- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Exploring the flavor structure of leptons via diffusion models

Satsuki Nishimura1,*, Hajime Otsuka1,2,†, and Haruki Uchiyama1,‡

  • 1Department of Physics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan
  • 2Quantum and Spacetime Research Institute (QuaSR), Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

  • *Contact author: nishimura.satsuki@phys.kyushu-u.ac.jp
  • †Contact author: otsuka.hajime@phys.kyushu-u.ac.jp
  • ‡Contact author: uc.haruki496ym@gmail.com
PDF

Phys. Rev. D 113, 055030 – Published 18 March, 2026

DOI: https://doi.org/10.1103/rtnd-vwt9

Abstract

We propose a method to explore the flavor structure of leptons using diffusion models, which are known as one of generative artificial intelligence (generative AI). We consider a simple extension of the Standard Model with the type I seesaw mechanism and train a neural network to generate the neutrino mass matrix. By utilizing transfer learning, the diffusion model generates 104 solutions that are consistent with the neutrino mass squared differences and the leptonic mixing angles. The distributions of the CP phases and the sums of neutrino masses, which are not included in the conditional labels but are calculated from the solutions, exhibit nontrivial tendencies. In addition, the effective mass in neutrinoless double beta decay is concentrated near the boundaries of the existing confidence intervals, allowing us to verify the obtained solutions through future experiments. An inverse approach using the diffusion model is expected to facilitate the experimental verification of flavor models from a perspective distinct from conventional analytical methods.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (46)

  1. C. D. Froggatt and H. B. Nielsen, Hierarchy of quark masses, Cabibbo angles and CP violation, Nucl. Phys. B147, 277 (1979).
  2. G. Altarelli and F. Feruglio, Discrete flavor symmetries and models of neutrino mixing, Rev. Mod. Phys. 82, 2701 (2010).
  3. H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada, and M. Tanimoto, Non-Abelian discrete symmetries in particle physics, Prog. Theor. Phys. Suppl. 183, 1 (2010).
  4. D. Hernandez and A. Y. Smirnov, Lepton mixing and discrete symmetries, Phys. Rev. D 86, 053014 (2012).
  5. S. F. King and C. Luhn, Neutrino mass and mixing with discrete symmetry, Rep. Prog. Phys. 76, 056201 (2013).
  6. S. F. King, A. Merle, S. Morisi, Y. Shimizu, and M. Tanimoto, Neutrino mass and mixing: From theory to experiment, New J. Phys. 16, 045018 (2014).
  7. S. T. Petcov, Discrete flavour symmetries, neutrino mixing and leptonic CP violation, Eur. Phys. J. C 78, 709 (2018).
  8. T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu, and M. Tanimoto, An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists (Springer, Berlin, Heidelberg, 2022), 10.1007/978-3-662-64679-3.
  9. 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 (2019), pp. 227–266, 10.1142/9789813238053_0012.
  10. T. Kobayashi and M. Tanimoto, Modular flavor symmetric models, Int. J. Mod. Phys. A 39, 2441012 (2024).
  11. G.-J. Ding and S. F. King, Neutrino mass and mixing with modular symmetry, Rep. Prog. Phys. 87, 084201 (2024).
  12. T. Kobayashi and H. Otsuka, Non-invertible flavor symmetries in magnetized extra dimensions, J. High Energy Phys. 11 (2024) 120.
  13. T. Kobayashi, H. Otsuka, and M. Tanimoto, Yukawa textures from non-invertible symmetries, J. High Energy Phys. 12 (2024) 117.
  14. J. Hisano, T. Moroi, K. Tobe, and M. Yamaguchi, Lepton flavor violation via right-handed neutrino Yukawa couplings in supersymmetric standard model, Phys. Rev. D 53, 2442 (1996).
  15. J. Hisano and D. Nomura, Solar and atmospheric neutrino oscillations and lepton flavor violation in supersymmetric models with the right-handed neutrinos, Phys. Rev. D 59, 116005 (1999).
  16. J. A. Casas and A. Ibarra, Oscillating neutrinos and μ→e,γ, Nucl. Phys. B618, 171 (2001).
  17. D. Falcone, Inverting the seesaw formula, Phys. Rev. D 68, 033002 (2003).
  18. S. Nishimura, C. Miyao, and H. Otsuka, Exploring the flavor structure of quarks and leptons with reinforcement learning, J. High Energy Phys. 12 (2023) 021.
  19. S. Nishimura, C. Miyao, and H. Otsuka, Reinforcement learning-based statistical search strategy for an axion model from flavor, J. High Energy Phys. 10 (2025) 043.
  20. P. Devlin, J.-W. Qiu, F. Ringer, and N. Sato, Diffusion model approach to simulating electron-proton scattering events, Phys. Rev. D 110, 016030 (2024).
  21. J. Sohl-Dickstein, E. A. Weiss, N. Maheswaranathan, and S. Ganguli, Deep unsupervised learning using nonequilibrium thermodynamics, arXiv:1503.03585.
  22. P. Minkowski, μ→eγ at a rate of one out of 109 muon decays?, Phys. Lett. 67B, 421 (1977).
  23. T. Yanagida, Horizontal gauge symmetry and masses of neutrinos, Conf. Proc. C 7902131, 95 (1979), https://inspirehep.net/files/19a4de25a11cc3ea4ec4e0bd0f45bdbd.
  24. M. Gell-Mann, P. Ramond, and R. Slansky, Complex spinors and unified theories, Conf. Proc. C 790927, 315 (1979), http://tuvalu.santafe.edu/~mgm/Site/Publications_files/MGM%2086.pdf.
  25. R. N. Mohapatra and G. Senjanovic, Neutrino mass and spontaneous parity nonconservation, Phys. Rev. Lett. 44, 912 (1980).
  26. L. J. Hall, H. Murayama, and N. Weiner, Neutrino mass anarchy, Phys. Rev. Lett. 84, 2572 (2000).
  27. Planck Collaboration, Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020).
  28. DESI Collaboration, DESI 2024 VI: Cosmological constraints from the measurements of baryon acoustic oscillations, J. Cosmol. Astropart. Phys. 02 (2025) 021.
  29. I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, J. a. P. Pinheiro, and T. Schwetz, NuFit-6.0: Updated global analysis of three-flavor neutrino oscillations, J. High Energy Phys. 12 (2024) 216.
  30. KamLAND-Zen Collaboration, Search for the Majorana nature of neutrinos in the inverted mass ordering region with KamLAND-Zen, Phys. Rev. Lett. 130, 051801 (2023).
  31. Katrin Collaboration, Direct neutrino-mass measurement based on 259 days of KATRIN data, Science 388, 180 (2025).
  32. Y. Lipman, R. T. Q. Chen, H. Ben-Hamu, M. Nickel, and M. Le, Flow matching for generative modeling, arXiv:2210.02747.
  33. C. Lu, Y. Zhou, F. Bao, J. Chen, C. Li, and J. Zhu, Dpm-solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps, arXiv:2206.00927.
  34. H. Shen, J. Zhang, B. Xiong, R. Hu, S. Chen, Z. Wan et al., Efficient diffusion models: A survey, arXiv:2502.06805.
  35. A. Nichol and P. Dhariwal, Improved denoising diffusion probabilistic models, arXiv:2102.09672.
  36. J. Song, C. Meng, and S. Ermon, Denoising diffusion implicit models, arXiv:2010.02502.
  37. L. T. Trinh and T. Hamagami, Latent denoising diffusion gan: Faster sampling, higher image quality, IEEE Access 12, 78161 (2024).
  38. Z. Tang, J. Bao, D. Chen, and B. Guo, Diffusion models without classifier-free guidance, arXiv:2502.12154.
  39. W. Feller, On the theory of stochastic processes, with particular reference to applications, in Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability (1949), pp. 403–432.
  40. J. Ho, A. Jain, and P. Abbeel, Denoising diffusion probabilistic models, arXiv:2006.11239.
  41. P. Dhariwal and A. Nichol, Diffusion models beat gans on image synthesis, arXiv:2105.05233.
  42. J. Ho and T. Salimans, Classifier-free diffusion guidance, arXiv:2207.12598.
  43. Y. Song and S. Ermon, Generative modeling by estimating gradients of the data distribution, arXiv:1907.05600.
  44. Y. Song and S. Ermon, Improved techniques for training score-based generative models, arXiv:2006.09011.
  45. Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole, Score-based generative modeling through stochastic differential equations, arXiv:2011.13456.
  46. H. Asmaul, M. Ethel, G. Jigmey, A. Zulfikar, A. Zeyar, and A. A. Mohammad, Transfer learning: A friendly introduction, J. Big Data 9, 102 (2022).

Outline

Information

Phys. Rev. D 113, 055030– Published 18 March, 2026
  • Received 11 October 2025
  • Accepted 23 February 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation