Quantum Field Theory - Problem Set 13 Solutions | PHY 396K, Assignments of Physics

Download Quantum Field Theory - Problem Set 13 Solutions | PHY 396K and more Assignments Physics in PDF only on Docsity! PHY–396 K. Problem set #13. Due December 2, 2004 (last class day). 1. Consider the Bhabha scattering e−e+ → e−e+. In QED, there are two tree-level Feynman diagrams contributing to this process. Note that their contributions must be added before squaring the amplitude and adding/averaging over spins, |M1 +M2|2 = |M1|2 + |M2|2 + 2 Re ( M∗1M2 ) 6= |M1|2 + |M2|2. (1) Your task is to calculate the un-polarized partial cross-section dσ/dΩ for the Bhabha scattering. For simplicity, assume E  me and neglect the electron’s mass throughout your calculation. You may find it convenient to use Mandelstam’s Lorentz-invariant kinematic variables s, t and u, see eq. (5.69) of the Peskin&Schroeder textbook for details. Notice s + t + u = 4m2e ≈ 0 in the me ≈ 0 approximation. The answer to this problem is simple: ( dσ dΩ ) c.m. = α2 2s [( t s )2 + (s t )2 + (u s + u t )2] (2) but the intermediate steps are quite complicated, so beware. 2. The Z0 (or simply Z) particle of the Standard Model is a massive (MZ ≈ 91 GeV) neutral vector boson. The Lagrangian for the Z0µ field is L[Z] = −14ZµνZ µν + 12M 2 Z ZµZ µ + g′ZµJ µ 0 + · · · (3) where Zµν = ∂µZν − ∂νZµ, Jµ0 is the neutral week current governing the Z0 coupling to the leptons and quarks (cf. the charged week currents Jµ± governing the W ± couplings to the fermions) and the ‘· · ·’ stand for additional interaction terms involving the W±, Higgs 1 and EM fields. The specific form of the neutral current is Jµ0 = ∑ fermions Ψ(gV + gAγ 5)γµΨ (4) where gV = (sin 2 θW − 14), gA = + 1 4 for charged leptons e −, µ− and τ−, gV = (+ 1 4), gA = − 1 4 for neutrinos νe, νµ and ντ , gV = ( 1 3 sin 2 θW − 14), gA = + 1 4 for charge − 1 3e quarks d, s and b, gV = (−23 sin 2 θW + 1 4), gA = − 1 4 for charge + 2 3e quarks u, c and t, g′ = e sin θW cos θW , sin2 θW ≈ 0.23. (5) The θW here is called the weak mixing angle; it is an experimentally determined parameter of the Standard Model. Write down the Feynman rules for the Z field and its interactions with the leptons and the quarks, and then calculate the total decay rate of the Z particle and the branching ratios B(Z → e+e−), B(Z → µ+µ−) and B(Z → qq̄ → hadrons). Use tree-level approximation. For your information, the top quark t is heavier than the Z particle while the other five quark flavors and all the leptons are so much lighter than the Z that you may neglect their masses altogether. You should also keep in mind that all quarks (but not the leptons) come in three colors. The color degree of freedom is fundamental to QCD but for the purposes of this calculation, it’s just an index taking three values. 3. Finally consider the e+e− → µ+µ− pair production in the Standard Model. There are two tree-level Feynman diagrams contributing to this process: one involving a virtual photon and the other a virtual Z particle. In QED, there is only the first diagram — which was discussed in class in much detail — but at high energies (which became available at the LEP e+e− collider) both diagrams are equally important. (a) Write down the combined tree-level amplitude M(e+e− → µ+µ−). 2