Soft photon approximation in a laser field: applications

We apply the soft photon approximation to QED processes in a strong plane-wave laser field. New compact expressions for the emission factors are presented, which simplify the generalization to multiple soft photon emission. The method is applied to nonlinear single, double, and $N$-photon Compton scattering. For single Compton scattering, we show that the soft photon approximation reproduces the exact spectrum with accuracy $\mathcal{O}(\omega/\varepsilon)$, significantly outperforming the classical radiation theory. For double Compton scattering, we derive analytical expressions for the differential probability in two kinematic regimes: when both photons are soft, and when one photon is soft while the other has arbitrary energy. We present a compact analytical expression for the amplitude of $N$-photon emission within the soft photon approximation. A numerical implementation in Wolfram Mathematica for single and double Compton scattering is provided.