the remaining integral in $\phi$ in Eq. (\[M\_op\_5\_C\]) would provide the $\delta$-function $\delta(K_{\phi})$, which, together with the other three $\delta$-functions, would imply the overall energy-momentum conservation $K^{\mu}=0$, as expected. In the presence of the laser field, a measure of the net number of photons exchanged with the laser field during the refractive QED process is determined by the quantity $K_{\phi}/\omega_0$, where $\omega_0$ is the central laser angular frequency. In order to estimate the net number of laser photons exchanged, we recall that the multiphoton nature of the process is controlled by the parameter $\xi=|e|E_0/m\omega_0$, where $E_0$ is the amplitude of the electric field of the laser [@Ritus_1985; @Di_Piazza_2012]. From the physical meaning of this parameter, in fact, it is not surprising that if $\xi\lesssim 1$, the net number of photons exchanged with the laser field is of the order of unity. This regime is the relevant one for present and future x-ray laser facilities [@Di_Piazza_2012], for which the parameter $\xi$ is not expected to exceed unity due to the relatively large photon energy ($\omega_0\gtrsim 1\;\text{KeV}$). Thus, we directly consider below the ultra-relativistic limit where $\xi\to \infty$, having in mind an optical laser system with $\omega_0\sim 1\;\text{eV}$. In order to further 