Rise and fall of laser-intensity effects in spectrally resolved Compton process

The spectrally resolved differential cross section of Compton scattering, $d sigma / d omega vert_{omega = const}$, rises from small towards larger laser intensity parameter $xi$, reaches a maximum, and falls towards the asymptotic strong-field region. Expressed by invariant quantities: $d sigma /du vert_{u = const}$ rises from small towards larger values of $xi$, reaches a maximum at $xi_{max} = frac49 {cal K} u m^2 / k cdot p$, ${cal K} = {cal O} (1)$, and falls at $xi > xi_{max}$ like $propto xi^{-3/2} exp left (- frac{2 u m^2}{3 xi , k cdot p} right )$ at $u ge 1$. [The quantity $u$ is the Ritus variable related to the light-front momentum-fraction $s = (1 + u)/u = k cdot k / k cdot p$ of the emitted photon (four-momentum $k$, frequency $omega$), and $k cdot p/m^2$ quantifies the invariant energy in the entrance channel of electron (four-momentum $p$, mass $m$) and laser (four-wave vector $k$).] Such a behavior of a differential observable is to be contrasted with the laser intensity dependence of the total probability, $lim_{chi = xi k cdot p/m^2, xi to infty} mathbb{P} propto alpha chi^{2/3} m^2 / k cdot p$, which is governed by the soft spectral part. We combine the hard-photon yield from Compton with the seeded Breit-Wheeler pair production in a folding model and obtain a rapidly increasing $e^+ e^-$ pair number at $xi lesssim 4$. Laser bandwidth effects are quantified in the weak-field limit of the related trident pair production.