Soft breaking of the µ ↔ τ symmetry by S4 ⊗ Z2


  • J. D. García-Aguilar CECyT No. 16, Instituto Politécnico Nacional
  • A. E. Pozas Ramírez ESFM-IPN
  • M. M. Suárez Castañeda ESFM-IPN
  • J. C. Gómez-Izquierdo CECyT No. 16, Instituto Politécnico Nacional



µ ↔ τ symmetry; CKM matrix


The µ ↔ τ symmetry has been ruled out by its predictions on the reactor and atmospheric angles, nevertheless, a breaking of this symmetry might provide correct values. For that reason, we build a non-renormalizable lepton model where the mixings arise from the spontaneous breaking of the S4 ⊗ Z2 discrete group, subsequently the µ ↔ τ symmetry is broken in the effective neutrino mass matrix, that comes from the type II see-saw mechanism. As main result, the reactor and atmospheric angles are corrected and their values are in good agreement with the experimental data for the inverted hierarchy. Furthermore, we point out a link between the atmospheric angle and reactor one. In the quark sector, under certain assumptions, the generalized Fritzsch textures shape to the quark mass matrices so that the CKM matrix values are guaranteed.


B. Abi et al. [Muon g-2], Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm, Phys. Rev. Lett. 126 (2021) 141801, [arXiv:2104.03281 [hep-ex]].

T. Aaltonen et al. [CDF], High-precision measurement of the W boson mass with the CDF II detector, Science 376 (2022) 170,

J. C. Romao, Supersymmetric Models for Neutrino Mass, [arXiv:0710.5730 [hep-ph]].

P. F. de Salas, D. V. Forero, S. Gariazzo, P. Martínez-Mirave, O. Mena, C. A. Ternes, M. Tortola and J. W. F. Valle, 2020 global reassessment of the neutrino oscillation picture, JHEP 02 (2021) 071, [arXiv:2006.11237 [hep-ph]].

I. Esteban, M. C. González-García, M. Maltoni, T. Schwetz and A. Zhou, The fate of hints: updated global analysis of three-flavor neutrino oscillations, JHEP 09 (2020) 178, [arXiv:2007.14792 [hep-ph]].

S. Gariazzo, M. Gerbino, T. Brinckmann, M. Lattanzi, O. Mena, T. Schwetz, S. Choudhury Roy, K. Freese, S. Hannestad and C. A. Ternes, et al. Neutrino mass and mass ordering: no conclusive evidence for normal ordering, JCAP 10 (2022) 010, [arXiv:2205.02195 [hep-ph]].

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 (2010) 1, [arXiv:1003.3552 [hep-th]].

W. Grimus and P. O. Ludl, Finite flavour groups of fermions, J. Phys. A 45 (2012) 233001, [arXiv:1110.6376 [hep-ph]].

H. Ishimori, T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu and M. Tanimoto, An introduction to non-Abelian discrete symmetries for particle physicists, Lect. Notes Phys. 858 (2012) 1,

S. F. King and C. Luhn, Neutrino Mass and Mixing with Discrete Symmetry, Rept. Prog. Phys. 76 (2013) 056201, [arXiv:1301.1340 [hep-ph]].

S. F. King, Models of Neutrino Mass, Mixing and CP Violation, J. Phys. G 42 (2015) 123001, [arXiv:1510.02091 [hep-ph

F. Feruglio and A. Romanino, Lepton flavor symmetries, Rev. Mod. Phys. 93 (2021) 015007, [arXiv:1912.06028 [hep-ph]].

Z. z. Xing, Flavor structures of charged fermions and massive neutrinos, Phys. Rept. 854 (2020) 1-147, [arXiv:1909.09610 [hep-ph]].

G. Chauhan et al., Discrete Flavor Symmetries and Lepton Masses and Mixings, [arXiv:2203.08105 [hep-ph]].

Z. z. Xing and Z. h. Zhao, A review of µ-τ flavor symmetry in neutrino physics, Rept. Prog. Phys. 79 (2016) 076201, [arXiv:1512.04207 [hep-ph]].

W. Grimus and L. Lavoura, A Discrete symmetry group for maximal atmospheric neutrino mixing, Phys. Lett. B 572 (2003) 189, [arXiv:hep-ph/0305046 [hepph]].

W. Grimus, A. S. Joshipura, S. Kaneko, L. Lavoura and M. Tanimoto, Lepton mixing angle θ13 = 0 with a horizontal symmetry D4, JHEP 07 (2004) 078, [arXiv:hepph/0407112 [hep-ph]].

A. Adulpravitchai, A. Blum and C. Hagedorn, A Supersymmetric D4 Model for mu-tau Symmetry, JHEP 03 (2009) 046, [arXiv:0812.3799 [hep-ph]].

H. Ishimori, T. Kobayashi, H. Ohki, Y. Omura, R. Takahashi and M. Tanimoto, D(4) Flavor Symmetry for Neutrino Masses and Mixing, Phys. Lett. B 662 (2008) 178, [arXiv:0802.2310 [hep-ph]].

C. Hagedorn and R. Ziegler, µ − τ Symmetry and Charged Lepton Mass Hierarchy in a Supersymmetric D4 Model, Phys. Rev. D 82 (2010) 053011, [arXiv:1007.1888 [hep-ph]].

J. C. Gómez-Izquierdo, ´ Non-minimal flavored S3 ⊗ Z2 left–right symmetric model, Eur. Phys. J. C 77 (2017) 551, [arXiv:1701.01747 [hep-ph]]. 22. E. A. Garces, J. C. Gómez-Izquierdo and F. González-Canales, Flavored non-minimal left–right symmetric model fermion masses and mixings, Eur. Phys. J. C 78 (2018) 812, [arXiv:1807.02727 [hep-ph]].

J. C. Gómez-Izquierdo and M. Mondragón, B–L Model with S3 symmetry: Nearest Neighbor Interaction Textures and Broken µ ↔ τ Symmetry, Eur. Phys. J. C 79 (2019) 285, [arXiv:1804.08746 [hep-ph]].

S. Morisi and E. Peinado, An S4 model for quarks and leptons with maximal atmospheric angle, Phys. Rev. D 81 (2010) 085015, [arXiv:1001.2265 [hep-ph]].

S. Morisi, K. M. Patel and E. Peinado, Model for T2K indication with maximal atmospheric angle and tri-maximal solar angle, Phys. Rev. D 84 (2011) 053002, [arXiv:1107.0696 [hep-ph]].

S. Gupta, A. S. Joshipura and K. M. Patel, Minimal extension of tri-bimaximal mixing and generalized Z2 → Z2 symmetries, Phys. Rev. D 85 (2012) 031903, [arXiv:1112.6113 [hep-ph]].

S. Gupta, A. S. Joshipura and K. M. Patel, How good is µ-τ symmetry after results on non-zero θ13?, JHEP 09 (2013) 035, [arXiv:1301.7130 [hep-ph]].

Z. h. Zhao, On the breaking of mu-tau flavor symmetry, 0017. [arXiv:1605.04498 [hep-ph]].

E. Becerra-García and A. Pérez-Lorenzana, Are neutrino oscillation mixings linked to the smallness of solar neutrino scale?, [arXiv:2202.00864 [hep-ph]].

Z. z. Xing and Z. h. Zhao, The minimal seesaw and leptogenesis models, Rept. Prog. Phys. 84 (2021) 066201, [arXiv:2008.12090 [hep-ph]].

L. Calibbi, M. L. López-Ibáñez, A. Melis and O. Vives, Implications of the Muon g-2 result on the flavour structure of the lepton mass matrix, Eur. Phys. J. C 81 (2021) 929, [arXiv:2104.03296 [hep-ph]].

R. L. Workman et al. [Particle Data Group], Review of Particle Physics, PTEP 2022 (2022) 083C01,

G. C. Branco, L. Lavoura and F. Mota, Nearest Neighbor Interactions and the Physical Content of Fritzsch Mass Matrices, Phys. Rev. D 39 (1989) 3443,

G. C. Branco and J. I. Silva-Marcos, NonHermitian Yukawa couplings?, Phys. Lett. B 331 (1994) 390,

K. Harayama and N. Okamura, Exact parametrization of the mass matrices and the KM matrix, Phys. Lett. B 387 (1996) 614, [arXiv:hep-ph/9605215 [hep-ph]].

K. Harayama, N. Okamura, A. I. Sanda and Z. Z. Xing, Getting at the quark mass matrices, Prog. Theor. Phys. 97 (1997) 781, [arXiv:hep-ph/9607461 [hep-ph]].

H. Fritzsch and Z. z. Xing, Mass and flavor mixing schemes of quarks and leptons, Prog. Part. Nucl. Phys. 45 (2000) 1, [arXiv:hep-ph/9912358 [hep-ph]].

J. Barranco, F. González Canales and A. Mondragón, Universal Mass Texture, CP violation and Quark-Lepton Complementarity, Phys. Rev. D 82 (2010) 073010, [arXiv:1004.3781 [hep-ph]].

H. Fritzsch, Neutrino Masses and Flavor Mixing, Mod. Phys. Lett. A 30 (2015) 1530012, [arXiv:1503.01857 [hep-ph]].

K. M. Patel, An SO(10)XS4 Model of Quark-Lepton Complementarity, Phys. Lett. B 695 (2011) 225, [arXiv:1008.5061 [hep-ph]].

P. V. Dong, H. N. Long, D. V. Soa and V. V. Vien, The 3-3-1 model with S4 flavor symmetry, Eur. Phys. J. C 71 (2011) 1544, [arXiv:1009.2328 [hep-ph]].

H. Ishimori, Y. Shimizu, M. Tanimoto and A. Watanabe, Neutrino masses and mixing from S4 flavor twisting, Phys. Rev. D 83 (2011) 033004, [arXiv:1010.3805 [hep-ph]].

R. N. Mohapatra and C. C. Nishi, S4 Flavored CP Symmetry for Neutrinos, Phys. Rev. D 86 (2012) 073007, [arXiv:1208.2875 [hep-ph]].

P. S. Bhupal Dev, B. Dutta, R. N. Mohapatra and M. Severson, θ13 and Proton Decay in a Minimal SO(10) × S4 model of Flavor, Phys. Rev. D 86 (2012) 035002, [arXiv:1202.4012 [hep-ph]].

V. V. Vien, H. N. Long and D. P. Khoi, Neutrino Mixing with Non-Zero θ13 and CP Violation in the 3-3-1 Model Based on S4 Flavor Symmetry, Int. J. Mod. Phys. A 30 (2015) 1550102, [arXiv:1506.06063 [hep-ph]].

V. V. Vien, H. N. Long and A. E. Cárcamo Hernández, Fermion Mass and Mixing in a Low-Scale Seesaw Model based on the S4 Flavor Symmetry, PTEP 2019 (2019) 113B04, [arXiv:1909.09532 [hep-ph]].

V. V. Vien and H. N. Long, Multiscalar B − L extension based on S4 flavor symmetry for neutrino masses and mixing, Chin. Phys. C 45 (2021) 043112, [arXiv:2012.01715 [hep-ph]].

V. V. Vien, H. N. Long and A. E. Cárcamo Hernández, Lepton masses and mixings, and muon anomalous magnetic moment in an extended B − L model with the type-I seesaw mechanism, PTEP 2022 (2022) 093B11, [arXiv:2206.06564 [hep-ph]].

S. Pakvasa and H. Sugawara, Discrete Symmetry and Cabibbo Angle, Phys. Lett. B 73 (1978) 61,

O. F. Beltran, M. Mondragon and E. Rodriguez-Jauregui, Conditions for vacuum stability in an S(3) extension of the standard model, J. Phys. Conf. Ser. 171 (2009) 012028,

D. Das and U. K. Dey, Analysis of an extended scalar sector with S3 symmetry, Phys. Rev. D 89 (2014) 095025, [erratum: Phys. Rev. D 91 (2015) 039905] [arXiv:1404.2491 [hep-ph]].

F. González Canales, A. Mondragón, M. Mondragón, U. J. Saldaña Salazar and L. Velasco-Sevilla, Quark sector of S3 models: classification and comparison with experimental data, Phys. Rev. D 88 (2013) 096004, [arXiv:1304.6644 [hep-ph]].

P. F. Harrison and W. G. Scott, Permutation symmetry, tri - bimaximal neutrino mixing and the S3 group characters, Phys. Lett. B 557 (2003) 76, [arXiv:hep-ph/0302025 [hepph]].

P. F. Harrison, D. H. Perkins and W. G. Scott, Tribimaximal mixing and the neutrino oscillation data, Phys. Lett. B 530 (2002) 167 [arXiv:hep-ph/0202074 [hepph]].

Z. z. Xing, Nearly tri bimaximal neutrino mixing and CP violation, Phys. Lett. B 533 (2002) 85, [arXiv:hepph/0204049 [hep-ph]].

G. Altarelli, F. Feruglio and L. Merlo, Tri-Bimaximal Neutrino Mixing and Discrete Flavour Symmetries, Fortsch. Phys. 61 (2013) 507, [arXiv:1205.5133 [hep-ph]].

J. C. Pati and A. Salam, Lepton Number as the Fourth Color, Phys. Rev. D 10 (1974) 275, [erratum: Phys. Rev. D 11 (1975) 703,]

R. N. Mohapatra and J. C. Pati, A Natural Left-Right Symmetry, Phys. Rev. D 11 (1975) 2558,

G. Senjanovic and R. N. Mohapatra, Exact Left-Right Symmetry and Spontaneous Violation of Parity, Phys. Rev. D 12 (1975) 1502,

G. Senjanovic, Spontaneous Breakdown of Parity in a Class of Gauge Theories, Nucl. Phys. B 153 (1979) 334,

M. Agostini et al. [GERDA], Results on Neutrinoless Double-β Decay of 76Ge from Phase I of the GERDA Experiment, Phys. Rev. Lett. 111 (2013) 122503, [arXiv:1307.4720 [nucl-ex]].

M. Agostini et al. [GERDA], Improved Limit on Neutrinoless Double-β Decay of 76Ge from GERDA Phase II, Phys. Rev. Lett. 120 (2018) 132503, [arXiv:1803.11100 [nuclex]].

Z. z. Xing, H. Zhang and S. Zhou, Updated Values of Running Quark and Lepton Masses, Phys. Rev. D 77 (2008) 113016, [arXiv:0712.1419 [hep-ph]].

R. N. Mohapatra and S. Nussinov, Bimaximal neutrino mixing and neutrino mass matrix, Phys. Rev. D 60 (1999) 013002, [arXiv:hep-ph/9809415 [hep-ph]].

C. S. Lam, A 2-3 symmetry in neutrino oscillations, Phys. Lett. B 507 (2001) 214, [arXiv:hep-ph/0104116 [hepph]].

T. Kitabayashi and M. Yasue, S2L permutation symmetry for left-handed µ and τ families and neutrino oscillations in an SU(3)L × SU(1)N gauge model, Phys. Rev. D 67 (2003) 015006, [arXiv:hep-ph/0209294 [hep-ph]].

Y. Koide, Universal texture of quark and lepton mass matrices with an extended flavor 2 ↔ 3 symmetry, Phys. Rev. D 69, (2004) 093001, [arXiv:hep-ph/0312207 [hep-ph]].

T. Fukuyama and H. Nishiura, Mass matrix of Majorana neutrinos, [arXiv:hep-ph/9702253 [hep-ph]].

B. D. Claude Cohen-Tannoudji and F. Laloe, Quantum Mechanics, volume II, pages 1093 − 1104, Wiley (1991) 70. F. F. González-Canales, Simetría Permutacional S3: Sabor y Ceros de Textura, Ph.D. thesis, Universidad Autónoma de México, Posgrado en Ciencias Físicas (2011).




How to Cite

J. D. García-Aguilar, A. E. Pozas Ramírez, M. M. Suárez Castañeda, and J. C. Gómez-Izquierdo, “Soft breaking of the µ ↔ τ symmetry by S4 ⊗ Z2”, Rev. Mex. Fís., vol. 69, no. 3 May-Jun, pp. 030802 1–, May 2023.