reason, the correct way of regularizing the vertex correction in the plane wave is to regularize the quantity $\Gamma^{\mu}(p,p',q;\phi)$ (see also Ref. [@Morozov_1981]). Since the divergence at hand is only logarithmic one first writes $\Gamma^{\mu}(p,p',q;\phi)=\Gamma^{\mu}(p,p',q;\phi)-\Gamma_0^{\mu}(p,p',q)+\Gamma_0^{\mu}(p,p',q)$, where $\Gamma_0^{\mu}(p,p',q)=\Gamma^{\mu}(p,p',q;\phi)|_{A^{\mu}(\phi)=0}$, and notices that $\Gamma^{\mu}(p,p',q;\phi)-\Gamma_0^{\mu}(p,p',q)$ is ultraviolet convergent. Then, one can regularize the vacuum expression $\Gamma_0^{\mu}(p,p',q)$ exactly as in the vacuum, i.e., by subtracting the same expression evaluated for $q^{\mu}=0$ and for $\hat{p}=\hat{p}'=m$ [@Itzykson_b_1980] (notice that the conservation laws in a plane wave already imply that $p^{\prime\mu}=p^{\mu}$ because these four-momenta are on-shell). In conclusion, by assuming that $e$ indicates the physical electron charge, we continue by investigating the regularized vertex function $\Gamma_R^{\mu}(p,p',q;\phi)=\Gamma^{\mu}(p,p',q;\phi)-\Gamma_0^{\mu}(p,p,0)|_{\hat{p}=m}$. By using the master integrals $$\begin{aligned}
\int\frac{d^4k}{(2\pi)^4}e^{iSk^2}&=-\frac{i}{16\pi^2S^2},\\
\int\frac{d^4k}{(2\pi)^4}k^{\mu}k^{\nu}e^{iSk^2}&=\frac{\eta^{\mu\nu}}{4}\int\frac{d^4k}{(2\pi)^4}k^2e^{iSk^2}=\frac{\eta^{\mu\nu}}{32\pi^2S^3},\end{aligned}$$ it is straightforward to take the integral in $d^4k$ in $\Gamma_0^{\mu}(p,p,0)|_{\hat{p}=m}$ and to obtain the result $$\Gamma_0^{\mu}(p,p,0)|_{\hat{p}=m}=-i\frac{\alpha}{2\pi}\gamma^{\mu}\int_0^{\infty}\frac{dsdudt}{S^3}\,e^{-i\kappa^2t-i\frac{(s+u)^2}{S}m^2}\left\{m^2\left[2t-\frac{(s+u)^2}{S}\right]+i\right\}.$$ From the derivations, it is clear that $\Gamma_0^{\mu}(p,p,0)|_{\hat{p}=m}$ has only components $\Gamma_{0,-}(p,p,0)|_{\hat{p}=m}$ and $\Gamma_{0,\perp,j}(p,p,0)|_{\hat{p}=m}$, and then that $\Gamma_{R,q}(p,p',q;\phi)=\Gamma_q(p,p',q;\phi)$, which, as we have mentioned, can be shown to vanish for $A^{\mu}(\phi)=0$ (see the appendix).

Now, we would like to investigate the convergence properties of the proper time integrals in $\Gamma_{R,-}(p,p',q;\phi)$ and $\Gamma_{R,\perp,j}(p,p',q;\phi)$. It is first convenient to use the following identity [@Schubert_2001] $$\begin{split}
\int_0^{\infty}ds\int_0^{\infty}du\int_0^{\infty}dt&=\int_0^{\infty}ds\int_0^{\infty}du\int_0^{\infty}dt\int_0^{\infty}dS\,\delta(S-s-u-t)\\
&=\int_0^{\infty}dS\int_0^Sds\int_0^Sdu\int_0^Sdt\,\delta(S-s-u-t)\\
&=\int_0^{\infty}dS\,S^2\int_0^1dx\int_0^1dy\int_0^1dz\,\delta(1-x-y-z),
\end{split}$$ where 