specify the physical regime, we have also to consider the parameters $\varkappa_j$ (see the discussion below Eq. (\[M\])). If $\varkappa_j$ largely exceeds unity, an electron-positron pair can be in principle created in the collision of the laser field and the $j$th external photon. The subsequent emission of radiation by such a pair would represent a background for the refractive QED process. Thus, we limit here to the case where the parameters $\varkappa_j$ are fixed and less or of the order of unity, such that electron-positron pair production from laser-external photons is negligible. Correspondingly, we also exclude the possibility that electron-positron pairs can be created only by the external photons, even though, as it will be clear below, the following considerations will not depend formally on this condition.

It is convenient to write explicitly $$\delta\pi^{\mu}_j(\phi,s'_j)\delta\pi_{j,\mu}(\phi,s'_j)=-2\delta\kappa_{j,X}\delta\kappa_{j,\phi}-[\delta\bm{\pi}_{j,\perp}(\phi,s'_j)]^2$$ and to shift the variable $\phi$ as $\phi\to\phi+\Phi$, with $\Phi$ such that $$\label{Shift}
K_{\phi}\Phi+2\sum_{j=1}^N\delta\kappa_{j,X}\delta\kappa_{j,\phi}s_j=0.$$ In this way, the the partial amplitude $M^{(1)}$ can be written in the convenient form $$\label{M_f}
\begin{split}
M^{(1)}=&\frac{i\pi}{2}(-ie)^N\delta(K_X)\delta^2(\bm{K}_{\perp})\int d\phi \int_0^{\infty} \frac{ds_1\cdots ds_N}{S^2}\, e^{-i[K_{\phi}\phi+F_{\phi}(\phi+\Phi,\{s\})]}\\
&\times\text{Tr}\bigg\langle\prod_{j=1}^N\{2[(p^{\mu}+\delta\pi_j^{\mu}(\phi+\Phi,s_j)) e_{j,\mu}]+\hat{k}_j\hat{e}_j\}\\
&\left.\times\left\{1+\frac{e}{2\delta\kappa_{j+1,X}}\hat{n}[\hat{A}(\phi+\Phi+\phi_{j+1})-\hat{A}(\phi+\Phi+\phi_j)]\right\}\right\rangle,
\end{split}$$ where $$\label{F_perp}
F_{\phi}(\phi+\Phi,\{s\})=\sum_{j=1}^N\int_0^{s_j}ds'_j\{[\delta\bm{\pi}_{j,\perp}(\phi+\Phi,s'_j)]^2+m^2-i\epsilon\}.$$ The advantage of this form with respect to that in Eq. (\[M\_op\_5\_C\]) is that all the $N$ integrands in $F_{\phi}(\phi+\Phi,\{s\})$ are strictly 