positive and therefore that $F_{\phi}(\phi+\Phi,\{s\})\ge 0$. This implies, in fact, that the integration region in $ds_1\cdots ds_N$ mainly contributing to the partial amplitude $M^{(1)}$ is confined to sufficiently small values of $s_j$ such that that $F_{\phi}(\phi+\Phi,\{s\})\lesssim 1$, as otherwise the function $\exp(-iF_{\phi}(\phi+\Phi,\{s\}))$ would be highly oscillating. From what we mentioned at the beginning of this section, this would already indicate that the net number of photon exchanged during the refractive QED process is of the order of unity. However, in order to complete the proof, we have still to analyze the pre-exponential function. In fact, if $N$ is small, then the different powers of the external field present in this function would not essentially change the net number of laser photons exchanged. However, this could in principle occur for large $N$s. In order to show that this is not the case, we recall that in the considered regime, the parameters $\eta_j=\varkappa_j/\xi$ are much smaller than unity and therefore, in the effective integration region with respect to the variables $s_1,\ldots,s_N$, it is $\omega_0|\delta\kappa_{j,X}|s_j\lesssim\omega_0|\delta\kappa_{j,X}|/m^2\ll 1$, where we used the fact that $s_j\lesssim 1/m^2$ (see Eq. (\[F\_perp\])). Consequently, it results that $\omega_0|\phi_j|,\omega_0|\phi'_j|\ll 1$ and, by assuming that $|k_{j,\phi}|\lesssim |K_{\phi}|$ for all $j$s, that 