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

The renormalization constant Z_J of the flavor-singlet axial-vector current with a non-anticommuting γ₅ in dimensional regularization is determined to order α_s³ in QCD with massless quarks. The result is obtained by computing the matrix elements of the operators appearing in the axial-anomaly equation [∂_μJ5^μ]R=α/_s4 π n_fT_F[F_{F ∼ ]R} between the vacuum and a state of two (off-shell) gluons to 4-loop order. Furthermore, through this computation, the equality between the MS ¯ renormalization constant Z_{F F ∼} associated with the operator [F_{F ∼ ]R} and that of α_s is verified explicitly to hold true at 4-loop order. This equality automatically ensures a relation between the respective anomalous dimensions, γ_J=α/_s4 π n_fT_Fγ_FJ, at order α_s⁴ given the validity of the axial-anomaly equation which was used to determine the non-MS ¯ piece of Z_J for the particular γ₅ prescription in use.