The local-constant-field approximation (LCFA) is an essential theoretical
tool for investigating strong-field QED phenomena in background electromagnetic
fields with complex spacetime structure. In our previous work
[Phys.~Rev.~A~\textbf{98}, 012134 (2018)] we have analyzed the shortcomings of
the LCFA in nonlinear Compton scattering at low emitted photon energies for the
case of a background plane-wave field. Here, we generalize that analysis to
background fields, which can feature a virtually arbitrary spacetime structure.
In addition, we provide an explicit and simple implementation of an improved
expression of the nonlinear Compton scattering differential probability that
solves the main shortcomings of the standard LCFA in the infrared region, and
is suitable for background electromagnetic fields with arbitrary spacetime
structure such as those occurring in particle-in-cell simulations. Finally, we
carry out a systematic procedure to calculate the probability of nonlinear
Compton scattering per unit of emitted photon light-cone energy and of
nonlinear Breit-Wheeler pair production per unit of produced positron
light-cone energy beyond the LCFA in a plane-wave background field, which
allows us to identify the limits of validity of this approximation
quantitatively.Comment: 15 pages, 3 figure
