An observation on Feynman diagrams with axial anomalous subgraphs in dimensional regularization with an anticommuting ${\gamma }_5$

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 anticommuting ${\gamma }_5$ defined in a Kreimer-scheme variant, we find that additional renormalization counter-terms proportional to the Chern-Simons current operator are needed starting from $\mathcal{O}({\alpha }_s^2)$ in QCD. This is in contrast to the well-known purely multiplicative renormalization of the singlet axial-current operator defined with a non-anticommuting ${\gamma }_5$. 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 $\mathcal{O}({\alpha }_s^3)$ 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 ${\gamma }_5$. Owing to the factorized form of these divergences, intimately related to the axial anomaly, we further performed a check, using concrete examples, that with ${\gamma }_5$ treated in this way, the axial-current operator needs no more additional renormalization in dimensional regularization but \textit{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.

arXiv commentRefined formuation of the algorithmic procedure to manipulate ${\gamma }_5$ with more discussions; a novel extension of the prescription proposed for applications to the Standard Model provided in the Appendix. Accepted for publication in JHEP

References in corpus (47)

• The Four-Loop QCD Beta-Function and Anomalous Dimensions
• The renormalization of the axial anomaly in dimensional regularization
• AMFlow: a Mathematica package for Feynman integrals computation via Auxiliary Mass Flow
• A Systematic and Efficient Method to Compute Multi-loop Master Integrals
• Four Loop Massless Propagators: an Algebraic Evaluation of All Master Integrals
• Facts of life with gamma(5)
• Four-loop renormalization of QCD: full set of renormalization constants and anomalous dimensions
• |Delta F| = 1 Nonleptonic Effective Hamiltonian in a Simpler Scheme
• Master Integrals for Four-Loop Massless Propagators up to Transcendentality Weight Twelve
• Renormalization of gauge invariant operators and anomalies in Yang-Mills theory
• Renormalization constants and beta functions for the gauge couplings of the Standard Model to three-loop order
• Two-Loop QCD Corrections to the Heavy Quark Form Factors: Anomaly Contributions
• Multiloop corrections for collider processes using auxiliary mass flow
• Anomalies to All Orders
• Determining Feynman integrals with only input from linear algebra
• The (alpha_s)^3 approximation of Quantum Chromodynamics to the Ellis-Jaffe sum rule
• The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements
• Four-loop strong coupling beta-function in the Standard Model
• Decays of Scalar and Pseudoscalar Higgs Bosons into Fermions: Two-loop QCD Corrections to the Higgs-Quark-Antiquark Amplitude
• Top-Yukawa effects on the $\beta$-function of the strong coupling in the SM at four-loop level
• A prescription for projectors to compute helicity amplitudes in D dimensions
• The ${\alpha }_s^3$ Corrections to the Effective Neutral Current and to the Z Decay Rate in the Heavy Top Quark Limit
• Calculation of Feynman loop integration and phase-space integration via auxiliary mass flow
• Weyl Consistency Conditions and ${\gamma }_5$
• Dimensional Regularization and Breitenlohner-Maison / 't Hooft-Veltman Scheme for ${\gamma }_5$ applied to Chiral YM Theories: Full One-Loop Counterterm and RGE Structure
• Gauge coupling beta functions to four-loop order in the Standard Model
• Mixed EW-QCD two-loop amplitudes for $q\bar{q}\to {\ell }^{+}{\ell }^{-}$ and ${\gamma }_5$ scheme independence of multi-loop corrections
• Feynman Rules for the Rational Part of the Standard Model One-loop Amplitudes in the 't Hooft-Veltman ${\gamma }_5$ Scheme
• Remarks on the renormalization of gauge invariant operators in Yang-Mills theory
• Anomaly and Anomaly-Free Treatment of QFTs Based on Symmetry-preserving Loop Regularization
• ${\gamma }_5$ in FDH
• Dimensional regularization vs methods in fixed dimension with and without ${\gamma }_5$
• Gauge Invariance and Finite Counterterms in Chiral Gauge Theories
• Explicit Calculation of the Renormalized Singlet Axial Anomaly
• Two-loop application of the Breitenlohner-Maison/'t Hooft-Veltman scheme with non-anticommuting ${\gamma }_5$: Full renormalization and symmetry-restoring counterterms in an abelian chiral gauge theory
• Gauge Invariant Treatment of ${\gamma }_5$ in the Scheme of 't Hooft and Veltman
• The complete singlet contribution to the massless quark form factor at three loops in QCD
• Renormalization of the flavor-singlet axial-vector current and its anomaly in dimensional regularization
• Fermion Traces Without Evanescence
• The VVA correlator at three loops in perturbative QCD
• Polarized $q\bar{q}\to Z+$Higgs amplitudes at two loops in QCD: the interplay between vector and axial vector form factors and a pitfall in applying a non-anticommuting ${\gamma }_5$
• The three-loop singlet contribution to the massless axial-vector quark form factor
• On the Bose symmetry and the left- and right-chiral anomalies
• A step towards a consistent treatment of chiral theories at higher loop order: the abelian case
• Renormalization of the topological charge density in QCD with dimensional regularization
• Renormalization of the axial current operator in dimensional regularization at four-loop in QCD
• Infrared Subtleties and Chiral Vertices at NLO: An Implicit Regularization Analysis

Cited by in corpus (8)

• Heavy-quark pair production at lepton colliders at NNNLO in QCD
• A Procedure g5anchor to Anchor ${\gamma }_5$ in Feynman Diagrams for the Standard Model
• Renormalization-group equations of the LEFT at two loops: dimension-six operators
• Do anomalies break the momentum routing invariance?
• Symmetry Restoration in the SMEFT: Finite Counterterms for a Non-Anticommuting ${\gamma }_5$
• Three-loop QCD Mass Relation between the $\overline{\mathrm{MS}}$ and Symmetric-momentum Subtraction Scheme Away from the Chiral Limit
• Two loops, four tops and two ${\gamma }_5$ schemes: a renormalization story
• Multi-parton contributions to $\bar{B}\to X_s\gamma$ at NLO