depend on $k_-$, implies that $k^{\mu}=k^{\prime\mu}-\tilde{G}^{\mu}/S$, where $$\label{tG}
\tilde{G}^{\mu}=\int_0^sds'\pi_{p'}^{\mu}(\tilde{\psi}_{s'})+\int_0^udu'\pi_p^{\mu}(\tilde{\psi}_{u'}),$$ such that $\tilde{G}_-=\tilde{F}_-=sp'_-+up_-$. Here, we have introduced the two shifted phases $$\begin{aligned}
\label{tpsi_p}
\tilde{\psi}_s&=\phi+2s\tau'_-+2sk_-,\\
\label{tpsi_pp}
\tilde{\psi}_u&=\phi-2u\tau_--2uk_-,\end{aligned}$$ where $$\begin{aligned}
\label{tau_p_m}
\tau'_-&=p'_--\frac{\tilde{G}_-}{S}=\frac{tp'_--uq_-}{S},
\\
\label{tau_m}
\tau_-&=p_--\frac{\tilde{G}_-}{S}=\frac{tp_-+sq_-}{S}.\end{aligned}$$ After the shift of the four-momentum $k^{\mu}$, we can write $\Gamma^{\mu}(p,p',q;\phi)$ in the form $$\Gamma^{\mu}(p,p',q;\phi)=e^2\int_0^{\infty}dsdudt\int\frac{d^4k}{(2\pi)^4}e^{-i\kappa^2t-i\frac{\tilde{G}^2}{S}+iSk^2}\tilde{L}(\tilde{Q}^{\prime\lambda}+\gamma^{\lambda}\hat{k})\tilde{C}^{\mu}(\tilde{Q}_{\lambda}+\hat{k}\gamma_{\lambda})\tilde{R},$$ where $$\begin{aligned}
\label{tL}
\tilde{L}&=1-\frac{e\hat{n}[\hat{A}(\tilde{\psi}_s)-\hat{A}(\phi)]}{2p'_-},\\
\tilde{Q}^{\lambda}&=2\pi_p^{\lambda}(\tilde{\psi}_u)+i\frac{e\hat{n}\hat{A}'(\tilde{\psi}_u)}{p_-}n^{\lambda}-\frac{\hat{\tilde{G}}}{S}\gamma^{\lambda},\\
\tilde{C}^{\mu}&=\left\{1+\frac{e\hat{n}[\hat{A}(\tilde{\psi}_s)-\hat{A}(\phi)]}{2(\tau'_-+k_-)}\right\}\gamma^{\mu}\left\{1-\frac{e\hat{n}[\hat{A}(\tilde{\psi}_u)-\hat{A}(\phi)]}{2(\tau_-+k_-)}\right\},\\
\tilde{Q}^{\prime\lambda}&=2\pi_{p'}^{\lambda}(\tilde{\psi}_s)+i\frac{e\hat{n}\hat{A}'(\tilde{\psi}_s)}{p'_-}n^{\lambda}-\gamma^{\lambda}\frac{\hat{\tilde{G}}}{S},\\
\label{tR}
\tilde{R}&=1+\frac{e\hat{n}[\hat{A}(\tilde{\psi}_u)-\hat{A}(\phi)]}{2p_-}.\end{aligned}$$ The integral in $d^4k$ in $\Gamma^{\mu}(p,p',q;\phi)$ is complicated by the fact that the variable $k_-$ is contained in the argument of the four-vector potential of the plane wave. Thus, we first take the integral in $d^2\bm{k}_{\perp}$, which is Gaussian: $$\Gamma^{\mu}(p,p',q;\phi)=-i\alpha\int_0^{\infty}\frac{dsdudt}{S}\int\frac{dk_-dk_+}{(2\pi)^2}e^{-i\kappa^2t-i\frac{\tilde{G}^2}{S}+2iSk_-k_+}\tilde{M}^{\mu}(k_-,k_+,s,u,t;\phi),$$ where $\alpha=e^2/4\pi\approx 1/137$ is the fine-structure constant and where $$\begin{split}
\tilde{M}^{\mu}(k_-,k_+,s,u,t;\phi)=&\tilde{L}\left[(\tilde{Q}^{\prime\lambda}+k_-\gamma^{\lambda}\hat{\tilde{n}})\tilde{C}^{\mu}(\tilde{Q}_{\lambda}+k_-\hat{\tilde{n}}\gamma_{\lambda})-\frac{i}{2S}\gamma^{\lambda}\gamma_{\perp,i}\tilde{C}^{\mu}\gamma_{\perp,i}\gamma_{\lambda}\right]\tilde{R}\\
&+k_+\tilde{L}[\gamma^{\lambda}\hat{n}\tilde{C}^{\mu}(\tilde{Q}_{\lambda}+k_-\hat{\tilde{n}}\gamma_{\lambda})+(\tilde{Q}^{\prime\lambda}+k_-\gamma^{\lambda}\hat{\tilde{n}})\tilde{C}^{\mu}\hat{n}\gamma_{\lambda}]\tilde{R}\\
&+k_+^2\tilde{L}\gamma^{\lambda}\hat{n}\gamma^{\mu}\hat{n}\gamma_{\lambda}\tilde{R}.
\end{split}$$ Finally, the integral in $dk_+$ results in a delta function and its first and second derivatives all evaluated at $2Sk_-$. This allows then also to take the integral in $dk_-$ and, after straightforward manipulations, the resulting expression of $\Gamma^{\mu}(p,p',q;\phi)$ can be written as $$\label{Gamma_f}
\begin{split}
\Gamma^{\mu}(p,p',q;\phi)&=-\frac{i\alpha}{4\pi}\int_0^{\infty}\frac{dsdudt}{S^3}\,e^{-i\kappa^2t}\bigg\{e^{-i\frac{\tilde{G}^2}{S}}\tilde{L}(S\tilde{Q}^{\prime\lambda}\tilde{C}^{\mu}\tilde{Q}_{\lambda}+2i\tilde{C}^{\mu})\tilde{R}\\
&\quad\left.\left.+\frac{i}{2}\frac{d}{dk_-}\left[e^{-i\frac{\tilde{G}^2}{S}}\tilde{L}(\gamma^{\lambda}\hat{n}\tilde{C}^{\mu}\tilde{Q}_{\lambda}+\tilde{Q}^{\prime\lambda}\tilde{C}^{\mu}\hat{n}\gamma_{\lambda})\tilde{R}\right]+\frac{\hat{n}}{S}n^{\mu}\frac{d^2}{dk^2_-}\left(e^{-i\frac{\tilde{G}^2}{S}}\right)\right\}\right\vert_{k_-=0}\\
&=-\frac{i\alpha}{4\pi}\int_0^{\infty}\frac{dsdudt}{S^3}\,e^{-i\kappa^2t-i\frac{\tilde{G}^2}{S}}\\
&\quad\times\Bigg\{\tilde{L}\left[S\tilde{Q}^{\prime\lambda}\tilde{C}^{\mu}\tilde{Q}_{\lambda}+2i\tilde{C}^{\mu}+\frac{1}{2S}\frac{d\tilde{G}^2}{dk_-}(\gamma^{\lambda}\hat{n}\tilde{C}^{\mu}\tilde{Q}_{\lambda}+\tilde{Q}^{\prime\lambda}\tilde{C}^{\mu}\hat{n}\gamma_{\lambda})\right]\tilde{R}\\
&\quad\left.\left.+\frac{i}{2}\frac{d}{dk_-}\left[\tilde{L}(\gamma^{\lambda}\hat{n}\tilde{C}^{\mu}\tilde{Q}_{\lambda}+\tilde{Q}^{\prime\lambda}\tilde{C}^{\mu}\hat{n}\gamma_{\lambda})\tilde{R}\right]-\frac{\hat{n}}{S^2}n^{\mu}\left[\frac{1}{S}\left(\frac{d\tilde{G}^2}{dk_-}\right)^2+i\frac{d^2\tilde{G}^2}{dk^2_-}\right]\right\}\right\vert_{k_-=0}.
\end{split}$$ This expression can be further manipulated especially to simplify its matrix structure. However, it is first convenient to make the following considerations related to the Ward identity to be fulfilled by $\Gamma^{\mu}(p,p',q;\phi)$ [@Itzykson_b_1980]. From now on we assume that 