---
abstract: 'Refractive processes in strong-field QED are pure quantum processes, which involve only external photons and the background electromagnetic field. We show analytically that such processes occurring in a plane-wave field and involving external real photons are all characterized by a surprisingly modest net exchange of energy and momentum with the laser field, corresponding to a few laser photons, even in the limit of ultra-relativistic laser intensities. We obtain this result by a direct calculation of the transition matrix element of an arbitrary refractive QED process and accounting exactly for the background plane-wave field. A simple physical explanation of this modest net exchange of laser photons is provided, based on the fact that the laser field couples with the external photons only indirectly through virtual electron-positron pairs. For stronger and stronger laser fields, the pairs cover a shorter and shorter distance before they annihilate again, such that the laser can transfer to them an energy corresponding to only a few photons. These results can be relevant for future experiments aiming to test strong-field QED at present and next-generation facilities.'
address: 'Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany'
author:
- 'A. Di Piazza'
bibliography:
- 'Refr\_AP.bib'
title: On refractive 