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

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

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

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

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 = 1 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 123 123 SS3 = Le SS4 = Lm 22/1/10 Particle Physics Lecture 4 Steve Playfer

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) t2 sin4 q*/2 ds = a2 (s2 + u2) with initial and final state factors dW 2s t2 Polarized cross-sections from: M ( ) = M ( ) = e2 u t M ( ) = M ( ) = e2 s 22/1/10 Particle Physics Lecture 4 Steve Playfer

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 4s Total cross-section: s = 4pa2 (decreases with increasing CM energy) 3s 22/1/10 Particle Physics Lecture 4 Steve Playfer

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

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

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

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

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

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

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

Accuracy of g-2 Measurements Anomalous magnetic moment of electron Anomalous magnetic moment of muon Experiment : g – 2 = 0.0011596521807(3) 2 Theory : g – 2 = 0.001159652183(8) Experiment : g – 2 = 0.0011659209(6) 2 Theory : g – 2 = 0.0011659183(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