of the vertex-correction function under gauge transformations of the radiation field and of the plane-wave background field are studied. In Sec. \[VC\_CP\] we show how to regularize and renormalize the vertex-correction function in the ultraviolet. The expression of the vertex-correction function within the LCFA is derived in Sec. \[VC\_LCFA\] and, finally, the main conclusions of the paper are reported in Sec. \[VC\_Conclusions\]. An appendix contains some technical considerations on a component of the vertex-correction function.

Notation {#Notation}
========

The notation employed below is the same as in Ref. [@Di_Piazza_2018_d] but it is convenient to report here the main definitions. As we have mentioned in the Introduction, the present paper focuses on studying radiative corrections in a general plane-wave field. The latter is described by the four-vector potential $A^{\mu}(\phi)$, which only depends on the light-cone time $\phi=t-\bm{n}\cdot \bm{x}$. Here, the unit vector $\bm{n}$ defines the propagation direction of the plane wave, which can be used to introduce two useful four-dimensional quantities: $n^{\mu}=(1,\bm{n})$ and $\tilde{n}^{\mu}=(1,-\bm{n})/2$ (note that $\phi=(nx)$). Assuming obvious differential properties of the four-vector potential $A^{\mu}(\phi)$ and its derivatives, it is clear that it is a solution of the free wave equation $\square A^{\mu}=0$, where $\square=\partial_{\nu}\partial^{\nu}$, and it is assumed to 