\begin{split}
G=&\frac{1}{\hat{\Pi}^2-m^2+i0}(\hat{\Pi}+m)=(-i)\int_0^{\infty}ds\, e^{-im^2s}\Big\{1+\frac{e}{2P_T}\hat{n}[\hat{A}(\Phi+2sP_T)-\hat{A}(\Phi)]\Big\}\\
&\times e^{-i\int_0^sds'[\bm{P}_{\perp}-e\bm{A}_{\perp}(\Phi+2s'P_T)]^2}e^{2isP_TP_{\phi}}(\hat{\Pi}+m).
\end{split}$$

General expression of the one-loop vertex correction {#VC_General}
====================================================

The one-loop vertex correction corresponds to the Feynman diagram in Fig. \[FD\_VC\], where we have implicitly assumed that the photon four-momentum $q^{\mu}$ is outgoing. Note that the two external electron lines correspond to real electrons, i.e., the four-momenta $p^{\mu}$ and $p^{\prime\,\mu}$ are on-shell ($p^2=p^{\prime\,2}=m^2$), whereas at the moment we make no assumptions about the outgoing photon, i.e., in particular, $q^2\neq 0$. If we denote by $s$ ($s'$) the spin quantum number of the incoming (outgoing) electron and by $l$ the polarization quantum number of the outgoing photon, the amplitude $-ie\Gamma_{s,s',l}(p,p',q)$ corresponding to the Feynman diagram in Fig. \[FD\_VC\] can be written as $$-ie\Gamma_{s,s',l}(p,p',q)=-e^3\int d^4x\, d^4y\, d^4z\,\bar{U}_{s'}(p',y)\gamma^{\lambda}G(y,z)\hat{e}^*_l(q)e^{i(qz)}G(z,x)\gamma^{\nu}U_s(p,x)D_{\lambda\nu}(x-y),$$ where $e^{\mu}_l(q)$ is the polarization four-vector of the outgoing photon. Here, we have introduced the photon propagator $D^{\lambda\nu}(x)$ and we work in the Lorenz gauge such that $$D^{\lambda\nu}(x)=\int\frac{d^4k}{(2\pi)^4}\frac{\eta^{\lambda\nu}}{k^2-\kappa^2+i0}e^{-i(kx)},$$ where $\kappa^2$ is the square of a fictitious photon mass, which has been introduced to avoid infrared divergences.

By using the completeness relation in Eq. (\[C\_x\]) and the translation properties in Eq. (\[Trans\_X\]), the amplitude can be written in the semi-operator form as $$\label{Gamma_2}
\begin{split}
-ie\Gamma_{s,s',l}(p,p',q)&=-e^3\int d^4x\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{k^2-\kappa^2+i0}\bar{U}_{s'}(p',x)e^{i(kx)}\gamma^{\lambda}G e^{i(qx)}\hat{e}^*_l(q)Ge^{-i(kx)}\gamma_{\lambda}U_s(p,x)\\
&=-e^3\int d^4x\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{k^2-\kappa^2+i0}\\
&\quad\times\bar{U}_{s'}(p',x)\gamma^{\lambda}\frac{1}{\hat{\Pi}(\phi)+\hat{k}-m+i0} 