Phenomenological implications of the Friedberg-Lee transformation in a neutrino mass model with μτ-flavored CP symmetry

  • Roopam SinhaEmail author
  • Sukannya Bhattacharya
  • Rome Samanta
Open Access
Regular Article - Theoretical Physics


We propose a neutrino mass model with μτ-flavored CP symmetry, where the effective light neutrino Lagrangian enjoys an additional invariance under a Friedberg-Lee (FL) transformation on the left-handed flavor neutrino fields that leads to a highly predictive and testable scenario. While both types of the light neutrino mass ordering, i.e., Normal Ordering (NO) as well as the Inverted Ordering (IO) are allowed, the absolute scale of neutrino masses is fixed by the vanishing determinant of light Majorana neutrino mass matrix Mν. We show that for both types of mass ordering, whilst the atmospheric mixing angle θ23 is in general nonmaximal (θ23π/4), the Dirac CP phase δ is exactly maximal (δ = π/2, 3π/2) for IO and nearly maximal for NO owing to cos δ ∝ sin θ13. For the NO, very tiny nonvanishing Majorana CP violation might appear through one of the Majorana phases β; otherwise the model predicts vanishing Majorana CP violation. Thus, despite the fact, that from the measurement of θ23, it is difficult to rule out the model, any large deviation of δ from its maximality, will surely falsify the scenario. For a comprehensive numerical analysis, beside fitting the neutrino oscillation global fit data, we also present a study on the νμνe oscillation which is expected to show up Dirac CP violation in different long baseline experiments. Finally, assuming purely astrophysical sources, we calculate the Ultra High Energy (UHE) neutrino flavor flux ratios at neutrino telescopes, such as IceCube, from which statements on the octant of θ23 could be made in our model.


CP violation Neutrino Physics 


Open Access

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  1. [1]
    S.F. King, Models of Neutrino Mass, Mixing and CP-violation, J. Phys. G 42 (2015) 123001 [arXiv:1510.02091] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    Planck collaboration, Planck intermediate results. XLVI. Reduction of large-scale systematic effects in HFI polarization maps and estimation of the reionization optical depth, Astron. Astrophys. 596 (2016) A107 [arXiv:1605.02985] [INSPIRE].
  3. [3]
    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].ADSCrossRefGoogle Scholar
  4. [4]
  5. [5]
    T2K collaboration, Combined Analysis of Neutrino and Antineutrino Oscillations at T2K, Phys. Rev. Lett. 118 (2017) 151801 [arXiv:1701.00432] [INSPIRE].
  6. [6]
    NOvA collaboration, Measurement of the neutrino mixing angle θ 23 in NOvA, Phys. Rev. Lett. 118 (2017) 151802 [arXiv:1701.05891] [INSPIRE].
  7. [7]
    NOvA collaboration, Constraints on Oscillation Parameters from ν e Appearance and ν μ Disappearance in NOvA, Phys. Rev. Lett. 118 (2017) 231801 [arXiv:1703.03328] [INSPIRE].
  8. [8]
    A. Himmel, New neutrino oscillation results from NOVA, (2018) [].
  9. [9]
    KamLAND-Zen collaboration, Search for double-beta decay of 136 Xe to excited states of 136 Ba with the KamLAND-Zen experiment, Nucl. Phys. A 946 (2016) 171 [arXiv:1509.03724] [INSPIRE].
  10. [10]
    GERDA collaboration, Results on Neutrinoless Double-β Decay of 76 Ge from Phase I of the GERDA Experiment, Phys. Rev. Lett. 111 (2013) 122503 [arXiv:1307.4720] [INSPIRE].
  11. [11]
    GERDA collaboration, The search for 0νββ decay with the GERDA experiment: Status and prospects, AIP Conf. Proc. 1672 (2015) 110003 [arXiv:1506.00415] [INSPIRE].
  12. [12]
    G. Altarelli and F. Feruglio, Discrete Flavor Symmetries and Models of Neutrino Mixing, Rev. Mod. Phys. 82 (2010) 2701 [arXiv:1002.0211] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    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
  14. [14]
    S.F. King, Unified Models of Neutrinos, Flavour and CP-violation, Prog. Part. Nucl. Phys. 94 (2017) 217 [arXiv:1701.04413] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    S.T. Petcov, Discrete Flavour Symmetries, Neutrino Mixing and Leptonic CP-violation, Eur. Phys. J. C 78 (2018) 709 [arXiv:1711.10806] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    R.N. Mohapatra and S. Nussinov, Bimaximal neutrino mixing and neutrino mass matrix, Phys. Rev. D 60 (1999) 013002 [hep-ph/9809415] [INSPIRE].
  17. [17]
    C.S. Lam, A 2-3 symmetry in neutrino oscillations, Phys. Lett. B 507 (2001) 214 [hep-ph/0104116] [INSPIRE].
  18. [18]
    E. Ma and M. Raidal, Neutrino mass, muon anomalous magnetic moment and lepton flavor nonconservation, Phys. Rev. Lett. 87 (2001) 011802 [Erratum ibid. 87 (2001) 159901] [hep-ph/0102255] [INSPIRE].
  19. [19]
    K.R.S. Balaji, W. Grimus and T. Schwetz, The Solar LMA neutrino oscillation solution in the Zee model, Phys. Lett. B 508 (2001) 301 [hep-ph/0104035] [INSPIRE].
  20. [20]
    T. Fukuyama and H. Nishiura, Mass matrix of Majorana neutrinos, hep-ph/9702253 [INSPIRE].
  21. [21]
    T. Fukuyama, Twenty years after the discovery of μ-τ symmetry, PTEP 2017 (2017) 033B11 [arXiv:1701.04985] [INSPIRE].
  22. [22]
    Daya Bay collaboration, New Measurement of Antineutrino Oscillation with the Full Detector Configuration at Daya Bay, Phys. Rev. Lett. 115 (2015) 111802 [arXiv:1505.03456] [INSPIRE].
  23. [23]
    G. Ecker, W. Grimus and H. Neufeld, A Standard Form for Generalized CP Transformations, J. Phys. A 20 (1987) L807 [INSPIRE].ADSGoogle Scholar
  24. [24]
    W. Grimus and M.N. Rebelo, Automorphisms in gauge theories and the definition of CP and P, Phys. Rept. 281 (1997) 239 [hep-ph/9506272] [INSPIRE].
  25. [25]
    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].
  26. [26]
    P.F. Harrison and W.G. Scott, μ-τ reflection symmetry in lepton mixing and neutrino oscillations, Phys. Lett. B 547 (2002) 219 [hep-ph/0210197] [INSPIRE].
  27. [27]
    R.N. Mohapatra and C.C. Nishi, S 4 Flavored CP Symmetry for Neutrinos, Phys. Rev. D 86 (2012) 073007 [arXiv:1208.2875] [INSPIRE].ADSGoogle Scholar
  28. [28]
    S. Gupta, A.S. Joshipura and K.M. Patel, Minimal extension of tri-bimaximal mixing and generalized Z 2 × Z 2 symmetries, Phys. Rev. D 85 (2012) 031903 [arXiv:1112.6113] [INSPIRE].ADSGoogle Scholar
  29. [29]
    F. Feruglio, C. Hagedorn and R. Ziegler, Lepton Mixing Parameters from Discrete and CP Symmetries, JHEP 07 (2013) 027 [arXiv:1211.5560] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    M. Holthausen, M. Lindner and M.A. Schmidt, CP and Discrete Flavour Symmetries, JHEP 04 (2013) 122 [arXiv:1211.6953] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    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].ADSCrossRefGoogle Scholar
  33. [33]
    F. Feruglio, C. Hagedorn and R. Ziegler, A realistic pattern of lepton mixing and masses from S 4 and CP, Eur. Phys. J. C 74 (2014) 2753 [arXiv:1303.7178] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    C.C. Nishi and B.L. Sánchez-Vega, Mu-tau reflection symmetry with a texture-zero, JHEP 01 (2017) 068 [arXiv:1611.08282] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    W. Rodejohann and X.-J. Xu, Trimaximal μ-τ reflection symmetry, Phys. Rev. D 96 (2017) 055039 [arXiv:1705.02027] [INSPIRE].ADSGoogle Scholar
  36. [36]
    J.T. Penedo, S.T. Petcov and A.V. Titov, Neutrino mixing and leptonic CP-violation from S 4 flavour and generalised CP symmetries, JHEP 12 (2017) 022 [arXiv:1705.00309] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    R. Samanta, P. Roy and A. Ghosal, Consequences of minimal seesaw with complex μτ antisymmetry of neutrinos, JHEP 06 (2018) 085 [arXiv:1712.06555] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    R. Sinha, P. Roy and A. Ghosal, CP transformed mixed μτ antisymmetry for neutrinos and its consequences, Phys. Rev. D 99 (2019) 033009 [arXiv:1809.06615] [INSPIRE].Google Scholar
  39. [39]
    R. Samanta, P. Roy and A. Ghosal, Extended scaling and residual flavor symmetry in the neutrino Majorana mass matrix, Eur. Phys. J. C 76 (2016) 662 [arXiv:1604.06731] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    R. Samanta, P. Roy and A. Ghosal, Complex Scaling in Neutrino Mass Matrix, Acta Phys. Polon. Supp. 9 (2016) 807 [arXiv:1604.01206] [INSPIRE].CrossRefGoogle Scholar
  41. [41]
    R. Samanta, M. Chakraborty, P. Roy and A. Ghosal, Baryon asymmetry via leptogenesis in a neutrino mass model with complex scaling, JCAP 03 (2017) 025 [arXiv:1610.10081] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    R. Sinha, R. Samanta and A. Ghosal, Generalized2 × ℤ2 in scaling neutrino Majorana mass matrix and baryogenesis via flavored leptogenesis, JHEP 12 (2017) 030 [arXiv:1706.00946] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    N. Nath, Z.-z. Xing and J. Zhang, μ-τ Reflection Symmetry Embedded in Minimal Seesaw, Eur. Phys. J. C 78 (2018) 289 [arXiv:1801.09931] [INSPIRE].
  44. [44]
    N. Nath, μ-τ Reflection Symmetry and Its Explicit Breaking for Leptogenesis in a Minimal Seesaw Model, arXiv:1808.05062 [INSPIRE].
  45. [45]
    C.C. Nishi, B.L. Sánchez-Vega and G. Souza Silva, μτ reflection symmetry with a high scale texture-zero, JHEP 09 (2018) 042 [arXiv:1806.07412] [INSPIRE].
  46. [46]
    M.H. Rahat, P. Ramond and B. Xu, Asymmetric tribimaximal texture, Phys. Rev. D 98 (2018) 055030 [arXiv:1805.10684] [INSPIRE].ADSGoogle Scholar
  47. [47]
    E. Ma, Neutrino mixing: A 4 variations, Phys. Lett. B 752 (2016) 198 [arXiv:1510.02501] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    R. Samanta, R. Sinha and A. Ghosal, Importance of generalized μτ symmetry and its CP extension on neutrino mixing and leptogenesis, arXiv:1805.10031 [INSPIRE].
  49. [49]
    P. Chen, G.-J. Ding, F. Gonzalez-Canales and J.W.F. Valle, Generalized μ-τ reflection symmetry and leptonic CP-violation, Phys. Lett. B 753 (2016) 644 [arXiv:1512.01551] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    R. Friedberg and T.D. Lee, A Possible Relation between the Neutrino Mass Matrix and the Neutrino Mapping Matrix, HEPNP 30 (2006) 591 [hep-ph/0606071] [INSPIRE].
  51. [51]
    Z.-z. Xing, H. Zhang and S. Zhou, Nearly Tri-bimaximal Neutrino Mixing and CP-violation from mu-tau Symmetry Breaking, Phys. Lett. B 641 (2006) 189 [hep-ph/0607091] [INSPIRE].
  52. [52]
    S. Luo and Z.-z. Xing, Friedberg-Lee Symmetry Breaking and Its Prediction for θ 13, Phys. Lett. B 646 (2007) 242 [hep-ph/0611360] [INSPIRE].
  53. [53]
    C.-S. Huang, T.-j. Li, W. Liao and S.-H. Zhu, Generalization of Friedberg-Lee Symmetry, Phys. Rev. D 78 (2008) 013005 [arXiv:0803.4124] [INSPIRE].ADSGoogle Scholar
  54. [54]
    X.-G. He and W. Liao, The Friedberg-Lee Symmetry and Minimal Seesaw Model, Phys. Lett. B 681 (2009) 253 [arXiv:0909.1463] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    Z.-h. Zhao, Modified Friedberg-Lee symmetry for neutrino mixing, Phys. Rev. D 92 (2015) 113001 [arXiv:1509.06915] [INSPIRE].ADSGoogle Scholar
  56. [56]
    T. Araki and R. Takahashi, Tri-Bimaximal Mixing from Twisted Friedberg-Lee Symmetry, Eur. Phys. J. C 63 (2009) 521 [arXiv:0811.0905] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    T. Araki and C.Q. Geng, Leptogenesis in model with Friedberg-Lee symmetry, Phys. Lett. B 680 (2009) 343 [arXiv:0906.1903] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    W. Grimus, A.S. Joshipura, S. Kaneko, L. Lavoura, H. Sawanaka and M. Tanimoto, Non-vanishing U e3 and cos2θ 23 from a broken Z 2 symmetry, Nucl. Phys. B 713 (2005) 151 [hep-ph/0408123] [INSPIRE].
  59. [59]
    IceCube collaboration, First observation of PeV-energy neutrinos with IceCube, Phys. Rev. Lett. 111 (2013) 021103 [arXiv:1304.5356] [INSPIRE].
  60. [60]
    IceCube collaboration, Evidence for High-Energy Extraterrestrial Neutrinos at the IceCube Detector, Science 342 (2013) 1242856 [arXiv:1311.5238] [INSPIRE].
  61. [61]
    IceCube collaboration, Observation of High-Energy Astrophysical Neutrinos in Three Years of IceCube Data, Phys. Rev. Lett. 113 (2014) 101101 [arXiv:1405.5303] [INSPIRE].
  62. [62]
    IceCube collaboration, The IceCube Neutrino ObservatoryContributions to ICRC 2015 Part II: Atmospheric and Astrophysical Diffuse Neutrino Searches of All Flavors, in Proceedings, 34th International Cosmic Ray Conference (ICRC 2015), The Hague, The Netherlands, July 30–August 6, 2015 (2015) [arXiv:1510.05223] [INSPIRE].
  63. [63]
    IceCube collaboration, The IceCube Neutrino ObservatoryContributions to ICRC 2017 Part II: Properties of the Atmospheric and Astrophysical Neutrino Flux, arXiv:1710.01191 [INSPIRE].
  64. [64]
  65. [65]
    J.G. Learned and S. Pakvasa, Detecting tau-neutrino oscillations at PeV energies, Astropart. Phys. 3 (1995) 267 [hep-ph/9405296] [INSPIRE].
  66. [66]
    S. Pakvasa, W. Rodejohann and T.J. Weiler, Flavor Ratios of Astrophysical Neutrinos: Implications for Precision Measurements, JHEP 02 (2008) 005 [arXiv:0711.4517] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    W. Rodejohann, Neutrino Mixing and Neutrino Telescopes, JCAP 01 (2007) 029 [hep-ph/0612047] [INSPIRE].
  68. [68]
    Z.-z. Xing and S. Zhou, Implications of Leptonic Unitarity Violation at Neutrino Telescopes, Phys. Lett. B 666 (2008) 166 [arXiv:0804.3512] [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    Particle Data Group collaboration, Review of Particle Physics, Phys. Rev. D 98 (2018) 030001 [INSPIRE].
  70. [70]
    Planck collaboration, Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209 [INSPIRE].
  71. [71]
    M. Agostini, G. Benato and J. Detwiler, Discovery probability of next-generation neutrinoless double-β decay experiments, Phys. Rev. D 96 (2017) 053001 [arXiv:1705.02996] [INSPIRE].ADSGoogle Scholar
  72. [72]
    CMB-S4 collaboration, CMB-S4 Science Book, First Edition, arXiv:1610.02743 [INSPIRE].
  73. [73]
    LSST Science and LSST Project collaborations, LSST Science Book, Version 2.0, arXiv:0912.0201 [INSPIRE].
  74. [74]
    M. Lattanzi and M. Gerbino, Status of neutrino properties and future prospectsCosmological and astrophysical constraints, Front. Phys. 5 (2018) 70 [arXiv:1712.07109] [INSPIRE].CrossRefGoogle Scholar
  75. [75]
    D. Spergel et al., Wide-Field InfrarRed Survey Telescope-Astrophysics Focused Telescope Assets WFIRST-AFTA 2015 Report, arXiv:1503.03757 [INSPIRE].
  76. [76]
    H. Nunokawa, S.J. Parke and J.W.F. Valle, CP Violation and Neutrino Oscillations, Prog. Part. Nucl. Phys. 60 (2008) 338 [arXiv:0710.0554] [INSPIRE].ADSCrossRefGoogle Scholar
  77. [77]
  78. [78]
    ANTARES and IceCube collaborations, The First Combined Search for Neutrino Point-sources in the Southern Hemisphere With the Antares and IceCube Neutrino Telescopes, Astrophys. J. 823 (2016) 65 [arXiv:1511.02149] [INSPIRE].
  79. [79]
    J.K. Becker, High-energy neutrinos in the context of multimessenger physics, Phys. Rept. 458 (2008) 173 [arXiv:0710.1557] [INSPIRE].ADSCrossRefGoogle Scholar
  80. [80]
    Y. Sui and P.S. Bhupal Dev, A Combined Astrophysical and Dark Matter Interpretation of the IceCube HESE and Throughgoing Muon Events, JCAP 07 (2018) 020 [arXiv:1804.04919] [INSPIRE].ADSCrossRefGoogle Scholar
  81. [81]
    M. Ahlers and F. Halzen, High-energy cosmic neutrino puzzle: a review, Rept. Prog. Phys. 78 (2015) 126901 [INSPIRE].ADSCrossRefGoogle Scholar
  82. [82]
    S. Hummer, M. Ruger, F. Spanier and W. Winter, Simplified models for photohadronic interactions in cosmic accelerators, Astrophys. J. 721 (2010) 630 [arXiv:1002.1310] [INSPIRE].ADSCrossRefGoogle Scholar
  83. [83]
    Z.-z. Xing, A further study of μτ symmetry breaking at neutrino telescopes after the Daya Bay and RENO measurements of θ 13, Phys. Lett. B 716 (2012) 220 [arXiv:1205.6532] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Roopam Sinha
    • 1
    Email author
  • Sukannya Bhattacharya
    • 1
  • Rome Samanta
    • 2
  1. 1.Saha Institute of Nuclear Physics, HBNIKolkataIndia
  2. 2.Physics and AstronomyUniversity of SouthamptonSouthamptonU.K.

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