Hopf-algebraic Renormalization of QED in the linear covariant Gauge

TL;DR

This paper explores the Hopf-algebraic structure of renormalization in massless QED within a linear covariant gauge, analyzing the role of invariant charges and the conditions for finite electron self-energy.

Contribution

It introduces a Hopf-algebraic framework for QED renormalization in linear covariant gauges, including explicit third-loop calculations and the conditions for finite electron self-energy.

Findings

Two invariant charges lead to different renormalization group functions.

Explicit third-loop order computations confirm the formulas.

A finite electron self-energy gauge exists only in quenched QED.

Abstract

In the context of massless quantum electrodynamics (QED) with a linear covariant gauge fixing, the connection between the counterterm and the Hopf-algebraic approach to renormalization is examined. The coproduct formula of Green's functions contains two invariant charges, which give rise to different renormalization group functions. All formulas are tested by explicit computations to third loop order. The possibility of a finite electron self-energy by fixing a generalized linear covariant gauge is discussed. An analysis of subdivergences leads to the conclusion that such a gauge only exists in quenched QED.