$$\label{Gamma_-}
\begin{split}
\Gamma_-(p,p',q;\phi)&=-\frac{i\alpha}{4\pi}\int_0^{\infty}\frac{dsdudt}{S^3}\,e^{-i\kappa^2t-i\frac{G^2}{S}}\left(SLQ^{\prime\lambda}\hat{n}Q_{\lambda}R+2i\hat{n}\right)\\
&=-\frac{i\alpha}{2\pi}\int_0^{\infty}\frac{dsdudt}{S^3}\,e^{-i\kappa^2t-i\frac{G^2}{S}}\left[\left(2S(\pi_s\pi_u)+\frac{G^2}{S}+i\right)\hat{n}-2G_-(\hat{\pi}_sR+L\hat{\pi}_u)\right.\\
&\left.\quad-\hat{G}\hat{\pi}_s\hat{n}-\hat{n}\hat{\pi}_u\hat{G}+2\tau_-L\hat{G}+2\tau'_-\hat{G}R+2\frac{G_-}{S}\hat{G}-\frac{G_-^2}{S}\frac{\hat{\Delta}_s\hat{n}\hat{\Delta}_u}{p_-p'_-}\right],
\end{split}$$ where $$\begin{aligned}
\pi_s^{\mu}&=\pi^{\mu}_{p'}(\psi_s),\quad \Delta^{\mu}_s=e[A^{\mu}(\psi_s)-A^{\mu}(\phi)], & L&=1-\frac{\hat{n}\hat{\Delta}_s}{2p'_-}, \quad \psi_s=\phi+2s\tau'_-,\\
\pi_u^{\mu}&=\pi^{\mu}_p(\psi_u),\quad \Delta^{\mu}_u=e[A^{\mu}(\psi_u)-A^{\mu}(\phi)], & R&=1+\frac{\hat{n}\hat{\Delta}_u}{2p_-},\quad \psi_u=\phi-2u\tau_-,\\
C^{\mu}&=\left(1+\frac{\hat{n}\hat{\Delta}_s}{2\tau'_-}\right)\gamma^{\mu}\left(1-\frac{\hat{n}\hat{\Delta}_u}{2\tau_-}\right), & G^{\mu}&=\int_0^sds'\pi^{\mu}_{p'}(\psi_{s'})+\int_0^udu'\pi^{\mu}_p(\psi_{u'}), \\
Q^{\lambda}&=2\pi_u^{\lambda}+i\frac{e\hat{n}\hat{A}'(\psi_u)}{p_-}n^{\lambda}-\frac{\hat{G}}{S}\gamma^{\lambda},&  
Q^{\prime\lambda}&=2\pi_s^{\lambda}+i\frac{e\hat{n}\hat{A}'(\psi_s)}{p'_-}n^{\lambda}-\gamma^{\lambda}\frac{\hat{G}}{S}.\end{aligned}$$ As we have mentioned, in order to compute the components $\Gamma_{\perp,j}(p,p',q;\phi)=-(\Gamma^{\mu}(p,p',q;\phi)a_{j,\mu})$ we can effectively assume that the matrix $\hat{n}$ anticommutes with $\gamma^{\mu}$. In the following four equations, with an abuse of notation, we use the equal symbol also for two matrices that are equal to each other up to terms proportional to $n^{\mu}$, which can anyway be ignored in the computation of $\Gamma_{\perp,j}(p,p',q;\phi)$. Going through the terms in Eq. (\[Gamma\_f\]) in order of complexity, one can easily show that $$\begin{aligned}
&LC^{\mu}R=\gamma^{\mu}+\frac{G_-}{2p'_-\tau'_-S}\hat{n}\hat{\Delta}_s\gamma^{\mu}-\frac{G_-}{2p_-\tau_-S}\gamma^{\mu}\hat{n}\hat{\Delta}_u,\\
\begin{split}
&L(\gamma^{\lambda}\hat{n}C^{\mu}Q_{\lambda}+Q^{\prime\lambda}C^{\mu}\hat{n}\gamma_{\lambda})R=-\frac{2\tau_-}{p'_-}\hat{n}\hat{\Delta}_s\gamma^{\mu}+\frac{2\tau'_-}{p_-}\gamma^{\mu}\hat{n}\hat{\Delta}_u-\frac{4G_-}{S}\gamma^{\mu}+\frac{4G^{\mu}}{S}\hat{n}\\
&\qquad-2\hat{n}\gamma^{\mu}\hat{\pi}_s-2\hat{\pi}_u\gamma^{\mu}\hat{n},
\end{split}\\
\begin{split}
&\frac{d}{dk_-}\left[\tilde{L}(\gamma^{\lambda}\hat{n}\tilde{C}^{\mu}\tilde{Q}_{\lambda}+\tilde{Q}^{\prime\lambda}\tilde{C}^{\mu}\hat{n}\gamma_{\lambda})\tilde{R}\right]_{k_-=0}=8\left(\frac{G_1^{\mu}}{S}-\frac{s\tau_-}{p'_-}\mathcal{A}_s^{\prime\,\mu}+\frac{u\tau'_-}{p_-}\mathcal{A}_u^{\prime\,\mu}\right)\hat{n}\\
&\qquad+4s\left(1+\frac{\tau_-}{p'_-}\right)\hat{n}\gamma^{\mu}\hat{\mathcal{A}}_s'-4u\left(1+\frac{\tau'_-}{p_-}\right)\hat{\mathcal{A}}_u'\gamma^{\mu}\hat{n},
\end{split}\\
\begin{split}
&LQ^{\prime\,\lambda}C^{\mu}Q_{\lambda}R=4(\pi_s\pi_u)\left(\gamma^{\mu}+\frac{G_-}{2p'_-\tau'_-S}\hat{n}\hat{\Delta}_s\gamma^{\mu}-\frac{G_-}{2p_-\tau_-S}\gamma^{\mu}\hat{n}\hat{\Delta}_u\right)\\
&\qquad+2i\frac{\tau_-}{p'_-}\hat{n}\hat{\mathcal{A}}'_s\gamma^{\mu}+2i\frac{\tau'_-}{p_-}\gamma^{\mu}\hat{n}\hat{\mathcal{A}}'_u-\frac{2}{S}LC^{\mu}\hat{G}\hat{\pi}_sR-\frac{2}{S}L\hat{\pi}_u\hat{G}C^{\mu}R\\
&\qquad-\frac{2}{S^2}L\hat{G}\left(\gamma^{\mu}+\frac{\gamma^{\mu}\hat{\Delta}_s\hat{n}}{2\tau'_-}-\frac{\hat{\Delta}_u\hat{n}\gamma^{\mu}}{2\tau_-}\right)\hat{G}R
\end{split}\end{aligned}$$ where $$\begin{split}
G_1^{\mu}&=\frac{d}{d\phi}\left[\int_0^sds'\,s'\pi^{\mu}_{p'}(\psi_{s'})-\int_0^udu'\,u'\pi^{\mu}_p(\psi_{u'})\right]\\
&=\frac{1}{2\tau'_-}\left[s\pi_{p'}^{\mu}(\psi_s)-\int_0^sds'\pi_{p'}^{\mu}(\psi_{s'})\right]+\frac{1}{2\tau_-}\left[u\pi_p^{\mu}(\psi_u)-\int_0^udu'\pi_p^{\mu}(\psi_{u'})\right]
\end{split}$$ and $$\mathcal{A}^{\mu}_{s/u}=eA^{\mu}(\psi_{s/u})$$ (the prime on these quantities indicates the derivative with respect to $\phi$).

In this way, we obtain the following expressions of the transverse components $\Gamma_{\perp,i}(p,p',q;\phi)$: $$\label{Gamma_j}
\begin{split}
\Gamma_{\perp,j}(p,p',q;\phi)&=\frac{i\alpha}{2\pi}\int_0^{\infty}\frac{dsdudt}{S^3}\,e^{-i\kappa^2t-i\frac{G^2}{S}}\\
&\quad\times\left\{(2S(\pi_s\pi_u)+i)\left(\hat{a}_j+\frac{G_-}{2p'_-\tau'_-S}\hat{n}\hat{\Delta}_s\hat{a}_j-\frac{G_-}{2p_-\tau_-S}\hat{a}_j\hat{n}\hat{\Delta}_u\right)\right.\\
&\qquad-L(Ca_j)\hat{G}\hat{\pi}_sR-L\hat{\pi}_u\hat{G}(Ca_j)R-\frac{1}{S}L\hat{G}\left(\hat{a}_j+\frac{\hat{a}_j\hat{\Delta}_s\hat{n}}{2\tau'_-}-\frac{\hat{\Delta}_u\hat{n}\hat{a}_j}{2\tau_-}\right)\hat{G}R\\
&\qquad-\frac{2(GG_1)}{S}\left(\frac{\tau_-}{p'_-}\hat{n}\hat{\Delta}_s\hat{a}_j-\frac{\tau'_-}{p_-}\hat{a}_j\hat{n}\hat{\Delta}_u+\frac{2G_-}{S}\hat{a}_j-\frac{2(Ga_j)}{S}\hat{n}+\hat{n}\hat{a}_j\hat{\pi}_s+\hat{\pi}_u\hat{a}_j\hat{n}\right)\\
&\qquad+2i\left(\frac{(G_1a_j)}{S}+s(\mathcal{A}'_sa_j)-u(\mathcal{A}'_ua_j)\right)\hat{n}\\
&\qquad\left.+i\left[s-(u+t)\frac{\tau_-}{p'_-}\right]\hat{\mathcal{A}}_s'\hat{n}\hat{a}_j-i\left[u-(s+t)\frac{\tau'_-}{p_-}\right]\hat{a}_j\hat{n}\hat{\mathcal{A}}_u'\right\}
\end{split}$$ Equations (\[Gamma\_exp\]), (\[Gamma\_-\]), and (\[Gamma\_j\]) are the main results of the paper and, as it can easily be shown, they reduce to the result in vacuum as, e.g., on page 339 of Ref. [@Itzykson_b_1980] \[as it is shown in the 