Theory of neutrino masses and mixing

  • W. Grimus


We motivate the usage of finite groups as symmetries of the Lagrangian. After a presentation of basic group-theoretical concepts, we introduce the notion of characters and character tables in the context of irreducible representations and discuss their applications. We exemplify these theoretical concepts with the groups S 4 and A 4. Finally, we discuss the relation between tensor products of irreducible representations and Yukawa couplings and describe a model for tri-bimaximal lepton mixing based on A 4.


Finite Group Yukawa Coupling Neutrino Mass Grand Unify Theory Neutrino Mass Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    R. Gatto, G. Sartori, and M. Tonin, “Weak Self-Masses, Cabibbo Angle, and Broken SU 2 × SU 3,” Phys. Lett. B 28, 128 (1968); N. Cabibbo and L. Maiani, “Dynamical Interrelations of Weak, Electromagnetic and Strong Interactions and the Value of θ,” Phys. Lett. B 28, 131 (1968).ADSCrossRefGoogle Scholar
  2. 2.
    P. F. Harrison, D. H. Perkins, and W. G. Scott, “Tri-Bimaximal Mixing and the Neutrino Oscillation Data,” Phys. Lett. B 530, 167 (2002), arXiv:hepph/0202074.ADSCrossRefGoogle Scholar
  3. 3.
    M. Maltoni, T. Schwetz, M. A. Tórtola, and J. W. F. Valle, “Status of Global Fits to Neutrino Oscillations,” New J. Phys. 6, 122 (2004), arXiv:hepph/0405172; G. L. Fogli, E. Lisi, A. Marrone, and A. Palazzo, “Global Analysis of Three-Flavor Neutrino Masses and Mixings,” Prog. Part. Nucl. Phys. 57, 742 (2006), arXiv:hep-ph/0506083; G. L. Fogli, E. Lisi, A. Marrone, A. Palazzo, and A. M. Rotunno, “Hints of θ13 > 0 from Global Neutrino Data Analysis,” Phys. Rev. Lett. 101, 141801 (2008), arXiv:0806.1649; T. Schwetz, M. Tórtola, and J. W. F. Valle, “Three-Flavour Neutrino Oscillation Update,” New J. Phys. 10, 113011 (2008), arXiv:0808.2016; M. C. Gonzalez-Garcia, M. Maltoni, and J. Salvado, “Updated Global Fit to Three Neutrino Mixing: Status of the Hints of θ13 > 0,” J. High Energy Phys. 04, 056 (2010), arXiv:1001.4524.ADSCrossRefGoogle Scholar
  4. 4.
    P. Minkowski, “μ → eγ at a Rate of One out of 109, Muon Decays?,” Phys. Lett. B 67, 421 (1977); T. Yanagida, in Proceedings of the Workshop on Unified Theory and Baryon Number in the Universe, Tsukuba, Japan, 1979, KEK Report Vol. 79-18; S. L. Glashow, Quarks and Leptons, Proceedings of the Advanced Study Institute, Cargèse, Corsica, 1979 (Plenum, New York, 1981); M. Gell-Mann, P. Ramond, and R. Slansky, Complex Spinors and Unified Theories. Supergravity (North-Holland, Amsterdam, 1979); R. N. Mohapatra and G. Senjanovic, “Neutrino Mass and Spontaneous Parity Violation,” Phys. Rev. Lett. 44, 912 (1980).ADSCrossRefGoogle Scholar
  5. 5.
    W. Grimus, “Neutrino Physics — Theory,” Lect. Notes Phys. 629, 169 (2004), arXiv:hep-ph/0307149.ADSCrossRefGoogle Scholar
  6. 6.
    G. Altarelli and F. Feruglio, “Discrete Flavor Symmetries and Models of Neutrino Mixing,” arXiv:1002.0211.Google Scholar
  7. 7.
    M. Hamermesh, Group Theory and its Application to Physical Problems (Addison-Wesley, Reading, MA, 1962).zbMATHGoogle Scholar
  8. 8.
    P. Ramond, Group Theory: A Physicists’s Survey (Cambridge Univ., Cambridge, UK, 2010).Google Scholar
  9. 9.
    P. O. Ludl, “Systematic Analysis of Finite Family Symmetry Groups and Their Applications to the Lepton Sector,” Diploma Thesis, arXiv:0907.5587.Google Scholar
  10. 10.
    H. Ishimori, T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu, and M. Tanimoto, “Non-Abelian Discrete Symmetries in Particle Physics,” Prog. Theor. Phys. Suppl. 183, 1 (2010), arXiv:1003.3552.ADSzbMATHCrossRefGoogle Scholar
  11. 11.
    W. Grimus and P. O. Ludl, “Principal Series of Finite Subgroups of SU(3),” J. Phys. A 43, 445209 (2010), arXiv:1006.0098.MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    P. O. Ludl, “On the Finite Subgroups of U(3) of Order Smaller than 512,” J. Phys. A 43, 395204 (2010) (arXiv:1006.1479).MathSciNetCrossRefGoogle Scholar
  13. 13.
    K. M. Parattu and A. Wingerter, “Tribimaximal Mixing from Small Groups,” arXiv:1012.2842.Google Scholar
  14. 14.
    W. Grimus, L. Lavoura, and D. Neubauer, “A Light Pseudoscalar in a Model with Lepton Family Symmetry O(2),” JHEP 07, 051 (2008), arXiv:0805.1175.ADSCrossRefGoogle Scholar
  15. 15.
    Y. Yamanaka, H. Sugawara, and S. Pakvasa, “Permutation Symmetries and the Fermion Mass Matrix,” Phys. Rev. D 25, 1895 (1982); Phys. Rev. D 29, 2135(E) (1984).ADSCrossRefGoogle Scholar
  16. 16.
    C. S. Lam, “Determining Horizontal Symmetry from Neutrino Mixing,” Phys. Rev. Lett. 101, 121602 (2008), arXiv:0804.2622.ADSCrossRefGoogle Scholar
  17. 17.
    W. Grimus, L. Lavoura, and P. O. Ludl, “Is S4 the Horizontal Symmetry of Tri-Bimaximal Lepton Mixing?” J. Phys. G 36, 115007 (2009), arXiv:0906.2689.ADSCrossRefGoogle Scholar
  18. 18.
    E. Ma and G. Rajasekaran, “Softly Broken A 4 Symmetry for Nearly Degenerate Neutrino Masses, Phys. Rev. D. 64, 113012 (2001), arXiv:hep-ph/0106291.ADSCrossRefGoogle Scholar
  19. 19.
    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, 207 (2003), arXiv:hep-ph/0206292; G. Altarelli and F. Feruglio, “Tri-Bimaximal Neutrino Mixing A 4 and the Modular Symmetry,” Nucl. Phys. B 741, 215 (2006), arXiv:hepph/0512103; 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, 153 (2007), arXiv:hepph/0512313.ADSCrossRefGoogle Scholar
  20. 20.
    X.-G. He, Y. Keum, and R. R. Volkas, “A4 Flavour Symmetry Breaking Scheme for Understanding Quark and Neutrino Mixing Angles,” JHEP 04, 039 (2006), arXiv:hep-ph/0601001.ADSCrossRefGoogle Scholar
  21. 21.
    D. Wyler, “Discrete Symmetries in the Six Quark SU(2) × U(1) Model,” Phys. Rev. D 19, 3369 (1979); G. C. Branco, H. P. Nilles, and V. Rittenberg, “Fermion Masses and Hierarchy in Symmetry Breaking,” Phys. Rev. D 21, 3417 (1980).ADSCrossRefGoogle Scholar
  22. 22.
    C. I. Low, “Abelian Family Symmetries and the Simplest Models that Give θ13 = 0° in the Neutrino Mixing Matrix,” Phys. Rev. D 71, 073007 (2005), arXiv:hepph/0501251.ADSCrossRefGoogle Scholar
  23. 23.
    W. Grimus, A. S. Joshipura, L. Lavoura, and M. Tanimoto, “Symmetry Realization of Texture Zeros,” Eur. Phys. J. C 36, 227 (2004), arXiv:hep-ph/0405016.ADSCrossRefGoogle Scholar
  24. 24.
    E. Ma, “Suitability of A 4 as a Family Symmetry in Grand Unification,” Mod. Phys. Lett. A 21, 2931 (2006), arXiv:hep-ph/0607190.ADSzbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  • W. Grimus
    • 1
  1. 1.Faculty of PhysicsUniversity of ViennaViennaAustria

Personalised recommendations