Four-loop Dyson-Schwinger-Johnson anatomy

TL;DR

This paper computes four-loop anomalous dimensions in quenched QED using Dyson-Schwinger equations, providing new high-order results and checks against QCD calculations, with a focus on scheme independence and methodological simplicity.

Contribution

It introduces a novel four-loop calculation method for anomalous dimensions in quenched QED based on Dyson-Schwinger equations, avoiding complex subtractions.

Findings

Calculated 4-loop anomalous dimensions for quenched QED.

Identified a 4-loop beta function with 24 unambiguous terms.

Discovered cancellations in the rational and zeta parts of the beta function.

Abstract

Dyson-Schwinger equations are used to evaluate the 4-loop anomalous dimensions of quenched QED in terms of finite, scheme-independent, 3-loop integrals. Three of the results serve as strong checks of terms from scheme-dependent 4-loop QCD calculations. The fourth, for the anomalous dimension of $\bar\psi\sigma_{\mu\nu}\psi$, was previously known only to 2-loop order. The 4-loop beta function has 24 unambiguous terms. Two of the simplest give $\beta_4=\frac13(22-160)=-46$. The rational, $\zeta_3$ and $\zeta_5$ parts of the other 22 miraculously sum to zero. Vertex anomalous dimensions have 40 terms, with no dramatic cancellations. Our methods come from work by the late Kenneth Johnson, done more than 30 years ago. They are entirely free of the subtractions and infrared rearrangements of later methods.