Abstract
We derive the finite one-loop counterterm required to restore the Ward Identities broken by the regularization scheme in chiral gauge theories. Our result is an analytic expression applicable to a wide class of regularizations satisfying a few general properties. We adopt the background field method, which ensures background gauge invariance in the quantized theory, and focus on renormalizable chiral theories with arbitrary gauge group and fermions in general representations. Our approach can be extended to theories involving scalars, such as the Standard Model, or to non-renormalizable theories, such as the SMEFT. As a concrete application, we work out the finite counterterm at one loop in the Standard Model, within dimensional regularization and the Breitenlohner-Maison-’t Hooft-Veltman prescription for γ5.
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R. Stora, Lagrangian field theory, contribution to Les Houches Summer School on Theoretical Physics, 21st Conference in the Les Houches Summer School series, Les Houches, France, 29 July–1 September 1971, pp. 1–80.
C. Becchi, A. Rouet and R. Stora, Renormalization of the Abelian Higgs-Kibble Model, Commun. Math. Phys. 42 (1975) 127 [INSPIRE].
C. Becchi, A. Rouet and R. Stora, Renormalization of Gauge Theories, Annals Phys. 98 (1976) 287 [INSPIRE].
G. Costa, J. Julve, T. Marinucci and M. Tonin, Nonabelian Gauge Theories and Triangle Anomalies, Nuovo Cim. A 38 (1977) 373 [INSPIRE].
O. Piguet and S.P. Sorella, Algebraic renormalization: Perturbative renormalization, symmetries and anomalies, Lect. Notes Phys. Monogr. 28 (1995) 1 [INSPIRE].
P.A. Grassi, Stability and renormalization of Yang-Mills theory with background field method: A Regularization independent proof, Nucl. Phys. B 462 (1996) 524 [hep-th/9505101] [INSPIRE].
P.A. Grassi, The Abelian anti-ghost equation for the standard model in the ’t Hooft background gauge, Nucl. Phys. B 537 (1999) 527 [hep-th/9804013] [INSPIRE].
E. Kraus, Renormalization of the Electroweak Standard Model to All Orders, Annals Phys. 262 (1998) 155 [hep-th/9709154] [INSPIRE].
R. Ferrari and P.A. Grassi, Constructive algebraic renormalization of the Abelian Higgs-Kibble model, Phys. Rev. D 60 (1999) 065010 [hep-th/9807191] [INSPIRE].
P.A. Grassi, T. Hurth and M. Steinhauser, Practical algebraic renormalization, Annals Phys. 288 (2001) 197 [hep-ph/9907426] [INSPIRE].
P.A. Grassi, Renormalization of nonsemisimple gauge models with the background field method, Nucl. Phys. B 560 (1999) 499 [hep-th/9908188] [INSPIRE].
C.M. Becchi, Renormalizable theories with symmetry breaking, arXiv e-prints [https://doi.org/10.48550/arXiv.1607.05458].
P.A. Grassi, T. Hurth and M. Steinhauser, The Algebraic method, Nucl. Phys. B 610 (2001) 215 [hep-ph/0102005] [INSPIRE].
S. Weinberg, High-energy behavior in quantum field theory, Phys. Rev. 118 (1960) 838 [INSPIRE].
W. Zimmermann, The power counting theorem for minkowski metric, Commun. Math. Phys. 11 (1968) 1 [INSPIRE].
J.H. Lowenstein, Differential vertex operations in Lagrangian field theory, Commun. Math. Phys. 24 (1971) 1 [INSPIRE].
Y.-M.P. Lam, Perturbation Lagrangian theory for scalar fields: Ward-Takahasi identity and current algebra, Phys. Rev. D 6 (1972) 2145 [INSPIRE].
T.E. Clark and J.H. Lowenstein, Generalization of Zimmermann’s Normal-Product Identity, Nucl. Phys. B 113 (1976) 109 [INSPIRE].
F. Brennecke and M. Duetsch, The Quantum Action Principle in the framework of Causal Perturbation Theory, in the proceedings of Conference on Recent Developments in Quantum Field Theory, (2008), pp. 177–196 [https://doi.org/10.1007/978-3-7643-8736-5_11] [arXiv:0801.1408] [INSPIRE].
O. Piguet and A. Rouet, Symmetries in Perturbative Quantum Field Theory, Phys. Rept. 76 (1981) 1 [INSPIRE].
S.L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev. 177 (1969) 2426 [INSPIRE].
J.S. Bell and R. Jackiw, A PCAC puzzle: π0 → γγ in the σ model, Nuovo Cim. A 60 (1969) 47 [INSPIRE].
G. Bonneau, Some Fundamental but Elementary Facts on Renormalization and Regularization: A Critical Review of the Eighties, Int. J. Mod. Phys. A 5 (1990) 3831 [INSPIRE].
C.P. Martin and D. Sanchez-Ruiz, Action principles, restoration of BRS symmetry and the renormalization group equation for chiral nonAbelian gauge theories in dimensional renormalization with a nonanticommuting γ5, Nucl. Phys. B 572 (2000) 387 [hep-th/9905076] [INSPIRE].
D. Sanchez-Ruiz, BRS symmetry restoration of chiral Abelian Higgs-Kibble theory in dimensional renormalization with a nonanticommuting γ5, Phys. Rev. D 68 (2003) 025009 [hep-th/0209023] [INSPIRE].
H. Bélusca-Maïto, A. Ilakovac, M. Mađor-Božinović and D. Stöckinger, Dimensional regularization and Breitenlohner-Maison/’t Hooft-Veltman scheme for γ5 applied to chiral YM theories: full one-loop counterterm and RGE structure, JHEP 08 (2020) 024 [arXiv:2004.14398] [INSPIRE].
H. Bélusca-Maïto et al., Two-loop application of the Breitenlohner-Maison/’t Hooft-Veltman scheme with non-anticommuting γ5: full renormalization and symmetry-restoring counterterms in an abelian chiral gauge theory, JHEP 11 (2021) 159 [arXiv:2109.11042] [INSPIRE].
J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95 [INSPIRE].
C.G. Bollini and J.J. Giambiagi, Dimensional Renormalization: The Number of Dimensions as a Regularizing Parameter, Nuovo Cim. B 12 (1972) 20 [INSPIRE].
G. ’t Hooft and M.J.G. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
Q. Bonnefoy et al., Comments on gauge anomalies at dimension-six in the Standard Model Effective Field Theory, JHEP 05 (2021) 153 [arXiv:2012.07740] [INSPIRE].
F. Feruglio, A Note on Gauge Anomaly Cancellation in Effective Field Theories, JHEP 03 (2021) 128 [arXiv:2012.13989] [INSPIRE].
G. Passarino, Veltman, Renormalizability, Calculability, Acta Phys. Polon. B 52 (2021) 533 [arXiv:2104.13569] [INSPIRE].
H. Kluberg-Stern and J.B. Zuber, Ward Identities and Some Clues to the Renormalization of Gauge Invariant Operators, Phys. Rev. D 12 (1975) 467 [INSPIRE].
H. Kluberg-Stern and J.B. Zuber, Renormalization of Nonabelian Gauge Theories in a Background Field Gauge. 1. Green Functions, Phys. Rev. D 12 (1975) 482 [INSPIRE].
L.F. Abbott, Introduction to the Background Field Method, Acta Phys. Polon. B 13 (1982) 33 [INSPIRE].
S. Ichinose and M. Omote, Renormalization Using the Background Field Method, Nucl. Phys. B 203 (1982) 221 [INSPIRE].
D.M. Capper and A. MacLean, The Background Field Method at Two Loops: A General Gauge Yang-Mills Calculation, Nucl. Phys. B 203 (1982) 413 [INSPIRE].
P. Breitenlohner and D. Maison, Dimensional Renormalization and the Action Principle, Commun. Math. Phys. 52 (1977) 11 [INSPIRE].
P. Breitenlohner and D. Maison, Dimensional Renormalization of Massless Yang-Mills Theories, MPI-PAE-PTH-26-75 (1975) [INSPIRE].
P. Breitenlohner and D. Maison, Dimensionally Renormalized Green’s Functions for Theories with Massless Particles. 1., Commun. Math. Phys. 52 (1977) 39 [INSPIRE].
P. Breitenlohner and D. Maison, Dimensionally Renormalized Green’s Functions for Theories with Massless Particles. 2., Commun. Math. Phys. 52 (1977) 55 [INSPIRE].
W. Grimus and M.N. Rebelo, Automorphisms in gauge theories and the definition of CP and P, Phys. Rept. 281 (1997) 239 [hep-ph/9506272] [INSPIRE].
H. Georgi and S.L. Glashow, Gauge theories without anomalies, Phys. Rev. D 6 (1972) 429 [INSPIRE].
G. Durieux, J. Gu, E. Vryonidou and C. Zhang, Probing top-quark couplings indirectly at Higgs factories, Chin. Phys. C 42 (2018) 123107 [arXiv:1809.03520] [INSPIRE].
C. Degrande et al., Automated one-loop computations in the standard model effective field theory, Phys. Rev. D 103 (2021) 096024 [arXiv:2008.11743] [INSPIRE].
J.G. Korner, D. Kreimer and K. Schilcher, A Practicable γ5 scheme in dimensional regularization, Z. Phys. C 54 (1992) 503 [INSPIRE].
H. Nicolai and P.K. Townsend, Anomalies and Supersymmetric Regularization by Dimensional Reduction, Phys. Lett. B 93 (1980) 111 [INSPIRE].
A.P. Balachandran, G. Marmo, V.P. Nair and C.G. Trahern, A Nonperturbative Proof of the Nonabelian Anomalies, Phys. Rev. D 25 (1982) 2713 [INSPIRE].
R. Delbourgo and G. Thompson, Anomalies, Dimensional Regularization and the Heat Kernel, Phys. Rev. D 32 (1985) 3300 [INSPIRE].
M.S. Chanowitz, M. Furman and I. Hinchliffe, The Axial Current in Dimensional Regularization, Nucl. Phys. B 159 (1979) 225 [INSPIRE].
D. Kreimer, The γ5 Problem and Anomalies: A Clifford Algebra Approach, Phys. Lett. B 237 (1990) 59 [INSPIRE].
F. Jegerlehner, Facts of life with γ5, Eur. Phys. J. C 18 (2001) 673 [hep-th/0005255] [INSPIRE].
B.S. DeWitt, Dynamical Theory of Groups and Fields, Gordon and Breach (1965).
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Cornella, C., Feruglio, F. & Vecchi, L. Gauge invariance and finite counterterms in chiral gauge theories. J. High Energ. Phys. 2023, 244 (2023). https://doi.org/10.1007/JHEP02(2023)244
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DOI: https://doi.org/10.1007/JHEP02(2023)244