Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 305 / 2 / ii

Write an amplitude with an external massless vector of momentum $k$ asChanging the polarization representative by ${ϵ}_{\mu }\mapsto {ϵ}_{\mu }+c,k_{\mu }$ is a gauge transformation. The physical amplitude must be unchanged, so the Ward identity isThus a longitudinal pure-gauge polarization decouples from the S-matrix. This on-shell statement is the amplitude counterpart of the off-shell Ward-Takahashi identity.