of the laser field, that the net number of photons exchanged with the laser field is of the order of unity. The general observation below Eq. (\[Exp\_2\]) indicates that also high-order terms in the laser field in the pre-exponential will not essentially increase the net number of laser photons exchanged during the refractive QED process. Note that the fact that only a low net number of laser photons are exchanged during a refractive QED effects implies that the strong background laser field is practically not altered by the process itself. This is in agreement with the use here of the Furry picture, which includes the external field as a “given” field.

Before discussing the obtained results, it is worth observing that in the special case where $N=2$ and with two external real photons the net exchange of laser photons is exactly zero, due to *kinematical* reasons [@Baier_1976_b; @Dittrich_b_2000]. Our results show that there is a *dynamical* reason such that the net exchange of laser photons is small also for arbitrary $N$.

Discussion {#Discussion}
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As we have already mentioned above, it is interesting to compare the low net exchange of laser photon in a refractive QED process with what happens 