that none of the external photons propagate along the same direction of the laser photons (of course, we exclude the trivial case of an external photon with vanishing energy).

In order to calculate the amplitude $M$, we employ below the operator technique, developed in [@Baier_1976_a; @Baier_1976_b] for the case of a background plane-wave laser field (the calculation of the amplitude can of course also be performed by employing the standard Feynman rules in the Furry picture [@Landau_b_4_1982], the advantage of the operator technique being to provide a more suitable expression of the amplitude to estimate the net number of photons exchanged with the laser field (see, in particular, the discussion below Eq. (\[Exp\_2\]))). In the operator technique the electron propagator in the laser field is written as $G(x,y|A)=\langle x|G(A)|y\rangle$, where $$G(A)=\frac{1}{\hat{\Pi}-m+i\epsilon},$$ with $\Pi^{\mu}=\Pi^{\mu}(A)=P^{\mu}-eA^{\mu}(\phi)$ and with $\epsilon$ being a positive infinitesimal quantity. Here, the four-vector $P^{\mu}$ is the four-momentum operator, satisfying the commutation rules $[x^{\mu},P^{\nu}]=-ig^{\mu\nu}$, where $g^{\mu\nu}=\text{diag}(+1,-1,-1,-1)$. By employing the above representation of the electron propagator and by using the cyclic property of the trace, the amplitude in Eq. (\[M\]) can be simply written as $$\label{M_op}
M=-e^N\int d^4x\,\text{Tr}\,\langle x\vert G_1(A)\cdots G_N(A)\vert x\rangle+\circlearrowleft,$$ where we have introduced the block operators $G_j(A)=G(A)\hat{e}_j\exp[-i(k_jx)]$. It 