Infrared divergences and the photon mass in QED

TL;DR

This work analyzes infrared aspects of QED through truncated Dyson–Schwinger equations for the fermion propagator, photon propagator, and photon–fermion vertex. It shows that requiring a finite photon self-energy for all momenta together with a smooth vertex guarantees a massless photon with $D(k^2)\sim 1/k^2$ in the infrared, independent of the coupling. It also assesses the Schwinger mechanism, identifying transverse form factors that could generate a gauge-invariant photon mass, though this decouples from the finite self-energy in the standard scenario. The fermion gap and the vertex remain infrared safe, and chiral fermions imply a modified UV behavior of the fermion propagator when the photon is massless. Overall, the IR issues seen in perturbation theory appear to be tied to the perturbative framework rather than intrinsic features of QED, with caveats for non-Abelian extensions like QCD.

Abstract

The infrared properties of QED are investigated within the framework of the Dyson-Schwinger equations. Our study finds that, independently of the value of the coupling constant, requiring the photon self-energy to be finite for any momenta, combined with a smooth behavior for the photon-fermion vertex, is equivalent to state that the photon is massless and that the photon propagator diverges at low momenta as $1/k^2$. Furthermore, the Schwinger mechanism to generate, in a gauge invariant way, a photon mass is investigated and the form factors that can be at the origin of a possible photon mass are identified. For the Schwinger mechanism the link between the finiteness of the photon self-energy and the masslessness of the photon is lost. The infrared behavior of the fermion gap equation and the vertex equation are found to be infrared safe integral equations. Moreover, by studying chiral fermions within QED it is observed that the requirement of the finiteness of the photon self-energy translates into a fermion propagator that behaves as $\slashed{p}/p^4$.