[@Landau_b_2_1975] $$p_{\phi}(\phi)=-\frac{m^2+[\bm{p}_{\perp}-e\bm{A}(\phi)]^2}{2p_X}.$$ By performing the change of variable $\phi'_j=2\delta\kappa_{j,X}s'_j$ in Eq. (\[F\_perp\]), we see that $F_{\phi}(\phi+\Phi,\{s\})$ qualitatively corresponds to the quantity $\sum_{j=1}^N\int_{\phi_{j-1}}^{\phi_j}d\phi'_j\mathcal{P}_{j,\phi}(\phi'_j)$, where $\phi_0=0$ and where $\mathcal{P}_{j,\phi}(\phi'_j)$ is the component $\phi$ of the four-momentum of the virtual electron/positron flowing between the $(j-1)$th vertex and the $j$th vertex. Thus, the condition $F_{\phi}(\phi+\Phi,\{s\})\lesssim 1$ corresponds to the fact that, according to Heisenberg uncertainty principle, the virtual electron-positron pair annihilates after an interval $\Delta\phi'_j$ in $\phi'_j$ given by $\Delta\phi'_j\sim 1/\mathcal{P}_{\phi,j}$, where $\mathcal{P}_{\phi,j}$ indicates the order of magnitude of the momentum flowing between the $(j-1)$th vertex and the $j$th vertex. This corroborates the interpretation that in a refractive QED process, the stronger is the laser field, the higher is the four-momentum flowing through the electron-positron loop. Accordingly, the virtual electron-positron pair propagates for a shorter distance inside the laser field, such that the net number of photons, that can be exchanged in the process is always of the order of unity.

This difference between the net number of photons exchanged with the laser field in a general refractive QED process, inferred here from the investigation of the amplitude of such processes, and in NBWPP could appear at first sight not to be compatible with 