Renormalization of the flavor-singlet axial-vector current and its anomaly in dimensional regularization

TL;DR

This paper calculates the renormalization constant of the flavor-singlet axial-vector current in QCD with massless quarks to four-loop order, verifying key anomaly relations and operator renormalization constants in dimensional regularization.

Contribution

It provides the first four-loop order calculation of the renormalization constant $Z_J$ for the flavor-singlet axial-vector current with a non-anticommuting $oldsymbol{ extgamma_5}$ in dimensional regularization, confirming conjectured operator relations.

Findings

Renormalization constant $Z_J$ determined to order $oldsymbol{ extalpha_s^3}$.

Verified equality of $Z_{F ilde{F}}$ and $oldsymbol{ extalpha_s}$ renormalization constants at four loops.

Established relation between anomalous dimensions $oldsymbol{ extgamma_J}$ and $oldsymbol{ extgamma_{FJ}}$ at order $oldsymbol{ extalpha_s^4}$.

Abstract

The renormalization constant $Z_J$ of the flavor-singlet axial-vector current with a non-anticommuting $\gamma_5$ in dimensional regularization is determined to order $\alpha_s^3$ in QCD with massless quarks. The result is obtained by computing the matrix elements of the operators appearing in the axial-anomaly equation $\big[\partial_{\mu} J^{\mu}_{5} \big]_{R} = \frac{\alpha_s}{4 \pi}\, n_f\, {\displaystyle \mathrm{T}_{F}} \, \big[F \tilde{F} \big]_{R}$ between the vacuum and a state of two (off-shell) gluons to 4-loop order. Furthermore, through this computation, the conjectured equality between the $\overline{\mathrm{MS}}$ renormalization constant $Z_{F\tilde{F}}$ associated with the operator $\big[F \tilde{F} \big]_{R}$ and that of $\alpha_s$ is verified explicitly to hold true at 4-loop order. This equality automatically ensures a relation between the respective anomalous dimensions, $\gamma_{{\scriptscriptstyle J}} = \frac{\alpha_s}{4 \pi}\, n_f\, {\displaystyle \mathrm{T}_{F}} \, \gamma_{{\scriptscriptstyle FJ}} $, at order $\alpha_s^4$ given the validity of the axial-anomaly equation which was used to determine the non-$\overline{\mathrm{MS}}$ piece of $Z_J$ for the particular $\gamma_5$ prescription in use.