Abstract
We present a comprehensive analysis of neutrino mass and lepton mixing in theories with A4 modular symmetry, where the only flavon field is the single modulus field τ, and all masses and Yukawa couplings are modular forms. Similar to previous analyses, we discuss all the simplest neutrino sectors arising from both the Weinberg operator and the type I seesaw mechanism, with lepton doublets and right-handed neutrinos assumed to be triplets of A4. Unlike previous analyses, we allow right-handed charged leptons to transform as all combinations of 1, 1′ and 1′′ representations of A4, using the simplest different modular weights to break the degeneracy, leading to ten different charged lepton Yukawa matrices, instead of the usual one. This implies ten different Weinberg models and thirty different type I seesaw models, which we analyse in detail. We find that fourteen models for both NO and IO neutrino mass ordering can accommodate the data, as compared to one in previous analyses, providing many new possibilities.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S.F. King and C. Luhn, Neutrino mass and mixing with discrete symmetry, Rept. Prog. Phys.76 (2013) 056201 [arXiv:1301.1340] [INSPIRE].
S.F. King, Unified models of neutrinos, flavour and CP-violation, Prog. Part. Nucl. Phys.94 (2017) 217 [arXiv:1701.04413] [INSPIRE].
Y. Koide, S 4flavor symmetry embedded into SU(3) and lepton masses and mixing, JHEP08 (2007) 086 [arXiv:0705.2275] [INSPIRE].
T. Banks and N. Seiberg, Symmetries and strings in field theory and gravity, Phys. Rev.D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
Y.-L. Wu, SU(3) gauge family symmetry and prediction for the lepton-flavor mixing and neutrino masses with maximal spontaneous CP-violation, Phys. Lett.B 714 (2012) 286 [arXiv:1203.2382] [INSPIRE].
A. Merle and R. Zwicky, Explicit and spontaneous breaking of SU(3) into its finite subgroups, JHEP02 (2012) 128 [arXiv:1110.4891] [INSPIRE].
B.L. Rachlin and T.W. Kephart, Spontaneous breaking of gauge groups to discrete symmetries, JHEP08 (2017) 110 [arXiv:1702.08073] [INSPIRE].
C. Luhn, Spontaneous breaking of SU(3) to finite family symmetries: a pedestrian’s approach, JHEP03 (2011) 108 [arXiv:1101.2417] [INSPIRE].
S.F. King and Y.-L. Zhou, Spontaneous breaking of SO(3) to finite family symmetries with supersymmetry — an A 4model, JHEP11 (2018) 173 [arXiv:1809.10292] [INSPIRE].
G. Altarelli, F. Feruglio and C. Hagedorn, A SUSY SU(5) grand unified model of tri-bimaximal mixing from A 4, JHEP03 (2008) 052 [arXiv:0802.0090] [INSPIRE].
T.J. Burrows and S.F. King, A 4family symmetry from SU(5) SUSY GUTs in 6d, Nucl. Phys.B 835 (2010) 174 [arXiv:0909.1433] [INSPIRE].
T.J. Burrows and S.F. King, A 4 × SU(5) SUSY GUT of flavour in 8d, Nucl. Phys.B 842 (2011) 107 [arXiv:1007.2310] [INSPIRE].
F.J. de Anda and S.F. King, An S 4 × SU(5) SUSY GUT of flavour in 6d, JHEP07 (2018) 057 [arXiv:1803.04978] [INSPIRE].
A. Adulpravitchai, A. Blum and M. Lindner, Non-Abelian discrete flavor symmetries from T 2/Z Norbifolds, JHEP07 (2009) 053 [arXiv:0906.0468] [INSPIRE].
T. Asaka, W. Buchmüller and L. Covi, Gauge unification in six-dimensions, Phys. Lett.B 523 (2001) 199 [hep-ph/0108021] [INSPIRE].
G. Altarelli, F. Feruglio and Y. Lin, Tri-bimaximal neutrino mixing from orbifolding, Nucl. Phys.B 775 (2007) 31 [hep-ph/0610165] [INSPIRE].
A. Adulpravitchai and M.A. Schmidt, Flavored orbifold GUT — an SO(10) × S 4model, JHEP01 (2011) 106 [arXiv:1001.3172] [INSPIRE].
T. Kobayashi, H.P. Nilles, F. Ploger, S. Raby and M. Ratz, Stringy origin of non-Abelian discrete flavor symmetries, Nucl. Phys.B 768 (2007) 135 [hep-ph/0611020] [INSPIRE].
F.J. de Anda and S.F. King, SU(3) × SO(10) in 6d, JHEP10 (2018) 128 [arXiv:1807.07078] [INSPIRE].
T. Kobayashi, S. Nagamoto, S. Takada, S. Tamba and T.H. Tatsuishi, Modular symmetry and non-Abelian discrete flavor symmetries in string compactification, Phys. Rev.D 97 (2018) 116002 [arXiv:1804.06644] [INSPIRE].
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].
A. Giveon, E. Rabinovici and G. Veneziano, Duality in string background space, Nucl. Phys.B 322 (1989) 167 [INSPIRE].
G. Altarelli and F. Feruglio, Tri-bimaximal neutrino mixing, A 4and the modular symmetry, Nucl. Phys.B 741 (2006) 215 [hep-ph/0512103] [INSPIRE].
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].
F. Feruglio, Are neutrino masses modular forms?, in From my vast repertoire... : Guido Altarelli’s legacy, A. Levy, S. Forte and G. Ridolfi eds., World Scientific, Singapore (2019), pg. 227 [arXiv:1706.08749] [INSPIRE].
J.C. Criado and F. Feruglio, Modular invariance faces precision neutrino data, SciPost Phys.5 (2018) 042 [arXiv:1807.01125] [INSPIRE].
T. Kobayashi, K. Tanaka and T.H. Tatsuishi, Neutrino mixing from finite modular groups, Phys. Rev.D 98 (2018) 016004 [arXiv:1803.10391] [INSPIRE].
T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, T.H. Tatsuishi and H. Uchida, Finite modular subgroups for fermion mass matrices and baryon/lepton number violation, Phys. Lett.B 794 (2019) 114 [arXiv:1812.11072] [INSPIRE].
T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T.H. Tatsuishi, Modular S 3invariant flavor model in SU(5) GUT, arXiv:1906.10341 [INSPIRE].
H. Okada and Y. Orikasa, A modular S 3symmetric radiative seesaw model, arXiv:1907.04716 [INSPIRE].
T. Kobayashi, N. Omoto, Y. Shimizu, K. Takagi, M. Tanimoto and T.H. Tatsuishi, Modular A 4invariance and neutrino mixing, JHEP11 (2018) 196 [arXiv:1808.03012] [INSPIRE].
H. Okada and M. Tanimoto, CP violation of quarks in A 4modular invariance, Phys. Lett.B 791 (2019) 54 [arXiv:1812.09677] [INSPIRE].
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].
T. Nomura and H. Okada, A two loop induced neutrino mass model with modular A 4symmetry, arXiv:1906.03927 [INSPIRE].
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].
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].
T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T.H. Tatsuishi, New A 4lepton flavor model from S 4modular symmetry, arXiv:1907.09141 [INSPIRE].
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].
G.-J. Ding, S.F. King and X.-G. Liu, Neutrino mass and mixing with A 5modular symmetry, arXiv:1903.12588 [INSPIRE].
F.J. de Anda, S.F. King and E. Perdomo, SU(5) grand unified theory with A 4modular symmetry, arXiv:1812.05620 [INSPIRE].
I. De Medeiros Varzielas, S.F. King and Y.-L. Zhou, Multiple modular symmetries as the origin of flavour, arXiv:1906.02208 [INSPIRE].
P.P. Novichkov, J.T. Penedo, S.T. Petcov and A.V. Titov, Generalised CP symmetry in modular-invariant models of flavour, JHEP07 (2019) 165 [arXiv:1905.11970] [INSPIRE].
X.-G. Liu and G.-J. Ding, Neutrino masses and mixing from double covering of finite modular groups, JHEP08 (2019) 134 [arXiv:1907.01488] [INSPIRE].
J.H. Bruinier, G. van der Geer, G. Harder and D. Zagier, The 1-2-3 of modular forms, Springer, Berlin Heidelberg, Germany (2008).
F. Diamond and J.M. Shurman, A first course in modular forms, Grad. Texts Math.228, Springer, New York, NY, U.S.A. (2005).
R.C. Gunning, Lectures on modular forms, Princeton University Press, Princeton, NJ, U.S.A. (1962).
I. Esteban, M.C. Gonzalez-Garcia, A. Hernandez-Cabezudo, M. Maltoni and T. Schwetz, Global analysis of three-flavour neutrino oscillations: synergies and tensions in the determination of θ 23, δ CPand the mass ordering, JHEP01 (2019) 106 [arXiv:1811.05487] [INSPIRE].
F. Feruglio, K.M. Patel and D. Vicino, Order and anarchy hand in hand in 5D SO(10), JHEP09 (2014) 095 [arXiv:1407.2913] [INSPIRE].
G. Ross and M. Serna, Unification and fermion mass structure, Phys. Lett.B 664 (2008) 97 [arXiv:0704.1248] [INSPIRE].
F. Feroz and M.P. Hobson, Multimodal nested sampling: an efficient and robust alternative to MCMC methods for astronomical data analysis, Mon. Not. Roy. Astron. Soc.384 (2008) 449 [arXiv:0704.3704] [INSPIRE].
F. Feroz, M.P. Hobson and M. Bridges, MultiNest: an efficient and robust Bayesian inference tool for cosmology and particle physics, Mon. Not. Roy. Astron. Soc.398 (2009) 1601 [arXiv:0809.3437] [INSPIRE].
Planck collaboration, Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209 [INSPIRE].
Particle Data Group collaboration, Review of particle physics, Phys. Rev.D 98 (2018) 030001 [INSPIRE].
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
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1907.11714
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/.
About this article
Cite this article
Ding, GJ., King, S.F. & Liu, XG. Modular A4 symmetry models of neutrinos and charged leptons. J. High Energ. Phys. 2019, 74 (2019). https://doi.org/10.1007/JHEP09(2019)074
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2019)074