in principle not justified for the remaining component $P_{\phi}$. However, we show in the Appendix B that gauge invariance implies that the four-dimensional scalar products $(Pe_j)$ actually do not involve the component $P_{\phi}$. The remaining matrix element can be calculated by employing the identity $$\begin{aligned}
e^{-i\phi_0P_\phi}|\phi\rangle=|\phi-\phi_0\rangle,\end{aligned}$$ where $\phi_0$ is a constant, and the fact that $\langle \phi|\phi'\rangle=\delta(\phi-\phi')$. The resulting $\delta$-function $\delta(2s_1(p_X+\kappa_{1,X})+\cdots+2s_N(p_X+\kappa_{N,X}))$ can be exploited to perform the integral in $p_X$ and the result is $$\label{M_op_4}
\begin{split}
M^{(1)}=&-\frac{(-ie)^N}{2}\delta(K_X)\delta^2(\bm{K}_{\perp})\int d\phi \int d^2 p_{\perp}\int_0^{\infty} \frac{ds_1\cdots ds_N}{S}\, e^{-i(m^2-i\epsilon)S} e^{-iK_{\phi}\phi}\\
&\times e^{-i\sum_{j=1}^N\int_0^{s_j}ds'_j\{\delta\kappa_{j,\phi}\delta\kappa_{j,X}+[\bm{p}_{\perp}+\bm{\pi}_{j,\perp}(\phi,s'_j)]^2\}}\\
&\times\text{Tr}\,\left\langle \{2[(p^{\mu}-eA^{\mu}(\phi)) e_{N,\mu}]+\hat{k}_N\hat{e}_N\}\left\{1+\frac{e}{2\delta\kappa_{1,X}}\hat{n}[\hat{A}(\phi+\phi_1)-\hat{A}(\phi)]\right\}\right.\\
&\times \{2[(p^{\mu}-eA^{\mu}(\phi+\phi_1)+\kappa^{\mu}_1) e_{1,\mu}]+\hat{k}_1\hat{e}_1\}\\
&\times\left\{1+\frac{e}{2\delta\kappa_{2,X}}\hat{n}[\hat{A}(\phi+\phi_2)-\hat{A}(\phi+\phi_1)]\right\}\\
&\cdots\times\{2[(p^{\mu}-eA^{\mu}(\phi+\phi_{N-1})+\kappa^{\mu}_{N-1}) e_{N-1,\mu}]+\hat{k}_{N-1}\hat{e}_{N-1}\}\\
&\left.\times\left\{1+\frac{e}{2\delta\kappa_{N,X}}\hat{n}[\hat{A}(\phi+\phi_N)-\hat{A}(\phi+\phi_{N-1})]\right\}\right\rangle.
\end{split}$$ In this expression we have simplified the notation by introducing the “average” $$\label{Av}
\bar{f}=\frac{1}{S}\sum_{j=1}^N\int_0^{s_j}ds'_jf_j(s'_j)$$ of $N$ arbitrary functions $f_j(s'_j)$, the residuals $$\label{Res}
\delta f_j(s'_j)=f_j(s'_j)-\bar{f},$$ and the quantities $$\label{phi_l}
\phi_j=2\sum_{i=1}^j\delta\kappa_{i,X}s_i$$ and $$\label{pi_j}
\pi^{\mu}_j(\phi,s'_j)=\kappa_j^{\mu}-eA^{\mu}(\phi+\phi'_j),$$ with $$\begin{aligned}
\label{phi_p_1}
\phi'_1&=2\delta\kappa_{1,X}s'_1\\
\label{phi_p_l}
\phi'_l&=2\sum_{i=1}^{l-1}\delta\kappa_{i,X}s_i+2\delta\kappa_{l,X}s'_l, &&l=2,\ldots,N.\end{aligned}$$ Note also that $p_X=-\bar{\kappa}_X$, that $\phi_N=0$ and that in our gauge $\pi_{j,X/\phi}(\phi,s'_j)=\kappa_{j,X/\phi}$. Moreover, in Eq. (\[M\_op\_4\]) and in the successive expressions of $M^{(1)}$, the quantity $p_X$ in the trace has to be interpreted as $-\bar{\kappa}_X$.

In order to take the integral in $\bm{p}_{\perp}$, it is convenient first to shift $\bm{p}_{\perp}$ as $\bm{p}_{\perp}\to\bm{p}_{\perp}-\bar{\bm{\pi}}_{\perp}(\phi,\{s\})$, where $\{s\}=s_1,\ldots, s_N$. In this way, the resulting expression of the 