a gauge transformation of the plane wave four-vector potential, as it can be proved by replacing $A^{\mu}(\phi)$ with $A^{\mu}(\phi)+\partial^{\mu}f(\phi)=A^{\mu}(\phi)+n^{\mu}f'(\phi)$, with $f(\phi)$ being an arbitrary function of $\phi$ \[we recall, in particular, that $\pi^{\mu}_p(\phi)$ is the kinetic four-momentum of an electron in a plane wave and it is therefore gauge invariant, see Eq. (\[pi\])\]. Now, concerning a gauge transformation of the radiation field and, in particular, of the external photon, we have already discussed that $\Gamma^{\mu}(p,p',q;\phi)$ is written in a form that automatically fulfills the Ward identity, in such a way that one-loop radiative corrections are gauge invariant. In addition, we study here the effect of the additional term $\delta\Gamma^{(\xi)}_{s,s',l}(p,p',q)$ brought about by considering the photon propagator $D^{(\xi)\,\lambda\nu}(x)$ [@Itzykson_b_1980] $$D^{(\xi)\,\lambda\nu}(x)=\int\frac{d^4k}{(2\pi)^4}\frac{e^{-i(kx)}}{k^2-\kappa^2+i0}\left[\eta^{\lambda\nu}+\left(1-\frac{1}{\xi}\right)\frac{k^{\lambda}k^{\nu}}{k^2-\kappa^2+i0}\right],$$ in an arbitrary gauge parametrized by the constant $\xi$ (the Lorenz gauge corresponds to $\xi=1$). It is clear from Eq. (\[Gamma\_2\]) that $$\begin{split}
\delta\Gamma^{(\xi)}_{s,s',l}(p,p',q)=&-ie^2\left(1-\frac{1}{\xi}\right)\int d^4x \int\frac{d^4k}{(2\pi)^4}\,\frac{1}{(k^2-\kappa^2+i0)^2}\\
&\times\bar{U}_{s'}(p',x)\hat{k}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}e^{i(qx)}\hat{e}^*_l(q)\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0}\hat{k}U_s(p,x).
\end{split}$$ Now, since $[\hat{\Pi}(\phi)-m]U_s(p,x)=[\hat{\Pi}(\phi)-m]U_{s'}(p',x)=0$, we have that $\bar{U}_{s'}(p',x)\hat{k}=\bar{U}_{s'}(p',x)[\hat{\Pi}(\phi)+\hat{k}-m]$ and analogously $\hat{k}U_s(p,x)=[\hat{\Pi}(\phi)+\hat{k}-m]U_s(p,x)$. Thus, the two electron propagators in $\delta\Gamma^{(\xi)}_{s,s',l}(p,p',q)$ simplify and this quantity can be written in the form $$\delta\Gamma^{(\xi)}_{s,s',l}(p,p',q)=Z^{(\xi)}\int d^4x\,e^{i(qx)}\bar{U}_{s'}(p',x)\hat{e}^*_l(q)U_s(p,x),$$ with $$Z^{(\xi)}=-ie^2\left(1-\frac{1}{\xi}\right)\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{(k^2-\kappa^2+i0)^2}$$ being a logarithmically divergent, gauge-dependent constant. However, since $\delta\Gamma^{(\xi)}_{s,s',l}(p,p',q)$ has exactly the same structure of the tree-level matrix element of (virtual) nonlinear 