One-loop vertex correction in a plane wave

We compute the general expression of the one-loop vertex correction in an arbitrary plane-wave background field for the case of two on-shell external electrons and an off-shell external photon. The properties of the vertex corrections under gauge transformations of the plane-wave background field and of the radiation field are studied. Concerning the divergences of the vertex correction, the infrared one is cured by assigning a finite mass to the photon, whereas the ultraviolet one is shown to be renormalized exactly as in vacuum. Finally, the corresponding expression of the vertex correction within the locally constant crossed field approximation is also derived and the high-field asymptotic is shown to scale according to the Ritus-Narozhny conjecture, although the terms featuring the leading-order dependence on the field are found not to contribute to physical transition amplitudes.