Quantum ElectroDynamics III

Dr.Farida TahirPhysics department

CIIT, Islamabad

Feynman diagram

Human Instinct

What? Why?

How?

Feynman diagrams Feynman diagrams

Feynman diagrams

What?Graphic way to represent exchange forces

Developed by Richard Feynmanwhen working on the development of

QED

Describing a variety of particle interactions

1942

Why ?

Decay rates and Scattering cross section

To calculate

Calculation of Decay rates and Scattering cross sections

Need two informationDynamical

Evaluation of relevant Feynman diagram todetermine the amplitude M for the process.

KinematicalThe phase space factor, it depends on the masses, energies, and momenta of the participants.

Ingredients (Rules)

Recipe (Structure)

How?

Feynman rules:Electron (e-) & Positron (e+)(Dirac eq.)Photons(Maxwell eq.)

Dirac Equation in electromagnetic field(concept of gauge symmetry)

The Feynman rules for

Electron (e-) Positron (e+)

021 =vv021 =uu Orthogonal

mcvv 2−=mcuu 2= Normalized

( )mcpvv s

s

s −=∑=

λλγ

2,1( )mcpuu

s

ss +=∑=

λλγ

2,1

Completeness

( ) 0=−mcPu λλγ ( ) 0=+mcPv λ

λγAdjoint

( ) 0=− umcPλλγ ( ) 0=+ vmcPλ

λγDirac

( ) ( ) ( )PuaeX sPXi .h

−

=ψ ( ) ( ) ( )PvaeXPXi .

h=ψFree

The Feynman rules for (γ)

Photon (γ)

( ) ( ) ( )saeXAPXi

μμ ε.

h

−

=

Orthogonal

Normalized

( ) ( ) jiijs

jsis pp ˆˆ2,1

)()( −=∑=

δεε Completeness

Lorentz condition

Free

0=μμε P

0)2(*)1( =μ

μ εε

1* =μμ εε

Complete Lagrangian for f & γ

Lagrangian density describing the fermionic field in the presence of an electromagnetic field is

( )[ ] λλ

λνλν

λλλγ AJFFXmqAiX −−Ψ−−∂Ψ=

41)()(L

[ ] ( ) λλλ

λνλν

λλ γγ AXXqJFFXmiX )()(

41)()( ΨΨ+−−Ψ−∂Ψ=L

Current produce by Dirac particle

Current coupled to Aλ, describe the interaction vertex.

Feynman diagram

Structure of Feynman Diagrams

Fermions represented by straight lines with arrows pointing in direction of time flow

Forward-facing arrows represent particles

Backward-facing arrows represent antiparticles

Photons and weak bosons, W- and W+ and Z0 are squiggly lines

Gluons are curly lines

Structure of Feynman Diagrams

Line types

External lines• Enter and leave diagram• Represent real “observable” particles

real particles and must have E2 = p2 + m2

Referred to as particle being on “mass shell”

Internal linesConnect vertices, called propagatorsRepresent “virtual” particles that cannotbe observedDo not have to obey relativistic mass,

energy, momentum relationship: (mc2)2= E2- (pc)2

Referred to as particle being off “mass shell”

Line types

At the point of emission (or absorption) the vertices gives a contradiction with Einstein relation between energy and mass.

Virtual

VerticesConserve: energy, momentum, & charge for all

types of interactionsDetermine order of perturbation contributes to the particular calculationSame number of arrows enter as leave

How to write currents by using F.D

For each QED vertex, write factor μγei

For each internal photon line having momentum k write factor

εαβ

αβ ikg

ikiD F += 2)(α β

For each internal lepton line having momentum p write factor

εε impmp

impipiS F +−

+=

+−= 22

1)(

For each external (initial) fermions)(pur

For each external (final) fermions)(pur

For each external (initial) anti-fermions)(pvr

For each external (final) anti - fermions)(pvr

αkFor each initial photon )(krαε

kαFor each final photon )(* krαε

ExamplesBasic vertices

ElectromagneticCharge particle enters, emits (or absorbs) a

photon and exits.

- ieγμ137

12

==c

eh

α

Current

)()( 22 puiepKuJ rr νν γ+=

+= μ

μ JJL

ε

μνμν

ikgikiD+

= 2)(

μ

ν

p'p

K

2p2pK +

)'()( pueipuJ rr μμ γ=

)()()'()( 222 puiepKuiK

gipuiepuM rrrrfi ν

μν

μ γε

γ +−

=

)()()'()( 222

2

pupKupupuiK

ei rrrrμ

μ γγε

+−

−=

Scattering

Elastic

Mott Møller

Bhabha

μμ ee → eeee →

+−+− → eeee

Inelastic

Compton

Pair production

Pair annihilation

Muon productionγγ→+−ee

+−+− → μμee

+−→ eeγγ

γγ ee →

Lowest Order Fundamental Processes

+−+− +→+ eeee

γγ +→+ +− ee+− +→+ eeγγ

γγ +→+ −− ee

Scattering

Møller scattering

Electron-electron scattering (Moller scattering)Two electron enters, a photon passes between them

eeee →

Bahabha scattering +−+− → eeee

Twist into any topological configuration

Particle line running backward in time is interpreted as the corresponding antiparticle going forward

Rule:

e-e+ annihilate to form a photon which produces a new e-e+

Bhabha & Moller scattering are related by cross symmetry

BADC

DBCA

DCBA

+→+

+→+

+→+

+−+− +→+ 4321 eeee

Compton scattering

Remember

Dominant contribution comes from tree level

Ignore higher order contribution

13712

==c

eh

α

QED InteractionsHigher order diagrams

Charge screening

Recall

Forces coupling Strength Range Particles

Strong αs 1 10-15 Gluons; m=0

Electromagnetic α 1/137

10-6

10-39

Weak αw

Photon; m=0∞

10-18 W &Z boson

∞ graviton; m=0Gravity αg