Infrared divergences and the photon mass in QED

TL;DR

This paper investigates the infrared behavior of QED using Dyson-Schwinger equations, establishing conditions for photon masslessness, exploring the Schwinger mechanism, and analyzing fermion propagator behavior.

Contribution

It provides a detailed analysis of infrared divergences in QED, linking photon self-energy finiteness to photon masslessness and examining mechanisms for photon mass generation.

Findings

Photon self-energy finiteness implies a massless photon with a $1/k^2$ propagator divergence.

The Schwinger mechanism can generate a photon mass, but breaks the link with self-energy finiteness.

Infrared safe equations are found for fermion gap and vertex functions.

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 $\slashed{p}/p^4$.