we have introduced the quantities $$\begin{aligned}
\label{phi_s}
\phi_s&=\phi+2s(p'_-+k_-),\\
\label{phi_u}
\phi_u&=\phi-2u(p_-+k_-),\end{aligned}$$ and the matrix $$\begin{split}
M^{\mu}(k,s,u;\phi)&=\left\{1-\frac{e\hat{n}[\hat{A}(\phi_s)-\hat{A}(\phi)]}{2p'_-}\right\}\left[2\pi_{p'}^{\lambda}(\phi_s)+i\frac{e\hat{n}\hat{A}'(\phi_s)}{p'_-}n^{\lambda}+\gamma^{\lambda}\hat{k}\right]\\
&\quad\times\left\{1+\frac{e\hat{n}[\hat{A}(\phi_s)-\hat{A}(\phi)]}{2(p'_-+k_-)}\right\}\gamma^{\mu}\left\{1-\frac{e\hat{n}[\hat{A}(\phi_u)-\hat{A}(\phi)]}{2(p_-+k_-)}\right\}\\
&\quad\times \left[2\pi_{p,\lambda}(\phi_u)+i\frac{e\hat{n}\hat{A}'(\phi_u)}{p_-}n_{\lambda}+\hat{k}\gamma_{\lambda}\right]\left\{1+\frac{e\hat{n}[\hat{A}(\phi_u)-\hat{A}(\phi)]}{2p_-}\right\}.
\end{split}$$ Note that the integrals in $T$ and $\bm{x}_{\perp}$ can be easily taken and enforce the conservation laws $p_-=p'_-+q_-$ and $\bm{p}_{\perp}=\bm{p}'_{\perp}+\bm{q}_{\perp}$, typical of problems in a plane-wave background field. As a related remark, it is clear that the quantity $\Gamma^{\mu}(p,p',q;\phi)$, unlike the corresponding vacuum expression, depends also on the plane-wave phase. Finally, the definition in Eq. (\[Gamma\^mu\]) is consistent with the idea that computing the amplitude of the vertex in a plane wave up to one loop, one can use the substitution rule $\gamma^{\mu}\to \gamma^{\mu}+\Gamma^{\mu}(p,p',q;\phi)$, as for the vertex correction in vacuum.

The phase in Eq. (\[Gamma\_3\]) can be written in a compact form by turning the integral from $\phi$ to $\phi_s$ (from $\phi$ to $\phi_u$) into an integral in $s'$ ($u'$) like that in the third line of Eq. (\[Gamma\_3\]). By exponentiating also the denominator $k^2-\kappa^2+i0$ in the photon propagator, the quantity $\Gamma^{\mu}(p,p',q;\phi)$ can be written as $$\Gamma^{\mu}(p,p',q;\phi)=e^2\int\frac{d^4k}{(2\pi)^4}\int_0^{\infty}ds\int_0^{\infty}du\int_0^{\infty}dt\,e^{iSk^2-i\kappa^2t+2i(k\tilde{F})} M^{\mu}(k,s,u;\phi),$$ where $S=u+s+t$ and $$\label{F}
\tilde{F}^{\mu}=\int_0^sds'\pi_{p'}^{\mu}(\phi_{s'})+\int_0^udu'\pi_p^{\mu}(\phi_{u'}).$$ As next step, we can perform the integrals in $d^4k$ analytically by shifting the four-momentum $k^{\mu}$ by setting $k^{\prime\mu}=k^{\mu}+\tilde{F}^{\mu}/S$, which, since all components of $\tilde{F}^{\mu}$ except $\tilde{F}_-$ 