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

Journal of High Energy Physics, vol. 2021

Abstract

Abstract The renormalization constantZJof the flavor-singlet axial-vector current with a non-anticommutingγ5in dimensional regularization is determined to order $$ {\alpha}_s^3 $$ αs3 in QCD with massless quarks. The result is obtained by computing the matrix elements of the operators appearing in the axial-anomaly equation $$ {\left[{\partial}_{\mu }{J}_5^{\mu}\right]}_R=\frac{\alpha_s}{4\pi }{n}_f{\mathrm{T}}_F{\left[F\tilde{F}\right]}_R $$ ∂μJ5μR=αs4πnfTFFF˜R between the vacuum and a state of two (off-shell) gluons to 4-loop order. Furthermore, through this computation, the equality between the $$ \overline{\mathrm{MS}} $$ MS¯ renormalization constant $$ {Z}_{F\tilde{F}} $$ ZFF˜ associated with the operator $$ {\left[F\tilde{F}\right]}_R $$ FF˜R and that ofαsis verified explicitly to hold true at 4-loop order. This equality automatically ensures a relation between the respective anomalous dimensions, $$ {\gamma}_J=\frac{\alpha_s}{4\pi }{n}_f{\mathrm{T}}_F{\gamma}_{FJ} $$ γJ=αs4πnfTFγFJ , at order $$ {\alpha}_s^4 $$ αs4 given the validity of the axial-anomaly equation which was used to determine the non- $$ \overline{\mathrm{MS}} $$ MS¯ piece ofZJfor the particularγ5prescription in use.

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