Abstract
Non-Abelian discrete family symmetries play a pivotal role in the formulation of models with tri-bimaximal lepton mixing. We discuss how to obtain symmetries such as \( {\mathcal{A}_4} \), \( {\mathcal{Z}_7} \) ⋊ \( {\mathcal{Z}_3} \) and Δ(27) from an underlying SU(3) gauge symmetry. Higher irreducible representations are required to achieve the spontaneous breaking of the continuous group. We present methods of identifying the required vacuum alignments and discuss in detail the symmetry breaking potentials.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Schwetz, M.A. Tortola and J.W.F. Valle, Three-flavour neutrino oscillation update, New J. Phys. 10 (2008) 113011 [arXiv:0808.2016] [SPIRES].
M.C. 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] [SPIRES].
P.F. Harrison, D.H. Perkins and W.G. Scott, Tri-bimaximal mixing and the neutrino oscillation data, Phys. Lett. B 530 (2002) 167 [hep-ph/0202074] [SPIRES].
P.F. Harrison and W.G. Scott, Symmetries and generalisations of tri-bimaximal neutrino mixing, Phys. Lett. B 535 (2002) 163 [hep-ph/0203209] [SPIRES].
E. Ma and G. Rajasekaran, Softly broken A 4 symmetry for nearly degenerate neutrino masses, Phys. Rev. D 64 (2001) 113012 [hep-ph/0106291] [SPIRES].
E. Ma, Neutrino mass matrix from S 4 symmetry, Phys. Lett. B 632 (2006) 352 [hep-ph/0508231] [SPIRES].
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] [SPIRES].
D.B. Kaplan and M. Schmaltz, Flavor unification and discrete non-Abelian symmetries, Phys. Rev. D 49 (1994) 3741 [hep-ph/9311281] [SPIRES].
M. Schmaltz, Neutrino oscillations from discrete non-Abelian family symmetries, Phys. Rev. D 52 (1995) 1643 [hep-ph/9411383] [SPIRES].
S.F. King and C. Luhn, On the origin of neutrino flavour symmetry, JHEP 10 (2009) 093 [arXiv:0908.1897] [SPIRES].
G. Altarelli and F. Feruglio, Discrete flavor symmetries and models of neutrino mixing, Rev. Mod. Phys. 82 (2010) 2701 [arXiv:1002.0211] [SPIRES].
L.M. Krauss and F. Wilczek, Discrete gauge symmetry in continuum theories, Phys. Rev. Lett. 62 (1989) 1221 [SPIRES].
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] [SPIRES].
L.E. Ibáñez and G.G. Ross, Discrete gauge symmetry anomalies, Phys. Lett. B 260 (1991) 291 [SPIRES].
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 [SPIRES].
H.K. Dreiner, C. Luhn and M. Thormeier, What is the discrete gauge symmetry of the MSSM?, Phys. Rev. D 73 (2006) 075007 [hep-ph/0512163] [SPIRES].
C. Luhn and M. Thormeier, Dirac neutrinos and anomaly-free discrete gauge symmetries, Phys. Rev. D 77 (2008) 056002 [arXiv:0711.0756] [SPIRES].
B.A. Ovrut, Isotropy subgroups of SO(3) and Higgs potentials, J. Math. Phys. 19 (1978) 418 [SPIRES].
G. Etesi, Spontaneous symmetry breaking in SO(3) gauge theory to discrete subgroups, J. Math. Phys. 37 (1996) 1596 [hep-th/9706029] [SPIRES].
M. Koca, M. Al-Barwani and R. Koc, Breaking SO(3) into its closed subgroups by Higgs mechanism, J. Phys. A 30 (1997) 2109 [SPIRES].
M. Koca, R. Koc and H. Tutunculer, Explicit breaking of SO(3) with Higgs fields in the representations L = 2 and L = 3, Int. J. Mod. Phys. A 18 (2003) 4817 [hep-ph/0410270] [SPIRES].
J. Berger and Y. Grossman, Model of leptons from SO(3) → A 4, JHEP 02 (2010) 071 [arXiv:0910.4392] [SPIRES].
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] [SPIRES].
C. Luhn and P. Ramond, Anomaly conditions for non-Abelian finite family symmetries, JHEP 07 (2008) 085 [arXiv:0805.1736] [SPIRES].
G.A. Miller, H.F. Blichfeldt, and L.E. Dickson, Theory and application of finite groups, John Wiley & Sons, New York U.S.A. (1916) [Dover edition (1961)].
W.M. Fairbairn, T. Fulton, W. H. Klink, Finite and disconnected subgroups of SU(3) and their application to the elementary-particle spectrum, J. Math. Phys. 5 (1964) 1038.
A. Bovier, M. L¨uling and D. Wyler, Finite subgroups of SU(3), J. Math. Phys. 22 (1981) 1543 [SPIRES].
C. Luhn, S. Nasri and P. Ramond, The flavor group Δ(3n 2), J. Math. Phys. 48 (2007) 073501 [hep-th/0701188] [SPIRES].
C. Luhn, S. Nasri and P. Ramond, Simple finite non-Abelian flavor groups, J. Math. Phys. 48 (2007) 123519 [arXiv:0709.1447] [SPIRES].
J.A. Escobar and C. Luhn, The flavor group Δ(6n 2), J. Math. Phys. 50 (2009) 013524 [arXiv:0809.0639] [SPIRES].
P.O. Ludl, Systematic analysis of finite family symmetry groups and their application to the lepton sector, arXiv:0907.5587 [SPIRES].
H. Ishimori et al., Non-Abelian discrete symmetries in particle physics, Prog. Theor. Phys. Suppl. 183 (2010) 1 [arXiv:1003.3552] [SPIRES].
W. Grimus and P.O. Ludl, Principal series of finite subgroups of SU(3), J. Phys. A 43 (2010) 445209 [arXiv:1006.0098] [SPIRES].
P. Ramond, Group theory: a physicist’s survey, Cambridge University Press, Cambridge U.K. (2010).
K.M. Parattu and A. Wingerter, Tribimaximal mixing from small groups, arXiv:1012.2842 [SPIRES].
P.O. Ludl, Comments on the classification of the finite subgroups of SU(3), arXiv:1101.2308 [SPIRES].
Z.G. Berezhiani and M.Y. Khlopov, Cosmology of spontaneously broken gauge family symmetry, Z. Phys. C 49 (1991) 73 [SPIRES].
S.F. King and C. Luhn, A supersymmetric grand unified theory of flavour with PSL(2, 7) × SO(10), Nucl. Phys. B 832 (2010) 414 [arXiv:0912.1344] [SPIRES];
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1101.2417
Rights and permissions
About this article
Cite this article
Luhn, C. Spontaneous breaking of SU(3) to finite family symmetries — a pedestrian’s approach. J. High Energ. Phys. 2011, 108 (2011). https://doi.org/10.1007/JHEP03(2011)108
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2011)108