energies of the final photons.

Calculation of the amplitude of a generic refractive QED process
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Refractive QED processes in a laser field involve in general $N_i$ incoming, $N_o$ outgoing photons, with $N_i+N_o>1$, and the laser photons (the special case $N_i+N_o=1$, corresponding to the tadpole diagram, is trivial in the case of a background plane-wave field [@Schwinger_1951] and it will not be considered here). However, for the sake of notational simplicity, we consider here the abstract case of only incoming photons ($N_o=0$) and we set $N_i=N$. The external photons have momenta $k_j^{\mu}$ and polarization four-vectors $e_j^{\mu}$, with $j=1,\ldots,N$ (see Fig. 1): the $j$th incoming photon can be “transformed” into an outgoing one via the substitutions $k_j^{\mu}\to -k_j^{\mu}$ and $e_j^{\mu}\to e_j^{\mu\,*}$ in the amplitude (see Eq. (\[M\]) below). As it will be clear below, the results of the paper are unaffected by this particular choice. Moreover, we limit here to the case of external real photons ($k_j^2=0$), although the analysis and the conclusions can be correspondingly extended to the case of off-shell external photons, as those representing external fields as, for example, a Coulomb field.

![Two typical Feynman diagrams relative to a generic refractive QED process in a laser field. The 