e_{N-1,\mu}]+\hat{k}_{N-1}\hat{e}_{N-1}\}D_N(A)\\
&\times e^{-ik_{N,\phi}\phi}[2(\Pi e_N)+\hat{k}_N\hat{e}_N]D(A)|x\rangle,
\end{split}$$ where $K^{\mu}=\sum_{j=1}^Nk_j^{\mu}$, $\kappa_j^{\mu}=\sum_{i=j}^Nk_i^{\mu}$ (note that $\kappa_1^{\mu}=K^{\mu}$), and $D_l(A)=D(A)\vert_{P_X\to P_X+\kappa_{l,X},\bm{P}_{\perp}\to \bm{P}_{\perp}+\bm{\kappa}_{l,\perp}}$, with $l=2,\ldots,N$. Now, the operators between the bra $\langle x|$ and the ket $|x\rangle$ do not contain the coordinates $X$ and $\bm{x}_{\perp}$, and the identities $$\begin{aligned}
\label{x_f_x}
\langle X|f(P_X)|X\rangle=\int\frac{dp_X}{2\pi}f(p_X), && \langle \bm{x}_{\perp}|g(\bm{P}_{\perp})|\bm{x}_{\perp}\rangle=\int\frac{d^2p_{\perp}}{(2\pi)^2}g(\bm{p}_{\perp})\end{aligned}$$ valid for arbitrary functions $f(P_X)$ and $g(\bm{P}_{\perp})$ can be applied (we assumed here that the eigenstates $|p\rangle$ of the four-momentum operator $P^{\mu}$, i.e., $P^{\mu}|p\rangle=p^{\mu}|p\rangle$, are such that $\langle x\vert p\rangle=e^{-i(px)}$ and $\langle p|p'\rangle=(2\pi)^4\delta^4(p-p')$). Moreover, the integrals in $X$ and $\bm{x}_{\perp}$ are easily taken and the partial amplitude $M^{(1)}$ becomes $$\label{M_op_3}
\begin{split}
M^{(1)}=&-(-ie)^N\delta(K_X)\delta^2(\bm{K}_{\perp})\int d\phi \int d p_X \int d^2 p_{\perp} \int_0^{\infty} ds_1\cdots ds_N\, e^{-i(m^2-i\epsilon)S} \\
&\times\text{Tr}\,\langle \phi|\{2[(p^{\mu}-eA^{\mu}(\phi)) e_{N,\mu}]+\hat{k}_N\hat{e}_N\}\\
&\times\left\{1+\frac{e}{2p_X}\hat{n}[\hat{A}(\phi+2s_1p_X))-\hat{A}(\phi)]\right\}\\
&\times e^{-i\int_0^{s_1}ds'_1[\bm{p}_{\perp}-e\bm{A}(\phi+2s'_1p_X)]^2}e^{-2is_1P_{\phi}p_X}e^{-i\kappa_{1,\phi}\phi}\\
&\times \{2[(p^{\mu}-eA^{\mu}(\phi)+\kappa^{\mu}_1) e_{1,\mu}]+\hat{k}_1\hat{e}_1\}\\
&\times\left\{1+\frac{e}{2(p_X+\kappa_{2,X})}\hat{n}[\hat{A}(\phi+2s_2(p_X+\kappa_{2,X}))-\hat{A}(\phi)]\right\}\\
&\times e^{-i\int_0^{s_2}ds'_2[\bm{p}_{\perp}+\bm{\kappa}_{2,\perp}-e\bm{A}(\phi+2s'_2(p_X+\kappa_{2,X}))]^2}e^{-2is_2P_{\phi}(p_X+\kappa_{2,X})}e^{-i\kappa_{2,\phi}\phi}\\
&\cdots\times \{2[(p^{\mu}-eA^{\mu}(\phi)+\kappa^{\mu}_{N-1}) e_{N-1,\mu}]+\hat{k}_{N-1}\hat{e}_{N-1}\}\\
&\times\left\{1+\frac{e}{2(p_X+\kappa_{N,X})}\hat{n}[\hat{A}(\phi+2s_N(p_X+\kappa_{N,X}))-\hat{A}(\phi)]\right\}\\
&\times e^{-i\int_0^{s_N}ds'_N[\bm{p}_{\perp}+\bm{\kappa}_{N,\perp}-e\bm{A}(\phi+2s'_N(p_X+\kappa_{N,X}))]^2}e^{-2is_NP_{\phi}(p_X+\kappa_{N,X})}e^{-i\kappa_{N,\phi}\phi}|\phi\rangle,
\end{split}$$ where $S=s_1+\cdots+s_N$. We note that in this expression of the amplitude, we have substituted the operator $P^{\mu}$ with the number $p^{\mu}+\kappa^{\mu}_j$ in the four-dimensional scalar products $(Pe_j)$. First, we observe that, since $(k_je_j)=0$, then it is $(\kappa_je_j)=(\kappa_{j+1}e_j)$, for $j=1,\ldots,N-1$ and $(\kappa_Ne_N)=0$. Moreover, although the substitution $(Pe_j)\to (p^{\mu}+\kappa^{\mu}_j)e_{j,\mu}$ is evident for the components $p_X$ and $\bm{p}_{\perp}$ (see Eq. (\[x\_f\_x\]) and the definition of the operators $D_l(A)$ below Eq. (\[M\_op\_2\])), it is 