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  • Open Access

Inverse seesaw model in nonholomorphic modular A4 flavor symmetry

Xianshuo Zhang* and Yakefu Reyimuaji†

  • *Contact author: zxianshuo@yeah.net
  • †Contact author: yreyi@hotmail.com

Phys. Rev. D 112, 075050 – Published 28 October, 2025

DOI: https://doi.org/10.1103/17p3-bw5r

Abstract

This paper investigates an inverse seesaw model of neutrino masses based on nonholomorphic modular A4 symmetry, extending the framework of modular-invariant flavor models beyond the conventional holomorphic paradigm. After the general theoretical framework is established, three concrete model realizations distinguished by their A4 representation assignments and modular weight configurations for the matter fields are analyzed. Focusing on these three specific realizations, a comprehensive analysis of neutrino phenomenology is performed. By constraining the modulus parameter τ to the fundamental domain and systematically scanning the parameter space, regions compatible with current neutrino oscillation data are identified. The numerical results provide predictions for currently unmeasured quantities, including the absolute neutrino mass scale, Dirac CP-violating phase, and Majorana phases. These predictions establish specific, testable signatures for upcoming neutrino experiments, particularly in neutrinoless double beta decay and precision oscillation measurements. The framework offers a well-defined target for future experimental verification or exclusion, while demonstrating the phenomenological viability of nonholomorphic modular symmetry approaches to flavor structure.

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References (85)

  1. R. N. Mohapatra, Phys. Rev. Lett. 56, 561 (1986).
  2. R. N. Mohapatra and J. W. F. Valle, Phys. Rev. D 34, 1642 (1986).
  3. F. Deppisch and J. W. F. Valle, Phys. Rev. D 72, 036001 (2005).
  4. P. S. B. Dev and R. N. Mohapatra, Phys. Rev. D 81, 013001 (2010).
  5. S. Centelles Chuliá, R. Srivastava, and A. Vicente, J. High Energy Phys. 03 (2021) 248.
  6. F. Feruglio, Are neutrino masses modular forms?, in From My Vast Repertoire ...: Guido Altarelli’s Legacy, edited by A. Levy, S. Forte, and G. Ridolfi (World Scientific, Singapore, 2019), pp. 227–266, 10.1142/9789813238053_0012.
  7. F. Feruglio and A. Romanino, Rev. Mod. Phys. 93, 015007 (2021).
  8. G.-J. Ding and S. F. King, Rep. Prog. Phys. 87, 084201 (2024).
  9. T. Kobayashi and M. Tanimoto, Int. J. Mod. Phys. A 39, 2441012 (2024).
  10. T. Kobayashi, K. Tanaka, and T. H. Tatsuishi, Phys. Rev. D 98, 016004 (2018).
  11. T. Kobayashi, N. Omoto, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi, J. High Energy Phys. 11 (2018) 196.
  12. P. P. Novichkov, S. T. Petcov, and M. Tanimoto, Phys. Lett. B 793, 247 (2019).
  13. T. Nomura and H. Okada, Nucl. Phys. B966, 115372 (2021).
  14. G.-J. Ding, S. F. King, and X.-G. Liu, J. High Energy Phys. 09 (2019) 074.
  15. T. Nomura, H. Okada, and O. Popov, Phys. Lett. B 803, 135294 (2020).
  16. G.-J. Ding, S. F. King, X.-G. Liu, and J.-N. Lu, J. High Energy Phys. 12 (2019) 030.
  17. D. Zhang, Nucl. Phys. B952, 114935 (2020).
  18. T. Nomura, H. Okada, and S. Patra, Nucl. Phys. B967, 115395 (2021).
  19. T. Kobayashi, T. Nomura, and T. Shimomura, Phys. Rev. D 102, 035019 (2020).
  20. X. Wang, Nucl. Phys. B957, 115105 (2020).
  21. G.-J. Ding and F. Feruglio, J. High Energy Phys. 06 (2020) 134.
  22. T. Nomura and H. Okada, arXiv:2007.15459.
  23. M. Aoki and D. Kaneko, Prog. Theor. Exp. Phys. 2021, 023B06 (2021).
  24. T. Asaka, Y. Heo, and T. Yoshida, Phys. Lett. B 811, 135956 (2020).
  25. H. Okada and M. Tanimoto, J. High Energy Phys. 03 (2021) 010.
  26. H. Okada, Y. Shimizu, M. Tanimoto, and T. Yoshida, J. High Energy Phys. 07 (2021) 184.
  27. I. de Medeiros Varzielas and J. a. Lourenço, Nucl. Phys. B979, 115793 (2022).
  28. T. Nomura, H. Okada, and Y.-h. Qi, Eur. Phys. J. C 85, 134 (2025).
  29. T. Kobayashi, H. Otsuka, M. Tanimoto, and K. Yamamoto, Phys. Rev. D 105, 055022 (2022).
  30. T. Kobayashi, H. Otsuka, M. Tanimoto, and K. Yamamoto, J. High Energy Phys. 08 (2022) 013.
  31. D. W. Kang, J. Kim, T. Nomura, and H. Okada, J. High Energy Phys. 07 (2022) 050.
  32. J. Gogoi, N. Gautam, and M. K. Das, Int. J. Mod. Phys. A 38, 2350022 (2023).
  33. M. Abbas and S. Khalil, Int. J. Mod. Phys. A 39, 2450137 (2024).
  34. M. R. Devi, arXiv:2303.04900.
  35. T. Nomura, H. Okada, and H. Otsuka, Nucl. Phys. B1004, 116579 (2024).
  36. T. Kobayashi, T. Nomura, H. Okada, and H. Otsuka, J. High Energy Phys. 01 (2024) 121.
  37. R. Kumar, P. Mishra, M. K. Behera, R. Mohanta, and R. Srivastava, Phys. Lett. B 853, 138635 (2024).
  38. G.-J. Ding, S.-Y. Jiang, S. F. King, J.-N. Lu, and B.-Y. Qu, J. High Energy Phys. 08 (2024) 134.
  39. T. Nomura and H. Okada, arXiv:2407.13167.
  40. L. Singh, M. Kashav, and S. Verma, Nucl. Phys. B1007, 116666 (2024).
  41. R. Kalita and M. Patgiri, Eur. Phys. J. C 85, 453 (2025).
  42. T. Nomura and H. Okada, arXiv:2409.10912.
  43. T. Nomura and H. Okada, Prog. Theor. Exp. Phys. 2025, 093B08 (2025).
  44. S. T. Petcov and M. Tanimoto, Eur. Phys. J. C 84, 914 (2024).
  45. G. Pathak, P. Das, and M. K. Das, Eur. Phys. J. C 85, 569 (2025).
  46. A. Moreno-Sánchez and A. Palavrić, Phys. Rev. D 112, 075002 (2025).
  47. G. Pathak and M. K. Das, arXiv:2505.03000.
  48. B.-Y. Qu and G.-J. Ding, J. High Energy Phys. 08 (2024) 136.
  49. T. Nomura, H. Okada, and O. Popov, Phys. Lett. B 860, 139171 (2025).
  50. T. Nomura and H. Okada, Phys. Lett. B 868, 139763 (2025).
  51. T. Nomura and H. Okada, Phys. Lett. B 867, 139618 (2025).
  52. T. Kobayashi, H. Okada, and Y. Orikasa, arXiv:2502.12662.
  53. T. Nomura, H. Okada, and X.-Y. Wang, J. High Energy Phys. 09 (2025) 163.
  54. M. A. Loualidi, M. Miskaoui, and S. Nasri, Phys. Rev. D 112, 015008 (2025).
  55. T. Nomura and H. Okada, arXiv:2506.02639.
  56. H. Okada and Y. Orikasa, arXiv:2501.15748.
  57. G.-J. Ding, J.-N. Lu, S. T. Petcov, and B.-Y. Qu, J. High Energy Phys. 01 (2025) 191.
  58. C.-C. Li, J.-N. Lu, and G.-J. Ding, J. High Energy Phys. 12 (2024) 189.
  59. G.-J. Ding, F. Feruglio, and X.-G. Liu, J. High Energy Phys. 01 (2021) 037.
  60. B. Kumar and M. K. Das, J. High Energy Phys. 09 (2025) 071.
  61. Z. Wang, Y. Reyimuaji, and N. Yalikun, Phys. Rev. D 112, 055041 (2025).
  62. P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov, J. High Energy Phys. 07 (2019) 165.
  63. I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, J. a. P. Pinheiro, and T. Schwetz, J. High Energy Phys. 12 (2024) 216.
  64. Z.-z. Xing, H. Zhang, and S. Zhou, Phys. Rev. D 77, 113016 (2008).
  65. A. Baur, flavorpy, Zenodo, 10.5281/zenodo.11060597 (2024).
  66. N. Aghanim et al. (Planck Collaboration), Astron. Astrophys. 641, A6 (2020); 652, C4(E) (2021).
  67. M. Aker et al. (Katrin Collaboration), Science 388, 180 (2025).
  68. S. Abe et al. (KamLAND-Zen Collaboration), arXiv:2406.11438.
  69. N. Abgrall et al. (LEGEND Collaboration), arXiv:2107.11462.
  70. G. Adhikari et al. (nEXO Collaboration), J. Phys. G 49, 015104 (2022).
  71. P. Stöcker et al. (GAMBIT Cosmology Workgroup), Phys. Rev. D 103, 123508 (2021).
  72. B. Abi et al. (DUNE Collaboration), arXiv:2002.03005.
  73. K. Abe et al. (Hyper-Kamiokande Collaboration), arXiv:1805.04163.
  74. M. Aker et al. (KATRIN Collaboration), J. Phys. G 49, 100501 (2022).
  75. A. A. Esfahani et al. (Project 8 Collaboration), in Snowmass 2021 (2022) arXiv:2203.07349.
  76. R. de Adelhart Toorop, F. Feruglio, and C. Hagedorn, Nucl. Phys. B858, 437 (2012).
  77. J. H. Bruinier, G. V. D. Geer, G. Harder, and D. Zagier, The 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid, Norway Universitext (Springer-Verlag, Berlin, Heidelberg, 2008).
  78. B.-Y. Qu, J.-N. Lu, and G.-J. Ding, arXiv:2506.19822.
  79. K. Ishiguro, T. Kobayashi, and H. Otsuka, J. High Energy Phys. 03 (2021) 161.
  80. E. Gonzalo, L. E. Ibáñez, and Á. M. Uranga, J. High Energy Phys. 05 (2019) 105.
  81. P. P. Novichkov, J. T. Penedo, and S. T. Petcov, J. High Energy Phys. 03 (2022) 149.
  82. S. Kachru, R. Kallosh, A. Linde, and S. P. Trivedi, Phys. Rev. D 68, 046005 (2003).
  83. V. Knapp-Perez, X.-G. Liu, H. P. Nilles, S. Ramos-Sanchez, and M. Ratz, Phys. Lett. B 844, 138106 (2023).
  84. J. M. Leedom, N. Righi, and A. Westphal, J. High Energy Phys. 02 (2023) 209.
  85. S. F. King and X. Wang, Phys. Rev. D 110, 076026 (2024).

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