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 γ5 defined in a Kreimer-scheme variant, we find that additional renormalization counter-terms proportional to the Chern-Simons current operator are needed starting from \( \mathcal{O} \)(\( {\alpha}_s^2 \)) in QCD. This is in contrast to the well-known purely multiplicative renormalization of the singlet axial-current operator defined with a non-anticommuting γ5. 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 \( \mathcal{O} \)(\( {\alpha}_s^3 \)) 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 γ5. Owing to the factorized form of these divergences, intimately related to the axial anomaly, we further performed a check, using concrete examples, that with γ5 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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Acknowledgments
The author is grateful to Y.Q. Ma for a chat that initiated the investigation presented in this work as well as his valuable feedback on the manuscript. The author is also grateful to M. Czakon for his careful reading of the manuscript as well as many helpful comments on it. This work was supported by the Natural Science Foundation of China under contract No. 12205171, No. 12235008 and No. 12321005.
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Chen, L. An observation on Feynman diagrams with axial anomalous subgraphs in dimensional regularization with an anticommuting γ5. J. High Energ. Phys. 2023, 30 (2023). https://doi.org/10.1007/JHEP11(2023)030
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DOI: https://doi.org/10.1007/JHEP11(2023)030