Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2020 / 3 / b / iv / Solution

Expand the gauge fixing square:The second term is a derivative cubic photon interaction; the third is a quartic photon contact interaction. Both arise from nonlinear Abelian gauge fixing, rather than an additional physical electromagnetic force.

For $\xi =1$, differentiating the quartic term with respect to four identical gauge fields gives eight copies for each pairing, henceWith the Fourier convention $A(x)=\int d^{4}p(2\pi {)}^{-4}{e}^{ipx}A(p)$ and all four-momenta incoming, the differentiated cubic term givesmatching the supplied Feynman rules. Reversing the Fourier convention reverses this cubic rule's sign; each exchange graph contains two such Feynman vertices, so its contribution is unchanged.