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 $[{\partial }_{\mu }{J}_5^{\mu }{]}_R=\frac{{\alpha }_s}{4\pi }{n}_f{\mathrm{T}}_F[F\tilde{F}{]}_R$ 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 $[F\tilde{F}{]}_R$ 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, ${\gamma }_J=\frac{{\alpha }_s}{4\pi }{n}_f{\mathrm{T}}_F{\gamma }_{FJ}$, 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.

Comments: 23 pages, 4 figures and 1 table, references updated, matched with the published version in JHEP