Skip to main content

Generalised CP symmetry in modular-invariant models of flavour

A preprint version of the article is available at arXiv.

Abstract

The formalism of combined finite modular and generalised CP (gCP) sym-metries for theories of flavour is developed. The corresponding consistency conditions for the two symmetry transformations acting on the modulus τ and on the matter fields are derived. The implications of gCP symmetry in theories of flavour based on modular invariance described by finite modular groups are illustrated with the example of a modular S4 model of lepton flavour. Due to the addition of the gCP symmetry, viable modular models turn out to be more constrained, with the modulus τ being the only source of CP violation.

References

  1. [1]

    G. Altarelli and F. Feruglio, Discrete flavor symmetries and models of neutrino mixing, Rev. Mod. Phys.82 (2010) 2701 [arXiv:1002.0211] [INSPIRE].

    ADS  Article  Google Scholar 

  2. [2]

    H. Ishimori et al., Non-abelian discrete symmetries in particle physics, Prog. Theor. Phys. Suppl.183 (2010) 1 [arXiv:1003.3552] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  3. [3]

    S.F. King and C. Luhn, Neutrino mass and mixing with discrete symmetry, Rept. Prog. Phys.76 (2013) 056201 [arXiv:1301.1340] [INSPIRE].

    ADS  Article  Google Scholar 

  4. [4]

    S.T. Petcov, Discrete flavour symmetries, neutrino mixing and leptonic CP-violation, Eur. Phys. J.C 78 (2018) 709 [arXiv:1711.10806] [INSPIRE].

    ADS  Article  Google Scholar 

  5. [5]

    S.M. Bilenky, J. Hosek and S.T. Petcov, On oscillations of neutrinos with Dirac and Majorana masses, Phys. Lett.B 94 (1980) 495.

    ADS  Article  Google Scholar 

  6. [6]

    F. Feruglio, C. Hagedorn and R. Ziegler, Lepton mixing parameters from discrete and CP symmetries, JHEP07 (2013) 027 [arXiv:1211.5560] [INSPIRE].

    ADS  Article  Google Scholar 

  7. [7]

    M. Holthausen, M. Lindner and M.A. Schmidt, CP and discrete flavour symmetries, JHEP04 (2013) 122 [arXiv:1211.6953] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  8. [8]

    G.-J. Ding, S.F. King and A.J. Stuart, Generalised CP and A 4family symmetry, JHEP12 (2013) 006 [arXiv:1307.4212] [INSPIRE].

    ADS  Article  Google Scholar 

  9. [9]

    I. Girardi, A. Meroni, S.T. Petcov and M. Spinrath, Generalised geometrical CP-violation in a Tlepton flavour model,JHEP02(2014) 050 [arXiv:1312.1966] [INSPIRE].

    ADS  Article  Google Scholar 

  10. [10]

    G.-J. Ding, S.F. King, C. Luhn and A.J. Stuart, Spontaneous CP-violation from vacuum alignment in S 4models of leptons, JHEP05 (2013) 084 [arXiv:1303.6180] [INSPIRE].

    ADS  Article  Google Scholar 

  11. [11]

    F. Feruglio, C. Hagedorn and R. Ziegler, A realistic pattern of lepton mixing and masses from S 4and CP, Eur. Phys. J.C 74 (2014) 2753 [arXiv:1303.7178] [INSPIRE].

    ADS  Article  Google Scholar 

  12. [12]

    C.-C. Li and G.-J. Ding, Generalised CP and trimaximal T M 1lepton mixing in S 4family symmetry, Nucl. Phys.B 881 (2014) 206 [arXiv:1312.4401] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  13. [13]

    C.-C. Li and G.-J. Ding, Deviation from bimaximal mixing and leptonic CP phases in S 4family symmetry and generalized CP, JHEP08 (2015) 017 [arXiv:1408.0785] [INSPIRE].

    ADS  Article  Google Scholar 

  14. [14]

    J.-N. Lu and G.-J. Ding, Alternative schemes of predicting lepton mixing parameters from discrete flavor and CP symmetry, Phys. Rev.D 95 (2017) 015012 [arXiv:1610.05682] [INSPIRE].

    ADS  Google Scholar 

  15. [15]

    J.T. Penedo, S.T. Petcov and A.V. Titov, Neutrino mixing and leptonic CP-violation from S 4flavour and generalised CP symmetries, JHEP12 (2017) 022 [arXiv:1705.00309] [INSPIRE].

    ADS  Article  Google Scholar 

  16. [16]

    C.-C. Li and G.-J. Ding, Lepton mixing in A 5family symmetry and generalized CP, JHEP05 (2015) 100 [arXiv:1503.03711] [INSPIRE].

    ADS  Article  Google Scholar 

  17. [17]

    A. Di Iura, C. Hagedorn and D. Meloni, Lepton mixing from the interplay of the alternating group A 5and CP, JHEP08 (2015) 037 [arXiv:1503.04140] [INSPIRE].

    Article  Google Scholar 

  18. [18]

    P. Ballett, S. Pascoli and J. Turner, Mixing angle and phase correlations from A 5with generalized CP and their prospects for discovery, Phys. Rev.D 92 (2015) 093008 [arXiv:1503.07543] [INSPIRE].

    ADS  Google Scholar 

  19. [19]

    J. Turner, Predictions for leptonic mixing angle correlations and nontrivial Dirac CP-violation from A 5with generalized CP symmetry, Phys. Rev.D 92 (2015) 116007 [arXiv:1507.06224] [INSPIRE].

    ADS  Google Scholar 

  20. [20]

    I. Girardi, S.T. Petcov and A.V. Titov, Predictions for the Majorana CP-violation phases in the neutrino mixing matrix and neutrinoless double β decay, Nucl. Phys.B 911 (2016) 754 [arXiv:1605.04172] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  21. [21]

    F. Feruglio, Are neutrino masses modular forms?, in From my vast repertoire: Guido Altarellis legacy, A. Levy et al. eds., World Scientific, Singapore (2019), arXiv:1706.08749 [INSPIRE].

  22. [22]

    R. de Adelhart Toorop, F. Feruglio and C. Hagedorn, Finite modular groups and lepton mixing, Nucl. Phys.B 858 (2012) 437 [arXiv:1112.1340] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  23. [23]

    T. Kobayashi, K. Tanaka and T.H. Tatsuishi, Neutrino mixing from finite modular groups, Phys. Rev.D 98 (2018) 016004 [arXiv:1803.10391] [INSPIRE].

    ADS  Google Scholar 

  24. [24]

    T. Kobayashi et al., Finite modular subgroups for fermion mass matrices and baryon/lepton number violation, Phys. Lett.B 794 (2019) 114 [arXiv:1812.11072] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  25. [25]

    J.C. Criado and F. Feruglio, Modular invariance faces precision neutrino data, SciPost Phys.5 (2018) 042 [arXiv:1807.01125] [INSPIRE].

    ADS  Article  Google Scholar 

  26. [26]

    T. Kobayashi et al., Modular A 4invariance and neutrino mixing, JHEP11 (2018) 196 [arXiv:1808.03012] [INSPIRE].

    ADS  Article  Google Scholar 

  27. [27]

    F.J. de Anda, S.F. King and E. Perdomo, SU(5) grand unified theory with A 4modular symmetry, arXiv:1812.05620 [INSPIRE].

  28. [28]

    H. Okada and M. Tanimoto, CP violation of quarks in A 4modular invariance, Phys. Lett.B 791 (2019) 54 [arXiv:1812.09677] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  29. [29]

    P.P. Novichkov, S.T. Petcov and M. Tanimoto, Trimaximal neutrino mixing from modular A 4invariance with residual symmetries, Phys. Lett.B 793 (2019) 247 [arXiv:1812.11289] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  30. [30]

    J.T. Penedo and S.T. Petcov, Lepton masses and mixing from modular S 4symmetry, Nucl. Phys.B 939 (2019) 292 [arXiv:1806.11040] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  31. [31]

    P.P. Novichkov, J.T. Penedo, S.T. Petcov and A.V. Titov, Modular S 4models of lepton masses and mixing, JHEP04 (2019) 005 [arXiv:1811.04933] [INSPIRE].

    ADS  Article  Google Scholar 

  32. [32]

    P.P. Novichkov, J.T. Penedo, S.T. Petcov and A.V. Titov, Modular A 5symmetry for flavour model building, JHEP04 (2019) 174 [arXiv:1812.02158] [INSPIRE].

    ADS  Article  Google Scholar 

  33. [33]

    G.-J. Ding, S.F. King and X.-G. Liu, Neutrino mass and mixing with A 5modular symmetry, arXiv:1903.12588 [INSPIRE].

  34. [34]

    T. Kobayashi et al., Modular symmetry and non-Abelian discrete flavor symmetries in string compactification, Phys. Rev.D 97 (2018) 116002 [arXiv:1804.06644] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  35. [35]

    T. Kobayashi and S. Tamba, Modular forms of finite modular subgroups from magnetized D-brane models, Phys. Rev.D 99 (2019) 046001 [arXiv:1811.11384] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  36. [36]

    A. Baur, H.P. Nilles, A. Trautner and P.K.S. Vaudrevange, Unification of flavor, CP and modular symmetries, Phys. Lett.B 795 (2019) 7 [arXiv:1901.03251] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  37. [37]

    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].

    ADS  Article  Google Scholar 

  38. [38]

    M.-C. Chen et al., CP violation from finite groups, Nucl. Phys.B 883 (2014) 267 [arXiv:1402.0507] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  39. [39]

    S. Ferrara, D. Lüst, A.D. Shapere and S. Theisen, Modular invariance in supersymmetric field theories, Phys. Lett.B 225 (1989) 363 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  40. [40]

    S. Ferrara, D. Lüst and S. Theisen, Target space modular invariance and low-energy couplings in orbifold compactifications, Phys. Lett.B 233 (1989) 147 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  41. [41]

    B.S. Acharya, D. Bailin, A. Love, W.A. Sabra and S. Thomas, Spontaneous breaking of CP symmetry by orbifold moduli, Phys. Lett.B 357 (1995) 387 [Erratum ibid.B 407 (1997) 451] [hep-th/9506143] [INSPIRE].

  42. [42]

    T. Dent, CP violation and modular symmetries, Phys. Rev.D 64 (2001) 056005 [hep-ph/0105285] [INSPIRE].

  43. [43]

    J. Giedt, CP violation and moduli stabilization in heterotic models, Mod. Phys. Lett.A 17 (2002) 1465 [hep-ph/0204017] [INSPIRE].

  44. [44]

    R.S. Kulkarni, An arithmetic-geometric method in the study of the subgroups of the modular group, Amer. J. Math.113 (1991) 1053.

    MathSciNet  MATH  Article  Google Scholar 

  45. [45]

    Particle Data Group collaboration, Review of particle physics, Phys. Rev.D 98 (2018) 030001 [INSPIRE].

  46. [46]

    B. Schoeneberg, Elliptic modular functions: an introduction, Springer, Germany (1974).

    MATH  Book  Google Scholar 

  47. [47]

    G. Altarelli, F. Feruglio and L. Merlo, Revisiting bimaximal neutrino mixing in a model with S 4discrete symmetry, JHEP05 (2009) 020 [arXiv:0903.1940] [INSPIRE].

    ADS  Article  Google Scholar 

  48. [48]

    G.-J. Ding, L.L. Everett and A.J. Stuart, Golden ratio neutrino mixing and A 5flavor symmetry, Nucl. Phys.B 857 (2012) 219 [arXiv:1110.1688] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  49. [49]

    H.M. Farkas and I. Kra, Theta constants, Riemann surfaces and the modular group, Graduate Studies in Mathematics volume 37, American Mathematical Society, U.S.A. (2001).

    Google Scholar 

  50. [50]

    S. Kharchev and A. Zabrodin, Theta vocabulary I, J. Geom. Phys.94 (2015) 19 [arXiv:1502.04603].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  51. [51]

    R.C. Gunning, Lectures on modular forms, Princeton University Press, Princeton U.S.A. (1962).

    MATH  Book  Google Scholar 

  52. [52]

    The Sage developers, SageMath, the Sage Mathematics Software System, version 8.4 (2018).

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to P.P. Novichkov.

Additional information

ArXiv ePrint: 1905.11970

Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Novichkov, P., Penedo, J., Petcov, S. et al. Generalised CP symmetry in modular-invariant models of flavour. J. High Energ. Phys. 2019, 165 (2019). https://doi.org/10.1007/JHEP07(2019)165

Download citation

Keywords

  • Beyond Standard Model
  • CP violation
  • Neutrino Physics