Modular A4 symmetry with three moduli and flavor problem

Abstract

The modular [Formula: see text] symmetry with three moduli is investigated. We assign different moduli to charged leptons, neutrinos, and quarks. We analyze these moduli at their fixed points where a residual symmetry exists. We consider two possibilities for right-handed neutrinos. First, they are assumed to be singlets under the modular symmetry. In this case, we show that the lepton masses and mixing can be obtained consistently with experimental observations. Second, they are assigned nontrivially under modular symmetry. We emphasize that a small deviation from their fixed point is required in this case. Finally, the quark masses and mixing are generated correctly around the fixed point of their modulus. In our analysis, we only consider the simple case of weight 2.