Tri-direct CP in the Littlest Seesaw playground

  • Gui-Jun Ding
  • Stephen F. King
  • Cai-Chang LiEmail author
Open Access
Regular Article - Theoretical Physics


We discuss spontaneously broken CP symmetry in two right-handed neutrino models based on the idea of having a different residual flavour symmetry, together with a different residual CP symmetry, associated with each of the two right-handed neutrinos. The charged lepton sector also has a different residual flavour symmetry. In such a tri-direct CP approach, we show that the combination of the three residual flavour and two residual CP symmetries provides a new way of fixing the parameters. To illustrate the approach, we revisit the Littlest Seesaw (LSS) model based on S4 and then propose new variants which have not so far appeared in the literature, with different predictions for each variant. We analyse numerically the predictions of the new variants, and then propose an explicit model which can realise one of the successful benchmark points, based on the atmospheric flavon vacuum alignment (1, ω2, ω) and the solar flavon vacuum alignment (1, −7/2, −7/2).


Discrete Symmetries Neutrino Physics 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    Particle Data Group collaboration, C. Patrignani et al., Review of Particle Physics, Chin. Phys. C 40 (2016) 100001 [INSPIRE].
  2. [2]
    F. Capozzi, E. Di Valentino, E. Lisi, A. Marrone, A. Melchiorri and A. Palazzo, Global constraints on absolute neutrino masses and their ordering, Phys. Rev. D 95 (2017) 096014 [arXiv:1703.04471] [INSPIRE].Google Scholar
  3. [3]
    P.F. de Salas, D.V. Forero, C.A. Ternes, M. Tortola and J.W.F. Valle, Status of neutrino oscillations 2018: 3σ hint for normal mass ordering and improved CP sensitivity, Phys. Lett. B 782 (2018) 633 [arXiv:1708.01186] [INSPIRE].CrossRefGoogle Scholar
  4. [4]
    I. Esteban, M.C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler and T. Schwetz, Updated fit to three neutrino mixing: exploring the accelerator-reactor complementarity, JHEP 01 (2017) 087 [arXiv:1611.01514] [INSPIRE].CrossRefGoogle Scholar
  5. [5]
    G. Altarelli and F. Feruglio, Discrete Flavor Symmetries and Models of Neutrino Mixing, Rev. Mod. Phys. 82 (2010) 2701 [arXiv:1002.0211] [INSPIRE].CrossRefGoogle Scholar
  6. [6]
    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].CrossRefzbMATHGoogle Scholar
  7. [7]
    S.F. King and C. Luhn, Neutrino Mass and Mixing with Discrete Symmetry, Rept. Prog. Phys. 76 (2013) 056201 [arXiv:1301.1340] [INSPIRE].CrossRefGoogle Scholar
  8. [8]
    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].CrossRefGoogle Scholar
  9. [9]
    S.F. King, Models of Neutrino Mass, Mixing and CP-violation, J. Phys. G 42 (2015) 123001 [arXiv:1510.02091] [INSPIRE].CrossRefGoogle Scholar
  10. [10]
    S.F. King, Unified Models of Neutrinos, Flavour and CP-violation, Prog. Part. Nucl. Phys. 94 (2017) 217 [arXiv:1701.04413] [INSPIRE].CrossRefGoogle Scholar
  11. [11]
    P. Minkowski, μeγ at a Rate of One Out of 109 Muon Decays?, Phys. Lett. B 67 (1977) 421 [INSPIRE].
  12. [12]
    M. Gell-Mann, P. Ramond and R. Slansky, Complex Spinors and Unified Theories, Conf. Proc. C 790927 (1979) 315 [arXiv:1306.4669] [INSPIRE].Google Scholar
  13. [13]
    T. Yanagida, Horizontal symmetry and masses of neutrinos, Conf. Proc. C 7902131 (1979) 95 [INSPIRE].Google Scholar
  14. [14]
    S.L. Glashow, The Future of Elementary Particle Physics, NATO Sci. Ser. B 61 (1980) 687 [INSPIRE].Google Scholar
  15. [15]
    R.N. Mohapatra and G. Senjanović, Neutrino Mass and Spontaneous Parity Violation, Phys. Rev. Lett. 44 (1980) 912 [INSPIRE].CrossRefzbMATHGoogle Scholar
  16. [16]
    J. Schechter and J.W.F. Valle, Neutrino Masses in SU(2) × U(1) Theories, Phys. Rev. D 22 (1980) 2227 [INSPIRE].Google Scholar
  17. [17]
    S.F. King, Atmospheric and solar neutrinos with a heavy singlet, Phys. Lett. B 439 (1998) 350 [hep-ph/9806440] [INSPIRE].
  18. [18]
    S.F. King, Large mixing angle MSW and atmospheric neutrinos from single right-handed neutrino dominance and U(1) family symmetry, Nucl. Phys. B 576 (2000) 85 [hep-ph/9912492] [INSPIRE].
  19. [19]
    S.F. King, Constructing the large mixing angle MNS matrix in seesaw models with right-handed neutrino dominance, JHEP 09 (2002) 011 [hep-ph/0204360] [INSPIRE].
  20. [20]
    P.H. Frampton, S.L. Glashow and T. Yanagida, Cosmological sign of neutrino CP-violation, Phys. Lett. B 548 (2002) 119 [hep-ph/0208157] [INSPIRE].
  21. [21]
    K. Harigaya, M. Ibe and T.T. Yanagida, Seesaw Mechanism with Occams Razor, Phys. Rev. D 86 (2012) 013002 [arXiv:1205.2198] [INSPIRE].Google Scholar
  22. [22]
    S.F. King, Minimal predictive see-saw model with normal neutrino mass hierarchy, JHEP 07 (2013) 137 [arXiv:1304.6264] [INSPIRE].CrossRefGoogle Scholar
  23. [23]
    S.F. King, Littlest Seesaw, JHEP 02 (2016) 085 [arXiv:1512.07531] [INSPIRE].CrossRefGoogle Scholar
  24. [24]
    S.F. King and C. Luhn, Littlest Seesaw model from S 4 × U(1), JHEP 09 (2016) 023 [arXiv:1607.05276] [INSPIRE].zbMATHGoogle Scholar
  25. [25]
    P. Ballett, S.F. King, S. Pascoli, N.W. Prouse and T. Wang, Precision neutrino experiments vs the Littlest Seesaw, JHEP 03 (2017) 110 [arXiv:1612.01999] [INSPIRE].CrossRefGoogle Scholar
  26. [26]
    F. Björkeroth, F.J. de Anda, I. de Medeiros Varzielas and S.F. King, Towards a complete A 4 × SU(5) SUSY GUT, JHEP 06 (2015) 141 [arXiv:1503.03306] [INSPIRE].CrossRefGoogle Scholar
  27. [27]
    F. Björkeroth, F.J. de Anda, I. de Medeiros Varzielas and S.F. King, Towards a complete Δ(27) × SO(10) SUSY GUT, Phys. Rev. D 94 (2016) 016006 [arXiv:1512.00850] [INSPIRE].Google Scholar
  28. [28]
    F. Björkeroth, F.J. de Anda, I. de Medeiros Varzielas and S.F. King, Leptogenesis in minimal predictive seesaw models, JHEP 10 (2015) 104 [arXiv:1505.05504] [INSPIRE].CrossRefGoogle Scholar
  29. [29]
    F. Björkeroth, F.J. de Anda, S.F. King and E. Perdomo, A natural S 4 × SO(10) model of flavour, JHEP 10 (2017) 148 [arXiv:1705.01555] [INSPIRE].CrossRefGoogle Scholar
  30. [30]
    G.-J. Ding, S.F. King and C.-C. Li, Golden Littlest Seesaw, Nucl. Phys. B 925 (2017) 470 [arXiv:1705.05307] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    F. Feruglio, C. Hagedorn and R. Ziegler, Lepton Mixing Parameters from Discrete and CP Symmetries, JHEP 07 (2013) 027 [arXiv:1211.5560] [INSPIRE].CrossRefGoogle Scholar
  32. [32]
    M. Holthausen, M. Lindner and M.A. Schmidt, CP and Discrete Flavour Symmetries, JHEP 04 (2013) 122 [arXiv:1211.6953] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    M.-C. Chen, M. Fallbacher, K.T. Mahanthappa, M. Ratz and A. Trautner, CP Violation from Finite Groups, Nucl. Phys. B 883 (2014) 267 [arXiv:1402.0507] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    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].CrossRefzbMATHGoogle Scholar
  35. [35]
    G.-J. Ding, S.F. King, C. Luhn and A.J. Stuart, Spontaneous CP-violation from vacuum alignment in S 4 models of leptons, JHEP 05 (2013) 084 [arXiv:1303.6180] [INSPIRE].CrossRefGoogle Scholar
  36. [36]
    G.-J. Ding, S.F. King and C.-C. Li, Lepton Mixing Predictions from S 4 in the Tri-Direct CP approach to Two Right-handed Neutrino Models, arXiv:1811.12340 [INSPIRE].
  37. [37]
    C.-C. Li and G.-J. Ding, Generalised CP and trimaximal TM 1 lepton mixing in S 4 family symmetry, Nucl. Phys. B 881 (2014) 206 [arXiv:1312.4401] [INSPIRE].CrossRefzbMATHGoogle Scholar
  38. [38]
    C.-Y. Yao and G.-J. Ding, CP Symmetry and Lepton Mixing from a Scan of Finite Discrete Groups, Phys. Rev. D 94 (2016) 073006 [arXiv:1606.05610] [INSPIRE].Google Scholar
  39. [39]
    G.-J. Ding, S.F. King and A.J. Stuart, Generalised CP and A 4 Family Symmetry, JHEP 12 (2013) 006 [arXiv:1307.4212] [INSPIRE].CrossRefGoogle Scholar
  40. [40]
    C. Jarlskog, Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP-violation, Phys. Rev. Lett. 55 (1985) 1039 [INSPIRE].CrossRefGoogle Scholar
  41. [41]
    G.C. Branco, L. Lavoura and M.N. Rebelo, Majorana Neutrinos and CP Violation in the Leptonic Sector, Phys. Lett. B 180 (1986) 264 [INSPIRE].CrossRefGoogle Scholar
  42. [42]
    J.F. Nieves and P.B. Pal, Minimal Rephasing Invariant CP Violating Parameters With Dirac and Majorana Fermions, Phys. Rev. D 36 (1987) 315 [INSPIRE].Google Scholar
  43. [43]
    E.E. Jenkins and A.V. Manohar, Rephasing Invariants of Quark and Lepton Mixing Matrices, Nucl. Phys. B 792 (2008) 187 [arXiv:0706.4313] [INSPIRE].CrossRefGoogle Scholar
  44. [44]
    G.C. Branco, R.G. Felipe and F.R. Joaquim, Leptonic CP-violation, Rev. Mod. Phys. 84 (2012) 515 [arXiv:1111.5332] [INSPIRE].CrossRefGoogle Scholar
  45. [45]
    JUNO collaboration, F. An et al., Neutrino Physics with JUNO, J. Phys. G 43 (2016) 030401 [arXiv:1507.05613] [INSPIRE].
  46. [46]
    T2K collaboration, K. Abe et al., Combined Analysis of Neutrino and Antineutrino Oscillations at T2K, Phys. Rev. Lett. 118 (2017) 151801 [arXiv:1701.00432] [INSPIRE].
  47. [47]
    NOvA collaboration, P. Adamson et al., Constraints on Oscillation Parameters from ν e Appearance and ν μ Disappearance in NOvA, Phys. Rev. Lett. 118 (2017) 231801 [arXiv:1703.03328] [INSPIRE].
  48. [48]
    S.F. King and C.C. Nishi, Mu-tau symmetry and the Littlest Seesaw, Phys. Lett. B 785 (2018) 391 [arXiv:1807.00023] [INSPIRE].CrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Interdisciplinary Center for Theoretical Study and Department of Modern PhysicsUniversity of Science and Technology of ChinaHefeiChina
  2. 2.Physics and AstronomyUniversity of SouthamptonSouthamptonU.K.

Personalised recommendations