$q_->0$. Thus, by using the three four-vectors $$\begin{aligned}
N^{\mu}&=q^{\mu}-\frac{q^2n^{\mu}}{2q_-},\\
\label{Lambda_i}
\Lambda^{\mu}_i&=a_i^{\mu}+\frac{q_{\perp,i}n^{\mu}}{q_-},\end{aligned}$$ with $i=1,2$ together with $n^{\mu}$, one can build a light-cone basis such that $$\eta^{\mu\nu}=\frac{N^{\mu}n^{\nu}+n^{\mu}N^{\nu}}{q_-}-\Lambda^{\mu}_1\Lambda^{\nu}_1-\Lambda^{\mu}_2\Lambda^{\nu}_2.$$ Then, the quantity $\Gamma^{\mu}(p,p',q;\phi)e^*_{l,\mu}(q)$, which is the one finally required here, can be written as \[recall that we work in the Lorenz gauge where $(qe^*_l(q))=0$\] $$\label{Gamma_exp}
\begin{split}
\Gamma^{\mu}(p,p',q;\phi)e^*_{l,\mu}(q)&=\frac{e^*_{l,-}(q)}{q_-}\Gamma_q(p,p',q;\phi)-\frac{q^2e^*_{l,-}(q)}{q_-^2}\Gamma_-(p,p',q;\phi)\\
&\quad-(\Gamma(p,p',q;\phi)\Lambda_1)(\Lambda_1e^*_l(q))-(\Gamma(p,p',q;\phi)\Lambda_2)(\Lambda_2e^*_l(q))\\
&=\frac{e^*_{l,-}(q)}{q_-}\Gamma_q(p,p',q;\phi)\\
&\quad-\left[\frac{q^2e^*_{l,-}(q)}{q_-^2}+\frac{q_{\perp,1}}{q_-}(\Lambda_1e^*_l(q))+\frac{q_{\perp,2}}{q_-}(\Lambda_2e^*_l(q))\right]\Gamma_-(p,p',q;\phi)\\
&\quad+\Gamma_{\perp,1}(p,p',q;\phi)(\Lambda_1e^*_l(q))+\Gamma_{\perp,2}(p,p',q;\phi)(\Lambda_2e^*_l(q))
\end{split}$$ where $\Gamma_q(p,p',q;\phi)=(\Gamma^{\mu}(p,p',q;\phi)q_{\mu})$ and all other symbols are defined in analogy to the definitions given in the introduction. Now, since the structure of the function $\Gamma^{\mu}(p,p',q;\phi)$ is complicated because of the presence of the plane wave, it is clear that the components $\Gamma_-(p,p',q;\phi)=(\Gamma^{\mu}(p,p',q;\phi)n_{\mu})$ and $\Gamma_{\perp,j}(p,p',q;\phi)=-(\Gamma^{\mu}(p,p',q;\phi)a_{j,\mu})$ are relatively easy to work out because the quantities $n^{\mu}$ and $a_i^{\mu}$ characterize the plane wave. For example, we observe that all the terms proportional to $n^{\mu}$ in $\Gamma^{\mu}(p,p',q;\phi)$ can be ignored in the computation of the components $\Gamma_-(p,p',q;\phi)$ and $\Gamma_{\perp,j}(p,p',q;\phi)$.

The apparently most complicated term is, therefore, $\Gamma_q(p,p',q;\phi)$, which is related to the fact that by itself the vertex-correction function is not gauge invariant. However, the gauge invariance of QED guarantees that the component $\Gamma_q(p,p',q;\phi)$ of the vertex-correction function does not to contribute to any transition amplitude involving on-shell external electrons/positrons. We explicitly prove this statement in 