Soft photon approximation in a laser field

In the present paper, we consider processes involving the emission of soft photons in the presence of a strong laser field. We demonstrate that the matrix element $S$ for a process $\text{i} \rightarrow \text{f} + \gamma$, with a soft photon $\gamma$, can be expressed in terms of the matrix element $S_0$ for the process $\text{i} \rightarrow \text{f}$ through a simple multiplicative factor in the integrand over $\phi$. This approximation enables a result that is exact in the phase and approximate in the prefactor to order $\mathcal{O}(\omega/\varepsilon_\text{char})$, where $\omega$ is the frequency of the soft photon and $\varepsilon_\text{char}$ is the characteristic energy of the $\text{i}\rightarrow\text{f}$ process. We demonstrate several important applications of this soft photon approximation. First, under soft photon approximation we compute the probabilities of nonlinear Compton scattering and photon emission in the superposition of a laser and atomic fields and compare obtained result with the exact one. Second, we demonstrate that the amplitude of $n$ soft photons emission has factorization, which corresponds to the independence of the emission of $n$ soft photons. Third, we use the discussed approximation to prove cancellation of real and virtual infrared divergences for nonlinear Compton scattering and derive the finite radiative corrections. The soft photon approximation is a useful tool for investigation of different QED processes in the presence of a strong laser field. Also, it can be widely used for computation of infrared part of radiative correction for some processes.