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Noninvertible symmetry as a solution to the strong CP problem in a GUT-inspired standard model

Tatsuo Kobayashi1,*, Hajime Otsuka2,3,†, and Tsutomu T. Yanagida4,5,‡

  • *Contact author: kobayashi@particle.sci.hokudai.ac.jp
  • †Contact author: otsuka.hajime@phys.kyushu-u.ac.jp
  • ‡Contact author: tsutomu.tyanagida@sjtu.edu.cn
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Phys. Rev. D 113, 055016 – Published 6 March, 2026

DOI: https://doi.org/10.1103/vbd5-5cp3

Abstract

We propose a three-zero texture for the down-quark mass matrix within a SU(5) grand unified theory (GUT)-inspired Standard Model, enforced by a noninvertible selection rule originating from Z2 gauging of ZN symmetries. Assuming CP invariance at the high-energy scale, our framework solves the strong CP problem while reproducing a realistic quark mass matrix. The CP symmetry is spontaneously broken down at an intermediate scale via complex vacuum expectation values of new scalar fields ηi, which generate the CP phase in the Cabibbo-Kobayashi-Maskawa matrix without inducing the quantum chromodynamics (QCD) θ¯ term. The present model incorporates three generations of matter, Higgs, and additional scalars with a common structure under the noninvertible symmetry, which may be naturally embedded into SO(10) GUTs at high energies.

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References (50)

  1. T. D. Brennan and S. Hong, Introduction to generalized global symmetries in QFT and particle physics, arXiv:2306.00912.
  2. S. Schafer-Nameki, ICTP lectures on (non-)invertible generalized symmetries, Phys. Rep. 1063, 1 (2024).
  3. S.-H. Shao, What’s done cannot be undone: TASI lectures on non-invertible symmetries, arXiv:2308.00747.
  4. J. J. Heckman, J. McNamara, M. Montero, A. Sharon, C. Vafa, and I. Valenzuela, Fate of stringy noninvertible symmetries, Phys. Rev. D 110, 106001 (2024).
  5. J. Kaidi, Y. Tachikawa, and H. Y. Zhang, On a class of selection rules without group actions in field theory and string theory, SciPost Phys. 17, 169 (2024).
  6. Y. Choi, H. T. Lam, and S.-H. Shao, Noninvertible global symmetries in the standard model, Phys. Rev. Lett. 129, 161601 (2022).
  7. C. Cordova and K. Ohmori, Noninvertible chiral symmetry and exponential hierarchies, Phys. Rev. X 13, 011034 (2023).
  8. C. Cordova, S. Hong, S. Koren, and K. Ohmori, Neutrino masses from generalized symmetry breaking, Phys. Rev. X 14, 031033 (2024).
  9. Y. Choi, M. Forslund, H. T. Lam, and S.-H. Shao, Quantization of axion-gauge couplings and noninvertible higher symmetries, Phys. Rev. Lett. 132, 121601 (2024).
  10. C. Cordova, S. Hong, and L.-T. Wang, Axion domain walls, small instantons, and non-invertible symmetry breaking, J. High Energy Phys. 05 (2024) 325.
  11. C. Cordova, S. Hong, and S. Koren, Non-invertible Peccei-Quinn symmetry and the massless quark solution to the strong CP problem, Phys. Rev. X 15, 031011 (2025).
  12. A. Delgado and S. Koren, Non-invertible Peccei-Quinn symmetry, natural 2HDM alignment, and the visible axion, J. High Energy Phys. 02 (2025) 178.
  13. T. Kobayashi and H. Otsuka, Non-invertible flavor symmetries in magnetized extra dimensions, J. High Energy Phys. 11 (2024) 120.
  14. T. Kobayashi, H. Otsuka, and M. Tanimoto, Yukawa textures from non-invertible symmetries, J. High Energy Phys. 12 (2024) 117.
  15. S. Funakoshi, T. Kobayashi, and H. Otsuka, Quantum aspects of non-invertible flavor symmetries in intersecting/magnetized D-brane models, J. High Energy Phys. 04 (2025) 183.
  16. T. Kobayashi, Y. Nishioka, H. Otsuka, and M. Tanimoto, More about quark Yukawa textures from selection rules without group actions, J. High Energy Phys. 05 (2025) 177.
  17. T. Kobayashi, H. Otsuka, M. Tanimoto, and H. Uchida, Lepton mass textures from non-invertible multiplication rules, J. High Energy Phys. 08 (2025) 189.
  18. M. Suzuki and L.-X. Xu, Phenomenological implications of a class of non-invertible selection rules, arXiv:2503.19964.
  19. T. Kobayashi, H. Okada, and H. Otsuka, Radiative neutrino mass models from non-invertible selection rules, J. High Energy Phys. 12 (2025) 111.
  20. T. Kobayashi, H. Mita, H. Otsuka, and R. Sakuma, Matter symmetries in supersymmetric standard models from non-invertible selection rules, arXiv:2506.10241.
  21. T. Nomura and O. Popov, No-group scotogenic model, arXiv:2507.10299.
  22. J. Chen, C.-Q. Geng, H. Okada, and J.-J. Wu, A radiative lepton model in a non-invertible fusion rule, arXiv:2507.11951.
  23. H. Okada and Y. Shigekami, Three-loop induced neutrino mass model in a non-invertible symmetry, arXiv:2507.16198.
  24. Q. Liang and T. T. Yanagida, Non-invertible symmetry as an axion-less solution to the strong CP problem, Phys. Lett. B 868, 139706 (2025).
  25. C. Abel et al., Measurement of the permanent electric dipole moment of the neutron, Phys. Rev. Lett. 124, 081803 (2020).
  26. M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies and Phenomenology (Cambridge University Press, Cambridge, 1988).
  27. T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, T. H. Tatsuishi, and H. Uchida, CP violation in modular invariant flavor models, Phys. Rev. D 101, 055046 (2020).
  28. T. Kobayashi and H. Otsuka, Challenge for spontaneous CP violation in Type IIB orientifolds with fluxes, Phys. Rev. D 102, 026004 (2020).
  29. K. Ishiguro, T. Kobayashi, and H. Otsuka, Spontaneous CP violation and symplectic modular symmetry in Calabi-Yau compactifications, Nucl. Phys. B973, 115598 (2021).
  30. P. P. Novichkov, J. T. Penedo, and S. T. Petcov, Modular flavour symmetries and modulus stabilisation, J. High Energy Phys. 03 (2022) 149.
  31. V. Knapp-Perez, X.-G. Liu, H. P. Nilles, S. Ramos-Sanchez, and M. Ratz, Matter matters in moduli fixing and modular flavor symmetries, Phys. Lett. B 844, 138106 (2023).
  32. T. Higaki, T. Kobayashi, K. Nasu, and H. Otsuka, Spontaneous CP violation and partially broken modular flavor symmetries, J. High Energy Phys. 09 (2024) 024.
  33. S. Antusch, M. Holthausen, M. A. Schmidt, and M. Spinrath, Solving the strong CP problem with discrete symmetries and the right unitarity triangle, Nucl. Phys. B877, 752 (2013).
  34. F. Feruglio, A. Strumia, and A. Titov, Modular invariance and the QCD angle, J. High Energy Phys. 07 (2023) 027.
  35. S. T. Petcov and M. Tanimoto, A4 modular invariance and the strong CP problem, Eur. Phys. J. C 84, 914 (2024).
  36. J. T. Penedo and S. T. Petcov, Finite modular symmetries and the strong CP problem, J. High Energy Phys. 10 (2024) 172.
  37. Q. Liang, R. Okabe, and T. T. Yanagida, Three-zero texture of quark-mass matrices as a solution to the strong CP problem, Phys. Lett. B 859, 139123 (2024).
  38. H. Abe, T. Kobayashi, and H. Ohki, Magnetized orbifold models, J. High Energy Phys. 09 (2008) 043.
  39. J. R. Ellis and M. K. Gaillard, Strong and weak CP violation, Nucl. Phys. B150, 141 (1979).
  40. H. Abe, K.-S. Choi, T. Kobayashi, and H. Ohki, Non-Abelian discrete flavor symmetries from Magnetized/Intersecting brane models, Nucl. Phys. B820, 317 (2009).
  41. M. Berasaluce-Gonzalez, P. G. Camara, F. Marchesano, D. Regalado, and A. M. Uranga, Non-Abelian discrete gauge symmetries in 4D string models, J. High Energy Phys. 09 (2012) 059.
  42. F. Marchesano, D. Regalado, and L. Vazquez-Mercado, Discrete flavor symmetries in D-brane models, J. High Energy Phys. 09 (2013) 028.
  43. J. Dong, T. Kobayashi, R. Nishida, S. Nishimura, and H. Otsuka, Coupling selection rules in heterotic Calabi-Yau compactifications, J. High Energy Phys. 09 (2025) 012.
  44. J. Dong, T. Jeric, T. Kobayashi, R. Nishida, and H. Otsuka, On discrete gauging and non-invertible selection rules, arXiv:2507.02375.
  45. M. Tanimoto and T. T. Yanagida, Occam’s razor in quark mass matrices, Prog. Theor. Exp. Phys. 2016, 043B03 (2016).
  46. M. Ibe, S. Shirai, M. Suzuki, and T. T. Yanagida, Novel GUT with apparently complete SU(5) multiplets, Phys. Rev. D 100, 055024 (2019).
  47. T. Hotta, K. I. Izawa, and T. Yanagida, Dynamical models for light Higgs doublets in supersymmetric grand unified theories, Phys. Rev. D 53, 3913 (1996).
  48. K. Nakayama, F. Takahashi, and T. T. Yanagida, A theory of extra radiation in the universe, Phys. Lett. B 697, 275 (2011).
  49. M. Fukugita and T. Yanagida, Baryogenesis without grand unification, Phys. Lett. B 174, 45 (1986).
  50. T. Kobayashi, H. Otsuka, M. Tanimoto, and T. T. Yanagida, GUT-motivated non-invertible symmetry as a solution to the strong CP problem and the neutrino CP-violating phase, arXiv:2510.01680.

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Phys. Rev. D 113, 055016– Published 6 March, 2026
  • Received 17 November 2025
  • Accepted 2 February 2026
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