The Four-Loop QCD Beta-Function and Anomalous Dimensions

TL;DR

This work provides an independent four‑loop verification of the QCD beta function and completes the four‑loop renormalization data for a general compact simple Lie group in the MSbar scheme within linear gauge. Using dimensional regularization and integration‑by‑parts reductions to fully massive tadpole master integrals solved via a Laporta algorithm, the authors derive the four‑loop anomalous dimensions for the quark, gluon, ghost fields, and the quark‑gluon vertex, and deduce the gauge‑invariant four‑loop beta coefficient $\beta_3$. The results align with previous SU(N) specializations (4lrenormalization) and furnish a comprehensive set of renormalization constants for general gauge groups, enhancing the reliability of high‑order RG calculations in QCD. This work also demonstrates the feasibility of cross‑checking complex multi‑loop computations through gauge invariance tests and independent computational pipelines, supporting future precision QCD studies.

Abstract

The four-loop beta-function of quantum chromodynamics is calculated and agreement is found with the previous result. The anomalous dimensions of the quark-gluon vertex, and quark, gluon and ghost fields are given for a general compact simple Lie group.

Figures (1)

• Figure 1: Completely massive four-loop tadpole master integrals. Dashed lines mark those integrals which are needed to one order higher in the $\epsilon$ expansion, i.e. up to finite parts for this calculation.