quantum electrodynamics iii · scattering cross sections need two information dynamical evaluation...
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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
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