Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2024 / 2 / b

Put $S(p)=i(/p+m)/(p^2-m^2+i0)$ for the Dirac propagator, and ${\int }_{\ell }={\mu }_R^{2ϵ}\int d^{4-2ϵ}\ell /(2\pi {)}^{4-2ϵ}$ for a translation-invariant dimensional regularization of the loop momentum integral. The two oriented scattering amplitudes areThe common minus sign is the closed fermion loop sign. There is no additional symmetry factor for either labelled orientation. Each expression is a trace over spinor indices; all external photon propagators are amputated. If $K(p)=(/p+m)/(p^2-m^2+i0)$, the common factor is $-(-ie{)}^3{i}^3=-e^3$, so, for example, $i{\mathcal{M}}_A=-e^3{\int }_{\ell }\mathrm{tr}[{\gamma }^{\mu }K(\ell +k_1){\gamma }^{\nu }K(\ell -k_3){\gamma }^{\rho }K(\ell )]$.