Building the Full Fermion-Photon Vertex of QED by Imposing Multiplicative Renormalizability of the Schwinger-Dyson Equations for the Fermion and Photon Propagators

TL;DR

This paper derives a non-perturbative, gauge-invariant fermion-photon vertex in massless QED that ensures consistent truncation of Schwinger-Dyson equations, improving the calculation of propagators beyond perturbation theory.

Contribution

It provides the first explicit form of the fermion-photon vertex satisfying gauge invariance and multiplicative renormalizability to all orders in leading logs, aiding non-perturbative QED studies.

Findings

Vertex form satisfies gauge invariance and renormalizability constraints.

Consistent truncation of Schwinger-Dyson equations achieved.

Matches perturbative results at order α.

Abstract

In principle, calculation of a full Green's function in any field theory requires knowledge of the infinite set of multi-point Green's functions, unless one can find some way of truncating the corresponding Schwinger-Dyson equations. For the fermion and boson propagators in QED this requires an {\it ansatz} for the full three point vertex. Here we illustrate how the properties of gauge invariance, gauge covariance and multiplicative renormalizability impose severe constraints on this fermion-boson interaction, allowing a consistent truncation of the propagator equations. We demonstrate how these conditions imply that the 3-point vertex {\bf in the propagator equations} is largely determined by the behaviour of the fermion propagator itself and not by knowledge of the many higher point functions. We give an explicit form for the fermion-photon vertex, which in the fermion and photon propagator fulfills these constraints to all orders in leading logarithms for massless QED, and accords with the weak coupling limit in perturbation theory at ${\cal O}(\alpha)$. This provides the first attempt to deduce non-perturbative Feynman rules for strong physics calculations of propagators in massless QED that ensures a more consistent truncation of the 2-point Schwinger-Dyson equations. The generalisation to next-to-leading order and masses will be described in a longer publication.