Abstract
The formalism of non-holomorphic modular flavor symmetry is developed, and the Yukawa couplings are level N polyharmonic Maaß forms satisfying the Laplacian condition. We find that the integer (even) weight polyharmonic Maaß forms of level N can be decomposed into multiplets of the finite modular group \( {\Gamma}_N^{\prime } \) (ΓN). The original modular invariance approach is extended by the presence of negative weight polyharmonic Maaß forms. The non-holomorphic modular flavor symmetry can be consistently combined with the generalized CP symmetry. We present three example models for lepton sector based on the Γ3 ≅ A4 modular symmetry, the charged lepton masses and the neutrino oscillation data can be accommodated very well, and the predictions for the leptonic CP violation phases and the effective Majorana neutrino mass are studied.
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Acknowledgments
We are grateful to professor Ferruccio Feruglio for sharing his note on non-holomorphic Yukawa couplings and insightful comments and questions. GJD would like to acknowledge all the participants of the workshop “Modular Invariance Approach to the Lepton and Quark Flavour Problems: from Bottom-up to Top-down” for stimulating discussions and helpful suggestions. GJD is grateful to the Mainz Institute for Theoretical Physics (MITP) of the Cluster of Excellence PRISMA+ (Project ID 390831469), for its hospitality and its partial support during the completion of this work. This work is supported by the National Natural Science Foundation of China under Grant No. 12375104.
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Qu, BY., Ding, GJ. Non-holomorphic modular flavor symmetry. J. High Energ. Phys. 2024, 136 (2024). https://doi.org/10.1007/JHEP08(2024)136
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DOI: https://doi.org/10.1007/JHEP08(2024)136