thin wavy lines indicate the external photons with four-momenta $k_1^{\mu},\ldots,k_N^{\mu}$ and polarization four-vectors $e_1^{\mu},\ldots,e_N^{\mu}$, respectively, and the thick plain lines indicate the laser-dressed electron propagators.](Figure_1.eps){width="0.9\linewidth"}

The mentioned process is described by the sum of all Feynman diagrams, which can be obtained from the one in the left side of Fig. 1 by permuting the labels in the photon legs. Among them, we consider here only the one in the right part of Fig. 1, and the treatment of the remaining diagrams can be performed in an analogous way (any diagram contributing to a refractive QED process can always be considered together with the other one, differing only in the direction of circulation of the four-momentum through the electron loop)[^2]. The reason for considering these two diagrams together is that this allows to formulate a simple set of substitution rules, which in turn clearly show the general structure of the amplitude of the process (see the discussions below Eq. (\[M\_s\]) here below and between Eqs. (\[T\_p\_3\]) and (\[T\_3\_T\]) in Appendix A). The amplitude $M$ corresponding to the diagrams in Fig. 1 is given by [@Landau_b_4_1982] $$\label{M}
\begin{split}
M=&-e^N\int d^4x_1\cdots d^4x_Ne^{-i[(k_1x_1)+\cdots+(k_Nx_N)]}\\
&\times\text{Tr}[\hat{e}_1G(x_1,x_2|A)\hat{e}_2G(x_2,x_3|A)\cdots\hat{e}_NG(x_N,x_1|A)]+\circlearrowleft,
\end{split}$$ where the “hat” indicates the contraction of a four-vector 