| Summary: | Abstract Through the calculation of the matrix element of the singlet axial-current operator between the vacuum and a pair of gluons in dimensional regularization with an anti-commuting γ 5 defined in a Kreimer-scheme variant, we find that additional renormalization counter-terms proportional to the Chern-Simons current operator are needed starting from O $$ \mathcal{O} $$ ( α s 2 $$ {\alpha}_s^2 $$ ) in QCD. This is in contrast to the well-known purely multiplicative renormalization of the singlet axial-current operator defined with a non-anticommuting γ 5. Consequently, without introducing compensation terms in the form of additional renormalization, the Adler-Bell-Jackiw anomaly equation does not hold automatically in the bare form in this kind of schemes. We determine the corresponding (gauge-dependent) coefficient to O $$ \mathcal{O} $$ ( α s 3 $$ {\alpha}_s^3 $$ ) in QCD, using a variant of the original Kreimer prescription which is implemented in our computation in terms of the standard cyclic trace together with a constructively-defined γ 5. Owing to the factorized form of these divergences, intimately related to the axial anomaly, we further performed a check, using concrete examples, that with γ 5 treated in this way, the axial-current operator needs no more additional renormalization in dimensional regularization but only for non-anomalous amplitudes in a perturbatively renormalizable theory. To be complete, we provide a few additional ingredients needed for a proposed extension of the algorithmic procedure formulated in the above analysis to potential applications to a renormalizable anomaly-free chiral gauge theory, i.e. the electroweak theory.
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