Note on Klein-Nishina effect in strong-field QED: the case of nonlinear Compton scattering

Suitably normalized differential probabilities of one-photon emission in external electromagnetic fields are compared to quantify the transit of nonlinear Compton scattering to linear Compton scattering, described by the Klein-Nishina formula, and to constant crossed field treatment. The known Klein-Nishina suppression at large energies is further enforced by increasing field intensity. In view of the Ritus-Narozhny conjecture, we demonstrate that different paths in the field intensity vs. energy plane towards large values of the quantum non-linearity parameter $\chi$ facilitate significantly different asymptotic dependencies, both in the Klein-Nishina regime and the constant crossed field regime and in between.