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CHAPTER 5 : EXAMPLES IN QUANTUM ELECTRODYNAMICS These are the processes that are calculated in chapter 5: ∎ e+ e- → μ+ μ- This is a basic example in QED. Also important because of the similar process e+ e- → q qbar, which proved the quark model and QCD by describing qqbar → hadrons; PETRA (DESY) and PEP (SLAC) accelerator experiments. ∎ e- + μ- → e- + μ- An example of crossing symmetry. Is scattering by a muon realistic? No, but the muon could be the projectile. also important because of the similar process e- + q → e- + q, which occurs in electron-proton deep-inelastic scattering (ep DIS); SLAC and HERA (DESY) experiments. ∎ γ e- → γ eCompton scattering. ∎ e+ e- → γ γ Pair annihilation into photons. Other examples ∎ e- e- → e- eMoller scattering. This is interesting as an example of identical particles. ∎ e+ e- → e+ eBhabha scattering. This process is used to determine the luminosity of e+ ecollisions. Provides the best upper limit on the radius of the electron. 1 THE FEYNMAN RULES FOR QED ● draw the topologically distinct connected diagrams. ● vertex factor = e γμ ; 4-momentum is conserved at the vertex. ● fermion propagator = SF(p) ● photon propagator = Dμν(q) ● external fermion leg = a Dirac spinor; u(p,s), u(p,s), v(p,s), v(p,s) ● external photon leg = a polarization 4vector; εμ(q) ● verify the relative signs ! E E-bar → Mu Mu-bar e-(p1) +e+(p2) → μ-(p3) +μ+(p4) p1μ + p2μ = p3μ + p4μ There is only one Feynman diagram Thus the transition matrix element is 2 ☆ For unpolarized annihilation, average over s1 and s2 , and sum over s3 and s4 . The goal is to calculate the cross section. Recall the steps in the calculation. ☆ Square M ☆ Calculate the traces 3 4 ☆ Substitute Mandelstam variables (See page 156) ☆ The center-of-mass differential cross section ☆ Result 5 6 We can approximate m = 0 because m << M and m << E. Then the result is PS equation 5.12. 7 8 Electron - positron annihilation to hadrons These were important experiments in the history of high-energy physics. From the particle data group, the figure shows the ratio R = σ ( e e → hadrons ) / σ ( e ebar → μ μbar) . The underlying process in hadron production is e- + e+ → q + qbar. Neglecting QCD interactions we would just have R = constant (green = naive quark model; red = QCD 3rd order). 9 The √{s} dependence is due to 2 effects: ● thresholds ● resonances But between thresholds we do indeed have R ≈ constant. And we can even estimate the constant: above the b-quark threshold, R = Σ eq2 / e2 = ( 1/9 + 4/9 + 1/9 + 4/9 + 1/9) × 3 = 11 /3 = 3.67 . This is a great success of QCD. Historically, measurements of R provided evidence for the quark model and QCD. 10