amplitude can be written as $$\label{M_op_5}
\begin{split}
M^{(1)}=&-\frac{(-ie)^N}{2}\delta(K_X)\delta^2(\bm{K}_{\perp})\int d\phi \int d^2 p_{\perp}\int_0^{\infty} \frac{ds_1\cdots ds_N}{S}\, e^{-i[K_{\phi}\phi-F(\phi,\{s\})]}\\
&\times e^{-iS\bm{p}_{\perp}^2}\text{Tr}\bigg\langle\prod_{j=1}^N\{2[(p^{\mu}+\delta\pi_j^{\mu}(\phi,s_j)) e_{j,\mu}]+\hat{k}_j\hat{e}_j\}\\
&\left.\times\left\{1+\frac{e}{2\delta\kappa_{j+1,X}}\hat{n}[\hat{A}(\phi+\phi_{j+1})-\hat{A}(\phi+\phi_j)]\right\}\right\rangle,
\end{split}$$ where $$\label{F}
F(\phi,\{s\})=\sum_{j=1}^N\int_0^{s_j}ds'_j[\delta\pi^{\mu}_j(\phi,s'_j)\delta\pi_{j,\mu}(\phi,s'_j)-m^2+i\epsilon],$$ where $\delta\kappa_{N+1}\equiv\delta\kappa_1$ and $\phi_{N+1}\equiv\phi_1$. The integral in $\bm{p}_{\perp}=(p_1,p_2)$ can be written as a sum of integrals of the form $$I_{n_1,n_2}=\int d^2p_{\perp}\, p_1^{n_1}p_2^{n_2}\, e^{-iS\bm{p}_{\perp}^2},$$ where $n_1$ and $n_2$ are two non-negative integers. The integral $I_{n_1,n_2}$ vanishes if $n_1$ and/or $n_2$ are odd, whereas it is equal to $$I_{n_1,n_2}=2\pi\frac{(n_1-1)!!(n_2-1)!!}{(2iS)^{(n_1+n_2+2)/2}}$$ if $n_1$ and $n_2$ are both even. In conclusion, we can write the partial amplitude $M^{(1)}$ in the compact form $$\label{M_op_5_C}
\begin{split}
M^{(1)}=&\frac{i\pi}{2}(-ie)^N\delta(K_X)\delta^2(\bm{K}_{\perp})\int d\phi \int_0^{\infty} \frac{ds_1\cdots ds_N}{S^2}\, e^{-i[K_{\phi}\phi-F(\phi,\{s\})]}\\
&\times\text{Tr}\bigg\langle\prod_{j=1}^N\{2[(p^{\mu}+\delta\pi_j^{\mu}(\phi,s_j)) e_{j,\mu}]+\hat{k}_j\hat{e}_j\}\\
&\left.\times\left\{1+\frac{e}{2\delta\kappa_{j+1,X}}\hat{n}[\hat{A}(\phi+\phi_{j+1})-\hat{A}(\phi+\phi_j)]\right\}\right\rangle,
\end{split}$$ where the substitution rules $$\begin{aligned}
\label{Sub_X}
p_X\to&-\bar{\kappa}_X\\
\label{Sub_perp}
\left(\frac{(pa_1)}{\sqrt{-a_1^2}}\right)^{n_1}\left(\frac{(pa_2)}{\sqrt{-a_2^2}}\right)^{n_2}\to&
\begin{cases}
0 & \text{if $n_1$ and/or $n_2$ are odd}\\
\frac{(n_1-1)!!(n_2-1)!!}{(2iS)^{(n_1+n_2)/2}} & \text{if $n_1$ and $n_2$ are even}
\end{cases}\end{aligned}$$ in the expression of the trace are understood. The amplitude in Eq. (\[M\_op\_5\_C\]) may diverge for $N<5$ [@Landau_b_4_1982; @Liang_2012]. The case $N=2$ (polarization operator) has been explicitly investigated in [@Baier_1976_b] and the case $N=3$ has been considered in [@Di_Piazza_2007; @Di_Piazza_2008_a]. The regularization procedure can be carried out by first subtracting and adding the corresponding amplitude $M_0^{(1)}$ at zero external field, i.e. by writing $M^{(1)}=(M^{(1)}-M_0^{(1)})+M_0^{(1)}$. Gauge invariance ensures that the quantity $M^{(1)}-M_0^{(1)}$ 