1. Lecture 4 – Quantum Electrodynamics (QED)
An introduction to the quantum field theory
of the electromagnetic interaction
22/1/10
Particle Physics Lecture 4 Steve Playfer

2. The Feynman Rules for QED
Incoming (outgoing) fermions have spinors u (u) (where u = u†g0)
Incoming (outgoing) antifermions have spinors v (v)
Incoming (outgoing) photons have polarization vectors em (em*)
Vertices have dimensionless coupling constants √a
At low four-momentum transfers (q2), a = e2/2phc = 1/137
Virtual photons have propagators 1/q2
Virtual fermions have propagators (gmqm + m)/(q2 – m2)
15/1/10
Particle Physics Lecture 2 Steve Playfer

3. Particle Physics Lecture 4 Steve Playfer
Summing Diagrams
The Matrix Element for a transition is the sum of all possible Feynman diagrams connecting the initial and final states
A minus sign is needed to asymmetrise between diagrams that differ by the interchange of two identical fermions
For an unpolarised cross-section need to average over initial state spins and sum over final state spins
For QED the sum of higher order diagrams converges.
There are more diagrams with higher numbers of vertices
… but for every two vertices you have a factor of a=1/137
The most precise QED calculations go up to O(a5) diagrams
22/1/10
Particle Physics Lecture 4 Steve Playfer

4. Electron Muon Scattering
There is only one lowest order diagram
Start from scattering of spinless particles
(Lecture 2) and replace plane waves with
electromagnetic currents and Dirac spinors √a
Vertex
Couplings m- m-
M = a (u3 gm u1 ) (u4 gm u2 ) q2
Photon
Propagator
Electron
current
Muon
current
15/1/10
Particle Physics Lecture 2 Steve Playfer

5. High Energy e-m scattering
In the relativistic limit E>>m it can be shown that fermion helicity is conserved during the scattering process:
There are only four possible spin configurations:
The unpolarized matrix element squared is:
|M|2 = a2 SS3,S4 (u3 gn u1 )* (u3 gm u1 ) (u4 gn u2 )*(u4 gm u2 )
(2S1+1)(2S2+1) q4
Le = 4[p3mp1n + p3np1m – (p3.p1 – me2)gmn] using trace theorems
Neglect lepton masses in relativistic limit:
|M|2 = 8e4 [ (p3.p4)(p1.p2) + (p3.p2)(p1.p4) ] q4
SS3 = Le SS4 = Lm
22/1/10
Particle Physics Lecture 4 Steve Playfer

6. Cross-section for e-m scattering
Can express matrix element squared in terms of Lorentz invariant Mandelstam variables s,t,u
or in terms of the CM scattering angle q*
|M|2 = 2e4 (s2 + u2) = 2e4 (1 + 4cos4 q*/2)
t sin4 q*/2
ds = a2 (s2 + u2) with initial and final state factors
dW s t2
Polarized cross-sections from:
M ( ) = M ( ) = e2 u t
M ( ) = M ( ) = e2 s
22/1/10
Particle Physics Lecture 4 Steve Playfer

7. Cross-section for e+e- ® m+m-
Related to e-m scattering by
crossing symmetry t s
(90o rotation of Feynman diagram) e- m+
|M|2 = 2e4 (t2 + u2) = e4 (1+cos2q) s2
Separated into allowed
spin configurations:
Differential cross-section: ds = a2 ( 1 + cos2q)
dW s
Total cross-section: s = 4pa (decreases with increasing CM energy) 3s
22/1/10
Particle Physics Lecture 4 Steve Playfer

8. Particle Physics Lecture 4 Steve Playfer
22/1/10
Particle Physics Lecture 4 Steve Playfer

9. Particle Physics Lecture 4 Steve Playfer
Halzen & Martin P.129
22/1/10
Particle Physics Lecture 4 Steve Playfer

10. Higher Order QED Diagrams - I
Two photon exchange diagrams (“box” diagrams)
These add two vertices with a factor of a = 1/137 p1 p1 p3 p3 k
p1+ k k
p1- k k
p3- k k
q - k
q - k
p2- k p2 p2 p4 p4
t channel
s channel
The four momentum k flowing round the loop can be anything!
Need to integrate over ∫ f(k) d4k
Unfortunately this integral gives ln(k) which diverges!
This is solved by “renormalisation” in which the infinities are
“miraculously swept up into redefinitions of mass and charge” (Aitchison & Hey P.51)
22/1/10
Particle Physics Lecture 4 Steve Playfer

11. Higher Order QED Diagrams - II
A virtual photon can be
emitted and reabsorbed
across a vertex.
This effectively changes
the coupling at the vertex.
A real (or virtual) fermion
can emit and reabsorb
a virtual photon.
This modifies the
fermion wavefunction.
When the photon is emitted
the fermion becomes virtual
and changes its 4-momentum.
In a virtual photon propagator
a fermion-antifermion pair
can be produced and
then annihilate.
This modifies the
photon propagator.
Any charged fermion is allowed
(quarks and leptons)!
Each of these diagram adds two vertices with a factor of a = 1/137
Again four momentum flowing round a loop can be anything!
22/1/10
Particle Physics Lecture 4 Steve Playfer

12. Particle Physics Lecture 4 Steve Playfer
Renormalisation
Impose a “cutoff” mass M on the four momentum inside a loop
This can be interpreted as a limit on the shortest range of the interaction
It can also be interpreted as possible substructure in pointlike fermions
Physical amplitudes should not depend on choice of M
Assume M2 >> q2 (but not infinity!)
ln(M2) terms appear in the amplitude
These terms are absorbed into fermion masses and vertex couplings
The masses m(q2) and couplings a(q2) are functions of q2
Renormalisation of electric charge:
eR = e (1 – e2 ln (M2/q2) )1/2
12p2
N.B. This formula assumes only one type of fermion/antifermion loop
Can be interpreted as a “screening” correction due to the production of
electron/positron pairs in a region round the primary vertex
22/1/10
Particle Physics Lecture 4 Steve Playfer

13. Running Coupling Constant
Converting from electric charge to fine structure constant
and allowing for all fermion types:
a(q2) = a(0) (1 + a(0) zf ln (-q2/M2) ) 3p
where zf = Sf Qf2 is a sum over the active fermion/antifermion charges (in units of e)
zf is a function of q2 : = 1 (1MeV), = 8/3 (1 GeV), = 38/9 (100GeV)
A trick is to replace the M2 dependence with a reference value m:
a(q2) = a(m2) (1 - a(m2) zf ln (-q2/m2) )-1 3p
Can choose any value of m.
For QED usual choice is m~1 MeV, a=1/137 (from atomic physics)
or alternatively m=MZ , a(MZ)=1/129
22/1/10
Particle Physics Lecture 4 Steve Playfer

14. Particle Physics Lecture 4 Steve Playfer
Gyromagnetic Ratio g
Measures the ratio of the magnetic moment to the spin
m = g mB S for an electron
with the Bohr magneton mB = eh/mec = 5.8x10-11 MeV/T
From the Dirac equation expect g=2 for a pointlike fermion
Higher order diagrams lead to an anomalous magnetic
moment described by g-2
The QED contributions have been calculated to O(a5)
The theoretical calculation is now limited by strong interactions
(corrections from the bubble diagrams with quarks)
22/1/10
Particle Physics Lecture 4 Steve Playfer

15. Accuracy of g-2 Measurements
Anomalous magnetic moment of electron
Anomalous magnetic moment of muon
Experiment : g – 2 = (3) 2
Theory : g – 2 = (8)
Experiment : g – 2 = (6) 2
Theory : g – 2 = (5)
Among the most precise tests of a theory in all of physics
Some argument about the 3s discrepancy in the muon g-2!
Could be evidence for physics beyond the Standard Model?
22/1/10
Particle Physics Lecture 4 Steve Playfer