appendix, the component $\Gamma_q(p,p',q;\phi)$ vanishes in vacuum\]. We notice that the pre-exponential matrices in all terms feature symmetry properties such that they can all be written as the sum of two classes of terms with the second one, being obtained from the first one by: 1) taking the Dirac conjugate, 2) swapping all indexes $s$ and $u$ in each quantity. We have exploited this symmetry in the computations presented below. Finally, we observe that all the terms in Eqs. (\[Gamma\_exp\]), (\[Gamma\_-\]), and (\[Gamma\_j\]) have at most three gamma matrices except the three terms on the third line of Eq. (\[Gamma\_j\]) [^1]. The three terms in the third line of Eq. (\[Gamma\_j\]) can be easily reduced to expressions containing at most five gamma matrices: $$\label{5_gamma_1}
\begin{split}
&L(Ca_j)\hat{G}\hat{\pi}_sR=\left(\hat{a}_j+\frac{G_-}{2Sp'_-\tau'_-}\hat{n}\hat{\Delta}_s\hat{a}_j
-\frac{G_-}{2Sp_-\tau_-}\hat{a}_j\hat{n}\hat{\Delta}_u\right)\hat{G}\hat{\pi}_s\\
&+\left(\hat{a}_j+\frac{G_-}{2Sp'_-\tau'_-}\hat{n}\hat{\Delta}_s\hat{a}_j\right)\left(p'_-\hat{G}-G_-\hat{\pi}_s\right)\frac{\hat{\Delta}_u}{p_-}\\
&+\frac{\hat{a}_j\hat{n}}{2p_-\tau_-}[2(G_-(\Delta_u\pi_s)-p'_-(\Delta_uG))\hat{\Delta}_u-(G_-\Delta_u^2+2\tau_-(\Delta_uG))\hat{\pi}_s+(p'_-\Delta_u^2+2\tau_-(\Delta_u\pi_s))\hat{G}],
\end{split}$$ $$\label{5_gamma_2}
\begin{split}
&L\hat{\pi}_u\hat{G}(Ca_j)R=\hat{\pi}_u\hat{G}\left(\hat{a}_j+\frac{G_-}{2Sp'_-\tau'_-}\hat{n}\hat{\Delta}_s\hat{a}_j
-\frac{G_-}{2Sp_-\tau_-}\hat{a}_j\hat{n}\hat{\Delta}_u\right)\\
&+\frac{\hat{\Delta}_s}{p'_-}\left(p_-\hat{G}-G_-\hat{\pi}_u\right)\left(\hat{a}_j-\frac{G_-}{2Sp_-\tau_-}\hat{a}_j\hat{n}\hat{\Delta}_u\right)\\
&+[2(G_-(\Delta_s\pi_u)-p_-(\Delta_sG))\hat{\Delta}_s-(G_-\Delta_s^2+2\tau'_-(\Delta_sG))\hat{\pi}_u+(p_-\Delta_s^2+2\tau'_-(\Delta_s\pi_u))\hat{G}]\frac{\hat{n}\hat{a}_j}{2p'_-\tau'_-},
\end{split}$$ $$\label{5_gamma_3}
\begin{split}
&L\hat{G}\left(\hat{a}_j+\frac{\hat{a}_j\hat{\Delta}_s\hat{n}}{2\tau'_-}-\frac{\hat{\Delta}_u\hat{n}\hat{a}_j}{2\tau_-}\right)\hat{G}R=\frac{G_-}{2}\left(\frac{\hat{G}\hat{a}_j\hat{\Delta}_s\hat{n}\hat{\Delta}_u}{p_-\tau'_-}+\frac{\hat{\Delta}_s\hat{n}\hat{\Delta}_u\hat{a}_j\hat{G}}{p'_-\tau_-}\right)+G_-\left(\frac{\hat{a}_j\hat{\Delta}_s\hat{G}}{\tau'_-}+\frac{\hat{G}\hat{\Delta}_u\hat{a}_j}{\tau_-}\right)\\
&+\left[(p'_-+\tau'_-)(Ga_j)+G_-(\Delta_sa_j)\right]\frac{\hat{\Delta}_s\hat{n}\hat{G}}{p'_-\tau'_-}+\left[(p_-+\tau_-)(Ga_j)+G_-(\Delta_ua_j)\right]\frac{\hat{G}\hat{n}\hat{\Delta}_u}{p_-\tau_-}-G^2\hat{a}_j\\
&-\left[(p'_-+\tau'_-)G^2+\frac{\tau'_-G^2_-}{p_-\tau_-}\Delta^2_u\right]\frac{\hat{\Delta}_s\hat{n}\hat{a}_j}{2p'_-\tau'_-}-\left[(p_-+\tau_-)G^2+\frac{\tau_-G^2_-}{p'_-\tau'_-}\Delta^2_s\right]\frac{\hat{a}_j\hat{n}\hat{\Delta}_u}{2p_-\tau_-}+2(Ga_j)\hat{G}\\
&-[G_-\Delta_s^2+2p'_-(G\Delta_s)]\frac{\hat{a}_j\hat{n}\hat{G}}{2p'_-\tau'_-}-[G_-\Delta_u^2+2p_-(G\Delta_u)]\frac{\hat{G}\hat{n}\hat{a}_j}{2p_-\tau_-}\\
&+\frac{G_-}{p_-p'_-}\left[(Ga_j)+\frac{G_-}{\tau'_-}(\Delta_sa_j)+\frac{G_-}{\tau_-}(\Delta_ua_j)\right]\hat{\Delta}_s\hat{n}\hat{\Delta}_u-G^2\left[\frac{(\Delta_sa_j)}{\tau'_-}+\frac{(\Delta_ua_j)}{\tau_-}\right]\hat{n},
\end{split}$$ Below, we will further investigate the structure of the vertex correction and discuss its divergences.

Gauge-invariance properties of the vertex-correction function {#VC_GI}
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The first aspect we would like to discuss is about the gauge invariance of the expression of $\Gamma_{s,s',l}(p,p',q)$ obtained above. On the one hand, it is clear that $\Gamma_{s,s',l}(p,p',q)$ is invariant under 