Implementing nonlinear Compton scattering beyond the local-constant-field approximation

In the calculation of probabilities of physical processes occurring in a background classical field, the local-constant-field approximation (LCFA) relies on the possibility of neglecting the space-time variation of the external field within the region of formation of the process. This approximation is widely employed in strong-field QED as it allows one to evaluate probabilities of processes occurring in arbitrary electromagnetic fields starting from the corresponding quantities computed in a constant electromagnetic field. Here, we scrutinize the validity of the LCFA in the case of nonlinear Compton scattering focusing on the role played by the energy of the emitted photon on the formation length of this process. In particular, we derive analytically the asymptotic behavior of the emission probability per unit of photon light-cone energy $k_{-}$ and show that it tends to a constant for $k_{-}\to 0$. With numerical codes being an essential tool for the interpretation of present and upcoming experiments in strong-field QED, we obtained an improved approximation for the photon emission probability, implemented it numerically, and showed that it amends the inaccurate behavior of the LCFA in the infrared region, such that it is in qualitative and good quantitative agreement with the full strong-field QED probability also in the infrared region.