Griffiths Introduction to Elementary Particles · Problem 7.34

Electrodynamics of Quarks and Hadrons — David J. Griffiths

Physics — QFT · electron-electron-scattering

Problem

Starting with equation (7.105), determine the spin-averaged amplitude, [analogous to equation (7.126)] for electron-electron scattering. Assume we're working at high energies, so that the mass of the electron can be ignored (i.e., set m=0m = 0). [Hint: You can read ⟨∣M1∣2⟩\langle|\mathcal{M}_1|^2\rangle and ⟨∣M2∣2⟩\langle|\mathcal{M}_2|^2\rangle from equation (7.126). For ⟨M1M2∗⟩\langle\mathcal{M}_1\mathcal{M}_2^*\rangle use the same strategy as Casimir's trick to get ⟨M1M2∗⟩=−ge44(p1−p3)2(p2−p4)2 Tr(γμ̸p1γν̸p4γμ̸p2γν̸p3)\langle\mathcal{M}_1\mathcal{M}_2^*\rangle = \frac{-g_e^4}{4(p_1-p_3)^2(p_2-p_4)^2}\,Tr(\gamma^\mu \not{p}_1 \gamma^\nu \not{p}_4 \gamma_\mu \not{p}_2 \gamma_\nu \not{p}_3) Then exploit the contraction theorems to evaluate the trace. Notice that for massless particles the conservation of momentum (p1+p2=p3+p4p_1+p_2 = p_3+p_4) implies that p1⋅p2=p3⋅p4p_1\cdot p_2 = p_3\cdot p_4, p1⋅p3=p2⋅p4p_1\cdot p_3 = p_2\cdot p_4, and p1⋅p4=p2⋅p3p_1\cdot p_4 = p_2\cdot p_3.] [Answer: ⟨∣M∣2⟩=2ge4(p1⋅p3)2(p1⋅p4)2[(p1⋅p2)4+(p1⋅p3)4+(p1⋅p4)4]]\left[\textit{Answer: }\langle|\mathcal{M}|^2\rangle = \frac{2g_e^4}{(p_1\cdot p_3)^2(p_1\cdot p_4)^2}\left[(p_1\cdot p_2)^4 + (p_1\cdot p_3)^4 + (p_1\cdot p_4)^4\right]\right]

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