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Chiral anomaly cancellation and neutral triple gauge boson vertices in the SM EFT

Dimitrios Beis* and Athanasios Dedes†

  • *Contact author: d.beis@uoi.gr
  • †Contact author: adedes@uoi.gr
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Phys. Rev. D 112, 056006 – Published 9 September, 2025

DOI: https://doi.org/10.1103/2tvx-gyj5

Abstract

We demonstrate the cancellation of chiral anomalies in the Standard Model (SM) effective field theory (EFT), achieved through a consistent choice of loop momentum routing in triangle diagrams with dimension-6 operator insertions. By enforcing gauge invariance and Bose symmetry, we show that Goldstone-boson contributions cancel anomalies arising from massive gauge-boson vertices, thereby preserving the consistency of the SM EFT. We compute neutral triple gauge-boson vertices at one loop, revealing dominant contributions from dimension-6 operators at all energies below the EFT cutoff. A UV-complete anomaly-free model with a heavy vectorlike electron validates our approach, illustrating how heavy fermion decoupling generates SM EFT operators while maintaining anomaly cancellation. Our results highlight the phenomenological relevance of these vertices for probing new physics at colliders, particularly through dimension-6 effects that scale as the inverse of the center of mass energy squared, 1/s, offering a viable pathway for experimental detection.

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References (85)

  1. J. S. Bell and R. Jackiw, A PCAC puzzle: π0→γγ in the σ model, Nuovo Cimento A 60, 47 (1969).
  2. S. L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev. 177, 2426 (1969).
  3. K. Fujikawa, Path integral measure for gauge invariant fermion theories, Phys. Rev. Lett. 42, 1195 (1979).
  4. C. Bouchiat, J. Iliopoulos, and P. Meyer, An anomaly free version of Weinberg’s model, Phys. Lett. 38B, 519 (1972).
  5. D. J. Gross and R. Jackiw, Effect of anomalies on quasirenormalizable theories, Phys. Rev. D 6, 477 (1972).
  6. C. P. Korthals Altes and M. Perrottet, Anomalous Ward-identities, gauge-variance and appearance of ghosts in Higgs-Kibble type theories, Phys. Lett. 39B, 546 (1972).
  7. I. Brivio and M. Trott, The Standard Model as an effective field theory, Phys. Rep. 793, 1 (2019).
  8. G. Isidori, F. Wilsch, and D. Wyler, The Standard Model effective field theory at work, Rev. Mod. Phys. 96, 015006 (2024).
  9. T. Appelquist and J. Carazzone, Infrared singularities and massive fields, Phys. Rev. D 11, 2856 (1975).
  10. O. Cata, W. Kilian, and N. Kreher, Gauge anomalies in the Standard-Model effective field theory, arXiv:2011.09976.
  11. Q. Bonnefoy, L. Di Luzio, C. Grojean, A. Paul, and A. N. Rossia, Comments on gauge anomalies at dimension-six in the Standard Model effective field theory, J. High Energy Phys. 05 (2021) 153.
  12. F. Feruglio, A note on gauge anomaly cancellation in effective field theories, J. High Energy Phys. 03 (2021) 128.
  13. G. Passarino, Veltman, renormalizability, calculability, Acta Phys. Pol. B 52, 533 (2021).
  14. T. Cohen, X. Lu, and Z. Zhang, Anomaly cancellation in effective field theories from the covariant derivative expansion, Phys. Rev. D 108, 056027 (2023).
  15. A. Dedes, W. Materkowska, M. Paraskevas, J. Rosiek, and K. Suxho, Feynman rules for the Standard Model effective field theory in Rξ-gauges, J. High Energy Phys. 06 (2017) 143.
  16. E. D’Hoker and E. Farhi, Decoupling a fermion whose mass is generated by a Yukawa coupling: The general case, Nucl. Phys. B248, 59 (1984).
  17. E. D’Hoker and E. Farhi, Decoupling a fermion in the standard electroweak theory, Nucl. Phys. B248, 77 (1984).
  18. A. Dedes and K. Suxho, Heavy fermion non-decoupling effects in triple gauge boson vertices, Phys. Rev. D 85, 095024 (2012).
  19. L. Michaels and F. Yu, Probing new U(1) gauge symmetries via exotic Z→Z′γ decays, J. High Energy Phys. 03 (2021) 120.
  20. G. D. Kribs, G. Lee, and A. Martin, Effective field theory of Stückelberg vector bosons, Phys. Rev. D 106, 055020 (2022).
  21. P. Anastasopoulos, M. Bianchi, E. Dudas, and E. Kiritsis, Anomalies, anomalous U(1)’s and generalized Chern-Simons terms, J. High Energy Phys. 11 (2006) 057.
  22. B. C. Allanach, J. Davighi, and S. Melville, An anomaly-free ATLAS: Charting the space of flavour-dependent gauged U(1) extensions of the Standard Model, J. High Energy Phys. 02 (2019) 082.
  23. P. Anastasopoulos, I. Antoniadis, K. Benakli, and F. Rondeau, Anomalous U(1) extension of the Standard Model, J. High Energy Phys. 07 (2024) 232.
  24. K. Hagiwara, R. D. Peccei, D. Zeppenfeld, and K. Hikasa, Probing the weak boson sector in e+e−→W+W−, Nucl. Phys. B282, 253 (1987).
  25. G. J. Gounaris, J. Layssac, and F. M. Renard, Signatures of the anomalous Zγ and ZZ production at the lepton and hadron colliders, Phys. Rev. D 61, 073013 (2000).
  26. B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, Dimension-six terms in the Standard Model Lagrangian, J. High Energy Phys. 10 (2010) 085.
  27. C. W. Murphy, Dimension-8 operators in the Standard Model effective field theory, J. High Energy Phys. 10 (2020) 174.
  28. OPAL Collaboration, Search for trilinear neutral gauge boson couplings in Z− gamma production at S(1/2)=189-GeV at LEP, Eur. Phys. J. C 17, 553 (2000).
  29. L3 Collaboration, Study of the e+e−→Zγ process at LEP and limits on triple neutral-gauge-boson couplings, Phys. Lett. B 597, 119 (2004).
  30. DELPHI Collaboration, Study of triple-gauge-boson couplings ZZZ, ZZγ and Zγγ LEP, Eur. Phys. J. C 51, 525 (2007).
  31. D0 Collaboration, Measurement of the Zγ→νν¯γ cross section and limits on anomalous ZZγ and Zγγ couplings in pp¯ collisions at s=1.96  TeV, Phys. Rev. Lett. 102, 201802 (2009).
  32. CDF Collaboration, Limits on anomalous trilinear gauge couplings in Zγ events from pp¯ collisions at s=1.96  TeV, Phys. Rev. Lett. 107, 051802 (2011).
  33. ATLAS Collaboration, Measurement of the Zγ→νν¯γ production cross section in pp collisions at s=13  TeV with the ATLAS detector and limits on anomalous triple gauge-boson couplings, J. High Energy Phys. 12 (2018) 010.
  34. CMS Collaboration, Measurement of the Zγ→νν¯γ production cross section in pp collisions at s=8  TeV and limits on anomalous ZZγ and Zγγ trilinear gauge boson couplings, Phys. Lett. B 760, 448 (2016).
  35. ATLAS Collaboration, Measurement of the Z(→ℓ+ℓ−)γ production cross-section in pp collisions at s=13  TeV with the ATLAS detector, J. High Energy Phys. 03 (2020) 054.
  36. A. Barroso, F. Boudjema, J. Cole, and N. Dombey, Electromagnetic properties of theZ boson. I, Z. Phys. C 28, 149 (1985).
  37. U. Baur and E. L. Berger, Probing the weak boson sector in Zγ production at hadron colliders, Phys. Rev. D 47, 4889 (1993).
  38. G. J. Gounaris, J. Layssac, and F. M. Renard, New and standard physics contributions to anomalous Z and gamma self-couplings, Phys. Rev. D 62, 073013 (2000).
  39. J. Alcaraz, On the experimental effects of the off-shell structure in anomalous neutral triple gauge vertices, Phys. Rev. D 65, 075020 (2002).
  40. O. Cata, Revisiting ZZ and γZ production with effective field theories, arXiv:1304.1008.
  41. C. Degrande, A basis of dimension-eight operators for anomalous neutral triple gauge boson interactions, J. High Energy Phys. 02 (2014) 101.
  42. A. Senol, S. Spor, E. Gurkanli, V. Cetinkaya, H. Denizli, and M. Köksal, Model-independent study on the anomalous ZZγ and Zγγ couplings at the future muon collider, Eur. Phys. J. Plus 137, 1354 (2022).
  43. J. Ellis, H.-J. He, and R.-Q. Xiao, Probing neutral triple gauge couplings with Z*γ(νν¯γ) production at hadron colliders, Phys. Rev. D 108, L111704 (2023).
  44. H. Novales-Sánchez and M. Salinas, Majorana neutrinos in the triple gauge boson coupling ZZZ*, Phys. Rev. D 108, 075032 (2023).
  45. A. Subba and R. K. Singh, Sensitivity of polarizations and spin correlations of Z boson to anomalous neutral triple gauge couplings at lepton collider with polarized beams, Phys. Rev. D 109, 055047 (2024).
  46. S. Jahedi, Optimal estimation of dimension-8 neutral triple gauge couplings at the e+e− colliders, J. High Energy Phys. 12 (2023) 031.
  47. R. Cepedello, F. Esser, M. Hirsch, and V. Sanz, Fermionic UV models for neutral triple gauge boson vertices, J. High Energy Phys. 07 (2024) 275.
  48. J. Ellis, H.-J. He, R.-Q. Xiao, S.-P. Zeng, and J. Zheng, UV completion of neutral triple gauge couplings, Phys. Rev. D 111, 015007 (2025).
  49. A. D. Medina, N. I. Mileo, A. Szynkman, S. A. Tanco, C. E. M. Wagner, and G. Zapata, Probing triple-gauge couplings in anomalous gauge theories at hadron and lepton colliders, Phys. Rev. D 111, 115010 (2025).
  50. E. Dudas, Y. Mambrini, S. Pokorski, and A. Romagnoni, Extra U(1) as natural source of a monochromatic gamma ray line, J. High Energy Phys. 10 (2012) 123.
  51. J. A. Dror, R. Lasenby, and M. Pospelov, Dark forces coupled to nonconserved currents, Phys. Rev. D 96, 075036 (2017).
  52. R. T. D’Agnolo, D. Liu, J. T. Ruderman, and P.-J. Wang, Forbidden dark matter annihilations into Standard Model particles, J. High Energy Phys. 06 (2021) 103.
  53. A. D. Medina, N. I. Mileo, A. Szynkman, and S. A. Tanco, Elusive muonic WIMP, Phys. Rev. D 106, 075018 (2022).
  54. J. M. Berryman, S. Gardner, and M. Zakeri, Neutron stars with baryon number violation probing dark sectors, Symmetry 14, 518 (2022).
  55. S. Cléry, P. Anastasopoulos, and Y. Mambrini, Reheating and leptogenesis after vector inflation, J. Cosmol. Astropart. Phys. 12 (2024) 035.
  56. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, Reading, MA, 1995), 10.1201/9780429503559.
  57. M. S. Chanowitz and M. K. Gaillard, The TeV physics of strongly interacting W’s and Z’s, Nucl. Phys. B261, 379 (1985).
  58. J. M. Cornwall, D. N. Levin, and G. Tiktopoulos, Uniqueness of spontaneously broken gauge theories, Phys. Rev. Lett. 30, 1268 (1973).
  59. J. M. Cornwall, D. N. Levin, and G. Tiktopoulos, Derivation of gauge invariance from high-energy unitarity bounds on the S-matrix, Phys. Rev. D 10, 1145 (1974).
  60. C. E. Vayonakis, Born helicity amplitudes and cross-sections in non-Abelian gauge theories, Lett. Nuovo Cimento Soc. Ital. Fis. 17, 383 (1976).
  61. B. W. Lee, C. Quigg, and H. B. Thacker, Weak interactions at very high-energies: The role of the Higgs boson mass, Phys. Rev. D 16, 1519 (1977).
  62. I. S. Gerstein and R. Jackiw, Anomalies in ward identities for three-point functions, Phys. Rev. 181, 1955 (1969).
  63. S. Weinberg, The Quantum Theory of Fields. Vol. 2: Modern Applications (Cambridge University Press, Cambridge, England, 2013), 10.1017/CBO9781139644174.
  64. H. K. Dreiner, H. E. Haber, and S. P. Martin, From Spinors to Supersymmetry (Cambridge University Press, Cambridge, England, 2023), 10.1017/9781139049740.
  65. A. Dedes and K. Suxho, Anatomy of the Higgs boson decay into two photons in the unitary gauge, Adv. High Energy Phys. 2013 631841 (2013).
  66. A. Dedes, J. Rosiek, M. Ryczkowski, K. Suxho, and L. Trifyllis, SmeftFR v3—Feynman rules generator for the Standard Model effective field theory, Comput. Phys. Commun. 294, 108943 (2024).
  67. Proceedings of the 13th Brandeis University Summer Institute in Theoretical Physics, Lectures on Elementary Particles and Quantum Field Theory, Waltham, MA, 1970, edited by S. D. Deser, M. T. Grisaru, and H. Pendleton (MIT, Cambridge, MA, 1970).
  68. L. Rosenberg, Electromagnetic interactions of neutrinos, Phys. Rev. 129, 2786 (1963).
  69. L. D. Landau, On the angular momentum of a system of two photons, Dokl. Akad. Nauk SSSR 60, 207 (1948).
  70. C.-N. Yang, Selection rules for the dematerialization of a particle into two photons, Phys. Rev. 77, 242 (1950).
  71. J. de Blas, J. Criado, M. Perez-Victoria, and J. Santiago, Effective description of general extensions of the Standard Model: The complete tree-level dictionary, J. High Energy Phys. 03 (2018) 109.
  72. A. Dedes and K. Mantzaropoulos, Universal scalar leptoquark action for matching, J. High Energy Phys. 11 (2021) 166.
  73. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, England, 1990).
  74. S. Weinberg, Effective gauge theories, Phys. Lett. 91B, 51 (1980).
  75. E. Witten, Short distance analysis of weak interactions, Nucl. Phys. B122, 109 (1977).
  76. Y. Kazama and Y.-P. Yao, Decoupling, effective Lagrangian, and gauge hierarchy in spontaneously broken non-Abelian gauge theories, Phys. Rev. D 25, 1605 (1982).
  77. M. S. Bilenky and A. Santamaria, One loop effective Lagrangian for a Standard Model with a heavy charged scalar singlet, Nucl. Phys. B420, 47 (1994).
  78. A. Broncano, M. B. Gavela, and E. E. Jenkins, The effective Lagrangian for the seesaw model of neutrino mass and leptogenesis, Phys. Lett. B 552, 177 (2003).
  79. E. Celada, T. Giani, J. ter Hoeve, L. Mantani, J. Rojo, A. N. Rossia, M. O. A. Thomas, and E. Vryonidou, Mapping the SMEFT at high-energy colliders: From LEP and the (HL-)LHC to the FCC-ee, J. High Energy Phys. 09 (2024) 091.
  80. D. Liu, R.-Q. Xiao, S. Li, J. Ellis, H.-J. He, and R. Yuan, Probing neutral triple gauge couplings via Zγ(ℓ+ℓ−γ) production at e+e− colliders, Front. Phys. 20, 015201 (2025).
  81. L. Lavoura and J. P. Silva, The oblique corrections from vector-like singlet and doublet quarks, Phys. Rev. D 47, 2046 (1993).
  82. ATLAS Collaboration, Search for third-generation vector-like leptons in pp collisions at s=13  TeV with the ATLAS detector, J. High Energy Phys. 07 (2023) 118.
  83. ATLAS Collaboration, Exploration at the high-energy frontier: ATLAS Run 2 searches investigating the exotic jungle beyond the Standard Model, Phys. Rep. 1116, 301 (2025).
  84. A. Adhikary, M. Olechowski, J. Rosiek, and M. Ryczkowski, Theoretical constraints on models with vectorlike fermions, Phys. Rev. D 110, 075029 (2024).
  85. D. Barducci, L. Di Luzio, M. Nardecchia, and C. Toni, Closing in on new chiral leptons at the LHC, J. High Energy Phys. 12 (2023) 154.

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Phys. Rev. D 112, 056006– Published 9 September, 2025
  • Received 30 March 2025
  • Accepted 5 August 2025
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