Nuclear and Particle Physics Franz Muheim 1

Quantum ElectrodynamicsQuantum Electrodynamics

OutlineOutlineClassical versus Quantum Theory

Force/interaction mediated by exchange of field quanta

Virtual ParticlesPropagator

Feynman DiagramsFeynman RulesMatrix ElementsCross sections

Electromagnetic vertexCoupling strength Coupling constantConservation laws

QED processesElectron-proton scatteringElectron-positron annihilation

Higher order DiagramsPrecision QED testsRunning of alpha

Dirac EquationAppendix

Nuclear and Particle Physics Franz Muheim 2

Quantum ElectrodynamicsQuantum Electrodynamics

Quantum Theory (QED)of Electromagnetic Interactions

Classical ElectromagnetismForces arise from Potentials V(r)act instantaneously at a distance

QED PictureForces described by exchange of virtual field quanta - photons

Matrix elementFull derivation in 2nd order perturbation theory Gives propagator term 1/(q2 - m2) for exchange bosonEquivalent to scattering in Yukawa potential

Propagator

( )22

2

mqgM fi −

=

Nuclear and Particle Physics Franz Muheim 3

Virtual ParticlesVirtual Particles

Electromagnetic InteractionForces between two charged particles are due to exchange of virtual photons

“Photon mediates electromagnetic interaction”No action at a distance required!

Virtual ParticlesDo not have mass of physical particle

known as “Off mass-shell”e.g. non-zero for photon4-momentum of virtual particle is energy and momentum transferbetween scattered particlesVirtual mass-squared q2 = EX

2 - pX2

q2 > 0 and q2 < 0 possiblePropagator - how far particle is off mass-shellInternal lines, not observable must observe ∆E ∆t ≈ ħ

Example: electron-electron scattering: e- e- → e- e-

),( qEq qr

=µ

q2

222XXX pEmr

−≠

Nuclear and Particle Physics Franz Muheim 4

Feynman DiagramsFeynman Diagrams

A Feynman diagram is a pictorial representation of aprocess corresponding to a particular transition amplitude Aitchison & Hey

“Gauge Theories in Particle Physics”

Basic PrincipleTransition amplitude for all processes - scattering, decay,absorption, emission - is describedby Feynman Diagrams

Feynman diagrams a most useful tool in modernparticle physics and will be used a lot in this course!

ConventionsTime flows from left to right (sometimes upwards)

Fermions are solid lines with arrowsBosons are wavy or dashed lines

Feynman RulesAllow to calculate quantitative results of transitionDerived from quantum field theory (QFT)

Nuclear and Particle Physics Franz Muheim 5

Electromagnetic VertexElectromagnetic Vertex

QED Feynman Diagram• Initial state fermion• Absorption or emission

of photon• Final state fermionExamples: e- → e- γ Bremsstrahlung

e- + γ→ e- Photoelectric effectAll electromagnetic interactions are described byvertex and photon propagator

Coupling StrengthTransition amplitude proportional to fermion charge Mfi ∝ eProbability/rate of γ emission or absorption

rate ∝ |Mfi|2

Rate proportional to coupling constant α

Coupling ConstantFine structure constantα ∝ e2

QED Conservation LawsMomentum and energy conserved at all verticesCharge is conserved in all QED verticesFermion flavour is conservede-→e-γ exists, but not c→ u γQED vertex also conserves parity

Coupling Constant Coupling Constant SI

1371

4 0

2

≅=c

ehπε

α

Nuclear and Particle Physics Franz Muheim 6

Basic QED ProcessesBasic QED Processes

Initial and final state particle satisfy relativisticfour-momentum conservation m2 = E2 - p2

In free space - Energy conservation violated for above diagrams if all particles are real

Nuclear and Particle Physics Franz Muheim 7

FeynmanFeynman Rules for QEDRules for QED

Each line and vertex in Feynman diagram corresponds to a term in the matrix element

Initial and final state fermionsFermion wave function Ψ (spinor when including spin)

Initial and final state bosonsBoson wave function includes polarisation

Internal virtual photonsPhoton propagator 1/(q2 - m2) = 1/q2

Internal fermionsSpinor propagator exchanged between charged particles similar in structure to photon propagator

VertexCoupling constant √α ~ e

Example:electron-muon scattering: e- µ- → e- µ-Transition amplitude

are 4x4 matrices account for spin-structure of electromagnetic interaction

currentelectron

propagator photon

currentmuon

13

2

24

uieu

qig

uieuiM fi

µ

µν

ν

γ

γ

−

=−

µνµγ gand

Nuclear and Particle Physics Franz Muheim 8

ElectronElectron--Proton ScatteringProton Scattering

Matrix Element e- p → e- pTransition Amplitudeuse Feynman rulessimple if neglecting spins

Cross sectionProbability for scattering

4-momentum transfer

when neglecting me

Rutherford ScatteringElastic Ef = Ei = E, neglect proton recoil

22

2

2

41qq

eeq

eM πα==∝

( )( )

⎟⎠⎞

⎜⎝⎛−=

−−=

⋅−+=−==

2sin4

cos22

2

2

2

2222

θ

θ

µµµ

µ

if

ifife

ififif

EE

ppEEm

ppppppqqqrr ( )

( )iii

fff

pEp

pEpr

r

,

,

=

=µ

µ

⎟⎠⎞

⎜⎝⎛

=Ω

2sin4 42

2

θασ

Edd

Lab

4

22

4

42 16

qqeM

dd απσ

=∝∝Ω

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=Ω 2

sin22

cos

2sin4

22

22

42

2 θθθ

ασ

pi

f

Lab Mq

EE

Edd

Rutherford Scattering Rutherford Scattering

Nuclear and Particle Physics Franz Muheim 9

e+ee+e-- AnnihilationAnnihilation

Matrix elementNeglecting spin effects

Cross sectionWork in centre-of-mass frame4-momentum transfer

Use Fermi’s Golden Rule, density of final state normalisation of wave function

Correct treatment of spins( ) s

EMdd 2

3

22

CoM 22 α

ππσ

==Ω

( ) ( ) sEppEEqqq ==+−+== 2221

221

2 )2(rr

µµ

( )

( ) [ ]2

2

22

CoM

GeV87

34

cos14

snb

see

sdd

==→

+=Ω

−+−+ παµµσ

θασ

q2

22

2

2

41qq

eeq

eM πα==∝

2

2242 2

suteM +

=

−+−+ → µµee

Nuclear and Particle Physics Franz Muheim 10

Higher Order DiagramsHigher Order Diagrams

QEDtime dependent perturbation theory

Lowest order

Second order

Higher ordersOrder n suppressed by α = 1/1372n

Lowest order dominates if coupling constant α is smallQED converges rapidly

442

1371

≈∝∝ ασ M

222

1371

≈∝∝ ασ M

Nuclear and Particle Physics Franz Muheim 11

QED Precision TestsQED Precision Tests

Magnetic momentCouples to spin of electron

Dirac Equation predicts gyromagnetic ratio g = 2for point-like particles

Anomalous magnetic momenta = (g-2)/2

Higher order correctionsBullet represents external electromagnetic field

muon spin-precession in magnetic field

2 loops – 7 Feynman diagrams3 loops – 72 Feynman diagrams4 loops – 891 Feynman diagrams

QED is most precise theory in physicsExperiment ae = (11596521.869 ± 0.041) ·10-10

Theory ae = (11596521.3 ± 0.3) ·10-10

cmeSg

eBB 2

where hrr== µµµ

muons

electrons

3101617.12

Vertex −⋅===πα

µaae

vertex vacuum polarisation

Nuclear and Particle Physics Franz Muheim 12

Running of Running of αα

Coupling constant αStrength of electromagnetic interactionα = e2/4π is not a constant at all distances

VacuumNot empty, around free electron creation and annihilationof virtual electron-positron pairs

ScreeningBare charge and mass of electron only visible at very short distancesα increases with with larger momentum transfer

Classical limitq2 = 0 α = 1/137

At short distancesq2 = (90 GeV)2 α = 1/128

−+−+ → µµee

Nuclear and Particle Physics Franz Muheim 13

DiracDirac EquationEquation

Schroedinger equation1st order in time derivative2nd order in space derivatives

Klein-Gordon equation2nd order in space and time derivatives

negative energies (E < 0) and negative probability densities (|Ψ|2 < 0)

Dirac equation

1st order in time and space derivatives

γµ are 4x4 matricesSolutions of Dirac equationWave function with 4-component spinor

2 positive energy solutions, E > 0 2 negative energy solutions, E < 0

( ) 0or00 =Ψ−∂=Ψ⎟⎠⎞

⎜⎝⎛ −∇⋅+

∂∂ mimit

i µµγγγ

rr

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

ΨΨΨΨ

=Ψ

4

3

2

1

( ) ( ) 22exp)(, mpExippuNtx +±=⇒−=Ψ νν

r

0or 222

222

2

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛+∇−

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∇+

∂∂

− ψψψ mt

mt

rr

Dirac Equation Dirac Equation

Dirac Equation not examinable