2893.[36] J. S. Gainer, J. Lykken, K. T. Matchev, S. Mrenna, and

M. Park (2014), 1403.4951.[37] Y. Chen, R. Harnik, and R. Vega-Morales,

Phys.Rev.Lett. 113, 191801 (2014), 1404.1336.[38] Tech. Rep. CMS-PAS-HIG-14-014, CERN, Geneva

(2014), CMS Collaboration.[39] V. Khachatryan et al. (CMS Collaboration) (2014),

1411.3441.[40] Y. Chen, E. Di Marco, J. Lykken, M. Spiropulu, R. Vega-

Morales, et al. (2014), 1410.4817.[41] Y. Chen (2014), Precision measurement of the 125 GeV

Higgs boson discovered in proton-proton collisions at√s = 7, 8 TeV with the CMS detector at the LHC, The-

sis in preparation for P.h.D. at California Institute ofTechnology.

[42] Y.Gao (2013), Talk given at ATLAS Higgs WG Work-shop, https://indico.cern.ch/event/280998/session/5/contribution/29/material/slides/0.pdf.

[43] R. Vega-Morales (2013), The Higgs Boson as a Windowto Beyond the Standard Model, Thesis in fulfillment ofP.h.D requirements, Northwestern University.

[44] B. Bradie, A friendly introduction to numerical analy-sis: with C and MATLAB materials on website; int. ed.(Prentice-Hall, Englewood Cliffs, NJ, 2006).

[45] P. Artoisenet and O. Mattelaer, PoS CHARGED2008,025 (2008).

[46] P. Artoisenet, V. Lemaitre, F. Maltoni, and O. Mattelaer,JHEP 1012, 068 (2010), 1007.3300.

[47] P. Artoisenet, P. de Aquino, F. Maltoni, and O. Matte-laer (2013), 1304.6414.

[48] A. Falkowski and R. Vega-Morales (2014), 1405.1095.[49] A. Bredenstein, A. Denner, S. Dittmaier, and M. Weber,

Phys.Rev. D74, 013004 (2006), hep-ph/0604011.[50] A. Bredenstein, A. Denner, S. Dittmaier, and M. Weber,

Nucl.Phys.Proc.Suppl. 160, 131 (2006), hep-ph/0607060.[51] N. Adam, T. Aziz, J. Andersen, A. Belyaev, T. Binoth,

et al., pp. 215–289 (2008), 0803.1154.[52] J. Neyman and E. S. Pearson (1933), Philosophical

Transactions of the Royal Society of London. Series AVol. 231, pp. 289-337.

[53] CMS (2006), CMS Physics: Technical Design ReportVolume 1: Detector Performance and Software.

[54] CMS (2013), https://twiki.cern.ch/twiki/bin/view/CMSPublic/EGMElectronsMoriond2013.

[55] CMS (2013), Electron performance with 19.6 fb−1 of datacollected at

√s = 8 TeV with the CMS detector.

[56] Y. Chen and R. Vega-Morales (2013), Website under con-struction:http://yichen.me/project/GoldenChannel/.

[57] N. D. Christensen and C. Duhr, Comput.Phys.Commun.180, 1614 (2009), 0806.4194.

[58] J. Alwall, P. Demin, S. de Visscher, R. Frederix, M. Her-quet, et al., JHEP 0709, 028 (2007), 0706.2334.

[59] Y. Chen, D. Stolarski, and R. Vega-Morales (2014), Workin preparation.

[60] A. Falkowski, F. Riva, and A. Urbano (2013), 1303.1812.[61] J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer, and

T. Stelzer, JHEP 1106, 128 (2011), 1106.0522.[62] T. Melia, P. Nason, R. Rontsch, and G. Zanderighi,

JHEP 1111, 078 (2011), 1107.5051.[63] F. James (1994), MINUIT Function Minimization and

Error Analysis: Reference Manual Version 94.1.

8D Likelihood Effective Higgs Couplings Extraction Framework in h → 4

Documents