in the last line we performed the changes of variables $s=xS$, $u=yS$, and $t=zS$. By setting $$\label{int_3d}
\int_{\delta} dxdydz=\int_0^1dx\int_0^1dy\int_0^1dz\,\delta(1-x-y-z),$$ it is instructive to report the expression of $\Gamma_0^{\mu}(p,p,0)|_{\hat{p}=m}$ in terms of the new variables: $$\Gamma_0^{\mu}(p,p,0)|_{\hat{p}=m}=-i\frac{\alpha}{2\pi}\gamma^{\mu}\int_0^{\infty}dS\int_{\delta}dxdydz\,e^{-i\kappa^2zS-im^2(x+y)^2S}\left\{m^2[2z-(x+y)^2]+\frac{i}{S}\right\},$$ because it clearly shows that only the term whose integrand is proportional to $i$ is (logarithmically) divergent (in the limit $S\to 0$). This divergence is related with the ultraviolet logarithmic divergence of the vertex-correction function. Keeping in mind that $z=1-x-y$ \[see Eq. (\[int\_3d\])\], another divergence for $x+y\to 0$ arises for a massless photon ($\kappa^2=0$), which corresponds to the infrared divergence of the vertex-correction function. By means of the above change of variables, we obtain $$\label{Gamma_R_-}
\begin{split}
&\Gamma_{R,-}(p,p',q;\phi)=\frac{\alpha}{2\pi}\hat{n}\int_0^{\infty}\frac{dS}{S}\int_{\delta} dxdydz\,e^{-i\kappa^2zS}\left[e^{-ig^2S}-e^{-im^2(x+y)^2S}\right]\\
&\quad-\frac{i\alpha}{2\pi}\int_0^{\infty}dS\int_{\delta} dxdydz\,e^{-i\kappa^2zS}\Bigg\{e^{-ig^2S}\Bigg[\left(2(\pi_s\pi_u)+g^2\right)\hat{n}-2g_-(\hat{\pi}_sR+L\hat{\pi}_u)-\hat{g}\hat{\pi}_s\hat{n}-\hat{n}\hat{\pi}_u\hat{g}\\
&\left.\quad+2\tau_-L\hat{g}+2\tau'_-\hat{g}R+2g_-\hat{g}-g_-^2\frac{\hat{\Delta}_s\hat{n}\hat{\Delta}_u}{p_-p'_-}\right]-m^2\hat{n}e^{-im^2(x+y)^2S}[2z-(x+y)^2]\Bigg\},
\end{split}$$ and $$\label{Gamma_R_j}
\begin{split}
&\Gamma_{R,\perp,j}(p,p',q;\phi)=-\frac{\alpha}{2\pi}\hat{a}_j\int_0^{\infty}\frac{dS}{S}\int_{\delta} dxdydz\,e^{-i\kappa^2zS}\left[e^{-ig^2S}-e^{-im^2(x+y)^2S}\right]\\
&\quad+\frac{i\alpha}{2\pi}\int_0^{\infty}dS\int_{\delta} dxdydz\,e^{-i\kappa^2zS}\left\langle e^{-ig^2S}\left\{2(\pi_s\pi_u)\left(\hat{a}_j+\frac{g_-}{2p'_-\tau'_-}\hat{n}\hat{\Delta}_s\hat{a}_j-\frac{g_-}{2p_-\tau_-}\hat{a}_j\hat{n}\hat{\Delta}_u\right)\right.\right.\\
&\quad+\frac{i}{S}\left(\frac{g_-}{2p'_-\tau'_-}\hat{n}\hat{\Delta}_s\hat{a}_j-\frac{g_-}{2p_-\tau_-}\hat{a}_j\hat{n}\hat{\Delta}_u\right)-L(Ca_j)\hat{g}\hat{\pi}_sR-L\hat{\pi}_u\hat{g}(Ca_j)R\\
&\quad-L\hat{g}\left(\hat{a}_j+\frac{\hat{a}_j\hat{\Delta}_s\hat{n}}{2\tau'_-}-\frac{\hat{\Delta}_u\hat{n}\hat{a}_j}{2\tau_-}\right)\hat{g}R-2S(gg_1)\left(\frac{\tau_-}{p'_-}\hat{n}\hat{\Delta}_s\hat{a}_j-\frac{\tau'_-}{p_-}\hat{a}_j\hat{n}\hat{\Delta}_u+2g_-\hat{a}_j-2(ga_j)\hat{n}\right.\\
&\quad+\hat{n}\hat{a}_j\hat{\pi}_s+\hat{\pi}_u\hat{a}_j\hat{n}\bigg)+2i\left((g_1a_j)+x(\mathcal{A}'_sa_j)-y(\mathcal{A}'_ua_j)\right)\hat{n}+i\left[x-(y+z)\frac{\tau_-}{p'_-}\right]\hat{\mathcal{A}}_s'\hat{n}\hat{a}_j\\
&\quad-i\left[y-(x+z)\frac{\tau'_-}{p_-}\right]\hat{a}_j\hat{n}\hat{\mathcal{A}}_u'\Bigg\}-m^2\hat{a}_je^{-im^2(x+y)^2S}[2z-(x+y)^2]\Bigg\rangle,
\end{split}$$ where $$\begin{split}
&L(Ca_j)\hat{g}\hat{\pi}_sR=\left(\hat{a}_j+\frac{g_-}{2p'_-\tau'_-}\hat{n}\hat{\Delta}_s\hat{a}_j
-\frac{g_-}{2p_-\tau_-}\hat{a}_j\hat{n}\hat{\Delta}_u\right)\hat{g}\hat{\pi}_s\\
&+\left(\hat{a}_j+\frac{g_-}{2p'_-\tau'_-}\hat{n}\hat{\Delta}_s\hat{a}_j\right)\left(p'_-\hat{g}-g_-\hat{\pi}_s\right)\frac{\hat{\Delta}_u}{p_-}\\
&+\frac{\hat{a}_j\hat{n}}{2p_-\tau_-}[2(g_-(\Delta_u\pi_s)-p'_-(\Delta_ug))\hat{\Delta}_u-(g_-\Delta_u^2+2\tau_-(\Delta_ug))\hat{\pi}_s+(p'_-\Delta_u^2+2\tau_-(\Delta_u\pi_s))\hat{g}],
\end{split}$$ $$\begin{split}
&L\hat{\pi}_u\hat{g}(Ca_j)R=\hat{\pi}_u\hat{g}\left(\hat{a}_j+\frac{g_-}{2p'_-\tau'_-}\hat{n}\hat{\Delta}_s\hat{a}_j-\frac{g_-}{2p_-\tau_-}\hat{a}_j\hat{n}\hat{\Delta}_u\right)
\\
&+\frac{\hat{\Delta}_s}{p'_-}\left(p_-\hat{g}-g_-\hat{\pi}_u\right)\left(\hat{a}_j-\frac{g_-}{2p_-\tau_-}\hat{a}_j\hat{n}\hat{\Delta}_u\right)\\
&+[2(g_-(\Delta_s\pi_u)-p_-(\Delta_sg))\hat{\Delta}_s-(g_-\Delta_s^2+2\tau'_-(\Delta_sg))\hat{\pi}_u+(p_-\Delta_s^2+2\tau'_-(\Delta_s\pi_u))\hat{g}]\frac{\hat{n}\hat{a}_j}{2p'_-\tau'_-},
\end{split}$$ $$\begin{split}
&L\hat{g}\left(\hat{a}_j+\frac{\hat{a}_j\hat{\Delta}_s\hat{n}}{2\tau'_-}-\frac{\hat{\Delta}_u\hat{n}\hat{a}_j}{2\tau_-}\right)\hat{g}R=\frac{g_-}{2}\left(\frac{\hat{g}\hat{a}_j\hat{\Delta}_s\hat{n}\hat{\Delta}_u}{p_-\tau'_-}+\frac{\hat{\Delta}_s\hat{n}\hat{\Delta}_u\hat{a}_j\hat{g}}{p'_-\tau_-}\right)+g_-\left(\frac{\hat{a}_j\hat{\Delta}_s\hat{g}}{\tau'_-}+\frac{\hat{g}\hat{\Delta}_u\hat{a}_j}{\tau_-}\right)\\
&+\left[(p'_-+\tau'_-)(ga_j)+g_-(\Delta_sa_j)\right]\frac{\hat{\Delta}_s\hat{n}\hat{g}}{p'_-\tau'_-}+\left[(p_-+\tau_-)(ga_j)+g_-(\Delta_ua_j)\right]\frac{\hat{g}\hat{n}\hat{\Delta}_u}{p_-\tau_-}-g^2\hat{a}_j\\
&-\left[(p'_-+\tau'_-)g^2+\frac{\tau'_-g^2_-}{p_-\tau_-}\Delta^2_u\right]\frac{\hat{\Delta}_s\hat{n}\hat{a}_j}{2p'_-\tau'_-}-\left[(p_-+\tau_-)g^2+\frac{\tau_-g^2_-}{p'_-\tau'_-}\Delta^2_s\right]\frac{\hat{a}_j\hat{n}\hat{\Delta}_u}{2p_-\tau_-}+2(ga_j)\hat{g}\\
&-[g_-\Delta_s^2+2p'_-(g\Delta_s)]\frac{\hat{a}_j\hat{n}\hat{g}}{2p'_-\tau'_-}-[g_-\Delta_u^2+2p_-(g\Delta_u)]\frac{\hat{g}\hat{n}\hat{a}_j}{2p_-\tau_-}\\
&+\frac{g_-}{p_-p'_-}\left[(ga_j)+\frac{g_-}{\tau'_-}(\Delta_sa_j)+\frac{g_-}{\tau_-}(\Delta_ua_j)\right]\hat{\Delta}_s\hat{n}\hat{\Delta}_u-g^2\left[\frac{(\Delta_sa_j)}{\tau'_-}+\frac{(\Delta_ua_j)}{\tau_-}\right]\hat{n},
\end{split}$$ and where it is clear that also in the case of $\Gamma_{R,\perp,j}(p,p',q;\phi)$ the only term requiring regularization is the one analogous to that in the first line of Eq. (\[Gamma\_R\_-\]). Due to the above change of variables, the various quantities appearing in $\Gamma_{R,-}(p,p',q;\phi)$, and $\Gamma_{R,\perp,j}(p,p',q;\phi)$ have to be interpreted as 