Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2021 / 1 / v

The scalar QED Ward identity equates the vertex renormalization and wave-function renormalization constants. Consequently, using part (iv),The renormalization constants describe gauge-dependent fields and off shell vertices, so their dependence on $\xi$ is permitted. A physical scattering amplitude includes all Feynman diagrams, the counterterms, and the external-state factors from LSZ reduction. The Ward identities ensure that the dependence on $\xi$ cancels for physical external states. Infrared-sensitive probabilities also require the usual inclusive treatment of unobserved soft photons. Thus no dependence on $\xi$ is implied for consistently calculated physical observables.