quantum electrodynamics iii · scattering cross sections need two information dynamical evaluation...
TRANSCRIPT
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Quantum ElectroDynamics III
Dr.Farida TahirPhysics department
CIIT, Islamabad
Feynman diagram
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Human Instinct
What? Why?
How?
Feynman diagrams Feynman diagrams
Feynman diagrams
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What?Graphic way to represent exchange forces
Developed by Richard Feynmanwhen working on the development of
QED
Describing a variety of particle interactions
1942
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Why ?
Decay rates and Scattering cross section
To calculate
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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.
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Ingredients (Rules)
Recipe (Structure)
How?
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Feynman rules:Electron (e-) & Positron (e+)(Dirac eq.)Photons(Maxwell eq.)
Dirac Equation in electromagnetic field(concept of gauge symmetry)
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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
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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
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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
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Structure of Feynman Diagrams
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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
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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”
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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
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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
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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)(
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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
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αkFor each initial photon )(krαε
kαFor each final photon )(* krαε
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ExamplesBasic vertices
ElectromagneticCharge particle enters, emits (or absorbs) a
photon and exits.
- ieγμ137
12
==c
eh
α
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Current
)()( 22 puiepKuJ rr νν γ+=
+= μ
μ JJL
ε
μνμν
ikgikiD+
= 2)(
μ
ν
p'p
K
2p2pK +
)'()( pueipuJ rr μμ γ=
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)()()'()( 222 puiepKuiK
gipuiepuM rrrrfi ν
μν
μ γε
γ +−
=
)()()'()( 222
2
pupKupupuiK
ei rrrrμ
μ γγε
+−
−=
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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
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+−+− +→+ eeee
γγ +→+ +− ee+− +→+ eeγγ
γγ +→+ −− ee
Scattering
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Møller scattering
Electron-electron scattering (Moller scattering)Two electron enters, a photon passes between them
eeee →
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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+
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Bhabha & Moller scattering are related by cross symmetry
BADC
DBCA
DCBA
+→+
+→+
+→+
+−+− +→+ 4321 eeee
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Compton scattering
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Remember
Dominant contribution comes from tree level
Ignore higher order contribution
13712
==c
eh
α
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QED InteractionsHigher order diagrams
Charge screening
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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