is finite and that only the vacuum-term $M_0^{(1)}$ needs to be regularized (see, in particular, [@Baier_1976_b]). The same procedure can be applied to the remaining case $N=4$, where the divergences are in general less severe than, e.g., for $N=2$. As it will be clear below, the present analysis is based essentially on the behavior of the field-dependent phase function $F(\phi,\{s\})$, then the conclusions, drawn starting from the unregularized amplitude $M^{(1)}$, also apply to the regularized one $M^{(1)}-M_0^{(1)}$. Since the regularization procedure is necessary only for $N<5$, in order to keep general the following formulas, we will still analyze the unregularized amplitude $M^{(1)}$, being understood, however, that for $N<5$, actually, the regularized amplitude $M^{(1)}-M_0^{(1)}$ has to be considered.

Before passing to the estimation of the net number of laser photon exchanged in a refractive QED process, we observe here that the integral representation $$\label{Prop}
\prod_{j=1}^N\frac{1}{p_j^2-m^2+i\epsilon}=(-i)^N\int_0^{\infty}ds_1\cdots ds_N\,e^{i\sum_{j=1}^N\int_0^{s_j}ds'_j(p_j^2-m^2+i\epsilon)},$$ of the electron propagator in vacuum in momentum space, suggests to interpret the quantity $\delta\pi^{\mu}_j(\phi,s'_j)$ as an “effective” instantaneous four-momentum of the virtual particle flowing between the $(j-1)$th and the $j$th vertex (see Eqs. (\[M\_op\_5\_C\]) and (\[F\])).

Estimation of the net number of exchanged laser photons {#Estimation}
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If there were no external laser field, 