Abstract
The Kähler potentials of modular symmetry models receive unsuppressed contributions which may be controlled by a flavor symmetry, where the combination of the two symmetry types is referred to as eclectic flavor symmetry. After briefly reviewing the consistency conditions of eclectic flavor symmetry models, including those with generalised (g)CP, we perform a comprehensive bottom-up study of eclectic flavor symmetry models based on Ω(1) ≅ ∆(27) ⋊ T′, consisting of the flavor symmetry ∆(27) in a semi-direct product with the modular symmetry T′. The modular transformations of different ∆(27) multiplets are given by solving the consistency condition. The eight nontrivial singlets of ∆(27) are related by T′ modular symmetry, and they have to be present or absent simultaneously in any Ω(1) model. The most general forms of the superpotential and Kähler potential invariant under Ω(1) are discussed, and the corresponding fermion mass matrices are presented. Based on the eclectic flavor group Ω(1), two concrete lepton models which can successfully describe the experimental data of lepton masses and mixing parameters are constructed. For the two models without gCP, all six mixing parameters vary in small regions. A nearly maximal atmospheric mixing angle θ23 and Dirac CP phase δCP are obtained in the first model. After considering the compatible gCP symmetry and the assumption of \( \mathfrak{R}\tau \) = 0 in the first model, the μ − τ reflection symmetry is preserved in the charged lepton diagonal basis. As a consequence, the atmospheric mixing angle and Dirac CP phase are predicted to be maximal, and two Majorana CP phases are predicted to be π.
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Acknowledgments
GJD and XGL are grateful to professor Ferruccio Feruglio for sharing the his insightful notes on eclectic flavor group. CCL is supported by the National Natural Science Foundation of China under Grant Nos. 12005167, 12247103. XGL is supported in part by U.S. National Science Foundation under Grant No. PHY-2210283. JNL is supported by the Grants No. NSFC-12147110 and the China Post-doctoral Science Foundation under Grant No. 2021M70. GJD is supported by the National Natural Science Foundation of China under Grant Nos. 11975224, 11835013. SFK acknowledges the STFC Consolidated Grant ST/L000296/1 and the European Union’s Horizon 2020 Research and Innovation programme under Marie Skłodowska-Curie grant agreement HIDDeN European ITN project (H2020-MSCA-ITN-2019//860881-HIDDeN).
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Ding, GJ., King, S.F., Li, CC. et al. Neutrino mass and mixing models with eclectic flavor symmetry ∆(27) ⋊ T′. J. High Energ. Phys. 2023, 144 (2023). https://doi.org/10.1007/JHEP05(2023)144
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DOI: https://doi.org/10.1007/JHEP05(2023)144