Skip to main content
SpringerLink
Log in

CP and discrete flavour symmetries

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We give a consistent definition of generalised CP transformations in the context of discrete flavour symmetries. Non-trivial consistency conditions imply that every generalised CP transformation can be interpreted as a representation of an automorphism of the discrete group. This allows us to give consistent generalised CP transformations of popular flavour groups. We are able to clear up issues concerning recent claims about geometrical CP violation in models based on T ′, clarify the origin of ”calculable phases” in Δ(27) and explain why apparently CP violating scalar potentials of A 4 result in a CP conserving ground state.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. DOUBLE-CHOOZ collaboration, Y. Abe et al., Indication for the disappearance of reactor electron antineutrinos in the Double CHOOZ experiment, Phys. Rev. Lett. 108 (2012) 131801 [arXiv:1112.6353] [INSPIRE].

    Article  ADS  Google Scholar 

  2. DAYA-BAY collaboration, F. An et al., Observation of electron-antineutrino disappearance at Daya Bay, Phys. Rev. Lett. 108 (2012) 171803 [arXiv:1203.1669] [INSPIRE].

    Article  ADS  Google Scholar 

  3. RENO collaboration, J. Ahn et al., Observation of Reactor Electron Antineutrino Disappearance in the RENO Experiment, Phys. Rev. Lett. 108 (2012) 191802 [arXiv:1204.0626] [INSPIRE].

    Article  ADS  Google Scholar 

  4. G. Branco, J. Gerard and W. Grimus, GEOMETRICAL T VIOLATION, Phys. Lett. B 136 (1984) 383 [INSPIRE].

    ADS  Google Scholar 

  5. I. de Medeiros Varzielas and D. Emmanuel-Costa, Geometrical CP-violation, Phys. Rev. D 84 (2011) 117901 [arXiv:1106.5477] [INSPIRE].

    ADS  Google Scholar 

  6. I. de Medeiros Varzielas, D. Emmanuel-Costa and P. Leser, Geometrical CP-violation from Non-Renormalisable Scalar Potentials, Phys. Lett. B 716 (2012) 193 [arXiv:1204.3633] [INSPIRE].

    Article  ADS  Google Scholar 

  7. I. de Medeiros Varzielas, Geometrical CP-violation in multi-Higgs models, JHEP 08 (2012) 055 [arXiv:1205.3780] [INSPIRE].

    Article  Google Scholar 

  8. G. Bhattacharyya, I. de Medeiros Varzielas and P. Leser, A common origin of fermion mixing and geometrical CP-violation and its test through Higgs physics at the LHC, Phys. Rev. Lett. 109 (2012) 241603 [arXiv:1210.0545] [INSPIRE].

    Article  ADS  Google Scholar 

  9. P. Ferreira, W. Grimus, L. Lavoura and P. Ludl, Maximal CP-violation in Lepton Mixing from a Model with Delta(27) flavour Symmetry, JHEP 09 (2012) 128 [arXiv:1206.7072] [INSPIRE].

    Article  ADS  Google Scholar 

  10. M.-C. Chen and K. Mahanthappa, Group Theoretical Origin of CP-violation, Phys. Lett. B 681 (2009) 444 [arXiv:0904.1721] [INSPIRE].

    Article  ADS  Google Scholar 

  11. A. Meroni, S. Petcov and M. Spinrath, A SUSY SU(5)xT’ Unified Model of Flavour with large θ 13, Phys. Rev. D 86 (2012) 113003 [arXiv:1205.5241] [INSPIRE].

    ADS  Google Scholar 

  12. T. Lee, A Theory of Spontaneous T Violation, Phys. Rev. D 8 (1973) 1226 [INSPIRE].

    ADS  Google Scholar 

  13. G.C. Branco, Spontaneous CP-violation in Theories with More Than Four Quarks, Phys. Rev. Lett. 44 (1980) 504 [INSPIRE].

    Article  ADS  Google Scholar 

  14. H.E. Haber and Z. Surujon, A Group-theoretic Condition for Spontaneous CP-violation, Phys. Rev. D 86 (2012) 075007 [arXiv:1201.1730] [INSPIRE].

    ADS  Google Scholar 

  15. G. Ecker, W. Grimus and W. Konetschny, Quark mass matrices in left-right symmetric gauge theories, Nucl. Phys. B 191 (1981) 465 [INSPIRE].

    Article  ADS  Google Scholar 

  16. G. Ecker, W. Grimus and H. Neufeld, Spontaneous CP-violation in left-right symmetric gauge theories, Nucl. Phys. B 247 (1984) 70 [INSPIRE].

    Article  ADS  Google Scholar 

  17. H. Neufeld, W. Grimus and G. Ecker, Generalized CP invariance, neutral flavor conservation and the structure of the mixing matrix, Int. J. Mod. Phys. A 3 (1988) 603 [INSPIRE].

    Article  ADS  Google Scholar 

  18. W. Grimus and M. Rebelo, Automorphisms in gauge theories and the definition of CP and P, Phys. Rept. 281 (1997) 239 [hep-ph/9506272] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  19. P. Harrison and W. Scott, μ − τ reflection symmetry in lepton mixing and neutrino oscillations, Phys. Lett. B 547 (2002) 219 [hep-ph/0210197] [INSPIRE].

    Article  ADS  Google Scholar 

  20. P. Harrison and W. Scott, Symmetries and generalizations of tri - bimaximal neutrino mixing, Phys. Lett. B 535 (2002) 163 [hep-ph/0203209] [INSPIRE].

    Article  ADS  Google Scholar 

  21. P. Harrison and W. Scott, The Simplest neutrino mass matrix, Phys. Lett. B 594 (2004) 324 [hep-ph/0403278] [INSPIRE].

    Article  ADS  Google Scholar 

  22. W. Grimus and L. Lavoura, A Nonstandard CP transformation leading to maximal atmospheric neutrino mixing, Phys. Lett. B 579 (2004) 113 [hep-ph/0305309] [INSPIRE].

    Article  ADS  Google Scholar 

  23. Y. Farzan and A.Y. Smirnov, Leptonic CP-violation: Zero, maximal or between the two extremes, JHEP 01 (2007) 059 [hep-ph/0610337] [INSPIRE].

    Article  ADS  Google Scholar 

  24. A.S. Joshipura, B.P. Kodrani and K.M. Patel, Fermion Masses and Mixings in a mu-tau symmetric SO(10), Phys. Rev. D 79 (2009) 115017 [arXiv:0903.2161] [INSPIRE].

    ADS  Google Scholar 

  25. W. Grimus and L. Lavoura, mu-tau Interchange symmetry and lepton mixing, arXiv:1207.1678 [INSPIRE].

  26. R. Mohapatra and C. Nishi, S 4 Flavored CP Symmetry for Neutrinos, Phys. Rev. D 86 (2012) 073007 [arXiv:1208.2875] [INSPIRE].

    ADS  Google Scholar 

  27. R. Krishnan, P. Harrison and W. Scott, Simplest Neutrino Mixing from S4 Symmetry, arXiv:1211.2000 [INSPIRE].

  28. F. Feruglio, C. Hagedorn and R. Ziegler, Lepton Mixing Parameters from Discrete and CP Symmetries, arXiv:1211.5560 [INSPIRE].

  29. K.S. Babu and J. Kubo, Dihedral families of quarks, leptons and Higgses, Phys. Rev. D 71 (2005) 056006 [hep-ph/0411226] [INSPIRE].

    ADS  Google Scholar 

  30. K. Babu, K. Kawashima and J. Kubo, Variations on the Supersymmetric Q 6 Model of Flavor, Phys. Rev. D 83 (2011) 095008 [arXiv:1103.1664] [INSPIRE].

    ADS  Google Scholar 

  31. GAP, GAP - Groups, Algorithms, and Programming, Version 4.5.5, The GAP Group (2012), http://www.gap-system.org).

  32. H.U.Besche, B.Eick, and E.O’Brien, SmallGroups - library of all ’small’ groups, GAP package, Version included in GAP 4.5.5, The GAP Group (2002), http://www.gap-system.org/Packages/sgl.html.

  33. W. Fairbairn and T. Fulton, Some comments on finite subgroups of SU(3), J. Math. Phys. 23 (1982) 1747 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. W. Grimus and P.O. Ludl, Finite flavour groups of fermions, J. Phys. A 45 (2012) 233001 [arXiv:1110.6376] [INSPIRE].

    ADS  Google Scholar 

  35. H. Ishimori et al., An introduction to non-Abelian discrete symmetries for particle physicists, Lect. Notes Phys. 858 (2012) 1.

    MathSciNet  Google Scholar 

  36. G. Branco et al., Theory and phenomenology of two-Higgs-doublet models, Phys. Rept. 516 (2012) 1 [arXiv:1106.0034] [INSPIRE].

    Article  ADS  Google Scholar 

  37. G. Ecker, W. Grimus and H. Neufeld, A standard form for generalized CP transformations, J. Phys. A 20 (1987) L807 [INSPIRE].

    ADS  Google Scholar 

  38. M. Gronau, A. Kfir and R. Loewy, Basis independent tests of CP-violation in fermion mass matrices, Phys. Rev. Lett. 56 (1986) 1538 [INSPIRE].

    Article  ADS  Google Scholar 

  39. J. Bernabeu, G. Branco and M. Gronau, CP restrictions on quark mass matrices, Phys. Lett. B 169 (1986) 243 [INSPIRE].

    Article  ADS  Google Scholar 

  40. G. Branco, L. Lavoura and M. Rebelo, Majorana neutrinos and CP-violation in the leptonic sector, Phys. Lett. B 180 (1986) 264 [INSPIRE].

    Article  ADS  Google Scholar 

  41. 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].

    ADS  Google Scholar 

  42. R. de Adelhart Toorop, F. Bazzocchi, L. Merlo and A. Paris, Constraining Flavour Symmetries At The EW Scale I: The A4 Higgs Potential, JHEP 03 (2011) 035 [Erratum ibid. 1301 (2013) 098] [arXiv:1012.1791] [INSPIRE].

  43. P. Ferreira and L. Lavoura, Seesaw Neutrino Masses from an A 4 Model with Two Equal Vacuum expectation values, arXiv:1111.5859 [INSPIRE].

  44. A. Machado, J. Montero and V. Pleitez, Three-Higgs-doublet model with A 4 symmetry, Phys. Lett. B 697 (2011) 318 [arXiv:1011.5855] [INSPIRE].

    Article  ADS  Google Scholar 

  45. R. de Adelhart Toorop, F. Bazzocchi, L. Merlo and A. Paris, Constraining Flavour Symmetries At The EW Scale I: The A4 Higgs Potential, JHEP 03 (2011) 035 [Erratum ibid. 1301 (2013) 098] [arXiv:1012.1791] [INSPIRE].

  46. 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].

    Article  MathSciNet  ADS  Google Scholar 

  47. R. de Adelhart Toorop, A Flavour of family symmetries in a family of flavour models, PhD. thesis, Nikhef (2012).

  48. I. Ivanov and E. Vdovin, Classification of finite reparametrization symmetry groups in the three-Higgs-doublet model, Eur. Phys. J. C 73 (2013) 2309 [arXiv:1210.6553] [INSPIRE].

    Article  ADS  Google Scholar 

  49. A. Degee, I.P. Ivanov and V. Keus, Geometric minimization of highly symmetric potentials, JHEP 02 (2013) 125 [arXiv:1211.4989] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  50. P. Frampton and T. Kephart, Simple nonAbelian finite flavor groups and fermion masses, Int. J. Mod. Phys. A 10 (1995) 4689 [hep-ph/9409330] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  51. 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].

  52. S. Antusch, S.F. King, C. Luhn and M. Spinrath, Right Unitarity Triangles and Tri-Bimaximal Mixing from Discrete Symmetries and Unification, Nucl. Phys. B 850 (2011) 477 [arXiv:1103.5930] [INSPIRE].

    Article  ADS  Google Scholar 

  53. 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].

    Article  ADS  Google Scholar 

  54. K.M. Parattu and A. Wingerter, Tribimaximal Mixing From Small Groups, Phys. Rev. D 84 (2011) 013011 [arXiv:1012.2842] [INSPIRE].

    ADS  Google Scholar 

  55. M. Holthausen and M.A. Schmidt, Natural Vacuum Alignment from Group Theory: The Minimal Case, JHEP 01 (2012) 126 [arXiv:1111.1730] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  56. M. Holthausen, M. Lindner and M.A. Schmidt, Lepton flavor at the electroweak scale: A complete A 4 model, Phys. Rev. D 87 (2013) 033006 [arXiv:1211.5143] [INSPIRE].

    ADS  Google Scholar 

  57. S. Pakvasa and H. Sugawara, Discrete Symmetry and Cabibbo Angle, Phys. Lett. B 73 (1978) 61 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  58. S. Pakvasa and H. Sugawara, Mass of the t Quark in SU(2) x U(1), Phys. Lett. B 82 (1979) 105 [INSPIRE].

    Article  ADS  Google Scholar 

  59. S.L. Adler, Model for particle masses, flavor mixing and CP-violation, based on spontaneously broken discrete chiral symmetry as the origin of families, Phys. Rev. D 59 (1999) 015012 [Erratum ibid. D 59 (1999) 099902] [hep-ph/9806518] [INSPIRE].

  60. 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].

    Article  ADS  Google Scholar 

  61. C. Hagedorn, M.A. Schmidt and A.Y. Smirnov, Lepton Mixing and Cancellation of the Dirac Mass Hierarchy in SO(10) GUTs with Flavor Symmetries T(7) and Sigma(81), Phys. Rev. D 79 (2009) 036002 [arXiv:0811.2955] [INSPIRE].

    ADS  Google Scholar 

  62. I. de Medeiros Varzielas, S. King and 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].

    Article  ADS  Google Scholar 

  63. C. Luhn, S. Nasri and P. Ramond, The Flavor group Δ(3n 2), J. Math. Phys. 48 (2007) 073501 [hep-th/0701188] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  64. J. Escobar and C. Luhn, The Flavor Group Δ(6n 2), J. Math. Phys. 50 (2009) 013524 [arXiv:0809.0639] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Max-Planck Institut für Kernphysik, Saupfercheckweg 1, 69117, Heidelberg, Germany

    Martin Holthausen & Manfred Lindner

  2. ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University of Melbourne, Victoria, 3010, Australia

    Michael A. Schmidt

Authors
  1. Martin Holthausen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manfred Lindner
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael A. Schmidt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Holthausen.

Additional information

ArXiv ePrint: 1211.6953

Rights and permissions

Reprints and permissions

About this article

Cite this article

Holthausen, M., Lindner, M. & Schmidt, M.A. CP and discrete flavour symmetries. J. High Energ. Phys. 2013, 122 (2013). https://doi.org/10.1007/JHEP04(2013)122

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP04(2013)122

Keywords

Search

Navigation