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

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.