Journal of High Energy Physics

, 2018:173 | Cite as

Spontaneous breaking of SO(3) to finite family symmetries with supersymmetry — an A4 model

  • Stephen F. King
  • Ye-Ling ZhouEmail author
Open Access
Regular Article - Theoretical Physics


We discuss the breaking of SO(3) down to finite family symmetries such as A4, S4 and A5 using supersymmetric potentials for the first time. We analyse in detail the case of supersymmetric A4 and its finite subgroups Z3 and Z2. We then propose a supersymmetric A4 model of leptons along these lines, originating from SO(3) × U(1), which leads to a phenomenologically acceptable pattern of lepton mixing and masses once subleading corrections are taken into account. We also discuss the phenomenological consequences of having a gauged SO(3), leading to massive gauge bosons, and show that all domain wall problems are resolved in this model.


Gauge Symmetry Discrete Symmetries Neutrino Physics Supersymmetric Effective Theories 


Open Access

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  1. [1]
    T. Ohlsson ed., Special Issue on “Neutrino Oscillations: Celebrating the Nobel Prize in Physics 2015”, Nucl. Phys. B 908 (2016) 1.Google Scholar
  2. [2]
    S.F. King, Neutrino mass models, Rept. Prog. Phys. 67 (2004) 107 [hep-ph/0310204] [INSPIRE].
  3. [3]
    S.F. King and G.G. Ross, Fermion masses and mixing angles from SU(3) family symmetry, Phys. Lett. B 520 (2001) 243 [hep-ph/0108112] [INSPIRE].
  4. [4]
    S.F. King and G.G. Ross, Fermion masses and mixing angles from SU(3) family symmetry and unification, Phys. Lett. B 574 (2003) 239 [hep-ph/0307190] [INSPIRE].
  5. [5]
    F.J. de Anda and S.F. King, SU(3) × SO(10) in 6d, JHEP 10 (2018) 128 [arXiv:1807.07078] [INSPIRE].
  6. [6]
    S.F. King, Predicting neutrino parameters from SO(3) family symmetry and quark-lepton unification, JHEP 08 (2005) 105 [hep-ph/0506297] [INSPIRE].
  7. [7]
    S.F. King and M. Malinsky, Towards a Complete Theory of Fermion Masses and Mixings with SO(3) Family Symmetry and 5-D SO(10) Unification, JHEP 11 (2006) 071 [hep-ph/0608021] [INSPIRE].
  8. [8]
    I. Masina, A Maximal atmospheric mixing from a maximal CP-violating phase, Phys. Lett. B 633 (2006) 134 [hep-ph/0508031] [INSPIRE].
  9. [9]
    E. Ma and G. Rajasekaran, Softly broken A 4 symmetry for nearly degenerate neutrino masses, Phys. Rev. D 64 (2001) 113012 [hep-ph/0106291] [INSPIRE].
  10. [10]
    K.S. Babu, E. Ma and J.W.F. Valle, Underlying A 4 symmetry for the neutrino mass matrix and the quark mixing matrix, Phys. Lett. B 552 (2003) 207 [hep-ph/0206292] [INSPIRE].
  11. [11]
    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] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  12. [12]
    G. Altarelli and F. Feruglio, Tri-bimaximal neutrino mixing from discrete symmetry in extra dimensions, Nucl. Phys. B 720 (2005) 64 [hep-ph/0504165] [INSPIRE].
  13. [13]
    G. Altarelli and F. Feruglio, Tri-bimaximal neutrino mixing, A 4 and the modular symmetry, Nucl. Phys. B 741 (2006) 215 [hep-ph/0512103] [INSPIRE].
  14. [14]
    I. de Medeiros Varzielas, S.F. King and G.G. Ross, Tri-bimaximal neutrino mixing from discrete subgroups of SU(3) and SO(3) family symmetry, Phys. Lett. B 644 (2007) 153 [hep-ph/0512313] [INSPIRE].
  15. [15]
    I. de Medeiros Varzielas, S.F. King and G.G. Ross, Neutrino tri-bi-maximal mixing from a non-Abelian discrete family symmetry, Phys. Lett. B 648 (2007) 201 [hep-ph/0607045] [INSPIRE].
  16. [16]
    S.F. King, Parametrizing the lepton mixing matrix in terms of deviations from tri-bimaximal mixing, Phys. Lett. B 659 (2008) 244 [arXiv:0710.0530] [INSPIRE].
  17. [17]
    S.F. King and C. Luhn, Neutrino Mass and Mixing with Discrete Symmetry, Rept. Prog. Phys. 76 (2013) 056201 [arXiv:1301.1340] [INSPIRE].
  18. [18]
    S.F. King, A. Merle, S. Morisi, Y. Shimizu and M. Tanimoto, Neutrino Mass and Mixing: from Theory to Experiment, New J. Phys. 16 (2014) 045018 [arXiv:1402.4271] [INSPIRE].
  19. [19]
    S.F. King, Models of Neutrino Mass, Mixing and CP-violation, J. Phys. G 42 (2015) 123001 [arXiv:1510.02091] [INSPIRE].
  20. [20]
    R.D. Peccei, Discrete and global symmetries in particle physics, Lect. Notes Phys. 521 (1999) 1 [hep-ph/9807516] [INSPIRE].
  21. [21]
    L.E. Ibáñez and G.G. Ross, Discrete gauge symmetries and the origin of baryon and lepton number conservation in supersymmetric versions of the standard model, Nucl. Phys. B 368 (1992) 3 [INSPIRE].
  22. [22]
    T. Kobayashi, H.P. Nilles, F. Ploger, S. Raby and M. Ratz, Stringy origin of non-Abelian discrete flavor symmetries, Nucl. Phys. B 768 (2007) 135 [hep-ph/0611020] [INSPIRE].
  23. [23]
    R. de Adelhart Toorop, F. Feruglio and C. Hagedorn, Finite Modular Groups and Lepton Mixing, Nucl. Phys. B 858 (2012) 437 [arXiv:1112.1340] [INSPIRE].
  24. [24]
    F. Feruglio, Are neutrino masses modular forms?, arXiv:1706.08749.
  25. [25]
    T. Kobayashi, K. Tanaka and T.H. Tatsuishi, Neutrino mixing from finite modular groups, Phys. Rev. D 98 (2018) 016004 [arXiv:1803.10391] [INSPIRE].
  26. [26]
    J.C. Criado and F. Feruglio, Modular Invariance Faces Precision Neutrino Data, SciPost Phys. 5 (2018) 042 [arXiv:1807.01125] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    J.T. Penedo and S.T. Petcov, Lepton Masses and Mixing from Modular S 4 Symmetry, arXiv:1806.11040 [INSPIRE].
  28. [28]
    T. Kobayashi, N. Omoto, Y. Shimizu, K. Takagi, M. Tanimoto and T.H. Tatsuishi, Modular A 4 invariance and neutrino mixing, arXiv:1808.03012 [INSPIRE].
  29. [29]
    G. Altarelli, F. Feruglio and Y. Lin, Tri-bimaximal neutrino mixing from orbifolding, Nucl. Phys. B 775 (2007) 31 [hep-ph/0610165] [INSPIRE].
  30. [30]
    T.J. Burrows and S.F. King, A 4 Family Symmetry from SU(5) SUSY GUTs in 6d, Nucl. Phys. B 835 (2010) 174 [arXiv:0909.1433] [INSPIRE].
  31. [31]
    T.J. Burrows and S.F. King, A 4 × SU(5) SUSY GUT of Flavour in 8d, Nucl. Phys. B 842 (2011) 107 [arXiv:1007.2310] [INSPIRE].
  32. [32]
    F.J. de Anda and S.F. King, An S 4 × SU(5) SUSY GUT of flavour in 6d, JHEP 07 (2018) 057 [arXiv:1803.04978] [INSPIRE].
  33. [33]
    T. Banks and M. Dine, Note on discrete gauge anomalies, Phys. Rev. D 45 (1992) 1424 [hep-th/9109045] [INSPIRE].
  34. [34]
    Ya.B. Zeldovich, I.Yu. Kobzarev and L.B. Okun, Cosmological Consequences of the Spontaneous Breakdown of Discrete Symmetry, Zh. Eksp. Teor. Fiz. 67 (1974) 3 [INSPIRE].
  35. [35]
    T.W.B. Kibble, Topology of Cosmic Domains and Strings, J. Phys. A 9 (1976) 1387 [INSPIRE].
  36. [36]
    A. Vilenkin, Cosmic Strings and Domain Walls, Phys. Rept. 121 (1985) 263 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    F. Riva, Low-Scale Leptogenesis and the Domain Wall Problem in Models with Discrete Flavor Symmetries, Phys. Lett. B 690 (2010) 443 [arXiv:1004.1177] [INSPIRE].
  38. [38]
    S. Antusch and D. Nolde, Matter inflation with A 4 flavour symmetry breaking, JCAP 10 (2013) 028 [arXiv:1306.3501] [INSPIRE].
  39. [39]
    J. Preskill, S.P. Trivedi, F. Wilczek and M.B. Wise, Cosmology and broken discrete symmetry, Nucl. Phys. B 363 (1991) 207 [INSPIRE].
  40. [40]
    S. Chigusa and K. Nakayama, Anomalous Discrete Flavor Symmetry and Domain Wall Problem, arXiv:1808.09601 [INSPIRE].
  41. [41]
    T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
  42. [42]
    B.A. Ovrut, Isotropy Subgroups of SO(3) and Higgs Potentials, J. Math. Phys. 19 (1978) 418 [INSPIRE].
  43. [43]
    G. Etesi, Spontaneous symmetry breaking in SO(3) gauge theory to discrete subgroups, J. Math. Phys. 37 (1996) 1596 [hep-th/9706029] [INSPIRE].
  44. [44]
    J. Berger and Y. Grossman, Model of leptons from SO(3) → A 4, JHEP 02 (2010) 071 [arXiv:0910.4392] [INSPIRE].
  45. [45]
    Y. Koide, S 4 flavor symmetry embedded into SU(3) and lepton masses and mixing, JHEP 08 (2007) 086 [arXiv:0705.2275] [INSPIRE].
  46. [46]
    Y.-L. Wu, SU(3) Gauge Family Symmetry and Prediction for the Lepton-Flavor Mixing and Neutrino Masses with Maximal Spontaneous CP-violation, Phys. Lett. B 714 (2012) 286 [arXiv:1203.2382] [INSPIRE].
  47. [47]
    R. Alonso, M.B. Gavela, D. Hernández, L. Merlo and S. Rigolin, Leptonic Dynamical Yukawa Couplings, JHEP 08 (2013) 069 [arXiv:1306.5922] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    R. Alonso, M.B. Gavela, G. Isidori and L. Maiani, Neutrino Mixing and Masses from a Minimum Principle, JHEP 11 (2013) 187 [arXiv:1306.5927] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    A. Adulpravitchai, A. Blum and M. Lindner, Non-Abelian Discrete Groups from the Breaking of Continuous Flavor Symmetries, JHEP 09 (2009) 018 [arXiv:0907.2332] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    W. Grimus and P.O. Ludl, Principal series of finite subgroups of SU(3), J. Phys. A 43 (2010) 445209 [arXiv:1006.0098] [INSPIRE].
  51. [51]
    C. Luhn, Spontaneous breaking of SU(3) to finite family symmetries: a pedestrian’s approach, JHEP 03 (2011) 108 [arXiv:1101.2417] [INSPIRE].
  52. [52]
    A. Merle and R. Zwicky, Explicit and spontaneous breaking of SU(3) into its finite subgroups, JHEP 02 (2012) 128 [arXiv:1110.4891] [INSPIRE].
  53. [53]
    B.L. Rachlin and T.W. Kephart, Spontaneous Breaking of Gauge Groups to Discrete Symmetries, JHEP 08 (2017) 110 [arXiv:1702.08073] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    A.E. Cárcamo Hernández, E. Cataño Mur and R. Martinez, Lepton masses and mixing in SU(3)C ⊗ SU(3)L ⊗ U(1)X models with a S 3 flavor symmetry, Phys. Rev. D 90 (2014) 073001 [arXiv:1407.5217] [INSPIRE].
  55. [55]
    S. Antusch, S.F. King and M. Malinsky, Solving the SUSY Flavour and CP Problems with SU(3) Family Symmetry, JHEP 06 (2008) 068 [arXiv:0708.1282] [INSPIRE].
  56. [56]
    G. Blankenburg, G. Isidori and J. Jones-Perez, Neutrino Masses and LFV from Minimal Breaking of U(3)5 and U(2)5 flavor Symmetries, Eur. Phys. J. C 72 (2012) 2126 [arXiv:1204.0688] [INSPIRE].
  57. [57]
    S. Pascoli and Y.-L. Zhou, The role of flavon cross couplings in leptonic flavour mixing, JHEP 06 (2016) 073 [arXiv:1604.00925] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    T. Morozumi, H. Okane, H. Sakamoto, Y. Shimizu, K. Takagi and H. Umeeda, Phenomenological Aspects of Possible Vacua of a Neutrino Flavor Model, Chin. Phys. C 42 (2018) 023102 [arXiv:1707.04028] [INSPIRE].
  59. [59]
    S. Antusch and S.F. King, Charged lepton corrections to neutrino mixing angles and CP phases revisited, Phys. Lett. B 631 (2005) 42 [hep-ph/0508044] [INSPIRE].
  60. [60]
    S. Antusch, P. Huber, S.F. King and T. Schwetz, Neutrino mixing sum rules and oscillation experiments, JHEP 04 (2007) 060 [hep-ph/0702286] [INSPIRE].
  61. [61]
    S.T. Petcov, Predicting the values of the leptonic CP-violation phases in theories with discrete flavour symmetries, Nucl. Phys. B 892 (2015) 400 [arXiv:1405.6006] [INSPIRE].
  62. [62]
    P. Ballett, S.F. King, C. Luhn, S. Pascoli and M.A. Schmidt, Testing solar lepton mixing sum rules in neutrino oscillation experiments, JHEP 12 (2014) 122 [arXiv:1410.7573] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    I. Girardi, S.T. Petcov and A.V. Titov, Determining the Dirac CP-violation Phase in the Neutrino Mixing Matrix from Sum Rules, Nucl. Phys. B 894 (2015) 733 [arXiv:1410.8056] [INSPIRE].
  64. [64]
    L.A. Delgadillo, L.L. Everett, R. Ramos and A.J. Stuart, Predictions for the Dirac CP-Violating Phase from Sum Rules, Phys. Rev. D 97 (2018) 095001 [arXiv:1801.06377] [INSPIRE].
  65. [65]
    I. de Medeiros Varzielas, T. Neder and Y.-L. Zhou, Effective alignments as building blocks of flavor models, Phys. Rev. D 97 (2018) 115033 [arXiv:1711.05716] [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.School of Physics and AstronomyUniversity of SouthamptonSouthamptonUnited Kingdom
  2. 2.Institute for Particle Physics Phenomenology, Department of PhysicsDurham UniversityDurhamUnited Kingdom

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