The five-loop beta function of Yang-Mills theory with fermions

TL;DR

This paper presents the calculation of the five-loop beta function for Yang-Mills theories with fermions, confirming previous results for QCD and QED and analyzing the numerical impact of these higher-order corrections.

Contribution

The authors computed the five-loop corrections to the beta function in Yang-Mills theories using advanced analytical techniques, extending previous lower-order calculations.

Findings

Five-loop corrections are smaller than four-loop ones in QCD.

The analytical results confirm previous QCD and QED calculations.

Higher-order corrections have minimal impact at low energy scales.

Abstract

We have computed the five-loop corrections to the scale dependence of the renormalized coupling constant for Quantum Chromodynamics (QCD), its generalization to non-Abelian gauge theories with a simple compact Lie group, and for Quantum Electrodynamics (QED). Our analytical result, obtained using the background field method, infrared rearrangement via a new diagram-by-diagram implementation of the R* operation and the Forcer program for massless four-loop propagators, confirms the QCD and QED results obtained by only one group before. The numerical size of the five-loop corrections is briefly discussed in the standard MSbar scheme for QCD with n_f flavours and for pure SU(N) Yang-Mills theory. Their effect in QCD is much smaller than the four-loop contributions, even at rather low scales.