Abstract
We have performed a systematical study of the eclectic flavor group ∆(27) ⋊ S3 which is the extension of the traditional flavor symmetry ∆(27) by the finite modular symmetry S3. Consistency between ∆(27) and S3 requires that the eight nontrivial singlet representations of ∆(27) should be arranged into four reducible doublets. The modular transformation matrices are determined for various ∆(27) multiplets, and the CP-like symmetry compatible with ∆(27) ⋊ S3 are discussed. We study the general form of the Kähler potential and superpotential invariant under ∆(27) ⋊ S3, and the corresponding fermion mass matrices are presented. We propose a bottom-up model for lepton masses and mixing based on ∆(27) ⋊ S3, a numerical analysis is performed and the experimental data can be accommodated.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Particle Data Group collaboration, Review of Particle Physics, PTEP 2022 (2022) 083C01 [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].
H. Ishimori et al., Non-Abelian Discrete Symmetries in Particle Physics, Prog. Theor. Phys. Suppl. 183 (2010) 1 [arXiv:1003.3552] [INSPIRE].
S.F. King, Unified Models of Neutrinos, Flavour and CP Violation, Prog. Part. Nucl. Phys. 94 (2017) 217 [arXiv:1701.04413] [INSPIRE].
F. Feruglio and A. Romanino, Lepton flavor symmetries, Rev. Mod. Phys. 93 (2021) 015007 [arXiv:1912.06028] [INSPIRE].
T. Kobayashi, S. Raby and R.-J. Zhang, Searching for realistic 4d string models with a Pati-Salam symmetry: Orbifold grand unified theories from heterotic string compactification on a Z6 orbifold, Nucl. Phys. B 704 (2005) 3 [hep-ph/0409098] [INSPIRE].
T. Kobayashi et al., Stringy origin of non-Abelian discrete flavor symmetries, Nucl. Phys. B 768 (2007) 135 [hep-ph/0611020] [INSPIRE].
H. Abe, K.-S. Choi, T. Kobayashi and H. Ohki, Non-Abelian Discrete Flavor Symmetries from Magnetized/Intersecting Brane Models, Nucl. Phys. B 820 (2009) 317 [arXiv:0904.2631] [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 [https://doi.org/10.1142/9789813238053_0012] [arXiv:1706.08749] [INSPIRE].
X.-G. Liu and G.-J. Ding, Modular flavor symmetry and vector-valued modular forms, JHEP 03 (2022) 123 [arXiv:2112.14761] [INSPIRE].
G.-J. Ding, X.-G. Liu, J.-N. Lu and M.-H. Weng, Modular binary octahedral symmetry for flavor structure of Standard Model, JHEP 11 (2023) 083 [arXiv:2307.14926] [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].
F. Feruglio, Fermion masses, critical behavior and universality, JHEP 03 (2023) 236 [arXiv:2302.11580] [INSPIRE].
T. Kobayashi and M. Tanimoto, Modular flavor symmetric models, arXiv:2307.03384 [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].
J.-N. Lu, X.-G. Liu and G.-J. Ding, Modular symmetry origin of texture zeros and quark lepton unification, Phys. Rev. D 101 (2020) 115020 [arXiv:1912.07573] [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. Baur, H.P. Nilles, A. Trautner and P.K.S. Vaudrevange, A String Theory of Flavor and \( \mathcal{CP} \), Nucl. Phys. B 947 (2019) 114737 [arXiv:1908.00805] [INSPIRE].
H.P. Nilles, S. Ramos-Sánchez and P.K.S. Vaudrevange, Eclectic Flavor Groups, JHEP 02 (2020) 045 [arXiv:2001.01736] [INSPIRE].
H.P. Nilles, S. Ramos-Sánchez and P.K.S. Vaudrevange, Lessons from eclectic flavor symmetries, Nucl. Phys. B 957 (2020) 115098 [arXiv:2004.05200] [INSPIRE].
H.P. Nilles, S. Ramos-Sánchez and P.K.S. Vaudrevange, Eclectic flavor scheme from ten-dimensional string theory — I. Basic results, Phys. Lett. B 808 (2020) 135615 [arXiv:2006.03059] [INSPIRE].
H.P. Nilles, S. Ramos-Sánchez and P.K.S. Vaudrevange, Eclectic flavor scheme from ten-dimensional string theory — II detailed technical analysis, Nucl. Phys. B 966 (2021) 115367 [arXiv:2010.13798] [INSPIRE].
M.-C. Chen et al., Quasi-eclectic modular flavor symmetries, Phys. Lett. B 824 (2022) 136843 [arXiv:2108.02240] [INSPIRE].
M.-C. Chen et al., CP Violation from Finite Groups, Nucl. Phys. B 883 (2014) 267 [arXiv:1402.0507] [INSPIRE].
M. Ratz and M.-C. Chen, Group-theoretical origin of CP violation, arXiv:1903.00792 [https://doi.org/10.31526/lhep.1.2019.125] [INSPIRE].
A. Baur et al., The first string-derived eclectic flavor model with realistic phenomenology, JHEP 09 (2022) 224 [arXiv:2207.10677] [INSPIRE].
G.-J. Ding et al., Neutrino mass and mixing models with eclectic flavor symmetry ∆(27) ⋊T′, JHEP 05 (2023) 144 [arXiv:2303.02071] [INSPIRE].
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].
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].
A. Reyes, Representation Theory of Semi-Direct Products, J. Lond. Math. Soc. s2-13 (1976) 281.
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].
G.-J. Ding, F. Feruglio and X.-G. Liu, CP symmetry and symplectic modular invariance, SciPost Phys. 10 (2021) 133 [arXiv:2102.06716] [INSPIRE].
The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.10.2, https://www.gap-system.org.
H.U. Besche, B. Eick, E. O’Brien and M. Horn, SmallGrp — a GAP package, Version 1.5.3, (2023) [https://gap-packages.github.io/smallgrp].
P.P. Novichkov, J.T. Penedo and S.T. Petcov, Modular flavour symmetries and modulus stabilisation, JHEP 03 (2022) 149 [arXiv:2201.02020] [INSPIRE].
M. Cvetic et al., Target space duality, supersymmetry breaking and the stability of classical string vacua, Nucl. Phys. B 361 (1991) 194 [INSPIRE].
E. Gonzalo, L.E. Ibáñez and Á.M. Uranga, Modular symmetries and the swampland conjectures, JHEP 05 (2019) 105 [arXiv:1812.06520] [INSPIRE].
I. Esteban et al., The fate of hints: updated global analysis of three-flavor neutrino oscillations, JHEP 09 (2020) 178 [arXiv:2007.14792] [INSPIRE].
S. Antusch and V. Maurer, Running quark and lepton parameters at various scales, JHEP 11 (2013) 115 [arXiv:1306.6879] [INSPIRE].
Planck collaboration, Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 (2020) A6 [Erratum ibid. 652 (2021) C4] [arXiv:1807.06209] [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].
LEGEND collaboration, The Large Enriched Germanium Experiment for Neutrinoless Double Beta Decay (LEGEND), AIP Conf. Proc. 1894 (2017) 020027 [arXiv:1709.01980] [INSPIRE].
nEXO collaboration, Sensitivity and Discovery Potential of nEXO to Neutrinoless Double Beta Decay, Phys. Rev. C 97 (2018) 065503 [arXiv:1710.05075] [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].
Acknowledgments
CCL is supported by the National Natural Science Foundation of China under Grant Nos. 12005167, 12247103, Natural Science Basic Research Program of Shaanxi (Program No. 2024JC-YBQN-0004) and the Young Talent Fund of Association for Science and Technology in Shaanxi, China. GJD is supported by the National Natural Science Foundation of China under Grant Nos. 11975224, 11835013.
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: 2308.16901
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
Li, CC., Ding, GJ. Eclectic flavor group ∆(27) ⋊ S3 and lepton model building. J. High Energ. Phys. 2024, 54 (2024). https://doi.org/10.1007/JHEP03(2024)054
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2024)054