Particle Physics 2

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Particle Physics 2

U23525, Particle Physics, Year 3 University of Portsmouth, 2013 - 2014

Prof. Glenn Patrick

𝝁𝑱 /𝝍

𝒁 𝟎

𝑾 ±

𝜸 𝑯𝟎 𝝉𝝂𝒆

𝒆𝝓 𝝎

𝑩𝑲 +¿¿𝒈

𝒒 ~𝝌

𝝂𝝁

𝝂𝝉

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Last LectureParticle Physics 1

Course OutlinePreliminaries - AssessmentPreliminaries - BooksPreliminaries - Course MaterialParticle Physics, Cosmology & Particle AstrophysicsNatural UnitsRationalised Heaviside-Lorentz EM UnitsSpecial Relativity and Lorentz InvarianceMandelstam Variables (s, t and u)Crossing Symmetry and s, t & u ChannelsSpin and Spin Statistics Theorem – Fermions and BosonsAddition of Angular MomentumNon-Relativistic Quantum Mechanics (Schrödinger Equation)Relativistic Quantum Mechanics (Klein-Gordon Equation)Feynman-Stückelberg Interpretation of Negative Energy States

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Today’s Plan Particle Physics 2

Dirac EquationDirac Interpretation of Negative StatesStückelberg & Feynman InterpretationsDiscovery: Positron & e+e- Pair ProductionDecays – LifetimesScattering – Cross-sectionsFeynman DiagramsQuantum Electrodynamics (QED)Higher Order DiagramsLEP exampleRunning Coupling Constant (QED RenormalisationElectron/Positron Annihilation & precisionLEP electroweak example

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Correction - Assessment40 hours of lectures across two teaching blocks

plus 8 hours of tutorial classes.

The main aim is to improve your understanding of fundamental physics.

However, we cannot forget the small matter of your degree….

1 Final written examination (2 hours) – 80%2 Coursework questions and problems – 20%

Main thing is that you enjoy the course. We will try and focus on understanding the underlying concepts. Extra material/maths shown mainly to aid understanding. Guidance will be given over essential knowledge needed for exam.

Not 3 hours as I said last week.

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Corrections - Timetable There is NO lecture this afternoon at 15:00 – 17:00. The

timetable was wrong.

There are also 2 extra weeks added in March to the end of teaching block 2 (weeks 34 and 35).

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Moodle Site

Slides - pptxSlides - pdf

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2013 Nobel Prize – Particle Physics

The Nobel Prize in Physics 2013 was awarded jointly to François Englert and Peter W. Higgs"for the theoretical discovery of a mechanism that contributes to our understanding of the

origin of mass of subatomic particles, and which recently was confirmed through the discovery of the predicted fundamental particle, by the ATLAS

and CMS experiments at CERN's Large Hadron Collider”.

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Dirac EquationThe Klein Gordon equation was believed to be sick, but today

we now understand it was telling us something about antiparticles.

To try to solve the problem of –ve energy solutions, Dirac: Wanted an equation first order in . Started by assuming a Hamiltonian, which is local and

linear in p and of the form:

mPPPmpHD 332211)(To also make the equation relativistically covariant, it has to be linear in :

)( mit

i

Solutions of this equation were also required to be solutions of the Klein Gordon equation.

Only true if:

Coefficients do not commute so they cannot be numbers – require 4 matrices.

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Dirac EquationIn two dimensions, a natural set of matrices would be the Pauli spin matrices

(from atomic physics for electron spin):

0110

1

00

2 ii

10

013

00

10

01

but there is no suitable fourth anti-commuting matrix for β.

Block matrix (2 x 2)

2 x 2 identity matrix

Convenient choice is to instead use the Dirac-Pauli representation:

0001001001001000

1

000000000

000

2

ii

ii

001100011000

0100

3

1011010100100001

These can be abbreviated as:(each element is a 2 x 2 matrix)

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Dirac EquationNow that and are matrices, the equation makes no sense unless the wave function is itself a matrix with four rows and one column.This is the Dirac Spinor:(a four componentwavefunction)

)( mit

i

),(),(),(),(

),(

4

3

2

1

trtrtrtr

tr

Plane wave solutions take the form: ).()(),( etrpieputr

There are four solutions: two with positive energy E = +Ep corresponding to the two possible spin states of a spin particle and two corresponding negative

energy solutions with E = - Ep.

where ) is also a four-component spinor satisfying the eigenvalue equation:)()()()( pEupumppuH p

Electron spin is therefore included in a natural way (rather than the previous ad-hoc attempts). All solutions also have positive probability density.

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Dirac InterpretationDirac interpreted the negative energy solutions by:

Postulating the existence of a “sea” of negative energy states, which are almost always filled – each with two electrons (spin “up” and spin “down”):

When an electron is added to the vacuum, it is confined to the positive energy region since all negative energy states are occupied (Pauli exclusion principle).

When energy is supplied to promote a negative energy electron to a positive energy level, an electron-hole pair is created. The hole is seen as a charge +e and E > 0 state.

A false start when Dirac identified negative energy electrons as

protons.

Eventually, persuaded by arguments of Weyl &

Oppenheimer that +ve particle had to have the same mass as

the electron.

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Dirac Interpretation

SeaProblems with this picture.

Bosons have no exclusion principle, so this picture does not work for them.

It implies the Universe has infinite negative energy!

Nonetheless, in 1931 Dirac postulated the existence of the positron as the electron’s antiparticle.

It was discovered 1 year later.

𝜸

Photon with excites electron from –ve energy state.

Leaves hole in vacuum corresponding to a state with more energy (less negative energy) and a positive charge wrt the

vacuum.

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Positron - First Antiparticle Discovered!

Positron = the anti-electron

Discovered in 1932 by Carl Anderson

photographing cosmic ray tracks in

a cloud chamber.

6 mm lead plate

e+

e+ e-

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Pair ProductionIn 1933, Blackett & Occhialini observed pair production in a

triggered cloud chamber and confirmed that Anderson’s particle was indeed Dirac’s positron.

𝜸+𝒏𝒖𝒄𝒍𝒆𝒖𝒔→𝒆+¿+𝒆−+𝒏𝒖𝒄𝒍𝒆𝒖𝒔 ¿

𝜸

𝒆+¿ ¿

𝒆−

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Stückelberg InterpretationAs early as 1941, Stückelberg cultivated the view that positrons may be

understood as electrons running backward in time.

E.C.G. Stueckelberg, Helvetica Physica Acta, 14 (1941), 588-594

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Feynman Interpretation

“The problem of the behavior of positrons and electrons in given external potentials, neglecting their mutual interaction, is analyzed by replacing the theory of holes by a reinterpretation of the solutions of the Dirac equation…..”.

…In this solution, the “negative energy states” appear in a form which may be pictured (as by Stückelberg) in space-time as waves traveling away from the external potential backwards in time. Experimentally, such a wave corresponds to a positron approaching the potential and annihilating the electron….

July1962

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Feynman Interpretation

Waves can proceed backward in time

Virtual pair production. Positron goes forward

in time (4 to 3) to be annihilated and electron

backwards (3 to 4)

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One electron in the entire Universe?

It was Feynman’s PhD supervisor, John Wheeler, who suggested that there may be only a single electron in the universe, propagating through space and time!

This obviously has a few problems! For example:• You would expect equal number of

electrons and positrons and yet we observe extremely few positrons.

• How do we account for the fact that electrons can be created/destroyed in weak interactions.

• This poor single electron would have had to traverse huge distances and be very ancient.

World-line ofsingle electron

Nonetheless, Feynman kept the idea that positrons could simply be represented as electrons going from the future to the past.

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Reminder: this is Applied Physics!This quantum theory is all very well, but what can we

physically measure?

AppliedPhysicistQuantum

Theorist

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Decays - Lifetimes In the case of particle decays, the most interesting physical quantity is the lifetime of the particle. Measured in the rest frame of the particle. We can define the decay rate, ,

as the probability per unit time the particle will decay.

𝑑𝑁=−Γ𝑁𝑑𝑡Similar to your Year 2nuclear physics 𝑁 (𝑡 )=𝑁 (0)𝑒−Γ 𝑡

Mean Lifetime (natural units) 𝐹𝑜𝑟 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 ,𝑢𝑠𝑒𝜏=ℏΓ

Most particles decay by several different routes. The total decay rate is then the sum of the individual decay rates: Γ 𝑡𝑜𝑡=∑

𝑖=1

𝑛

Γ 𝑖

The lifetime of the particle is then:

The different final states are known as decay modes.

The branching ratio for i’th decay mode is:

𝜏= 1Γ 𝑡𝑜𝑡

h𝐵𝑟𝑎𝑛𝑐 𝑖𝑛𝑔 𝑅𝑎𝑡𝑖𝑜=Γ 𝑖

Γ𝑡𝑜𝑡

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Meson Example

Particle Data Group: http://pdg.lbl.gov/

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Scattering - Cross-Section

)()()()( dLdddFnddnvnddN bbaa

na = no. of beam particlesva = velocity of beam particlesnb = no. target particles/area

Incident flux F=nava dN = no. scattered particles in solid angle dΩ

d ddTotal

cross-section

ddN

Ldd 1Differential

cross-section

.d.ddΩ sin

Measured in barns. 1 b = 10-24 cm-2

LN rateEvent

Luminosity L = flux x no. targets (cm-2s-1)

Cross-section quantifies rate of reaction. Depends

on underlying physics.

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Feynman Diagrams

“Like the silicon chip of more recent years, the Feynman diagram was bringing computation to the masses”. Julian Schwinger

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Feynman Diagrams – The Problem

𝑒− 𝑒+¿ ¿

𝜇−

𝜇+¿¿

𝜃𝒑𝒑 ′

𝒌

𝒌 ′

In the Centre of Mass frame:

where are momenta

𝒆+¿𝒆−→𝝁 +¿ 𝝁−¿ ¿Consider the calculation of thecross-section of one of the simplest

QED processes.

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22641section -Cross alDifferenti

CMEdd

Bad News! Even for this simple process the exact expression is not known.Best that can be done is to obtain a formal expression for as a perturbation

series in the strength of the EM interaction & evaluate the first few terms.Feynman invented a beautiful way to organise, visualise and thereby calculate

the perturbation series.

= QM amplitude for process.

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Quantum Field Theory - Basics• In Quantum Field Theory, the scattering and decay of particles

is described in terms of transition amplitudes.

• For a transition process , the transition amplitude is written as , where S is an operator known as the Scattering Matrix or S-Matrix.

• Exact calculations of are not possible, but it is possible to use perturbation theory which allow approximate calculations of the transition amplitude.

• A bit like a binomial/Taylor series expansion: e.g. and then keeping the first 2 terms in the expansion (for small values of x).

• Feynman diagrams are a technique to solve quantum field theory by calculating the amplitude for a state with specified incoming and outgoing particles.

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Feynman Diagrams - Basics

• Time runs from left to right.• Particles point in +ve time direction.• Anti-particles point in –ve time direction.• Charge, energy, momentum, angular momentum, baryon no. & lepton no.

conserved at interaction vertices. Quark flavour for strong & EM interactions.

time

vertexvertex

fermion (solid line)

boson (wavy line)Initial State

Final State

Annihilation Diagram

Exchange Diagram

Virtual Particle

External legs represent amplitudes of initial & final state particles.

Internal lines (propagators) represent amplitude of exchanged particle.

fermion

anti-fermion

photons,W, Z bosons

gluons

H bosons

CONVENTION

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Feynman Diagrams - Calculation Perturbation theory. Expand and keep the most important terms for

calculations. Associate each vertex with the square root of the appropriate coupling

constant, i.e. . When the amplitude is squared to yield a cross-section there will be a

factor , where n is the number of vertices (known as the “order” of the diagram).

1371

4

2

HL

e

Lowest order Second order

Add the amplitudes from all possible diagrams to get the total amplitude, M, for a process transition probability.

For QED:

)space phase(2Rate Transition 2 M

Fermi’s Golden Rule

Contains the fundamental physics “Just” kinematics

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Phase SpaceIntroduced by Willard Gibbs in 1901. Defines the space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space.

Example: 3 Body DecayPhase space illustrated by this 2-D plot of a three-body decay (a so-called “Dalitz plot”).The contour line shows the boundary of what is kinematically possible - i.e. the edge of phase space.

MeV 2050 000 spp

Crystal Barrel Experiment

Contour showing limitof kinematically available

phase space.

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Quantum Electrodynamics (QED)

Julian SchwingerSin-Itiro Tomonaga

Early relativistic quantum theories

had to be rethought after the

discovery of the Lamb Shift in

hydrogen.

Willis Lamb, Robert Retherford,

1947

Detailed results disagreed with the Dirac Equation.Due to self-energy of the electron.

Led to concept of renormalisation (Bethe).

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The Most Accurate Theory

Anomalous magnetic moment of electron (g-2).

[0.24ppb] [0.67ppb]

Calculated from12,672 Feynman diagrams!

T. Aoyami et al, Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant, ArXiv:1205.5368

2)2(

gae

Dirac theory predicts g=2 exactly, but this is modified by quantum loops and the difference is defined as ………….

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QED Vertex Factor

𝒆−

𝜸

𝒆− 𝒆−

√𝜶

𝒆−

𝜸

√𝜶

Coupling constant, , specifies the strength between the 3 particles at each vertex.This is a measure of the probability of spin fermion emitting or absorbing a photon.

Because , often we simply use in place of

Taking the 2 vertices, we get a total factor of

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Basic QED Processes and Vertices (8)

ee

ee

ee

ee vacuum

ee

ee

ee

vacuum ee

𝒆−

𝒆−

𝒆−

𝒆−

𝒆−

𝒆−

𝒆−

𝒆−

𝒆+¿ ¿

𝒆+¿ ¿

𝒆+¿ ¿

𝒆+¿ ¿

𝒆+¿ ¿

𝒆+¿ ¿

𝒆+¿ ¿

𝒆+¿ ¿

Pair Production

Electron Bremsstrahlung

Positron Bremsstrahlung

Photon Absorption

Photon Absorption

Annihilation

VacuumProduction

VacuumExtinction

time

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Charged LeptonsAs well as electrons and positrons interacting with the EM field (i.e. the photon),

QED also includes the interactions of the other charged leptons – the muon ( and the tau lepton (.

The Charged Leptons

Lepton UniversalityElectromagnetic properties of muons and tau leptons are identical with those of electrons provided the mass difference is taken into account.

All properties shared except for their very different masses and lifetimes.

ExampleDirac magnetic moment

Swhere is the mass of the lepton.Both the and decay by the weak interaction and we

will return to this in a later lecture.

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Also Charged Leptons and Quarks

𝜸

𝒆− 𝒆−

√𝜶𝝁− 𝝁−

√𝜶𝝉− 𝝉−

√𝜶

𝜸

Same interaction strength for all charged leptons – QED only cares about charge.

𝜸

u u𝟐𝟑 √𝜶

𝜸

𝜸

d d𝟏𝟑 √𝜶

Coupling less for quarks due to fractional charge.

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QED Propagator Factor

• Virtual photons have propagators proportional to , where is the four-momentum transfer between the vertices (or the four-momentum of the exchanged particle).

• Heavy bosons with mass have propagators

Virtual ParticlesDo not have mass of a physical particle.

Known as “off –mass shell” (e.g. not zero for photon)

2222XXX pEmq Propagator

𝜸

𝒆− 𝒆−

√𝜶

𝒆−𝒆−√𝜶

Propagator factor tells us about the contribution to the amplitude from an intermediate (or virtual) particle travelling through space and time.

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Feynman Factors: Summary Each part of a Feynman diagram has factors associated with it.

Multiply them all together to get matrix element . Initial and final state particles use wavefunction currents:

Spin-0 bosons are plane waves. Spin-1/2 fermions have Dirac spinors. Spin-1 bosons have polarisation vectors .

Vertices have dimensionless coupling constants. In the electromagnetic case, . In the strong interaction, (more later).

Vertices have propagators, , which is the momentum transferred by boson. Virtual photon propagator is Virtual W/Z boson propagator is or Virtual fermion propagator is

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Feynman Rules for QEDIncoming particle 𝑢(𝑝)Outgoing particle 𝑢(𝑝)Incoming antiparticle 𝑣 (𝑝 )Outgoing antiparticle 𝑣 (𝑝 )

External Lines

Spin

Internal Lines (propagators)

Incoming photon 𝜀𝜇(𝑝 )Outgoing photon 𝜀𝜇(𝑝 )∗

Spin

Photon 𝜇 𝜈−𝑖𝑔𝜇𝜈

𝑞2

Fermion

Spin

Spin −𝑖 (𝛾𝜇𝑞𝜇+𝑚 )𝑞2−𝑚2

Vertex FactorsSpin Fermion (charge ) −𝑖𝑒𝛾𝜇

Matrix Element Product of all factors−𝑖ℳ=∑𝑛=1

∞

ℳ𝑛

Taken fromThomson,page 124

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Example: scattering

𝜸

𝒆− 𝒆−

𝝁

𝝉−𝝉−𝝂

𝑝1 𝑝3

𝑝2 𝑝4

𝑞=𝑝1−𝑝3

𝑢 (𝑝3 ) [𝑖𝑒𝛾𝜇 ]𝑢 (𝑝1 )

𝑢 (𝑝4 ) [𝑖𝑒𝛾𝜈 ]𝑢 (𝑝2 )

−𝑖𝑔𝜇𝜈

𝑞2

−𝑖ℳ=[𝑢 (𝑝3 ) {𝑖𝑒𝛾𝜇}𝑢 (𝑝1 ) ]−𝑖𝑔𝜇𝜈

𝑞2 [𝑢 (𝑝4 ) {𝑖𝑒𝛾𝜈 }𝑢 (𝑝2 ) ]

propagator

electroncurrent

taucurrent

In lowest order, the amplitude by applying the Feynman rules to the above diagram is therefore:

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Higher Order DiagramsTree level Processes (or Born Diagrams) = diagrams that contain no loops

Higher Order Corrections (or Radiative Corrections) = loop diagrams

Order = number of vertices in each diagram

Any diagram of order gives a contribution of

Lowest order diagram𝓜∝𝒆𝟐∝𝜶𝜸𝒆+¿ ¿

𝒆−

𝝁+¿¿

𝝁−

+ +..

Second order diagrams:

Total amplitude:

√𝜶√𝜶

𝒆+¿ ¿

𝒆−

𝝁+¿¿

𝝁−

√𝜶√𝜶 √𝜶

√𝜶𝜸𝒆+¿ ¿

𝒆−

𝝁+¿¿

𝝁−

√𝜶 √𝜶√𝜶

√𝜶

Jargon Leading Order (LO) + Next-to-Leading-Order (NLO)

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LEP ExampleBorn level diagrams

+

Radiative corrections

Different for eachexperiment dueto phase space

Excellent agreementwith QED

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Alpha – Fine Structure Constanto Need to take care with the Fine Structure Constant, and its role as a

coupling constant measuring the strength of the EM interaction.o was introduced by A. Sommerfield in 1916 to explain the fine structure of

the energy levels of the hydrogen atom. In particle physics, it is not really a constant.

o This is because in QED an electron can emit virtual photons, which form virtual e+/e- pairs which “screen” the electron.

Alpha not really a constant!At Q2=0, At Q2mW

2,

𝒆−

𝒆+¿ ¿

𝒆−

𝒆−

𝒆+¿ ¿

𝒆−

𝒆+¿ ¿

OPAL, CERN-EP/98-108

)(1)0()( 2

2

QQ

Vacuum polarisationloops. Correction:

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QED RenormalisationThe strength of the coupling between a photon and an electron is determined by the coupling at the QED vertex, which until now we have assumed constant with value .• The value is obtained from measurements of the static Coulomb potential in

atomic physics.• This is not the same as the strength of the coupling in Feynman diagrams,

which can be written as (and called the bare electron charge).• The experimentally measured value of is the effective strength of the

interaction which results from summing over all relevant higher order diagrams.

• There is an infinite set of higher-order corrections, including the ones below…

Lowest order corrections to QED vertex

Correction to propagator

Corrections to electron four vector current

(a) (b) (c) (d) (e)

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QED Renormalisation

Techniques to deal with this are beyond scope of this course, but basically…..1. Use cut-off procedures in integrals. It turns out that this enables the

calculations to be separated into two parts: a finite term and one which blows up.

2. Amazingly, all the divergent terms then appear as additions to the bare parameters such as mass (), charge () and coupling constant (.i.e.

3. The strategy is then to absorb the infinities into renormalisable masses and coupling constants. i.e. it means that we use the physical values as determined by experiment and NOT the values () that appeared in the original Feynman rules that we wrote down.

𝛼 (𝑞2 )= 𝛼 (𝜇2)

1−𝛼 (𝜇2) 13𝜋 ln (𝑞

2

𝜇2 )

There are no restrictions on the momentum, , of the virtual particles and the self-energy terms include integrals of the form , which is logarithmically divergent.

As already discussed, oneconsequence is that the coupling constant depends logarithmically onenergy scale

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CLIC after LHC at CERN?

CLIC= Compact Linear (e+e-) Collider

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Electron Positron AnnihilationElectron-positron colliders have been central to the

development and understanding of the Standard Model.

WHY?• It is easier to accelerate protons to very high energies than

leptons, but the detailed collision process of protons cannot be well controlled or selected.

• Electron positron colliders offer a well-defined initial state.• The collision energy is known and it is tuneable (e.g. for

scanning thresholds of particle production).• Polarisation of electrons/positrons is possible.• In proton collisions, the rate of unwanted collision processes is

very high, whereas the point-like nature of leptons results in low backgrounds.

• Scattering of point-like particles can be calculated to very high precision in theory.

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LEP Electroweak Measurements Example

Cross-sections of electroweak Standard Model (SM) processes. Dots with error bars show the measurements, while curves show theoretical predictions

based on SM.

Electroweak measurements in electron-positron collisions at W-boson-pair energies at LEP, Physics Reports, 2013, in press

Precision results on fundamental properties of W boson & EW theory

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CONTACTProfessor Glenn Patrickemail: [email protected]: [email protected]

End