Abstract
We perform a comprehensive analysis of the homogeneous finite modular group $\Gamma'_4\equiv S'_4$ which is the double covering of $S_4$ group. The weight 1 modular forms of level 4 are constructed in terms of Dedekind eta function, and they transform as a triplet $\mathbf{\hat{3}'}$ of $S'_4$. The integral weight modular forms until weight 6 are built from the tensor products of weight 1 modular forms. We perform a systematical classification of $S'_4$ modular models for lepton masses and mixing with/without generalized CP, where the left-handed leptons are assigned to triplet of $S'_4$ and right-handed charged leptons transform as singlets under $S'_4$, and we consider both scenarios where the neutrino masses arise from Weinberg operator or type I seesaw mechanism. The phenomenological implications of the minimal models for lepton masses, mixing angles, CP violation phases and neutrinoless double decay are discussed. The $S'_4$ modular symmetry is extended to quark sector, we present several predictive models which use nine or ten free parameters including real and imaginary parts of $\tau$ to describe quark masses and Cabibbo-Kobayashi-Maskawa mixing matrix. We give a quark-lepton unified model which can explain the flavor structure of quarks and leptons simultaneously for a common value of $\tau$.
Highlights
The origin of fermion masses and the mixing matrices is one of the greatest challenges for modern particle physics
We perform a systematical classification of S04 modular models for lepton masses and mixing with and without generalized CP, where the left-handed leptons are assigned to a triplet of S04 and right-handed charged leptons transform as singlets under S04, and we consider both scenarios where the neutrino masses arise from a Weinberg operator or type-I seesaw mechanism
Given the successful use of symmetries in various fields of physics, it was conceived that the flavor structure of quarks and leptons is dictated by certain flavor symmetry, and different kinds of flavor symmetry groups
Summary
The origin of fermion masses and the mixing matrices is one of the greatest challenges for modern particle physics. Aiming at a minimal and predictive model for quarks and leptons, we impose both S04 modular symmetry and the generalized CP symmetry which are spontaneously broken by the VEV of the modulus field τ. After comprehensively scanning the possible weight and representation assignments for the quark and lepton fields, we find a model which can describe the flavor structure of quarks and leptons simultaneously for a common value of τ This model has 15 real parameters to explain the 22 observables: six quark masses, three quark mixing angles, one quark. The models based on another two possible assignments of right-handed charged leptons are discussed in Appendix C
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