\begin{document}$S_{3L} \times S_{3R} \rightarrow S_{2L} \times S_{2R} \rightarrow \emptyset$\end{document} within both charged-lepton and neutrino sectors at the mass-matrix level. For the three distinct residual subgroups \begin{document}$S_{2L}^{(23)} \times S_{2R}^{(23)}$\end{document}, \begin{document}$S_{2L}^{(13)} \times S_{2R}^{(13)}$\end{document} and \begin{document}$S_{2L}^{(12)} \times S_{2R}^{(12)}$\end{document} under consideration, we systematically analyze the various parameterizations of the lepton mass matrices. It is shown that all the three scenarios are in good agreement with current neutrino oscillation data. Notably, within the latest best-fit values of neutrino oscillation parameters, the predicted Dirac CP-violating phase δ is calculated to be \begin{document}$294.6^\circ$\end{document}, \begin{document}$302.3^\circ$\end{document} and \begin{document}$287.0^\circ$\end{document}, respectively. To further assess the viability of the model, a comprehensive numerical analysis is performed by utilizing neutrino oscillation parameters at the \begin{document}$3\sigma$\end{document} level. It is found that the allowed range of δ is \begin{document}$281.2^\circ \rightarrow 338.7^\circ$\end{document}, \begin{document}$287.0^\circ \rightarrow 342.2^\circ$\end{document} and \begin{document}$282.7^\circ \rightarrow 297.0^\circ$\end{document}, all fall within its \begin{document}$3\sigma$\end{document} range. These results indicate that the proposed symmetry-breaking scheme \begin{document}$S_{3L} \times S_{3R} \rightarrow S_{2L} \times S_{2R} \rightarrow \emptyset$\end{document} can naturally explain the realistic lepton mass hierarchy and mixing pattern, thereby providing valuable theoretical perspectives for future research."> Further study on the lepton mass spectra and flavor mixing with <i>S</i><sub>3<i>L</i></sub> × <i>S</i><sub>3<i>R</i></sub> flavor symmetry -
  • [1]

    S. L. Glashow, Nucl. Phys.22, 579 (1961)

  • [2]

    S. Weinberg, Phys. Rev. Lett.19, 1264 (1967)

  • [3]

    A. Salam, Conf. Proc. C680519, 367 (1968)

  • [4]

    P. Di Bari, Prog. Part. Nucl. Phys.122, 103913 (2022)

  • [5]

    A. Arbey and F. Mahmoudi, Prog. Part. Nucl. Phys.119, 103865 (2021)

  • [6]

    A. Crivellin and B. Mellado, Nature Rev. Phys.6(5), 294 (2024)

  • [7]

    M. Losada, Y. Nir, G. Perez and Y. Shpilman, JHEP04, 030 (2022)

  • [8]

    M. Sajjad Athar, S. W. Barwick, T. Brunner, J. Cao, M. Danilov, K. Inoue, T. Kajita, M. Kowalski, M. Lindner and K. R. Long,et al., Prog. Part. Nucl. Phys.124, 103947 (2022)

  • [9]

    P. B. Denton, arXiv: 2501.08374[hep-ph].

  • [10]

    M. Abreuet al. [SNO+], Phys. Rev. Lett.135(12), 121801 (2025)

  • [11]

    K. Abeet al. [Super-Kamiokande], Phys. Rev. D109(9), 092001 (2024)

  • [12]

    K. Abeet al. [T2K], Phys. Rev. D108(7), 072011 (2023)

  • [13]

    S. Abubakaret al.[NOvA], arXiv: 2509.04361[hep-ex].

  • [14]

    I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, J. P. Pinheiro and T. Schwetz, JHEP12, 216 (2024)

  • [15]

    F. Capozzi, W. Giarè, E. Lisi, A. Marrone, A. Melchiorri and A. Palazzo, Phys. Rev. D111(9), 093006 (2025)

  • [16]

    P. F. de Salas, D. V. Forero, S. Gariazzo, P. Martínez-Miravé, O. Mena, C. A. Ternes, M. Tórtola and J. W. F. Valle, JHEP02, 071 (2021)

  • [17]

    M. Akeret al. [KATRIN], Science388(6743), adq9592 (2025)

  • [18]

    P. Minkowski, Phys. Lett. B67, 421 (1977)

  • [19]

    T. Yanagida, Conf. Proc. C7902131, 95 (1979)

  • [20]

    M. Gell-Mann, P. Ramond and R. Slansky, Conf. Proc. C790927, 315 (1979)

  • [21]

    S. L. Glashow, NATO Sci. Ser. B61, 687 (1980)

  • [22]

    R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett.44, 912 (1980)

  • [23]

    D. Wyler and L. Wolfenstein, Nucl. Phys. B218, 205 (1983)

  • [24]

    R. N. Mohapatra and J. W. F. Valle, Phys. Rev. D34, 1642 (1986)

  • [25]

    G. C. Branco and G. Senjanovic, Phys. Rev. D18, 1621 (1978)

  • [26]

    D. Chang and R. N. Mohapatra, Phys. Rev. Lett.58, 1600 (1987)

  • [27]

    K. S. Babu and X. G. He, Mod. Phys. Lett. A4, 61 (1989)

  • [28]

    P. Q. Hung, Phys. Rev. D59, 113008 (1999)

  • [29]

    Z. z. Xing, Phys. Rept.854, 1 (2020)

  • [30]

    S. F. King and C. Luhn, Rept. Prog. Phys.76, 056201 (2013)

  • [31]

    H. Harari, H. Haut and J. Weyers, Phys. Lett. B78, 459 (1978)

  • [32]

    C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B147, 277 (1979)

  • [33]

    Y. Koide, Z. Phys. C45, 39 (1989)

  • [34]

    M. Tanimoto, Phys. Rev. D41, 1586 (1990)

  • [35]

    P. Kaus and S. Meshkov, Phys. Rev. D42, 1863 (1990)

  • [36]

    G. C. Branco, J. I. Silva-Marcos and M. N. Rebelo, Phys. Lett. B237, 446 (1990)

  • [37]

    H. Fritzsch and J. Plankl, Phys. Lett. B237, 451 (1990)

  • [38]

    H. Fritzsch and D. Holtmannspotter, Phys. Lett. B338, 290 (1994)

  • [39]

    G. C. Branco and J. I. Silva-Marcos, Phys. Lett. B359, 166 (1995)

  • [40]

    H. Fritzsch and Z. Z. Xing, Phys. Lett. B372, 265 (1996)

  • [41]

    Z z. Xing, J. Phys. G23, 1563 (1997)

  • [42]

    A. Mondragon and E. Rodriguez-Jauregui, Phys. Rev. D59, 093009 (1999)

  • [43]

    H. Fritzsch and Z. z. Xing, Phys. Lett. B440, 313 (1998)

  • [44]

    N. Haba, Y. Matsui, N. Okamura and T. Suzuki, Phys. Lett. B489, 184 (2000)

  • [45]

    G. C. Branco and J. I. Silva-Marcos, Phys. Lett. B526, 104 (2002)

  • [46]

    M. Fujii, K. Hamaguchi and T. Yanagida, Phys. Rev. D65, 115012 (2002)

  • [47]

    H. Fritzsch and Z. z. Xing, Phys. Lett. B598, 237 (2004)

  • [48]

    W. Rodejohann and Z. z. Xing, Phys. Lett. B601, 176 (2004)

  • [49]

    T. Teshima, Phys. Rev. D73, 045019 (2006)

  • [50]

    G. Altarelli and F. Feruglio, Rev. Mod. Phys.82, 2701 (2010)

  • [51]

    H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada and M. Tanimoto, Prog. Theor. Phys. Suppl.183, 1 (2010)

  • [52]

    Z. z. Xing, D. Yang and S. Zhou, Phys. Lett. B690, 304 (2010)

  • [53]

    S. Zhou, Phys. Lett. B704, 291 (2011)

  • [54]

    S. Dev, R. R. Gautam and L. Singh, Phys. Lett. B708, 284 (2012)

  • [55]

    F. Gonzalez Canales, A. Mondragon and M. Mondragon, Fortsch. Phys.61, 546 (2013)

  • [56]

    R. Jora, J. Schechter and M. N. Shahid, Int. J. Mod. Phys. A28, 1350028 (2013)

  • [57]

    H. Ishimori, T. Kobayashi, Y. Shimizu, H. Ohki, H. Okada and M. Tanimoto, Fortsch. Phys.61, 441 (2013)

  • [58]

    M. J. S. Yang, Phys. Lett. B760, 747 (2016)

  • [59]

    H. Fritzsch, Z. z. Xing and D. Zhang, Chin. Phys. C41(9), 093104 (2017)

  • [60]

    Z. g. Si, X. h. Yang and S. Zhou, Chin. Phys. C41(11), 113105 (2017)

  • [61]

    S. Pramanick, Phys. Rev. D100(3), 035009 (2019)

  • [62]

    S. Mishra and A. Giri, J. Phys. G47(5), 055008 (2020)

  • [63]

    S. Mishra, Eur. Phys. J. Plus135(6), 485 (2020)

  • [64]

    J. D. García-Aguilar and J. C. Gómez-Izquierdo, Rev. Mex. Fis.68(4), 040801 (2022)

  • [65]

    K. S. Babu, Y. Wu and S. Xu, JHEP12, 166 (2024)

  • [66]

    C. C. Li, J. N. Lu and G. J. Ding, JHEP12, 015 (2024)

  • [67]

    J. C. Gómez-Izquierdo, C. Espinoza, L. E. G. Luna and M. Mondragón, Nucl. Phys. B1018, 117027 (2025)

  • [68]

    N. Gresnigt and L. Gourlay, J. Phys. Conf. Ser.2912(1), 012019 (2024)

  • [69]

    S. Navaset al. [Particle Data Group], Phys. Rev. D110(3), 030001 (2024)

  • [70]

    N. Aghanimet al.[Planck], Astron. Astrophys.641, A6 (2020) [erratum: Astron. Astrophys.652, C4 (2021)].

  • [71]

    G. y. Huang and S. Zhou, Phys. Rev. D103(1), 016010 (2021)

  • [72]

    C. Jarlskog, Phys. Rev. Lett.55, 1039 (1985)

  • [73]

    D. d. Wu, Phys. Rev. D33, 860 (1986)

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