Neutrino masses and mixing from double covering of finite modular groups

Abstract

We extend the even weight modular forms of modular invariant approach to general integral weight modular forms. We find that the modular forms of integral weights and level N can be arranged into irreducible representations of the homogeneous finite modular group Γ N which is the double covering of Γ N . The lowest weight 1 modular forms of level 3 are constructed in terms of Dedekind eta-function, and they transform as a doublet of Γ 3 ∼ = T . The modular forms of weights 2, 3, 4, 5 and 6 are presented.We build a model of lepton masses and mixing based on T modular symmetry. *

Modular S4 and A4 symmetries and their fixed points: new predictive examples of lepton mixing

“…(11) that the linear space M k (Γ(4)) has dimension 2k + 1. As shown in [12,32], the even weight modular forms of level 4 can be decomposed into irreducible representations of the inhomogeneous finite modular groups Γ 4 ∼ = S 4 , while odd weight modular forms can be arranged into irreducible representations of the homogeneous finite modular groups Γ 4 which is the double covering group of S 4 . In the present work, we shall focus on the modular forms of even weights.…”

“…It is remarkable that even the A 4 modular models can reproduce the measured neutrino masses and mixing angles [12,19,23]. The modular invariance approach has been extended to include odd weight modular forms which can be decomposed into irreducible representations of the the homogeneous finite modular group Γ N [32], and the modular symmetry Γ 3 ∼ = T has been discussed. It has been shown that the modular symmetry can be consistently combined with generalized CP symmetry, and the modulus τ transforms as τ → −τ * under the CP transformation [33][34][35][36][37].…”

“…The modular forms of weight k and level N span a linear space of finite dimension. It is always possible to choose a basis in this linear space such that the modular forms can be arranged into some modular multiplets f r ≡ (f 1 (τ ), f 2 (τ ), ...) T which transform as irreducible representation r of the finite modular group Γ N for even k [12,32], i.e.…”

Modular A4 symmetry models of neutrinos and charged leptons

“…The generalized CP symmetry in modular invariant models are studied in [42]. The formalism of modular invariant approach has extended to include odd weight modular forms [43].…”

“…The modular forms of weight k and level N span a linear space M k (Γ(N )) with finite dimension. As has been shown in [25,43], we can choose the basis vectors of M k (Γ(N )) such that they can be organized into multiplets of modular forms F r (τ ) ≡ (f 1 (τ ), f 2 (τ ), . .…”