# Spontaneous breaking of SO(3) to finite family symmetries with supersymmetry — an *A*_{4} model

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## Abstract

We discuss the breaking of SO(3) down to finite family symmetries such as *A*_{4}, *S*_{4} and *A*_{5} using supersymmetric potentials for the first time. We analyse in detail the case of supersymmetric *A*_{4} and its finite subgroups *Z*_{3} and *Z*_{2}. We then propose a supersymmetric *A*_{4} 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.

## Keywords

Gauge Symmetry Discrete Symmetries Neutrino Physics Supersymmetric Effective Theories## Notes

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## References

- [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]
- [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]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]
- [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]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]I. Masina,
*A Maximal atmospheric mixing from a maximal CP-violating phase*,*Phys. Lett.***B 633**(2006) 134 [hep-ph/0508031] [INSPIRE]. - [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]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]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]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]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]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]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]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]S.F. King and C. Luhn,
*Neutrino Mass and Mixing with Discrete Symmetry*,*Rept. Prog. Phys.***76**(2013) 056201 [arXiv:1301.1340] [INSPIRE]. - [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]S.F. King,
*Models of Neutrino Mass, Mixing and CP-violation*,*J. Phys.***G 42**(2015) 123001 [arXiv:1510.02091] [INSPIRE]. - [20]R.D. Peccei,
*Discrete and global symmetries in particle physics*,*Lect. Notes Phys.***521**(1999) 1 [hep-ph/9807516] [INSPIRE]. - [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]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]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]F. Feruglio,
*Are neutrino masses modular forms?*, arXiv:1706.08749. - [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]J.C. Criado and F. Feruglio,
*Modular Invariance Faces Precision Neutrino Data*,*SciPost Phys.***5**(2018) 042 [arXiv:1807.01125] [INSPIRE].ADSCrossRefGoogle Scholar - [27]J.T. Penedo and S.T. Petcov,
*Lepton Masses and Mixing from Modular S*_{4}*Symmetry*, arXiv:1806.11040 [INSPIRE]. - [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]G. Altarelli, F. Feruglio and Y. Lin,
*Tri-bimaximal neutrino mixing from orbifolding*,*Nucl. Phys.***B 775**(2007) 31 [hep-ph/0610165] [INSPIRE]. - [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]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]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]T. Banks and M. Dine,
*Note on discrete gauge anomalies*,*Phys. Rev.***D 45**(1992) 1424 [hep-th/9109045] [INSPIRE]. - [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]
- [36]A. Vilenkin,
*Cosmic Strings and Domain Walls*,*Phys. Rept.***121**(1985) 263 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [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]S. Antusch and D. Nolde,
*Matter inflation with A*_{4}*flavour symmetry breaking*,*JCAP***10**(2013) 028 [arXiv:1306.3501] [INSPIRE]. - [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]S. Chigusa and K. Nakayama,
*Anomalous Discrete Flavor Symmetry and Domain Wall Problem*, arXiv:1808.09601 [INSPIRE]. - [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]B.A. Ovrut,
*Isotropy Subgroups of*SO(3)*and Higgs Potentials*,*J. Math. Phys.***19**(1978) 418 [INSPIRE]. - [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]J. Berger and Y. Grossman,
*Model of leptons from*SO(3) →*A*_{4},*JHEP***02**(2010) 071 [arXiv:0910.4392] [INSPIRE]. - [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]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]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]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]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]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]C. Luhn,
*Spontaneous breaking of*SU(3)*to finite family symmetries: a pedestrian’s approach*,*JHEP***03**(2011) 108 [arXiv:1101.2417] [INSPIRE]. - [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]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]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]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]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]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]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]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]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]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]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]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]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]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].