Inverse Compton Cross Section Revisited

The design of advanced machines working in the quantum regime (ELI-NP, IRIDE, e − γ and γ − γ colliders) requires to set the fundamentals needed to have an accurate prediction of the radiation qualities after the Compton scattering. Due to the high energy of the electron beam in the cases above mentioned, the quantum effects, referred as inverse Compton, which occur during the collision with the laser radiation, are not negligible.We present a rigorous method to obtain the inverse Compton cross section in the general case of not null initial momentum of the electrons from a pure QED calculation, avoiding the usual approaches based on the derivation of this cross section either from the Klein and Nishina formula and the Lorentz transformations or throught Feynman diagrams and Mandelstam invariants. In the derivation of the cross section from the transition amplitude we pay particular attention to the long time behavior of the system evolution. Proceeding in this way we obtain the transition probability in the time unit, which integrated over the solid angle of emission defines spectrum and number of the scattered photons.