Renormalized perturbation theory in an intense background electromagnetic field

Abstract

Quantum electrodynamics in strong background electromagnetic fields or strong-field QED (SFQED) has been investigated in great detail at the tree level. The study of SFQED at higher loops has not been carried out in a correspondingly systematic way, with the notable exception of strong background atomic fields. Here, we investigate the renormalization of SFQED by writing the standard unrenormalized SFQED Lagrangian density, which differs from the vacuum-QED Lagrangian density by the additional interaction term of the Dirac four-current density with the background four-vector potential, in terms of renormalized quantities and counterterms. Within this framework, we confirm the necessity of renormalizing the background field as an electric charge rather than as a photon field. We compare this approach with an alternative one which features a different renormalized Lagrangian density and we show that they are physically equivalent. As a byproduct, it will become clear that the renormalizability of SFQED directly derives from the renormalizability of vacuum QED. Also, we obtain that the renormalized SFQED Lagrangian density features a new counterterm as compared to the vacuum Lagrangian density. This counterterm is necessary to renormalize the electron self energy, which undergoes a new contribution as compared to vacuum QED, physically due to the electromagnetic field produced by the four-current density induced in the vacuum by the background electromagnetic field. The vacuum-induced electromagnetic field diverges and needs to be renormalized. We show how the renormalization of the vacuum-induced electromagnetic field is implemented explicitly at one loop and at two loops. The renormalization at all loops is shown to be more easily deduced at the level of the Lagrangian density. Finally, some subtleties concerning the renormalization procedure in the case of a free background field.