Abstract
We present minimalist constructions for lepton masses and mixing based on flavour symmetry under the modular group ΓN of lowest level N = 2. As opposed to the only existing model of Γ2 ≅ S3 formulated in a SUSY framework, the only non-SM field is the modulus τ, and a generalised CP symmetry is implemented. Charged-leptons masses are reproduced through symmetry arguments, without requiring fine-tuning of the free parameters. As a result, all lepton observables (masses and mixing) are reproduced within 1σ experimental range using a minimum of nine free real parameters (including the real and imaginary parts of the modulus). A normal ordering for the neutrino masses is predicted. We also obtain predictions for the CP violating phases: the Dirac CP phase is predicted around 1.6π, the Majorana phases lie in narrow regions near ±π. The sum of neutrino masses is within the current bound at ∼ 0.09 eV. Furthermore, we provide predictions for the neutrinoless double beta decay and tritium decay effective masses, around 20 meV. Given the reduced number of free input parameters as compared to the existing literature on modular S3, this work renews interest for a unified predictive model of quark-lepton sectors based on Γ2 ≅ S3.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C.D. Froggatt and H.B. Nielsen, Hierarchy of Quark Masses, Cabibbo Angles and CP Violation, Nucl. Phys. B 147 (1979) 277 [INSPIRE].
Z. Maki, M. Nakagawa and S. Sakata, Remarks on the unified model of elementary particles, Prog. Theor. Phys. 28 (1962) 870 [INSPIRE].
B. Pontecorvo, Inverse beta processes and nonconservation of lepton charge, Zh. Eksp. Teor. Fiz. 34 (1957) 247 [INSPIRE].
G. Altarelli and F. Feruglio, Discrete Flavor Symmetries and Models of Neutrino Mixing, Rev. Mod. Phys. 82 (2010) 2701 [arXiv:1002.0211] [INSPIRE].
F. Feruglio and Y. Lin, Fermion Mass Hierarchies and Flavour Mixing from a Minimal Discrete Symmetry, Nucl. Phys. B 800 (2008) 77 [arXiv:0712.1528] [INSPIRE].
G. Altarelli and F. Feruglio, Tri-bimaximal neutrino mixing, A4 and the modular symmetry, Nucl. Phys. B 741 (2006) 215 [hep-ph/0512103] [INSPIRE].
C.S. Lam, The Unique Horizontal Symmetry of Leptons, Phys. Rev. D 78 (2008) 073015 [arXiv:0809.1185] [INSPIRE].
F. Feruglio, Are neutrino masses modular forms?, in From My Vast Repertoire . . .: Guido Altarelli’s Legacy, A. Levy et al. eds., World Scientific (2019), p. 227–266 [10.1142/9789813238053_0012] [arXiv:1706.08749] [INSPIRE].
J. Lauer, J. Mas and H.P. Nilles, Duality and the Role of Nonperturbative Effects on the World Sheet, Phys. Lett. B 226 (1989) 251 [INSPIRE].
J. Lauer, J. Mas and H.P. Nilles, Twisted sector representations of discrete background symmetries for two-dimensional orbifolds, Nucl. Phys. B 351 (1991) 353 [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 et al., Finite modular subgroups for fermion mass matrices and baryon/lepton number violation, Phys. Lett. B 794 (2019) 114 [arXiv:1812.11072] [INSPIRE].
J.C. Criado and F. Feruglio, Modular Invariance Faces Precision Neutrino Data, SciPost Phys. 5 (2018) 042 [arXiv:1807.01125] [INSPIRE].
T. Kobayashi et al., Modular A4 invariance and neutrino mixing, JHEP 11 (2018) 196 [arXiv:1808.03012] [INSPIRE].
H. Okada and M. Tanimoto, CP violation of quarks in A4 modular invariance, Phys. Lett. B 791 (2019) 54 [arXiv:1812.09677] [INSPIRE].
H. Okada and M. Tanimoto, Towards unification of quark and lepton flavors in A4 modular invariance, Eur. Phys. J. C 81 (2021) 52 [arXiv:1905.13421] [INSPIRE].
G.-J. Ding, S.F. King and X.-G. Liu, Modular A4 symmetry models of neutrinos and charged leptons, JHEP 09 (2019) 074 [arXiv:1907.11714] [INSPIRE].
T. Kobayashi et al., A4 lepton flavor model and modulus stabilization from S4 modular symmetry, Phys. Rev. D 100 (2019) 115045 [Erratum ibid. 101 (2020) 039904] [arXiv:1909.05139] [INSPIRE].
T. Asaka, Y. Heo, T.H. Tatsuishi and T. Yoshida, Modular A4 invariance and leptogenesis, JHEP 01 (2020) 144 [arXiv:1909.06520] [INSPIRE].
G.-J. Ding, S.F. King, X.-G. Liu and J.-N. Lu, Modular S4 and A4 symmetries and their fixed points: new predictive examples of lepton mixing, JHEP 12 (2019) 030 [arXiv:1910.03460] [INSPIRE].
D. Zhang, A modular A4 symmetry realization of two-zero textures of the Majorana neutrino mass matrix, Nucl. Phys. B 952 (2020) 114935 [arXiv:1910.07869] [INSPIRE].
S.J.D. King and S.F. King, Fermion mass hierarchies from modular symmetry, JHEP 09 (2020) 043 [arXiv:2002.00969] [INSPIRE].
G.-J. Ding and F. Feruglio, Testing Moduli and Flavon Dynamics with Neutrino Oscillations, JHEP 06 (2020) 134 [arXiv:2003.13448] [INSPIRE].
T. Asaka, Y. Heo and T. Yoshida, Lepton flavor model with modular A4 symmetry in large volume limit, Phys. Lett. B 811 (2020) 135956 [arXiv:2009.12120] [INSPIRE].
H. Okada and M. Tanimoto, Spontaneous CP violation by modulus τ in A4 model of lepton flavors, JHEP 03 (2021) 010 [arXiv:2012.01688] [INSPIRE].
C.-Y. Yao, J.-N. Lu and G.-J. Ding, Modular Invariant A4 Models for Quarks and Leptons with Generalized CP Symmetry, JHEP 05 (2021) 102 [arXiv:2012.13390] [INSPIRE].
H. Okada, Y. Shimizu, M. Tanimoto and T. Yoshida, Modulus τ linking leptonic CP violation to baryon asymmetry in A4 modular invariant flavor model, JHEP 07 (2021) 184 [arXiv:2105.14292] [INSPIRE].
T. Nomura, H. Okada and Y. Orikasa, Quark and lepton flavor model with leptoquarks in a modular A4 symmetry, Eur. Phys. J. C 81 (2021) 947 [arXiv:2106.12375] [INSPIRE].
M.-C. Chen et al., Quasi-eclectic modular flavor symmetries, Phys. Lett. B 824 (2022) 136843 [arXiv:2108.02240] [INSPIRE].
T. Nomura, H. Okada and Y. Shoji, SU(4)C × SU(2)L × U(1)R models with modular A4 symmetry, arXiv:2206.04466 [INSPIRE].
Y. Gunji, K. Ishiwata and T. Yoshida, Subcritical regime of hybrid inflation with modular A4 symmetry, JHEP 11 (2022) 002 [arXiv:2208.10086] [INSPIRE].
J.T. Penedo and S.T. Petcov, Lepton Masses and Mixing from Modular S4 Symmetry, Nucl. Phys. B 939 (2019) 292 [arXiv:1806.11040] [INSPIRE].
P.P. Novichkov, J.T. Penedo, S.T. Petcov and A.V. Titov, Modular S4 models of lepton masses and mixing, JHEP 04 (2019) 005 [arXiv:1811.04933] [INSPIRE].
I. de Medeiros Varzielas, S.F. King and Y.-L. Zhou, Multiple modular symmetries as the origin of flavor, Phys. Rev. D 101 (2020) 055033 [arXiv:1906.02208] [INSPIRE].
T. Kobayashi et al., New A4 lepton flavor model from S4 modular symmetry, JHEP 02 (2020) 097 [arXiv:1907.09141] [INSPIRE].
S.F. King and Y.-L. Zhou, Trimaximal TM1 mixing with two modular S4 groups, Phys. Rev. D 101 (2020) 015001 [arXiv:1908.02770] [INSPIRE].
J.C. Criado, F. Feruglio and S.J.D. King, Modular Invariant Models of Lepton Masses at Levels 4 and 5, JHEP 02 (2020) 001 [arXiv:1908.11867] [INSPIRE].
X. Wang and S. Zhou, The minimal seesaw model with a modular S4 symmetry, JHEP 05 (2020) 017 [arXiv:1910.09473] [INSPIRE].
X. Wang, Dirac neutrino mass models with a modular S4 symmetry, Nucl. Phys. B 962 (2021) 115247 [arXiv:2007.05913] [INSPIRE].
B.-Y. Qu, X.-G. Liu, P.-T. Chen and G.-J. Ding, Flavor mixing and CP violation from the interplay of an S4 modular group and a generalized CP symmetry, Phys. Rev. D 104 (2021) 076001 [arXiv:2106.11659] [INSPIRE].
P.P. Novichkov, J.T. Penedo, S.T. Petcov and A.V. Titov, Modular A5 symmetry for flavour model building, JHEP 04 (2019) 174 [arXiv:1812.02158] [INSPIRE].
G.-J. Ding, S.F. King and X.-G. Liu, Neutrino mass and mixing with A5 modular symmetry, Phys. Rev. D 100 (2019) 115005 [arXiv:1903.12588] [INSPIRE].
F. Feruglio and A. Romanino, Lepton flavor symmetries, Rev. Mod. Phys. 93 (2021) 015007 [arXiv:1912.06028] [INSPIRE].
P.P. Novichkov, J.T. Penedo, S.T. Petcov and A.V. Titov, Generalised CP Symmetry in Modular-Invariant Models of Flavour, JHEP 07 (2019) 165 [arXiv:1905.11970] [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].
B.S. Acharya et al., Spontaneous breaking of CP symmetry by orbifold moduli, Phys. Lett. B 357 (1995) 387 [hep-th/9506143] [INSPIRE].
T. Dent, CP violation and modular symmetries, Phys. Rev. D 64 (2001) 056005 [hep-ph/0105285] [INSPIRE].
J. Giedt, CP violation and moduli stabilization in heterotic models, Mod. Phys. Lett. A 17 (2002) 1465 [hep-ph/0204017] [INSPIRE].
P.P. Novichkov, J.T. Penedo and S.T. Petcov, Fermion mass hierarchies, large lepton mixing and residual modular symmetries, JHEP 04 (2021) 206 [arXiv:2102.07488] [INSPIRE].
P. Beneš, H. Okada and Y. Orikasa, Towards unification of lepton and quark mass matrices from double covering of modular A4 flavor symmetry, arXiv:2212.07245 [INSPIRE].
M.-C. Chen, S. Ramos-Sánchez and M. Ratz, A note on the predictions of models with modular flavor symmetries, Phys. Lett. B 801 (2020) 135153 [arXiv:1909.06910] [INSPIRE].
X. Du and F. Wang, SUSY breaking constraints on modular flavor S3 invariant SU(5) GUT model, JHEP 02 (2021) 221 [arXiv:2012.01397] [INSPIRE].
T. Kobayashi et al., Modular S3-invariant flavor model in SU(5) grand unified theory, PTEP 2020 (2020) 053B05 [arXiv:1906.10341] [INSPIRE].
H. Okada and Y. Orikasa, Modular S3 symmetric radiative seesaw model, Phys. Rev. D 100 (2019) 115037 [arXiv:1907.04716] [INSPIRE].
S. Mishra, Neutrino mixing and Leptogenesis with modular S3 symmetry in the framework of type III seesaw, arXiv:2008.02095 [INSPIRE].
F. Capozzi et al., Unfinished fabric of the three neutrino paradigm, Phys. Rev. D 104 (2021) 083031 [arXiv:2107.00532] [INSPIRE].
eBOSS collaboration, Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cosmological implications from two decades of spectroscopic surveys at the Apache Point Observatory, Phys. Rev. D 103 (2021) 083533 [arXiv:2007.08991] [INSPIRE].
KamLAND-Zen collaboration, Search for the Majorana Nature of Neutrinos in the Inverted Mass Ordering Region with KamLAND-Zen, Phys. Rev. Lett. 130 (2023) 051801 [arXiv:2203.02139] [INSPIRE].
KATRIN collaboration, Direct neutrino-mass measurement with sub-electronvolt sensitivity, Nature Phys. 18 (2022) 160 [arXiv:2105.08533] [INSPIRE].
R.C. Gunning, Lectures on Modular Forms. (AM-48), Princeton University Press (1962) [https://doi.org/10.1515/9781400881666].
H. Cohen and F. Strömberg, Modular Forms: A Classical Approach, American Mathematical Society (2017).
X.-G. Liu and G.-J. Ding, Neutrino Masses and Mixing from Double Covering of Finite Modular Groups, JHEP 08 (2019) 134 [arXiv:1907.01488] [INSPIRE].
X.-G. Liu, C.-Y. Yao and G.-J. Ding, Modular invariant quark and lepton models in double covering of S4 modular group, Phys. Rev. D 103 (2021) 056013 [arXiv:2006.10722] [INSPIRE].
X. Wang, B. Yu and S. Zhou, Double covering of the modular A5 group and lepton flavor mixing in the minimal seesaw model, Phys. Rev. D 103 (2021) 076005 [arXiv:2010.10159] [INSPIRE].
C.-Y. Yao, X.-G. Liu and G.-J. Ding, Fermion masses and mixing from the double cover and metaplectic cover of the A5 modular group, Phys. Rev. D 103 (2021) 095013 [arXiv:2011.03501] [INSPIRE].
P.P. Novichkov, J.T. Penedo and S.T. Petcov, Double cover of modular S4 for flavour model building, Nucl. Phys. B 963 (2021) 115301 [arXiv:2006.03058] [INSPIRE].
I. de Medeiros Varzielas, S.F. King and M. Levy, Littlest modular seesaw, JHEP 02 (2023) 143 [arXiv:2211.00654] [INSPIRE].
G.-J. Ding, X.-G. Liu and C.-Y. Yao, A minimal modular invariant neutrino model, JHEP 01 (2023) 125 [arXiv:2211.04546] [INSPIRE].
P.P. Novichkov, J.T. Penedo and S.T. Petcov, Modular flavour symmetries and modulus stabilisation, JHEP 03 (2022) 149 [arXiv:2201.02020] [INSPIRE].
V. Knapp-Perez et al., Matter matters in moduli fixing and modular flavor symmetries, arXiv:2304.14437 [https://doi.org/10.1016/j.physletb.2023.138106] [INSPIRE].
K. Ishiguro, T. Kobayashi and H. Otsuka, Landscape of Modular Symmetric Flavor Models, JHEP 03 (2021) 161 [arXiv:2011.09154] [INSPIRE].
K. Ishiguro, H. Okada and H. Otsuka, Residual flavor symmetry breaking in the landscape of modular flavor models, JHEP 09 (2022) 072 [arXiv:2206.04313] [INSPIRE].
X.-G. Liu and G.-J. Ding, Modular flavor symmetry and vector-valued modular forms, JHEP 03 (2022) 123 [arXiv:2112.14761] [INSPIRE].
S. Weinberg, Baryon and Lepton Nonconserving Processes, Phys. Rev. Lett. 43 (1979) 1566 [INSPIRE].
H. Kuranaga, H. Ohki and S. Uemura, Modular origin of mass hierarchy: Froggatt-Nielsen like mechanism, JHEP 07 (2021) 068 [arXiv:2105.06237] [INSPIRE].
F. Feruglio, V. Gherardi, A. Romanino and A. Titov, Modular invariant dynamics and fermion mass hierarchies around τ = i, JHEP 05 (2021) 242 [arXiv:2101.08718] [INSPIRE].
F. Feruglio, Universal Predictions of Modular Invariant Flavor Models near the Self-Dual Point, Phys. Rev. Lett. 130 (2023) 101801 [arXiv:2211.00659] [INSPIRE].
P. Novichkov, Aspects of the Modular Symmetry Approach to Lepton Flavour, Ph.D. Thesis, SISSA, Trieste, Italy (2021) [INSPIRE].
S. Antusch et al., Running neutrino mass parameters in see-saw scenarios, JHEP 03 (2005) 024 [hep-ph/0501272] [INSPIRE].
G. Altarelli and G. Blankenburg, Different SO(10) Paths to Fermion Masses and Mixings, JHEP 03 (2011) 133 [arXiv:1012.2697] [INSPIRE].
DUNE collaboration, Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE): Conceptual Design Report, Volume 3: Long-Baseline Neutrino Facility for DUNE, arXiv:1601.05823 [https://doi.org/10.2172/1250880] [INSPIRE].
Hyper-Kamiokande Proto- collaboration, Physics potential of a long-baseline neutrino oscillation experiment using a J-PARC neutrino beam and Hyper-Kamiokande, PTEP 2015 (2015) 053C02 [arXiv:1502.05199] [INSPIRE].
nEXO collaboration, nEXO Pre-Conceptual Design Report, arXiv:1805.11142 [INSPIRE].
S. Antusch, J. Kersten, M. Lindner and M. Ratz, Running neutrino masses, mixings and CP phases: Analytical results and phenomenological consequences, Nucl. Phys. B 674 (2003) 401 [hep-ph/0305273] [INSPIRE].
A. Ereditato, The State of the Art of Neutrino Physics, World Scientific (2018) [https://doi.org/10.1142/10600] [INSPIRE].
H. Ishimori et al., An introduction to non-Abelian discrete symmetries for particle physicists, Springer (2012) [https://doi.org/10.1007/978-3-642-30805-5] [INSPIRE].
Acknowledgments
We thank J. Penedo and A. Titov for very useful discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2306.09028
Rights and permissions
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.
About this article
Cite this article
Meloni, D., Parriciatu, M. A simplest modular S3 model for leptons. J. High Energ. Phys. 2023, 43 (2023). https://doi.org/10.1007/JHEP09(2023)043
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2023)043