Journal of High Energy Physics

, 2012:126 | Cite as

Natural vacuum alignment from group theory: the minimal case

  • Martin Holthausen
  • Michael A. Schmidt


Discrete flavour symmetries have been proven successful in explaining the leptonic flavour structure. To account for the observed mixing pattern, the flavour symmetry has to be broken to different subgroups in the charged and neutral lepton sector. However, cross-couplings via non-trivial contractions in the scalar potential force the group to break to the same subgroup. We present a solution to this problem by extending the flavour group in such a way that it preserves the flavour structure, but leads to an ’accidental’ symmetry in the flavon potential.

We have searched for symmetry groups up to order 1000, which forbid all dangerous cross-couplings and extend one of the interesting groups A 4, T 7, S 4, T′ or Δ(27). We have found a number of candidate groups and present a model based on one of the smallest extensions of A 4, namely \( {{Q}_8} \rtimes {{A}_4} \). We show that the most general nonsupersymmetric potential allows for the correct vacuum alignment. We investigate the effects of higher dimensional operators on the vacuum configuration and mixing angles, and give a see-saw-like UV completion. Finally, we discuss the supersymmetrization of the model. Additionally, we release the Mathematica package Discrete providing various useful tools for model building such as easily calculating invariants of discrete groups and flavon potentials.


Discrete and Finite Symmetries Beyond Standard Model Neutrino Physics 


  1. [1]
    T. Schwetz, M. Tortola and J. Valle, Where we are on θ 13 : addendum to ’Global neutrino data and recent reactor fluxes: status of three-flavour oscillation parameters’, New J. Phys. 13 (2011) 109401 [arXiv:1108.1376] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    T. Schwetz, M. Tortola and J. Valle, Global neutrino data and recent reactor fluxes: status of three-flavour oscillation parameters, New J. Phys. 13 (2011) 063004 [arXiv:1103.0734] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    G. Fogli, E. Lisi, A. Marrone, A. Palazzo and A. Rotunno, Evidence of θ 13 > 0 from global neutrino data analysis, Phys. Rev. D 84 (2011) 053007 [arXiv:1106.6028] [INSPIRE].ADSGoogle Scholar
  4. [4]
    M. Gonzalez-Garcia, M. Maltoni and J. Salvado, Updated global fit to three neutrino mixing: status of the hints of θ 13 > 0, JHEP 04 (2010) 056 [arXiv:1001.4524] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    T2K collaboration, K. Abe et al., Indication of electron neutrino appearance from an accelerator-produced off-axis muon neutrino beam, Phys. Rev. Lett. 107 (2011) 041801 [arXiv:1106.2822] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    CHOOZ collaboration, M. Apollonio et al., Search for neutrino oscillations on a long baseline at the CHOOZ nuclear power station, Eur. Phys. J. C 27 (2003) 331 [hep-ex/0301017] [INSPIRE].ADSGoogle Scholar
  7. [7]
    P. Harrison, D. Perkins and W. Scott, A redetermination of the neutrino mass squared difference in tri-maximal mixing with terrestrial matter effects, Phys. Lett. B 458 (1999) 79 [hep-ph/9904297] [INSPIRE].ADSGoogle Scholar
  8. [8]
    P. Harrison, D. Perkins and W. Scott, Tri-bimaximal mixing and the neutrino oscillation data, Phys. Lett. B 530 (2002) 167 [hep-ph/0202074] [INSPIRE].ADSGoogle Scholar
  9. [9]
    P. Harrison and W. Scott, Symmetries and generalizations of tri-bimaximal neutrino mixing, Phys. Lett. B 535 (2002) 163 [hep-ph/0203209] [INSPIRE].ADSGoogle Scholar
  10. [10]
    MINOS collaboration, P. Adamson et al., Improved search for muon-neutrino to electron-neutrino oscillations in MINOS, Phys. Rev. Lett. 107 (2011) 181802 [arXiv:1108.0015] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    C.D. Froggatt H.B. Nielsen, Hierarchy of quark masses, Cabibbo angles and CP violation, Nucl. Phys. B 147 (1979) 277 [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    K. Babu and X.-G. He, Model of geometric neutrino mixing, hep-ph/0507217 [INSPIRE].
  13. [13]
    E. Ma, A 4 symmetry and neutrinos with very different masses, Phys. Rev. D 70 (2004) 031901 [hep-ph/0404199] [INSPIRE].ADSGoogle Scholar
  14. [14]
    K. Babu, E. Ma and J. 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].ADSGoogle Scholar
  15. [15]
    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].ADSGoogle Scholar
  16. [16]
    X.-G. He, Y.-Y. Keum and R.R. Volkas, A 4 flavor symmetry breaking scheme for understanding quark and neutrino mixing angles, JHEP 04 (2006) 039 [hep-ph/0601001] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    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].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    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].ADSCrossRefGoogle Scholar
  19. [19]
    C. Luhn, S. Nasri and P. Ramond, Tri-bimaximal neutrino mixing and the family symmetry semidirect product of Z(7) and Z(3), Phys. Lett. B 652 (2007) 27 [arXiv:0706.2341] [INSPIRE].ADSGoogle Scholar
  20. [20]
    S. Pakvasa and H. Sugawara, Mass of the t quark in SU(2) × U(1), Phys. Lett. B 82 (1979) 105 [INSPIRE].ADSGoogle Scholar
  21. [21]
    Y. Yamanaka and H. Sugawara, Permutation symmetries and the fermion mass matrix, Phys. Rev. D 25 (1982) 1895 [Erratum ibid. D29 (1984) 2135+ [INSPIRE].ADSGoogle Scholar
  22. [22]
    T. Brown et al., CP nonconservation and rare processes in S 4 model of permutation symmetry, Phys. Lett. B 141 (1984) 95 [INSPIRE].ADSGoogle Scholar
  23. [23]
    T. Brown et al., Neutrino masses, mixing and oscillations in S 4 model of permutation symmetry, Phys. Rev. D 30 (1984) 255 [INSPIRE]ADSGoogle Scholar
  24. [24]
    D.-G. Lee and R. Mohapatra, An SO(10) × S 4 scenario for naturally degenerate neutrinos, Phys. Lett. B 329 (1994) 463 [hep-ph/9403201] [INSPIRE].ADSGoogle Scholar
  25. [25]
    E. Ma, Neutrino mass matrix from S 4 symmetry, Phys. Lett. B 632 (2006) 352 [hep-ph/0508231] [INSPIRE].ADSGoogle Scholar
  26. [26]
    C. Hagedorn, M. Lindner and R. Mohapatra, S 4 flavor symmetry and fermion masses: towards a grand unified theory of flavor, JHEP 06 (2006) 042 [hep-ph/0602244] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    Y. Cai and H.-B. Yu, A SO(10) GUT model with S 4 flavor symmetry, Phys. Rev. D 74 (2006) 115005 [hep-ph/0608022] [INSPIRE].ADSGoogle Scholar
  28. [28]
    F. Caravaglios and S. Morisi, Gauge boson families in grand unified theories of fermion masses: \( E_6^4 \times {{S}_4} \), Int. J. Mod. Phys. A 22 (2007) 2469 [hep-ph/0611078] [INSPIRE].MathSciNetADSGoogle Scholar
  29. [29]
    H. Zhang, Flavor S 4 × Z(2) symmetry and neutrino mixing, Phys. Lett. B 655 (2007) 132 [hep-ph/0612214] [INSPIRE].ADSGoogle Scholar
  30. [30]
    Y. Koide, S 4 flavor symmetry embedded into SU(3) and lepton masses and mixing, JHEP 08 (2007) 086 [arXiv:0705.2275] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    M. Parida, Intermediate left-right gauge symmetry, unification of couplings and fermion masses in SUSY SO(10) × S 4, Phys. Rev. D 78 (2008) 053004 [arXiv:0804.4571] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    F. Bazzocchi and S. Morisi, S 4 as a natural flavor symmetry for lepton mixing, Phys. Rev. D 80 (2009) 096005 [arXiv:0811.0345] [INSPIRE].ADSGoogle Scholar
  33. [33]
    H. Ishimori, Y. Shimizu and M. Tanimoto, S 4 flavor symmetry of quarks and leptons in SU(5) GUT, Prog. Theor. Phys. 121 (2009) 769 [arXiv:0812.5031] [INSPIRE].ADSMATHCrossRefGoogle Scholar
  34. [34]
    F. Bazzocchi, L. Merlo and S. Morisi, Phenomenological consequences of see-saw in S 4 based models, Phys. Rev. D 80 (2009) 053003 [arXiv:0902.2849] [INSPIRE].ADSGoogle Scholar
  35. [35]
    G. Altarelli, F. Feruglio and L. Merlo, Revisiting bimaximal neutrino mixing in a model with S 4 discrete symmetry, JHEP 05 (2009) 020 [arXiv:0903.1940] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    H. Ishimori, Y. Shimizu and M. Tanimoto, S 4 flavor model of quarks and leptons, Prog. Theor. Phys. Suppl. 180 (2010) 61 [arXiv:0904.2450] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    W. Grimus, L. Lavoura and P. Ludl, Is S 4 the horizontal symmetry of tri-bimaximal lepton mixing?, J. Phys. G 36 (2009) 115007 [arXiv:0906.2689] [INSPIRE].ADSGoogle Scholar
  38. [38]
    G.-J. Ding, Fermion masses and flavor mixings in a model with S 4 flavor symmetry, Nucl. Phys. B 827 (2010) 82 [arXiv:0909.2210] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    D. Meloni, A see-saw S 4 model for fermion masses and mixings, J. Phys. G 37 (2010) 055201 [arXiv:0911.3591] [INSPIRE].ADSGoogle Scholar
  40. [40]
    S. Morisi and E. Peinado, An S 4 model for quarks and leptons with maximal atmospheric angle, Phys. Rev. D 81 (2010) 085015 [arXiv:1001.2265] [INSPIRE].ADSGoogle Scholar
  41. [41]
    B. Dutta, Y. Mimura and R. Mohapatra, An SO(10) grand unified theory of flavor, JHEP 05 (2010) 034 [arXiv:0911.2242] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    C. Lam, The unique horizontal symmetry of leptons, Phys. Rev. D 78 (2008) 073015 [arXiv:0809.1185] [INSPIRE].ADSGoogle Scholar
  43. [43]
    R. Mohapatra, M. Parida and G. Rajasekaran, High scale mixing unification and large neutrino mixing angles, Phys. Rev. D 69 (2004) 053007 [hep-ph/0301234] [INSPIRE].ADSGoogle Scholar
  44. [44]
    G.-J. Ding, Fermion mass hierarchies and flavor mixing from T -prime symmetry, Phys. Rev. D 78 (2008) 036011 [arXiv:0803.2278] [INSPIRE].ADSGoogle Scholar
  45. [45]
    P.H. Frampton and S. Matsuzaki, T -prime predictions of PMNS and CKM angles, Phys. Lett. B 679 (2009) 347 [arXiv:0902.1140] [INSPIRE].ADSGoogle Scholar
  46. [46]
    P.H. Frampton and T.W. Kephart, Flavor symmetry for quarks and leptons, JHEP 09 (2007) 110 [arXiv:0706.1186] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    A. Aranda, Neutrino mixing from the double tetrahedral group T -prime, Phys. Rev. D 76 (2007) 111301 [arXiv:0707.3661] [INSPIRE].ADSGoogle Scholar
  48. [48]
    P.D. Carr and P.H. Frampton, Group theoretic bases for tribimaximal mixing, hep-ph/0701034 [INSPIRE].
  49. [49]
    F. Feruglio, C. Hagedorn, Y. Lin and L. Merlo, Tri-bimaximal neutrino mixing and quark masses from a discrete flavour symmetry, Nucl. Phys. B 775 (2007) 120 [Erratum ibid. 836 (2010) 127-128] [hep-ph/0702194] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    M.-C. Chen and K. Mahanthappa, CKM and tri-bimaximal MNS matrices in a SU(5) × (d) T model, Phys. Lett. B 652 (2007) 34 [arXiv:0705.0714] [INSPIRE].ADSGoogle Scholar
  51. [51]
    F. Bazzocchi and I. de Medeiros Varzielas, Tri-bi-maximal mixing in viable family symmetry unified model with extended seesaw, Phys. Rev. D 79 (2009) 093001 [arXiv:0902.3250] [INSPIRE].ADSGoogle Scholar
  52. [52]
    C. Luhn, S. Nasri and P. Ramond, The flavor group Δ(3n 2), J. Math. Phys. 48 (2007) 073501 [hep-th/0701188] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  53. [53]
    W. Grimus and L. Lavoura, A model for trimaximal lepton mixing, JHEP 09 (2008) 106 [arXiv:0809.0226] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    I. de Medeiros Varzielas, S. King and G. Ross, Neutrino tri-bi-maximal mixing from a non-abelian discrete family symmetry, Phys. Lett. B 648 (2007) 201 [hep-ph/0607045] [INSPIRE].ADSGoogle Scholar
  55. [55]
    Y. Shimizu, M. Tanimoto and A. Watanabe, Breaking tri-bimaximal mixing and large θ 13, Prog. Theor. Phys. 126 (2011) 81 [arXiv:1105.2929] [INSPIRE].ADSMATHCrossRefGoogle Scholar
  56. [56]
    R.d.A. Toorop, F. Feruglio and C. Hagedorn, Discrete flavour symmetries in light of T2K, Phys. Lett. B 703 (2011) 447 [arXiv:1107.3486] [INSPIRE].ADSGoogle Scholar
  57. [57]
    T. Kobayashi, Y. Omura and K. Yoshioka, Flavor symmetry breaking and vacuum alignment on orbifolds, Phys. Rev. D 78 (2008) 115006 [arXiv:0809.3064] [INSPIRE].ADSGoogle Scholar
  58. [58]
    K. Babu and S. Gabriel, Semidirect product groups, vacuum alignment and tribimaximal neutrino mixing, Phys. Rev. D 82 (2010) 073014 [arXiv:1006.0203] [INSPIRE].ADSGoogle Scholar
  59. [59]
    GAP group, GAP — Groups, Algorithms, and Programming. Version 4.4.12,
  60. [60]
    H.U. Besche, B. Eick and E.O’Brien, SmallGroups — Library of all ’small’ groups. GAP package, version included in GAP 4.4.12
  61. [61]
    K.M. Parattu and A. Wingerter, Tribimaximal mixing from small groups, Phys. Rev. D 84 (2011) 013011 [arXiv:1012.2842] [INSPIRE].ADSGoogle Scholar
  62. [62]
    W. Grimus and P.O. Ludl, Finite flavour groups of fermions, arXiv:1110.6376 [INSPIRE].
  63. [63]
    S.F. King and C. Luhn, On the origin of neutrino flavour symmetry, JHEP 10 (2009) 093 [arXiv:0908.1897] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    B. Brahmachari, S. Choubey and M. Mitra, The A 4 flavor symmetry and neutrino phenomenology, Phys. Rev. D 77 (2008) 073008 [Erratum ibid. D 77 (2008) 119901] [arXiv:0801.3554] [INSPIRE].ADSGoogle Scholar
  65. [65]
    J. Barry and W. Rodejohann, Deviations from tribimaximal mixing due to the vacuum expectation value misalignment in A 4 models, Phys. Rev. D 81 (2010) 093002 [Erratum ibid. D 81 (2010) 119901] [arXiv:1003.2385] [INSPIRE].ADSGoogle Scholar
  66. [66]
    G. Altarelli, F. Feruglio and Y. Lin, Tri-bimaximal neutrino mixing from orbifolding, Nucl. Phys. B 775 (2007) 31 [hep-ph/0610165] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    M. Honda and M. Tanimoto, Deviation from tri-bimaximal neutrino mixing in A 4 flavor symmetry, Prog. Theor. Phys. 119 (2008) 583 [arXiv:0801.0181] [INSPIRE].ADSMATHCrossRefGoogle Scholar
  68. [68]
    S. Antusch, J. Kersten, M. Lindner, M. Ratz and M.A. Schmidt, Running neutrino mass parameters in see-saw scenarios, JHEP 03 (2005) 024 [hep-ph/0501272] [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    M. Lattanzi and J. Valle, Decaying warm dark matter and neutrino masses, Phys. Rev. Lett. 99 (2007) 121301 [arXiv:0705.2406] [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    P. Minkowski, μeγ at a rate of one out of 1-billion muon decays?, Phys. Lett. B 67 (1977) 421 [INSPIRE].ADSGoogle Scholar
  71. [71]
    T. Yanagida, Horizontal symmetry and masses of neutrinos, in the proceedings of the Workshop on the unified theory and the baryon number in the universe, O. Sawada ed., KEK, Tsukuba Japan (1979).Google Scholar
  72. [72]
    S.L. Glashow, The future of elementary particle physics, in the proceedings of the 1979 Cargèse summer institute on quarks and leptons, M. Levy et al. eds., Plenum Press, New York U.S.A. (1980).Google Scholar
  73. [73]
    M. Gell-Mann, P. Ramond and R. Slansky, Complex spinors and unified theories, in Supergravity, P. van Nieuwenhuizen and D.Z. Freedman eds., North Holland, Amsterdam The Netherlands (1979).Google Scholar
  74. [74]
    R.N. Mohapatra and G. Senjanović, Neutrino mass and spontaneous parity violation, Phys. Rev. Lett. 44 (1980) 912 [INSPIRE].ADSCrossRefGoogle Scholar
  75. [75]
    J. Schechter and J.W.F. Valle, Neutrino masses in SU(2) × U(1) theories, Phys. Rev. D 22 (1980) 2227 [INSPIRE].ADSGoogle Scholar
  76. [76]
    J. Schechter and J.W.F. Valle, Neutrino decay and spontaneous violation of lepton number, Phys. Rev. D 25 (1982) 774 [INSPIRE].ADSGoogle Scholar
  77. [77]
    M. Malinsky, J. Romao and J. Valle, Novel supersymmetric SO(10) seesaw mechanism, Phys. Rev. Lett. 95 (2005) 161801 [hep-ph/0506296] [INSPIRE].ADSCrossRefGoogle Scholar
  78. [78]
    F. Feruglio, C. Hagedorn and L. Merlo, Vacuum alignment in SUSY A 4 models, JHEP 03 (2010) 084 [arXiv:0910.4058] [INSPIRE].ADSCrossRefGoogle Scholar
  79. [79]
    G. Giudice and R. Rattazzi, Theories with gauge mediated supersymmetry breaking, Phys. Rept. 322 (1999) 419 [hep-ph/9801271] [INSPIRE].ADSCrossRefGoogle Scholar
  80. [80]
    S. Antusch, S.F. King, M. Malinsky and G.G. Ross, Solving the SUSY flavour and CP problems with non-abelian family symmetry and supergravity, Phys. Lett. B 670 (2009) 383 [arXiv:0807.5047] [INSPIRE].ADSGoogle Scholar
  81. [81]
    V. Dabbaghian, REPSN — For constructing representations of finite groups, GAP package, Version 3.0.2,
  82. [82]
    A. Merle and R. Zwicky, Explicit and spontaneous breaking of SU(3) into its finite subgroups, arXiv:1110.4891 [INSPIRE].
  83. [83]
    P.M. van Den Broek and J.F. Cornwell, Clebsch-Gordan coefficients of symmetry groups, Phys. Status Solidi B 90 (1978) 211.ADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.Max-Planck Institut für KernphysikHeidelbergGermany
  2. 2.Institute for Particle Physics Phenomenology (IPPP)University of DurhamDurhamU.K.
  3. 3.ARC Centre of Excellence for Particle Physics at the Terascale, School of PhysicsThe University of MelbourneVictoriaAustralia

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