Neutrino masses and mixing from double covering of finite modular groups

Xiang-Gan Liu,Gui-Jun Ding

doi:10.1007/jhep08(2019)134

Xiang-Gan Liu, Gui-Jun Ding

Open Access

https://doi.org/10.1007/jhep08(2019)134

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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 {Gamma}_N^{prime } which is the double covering of ΓN. The lowest weight 1 modular forms of level 3 are constructed in terms of Dedekind eta-function, and they transform as a doublet of {Gamma}_3^{prime } ≅ 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.

Highlights

A new approach of modular invariance as flavor symmetry was proposed to solve the flavor problem of SM [15]
We find that the modular forms of integral weights and level N can be arranged into irreducible representations of the homogeneous finite modular group ΓN which is the double covering of ΓN
It is notable that the flavon fields could not be needed and the flavor symmetry could be completely broken by the vacuum expectation value (VEV) of the modulus τ in the supersymmetric modular invariant models

Summary

Modular symmetry and double covering of finite modular group

The modular group Γ is the linear fraction transformation of the upper half complex plane H = {τ ∈ C | Im τ > 0}, and it has the following form aτ + b ab τ → γτ ≡. The modular group Γ is isomorphic to the projective special linear group PSL(2, Z) = SL(2, Z)/{I, −I}, where I is the two-dimensional unit element. It’s well known that modular form f (τ ) of weight k and level N is a holomorphic function of the complex variable τ , and under Γ(N ) it should transform in the following way f aτ + b = (cτ + d)kf (τ ) for ∀ γ = a b ∈ Γ(N ) , cτ + d cd (2.10). Modular forms of even weights have been used to build models of quark and lepton flavors so far, we shall extend the formalism of modular invariance to general integral modular forms in the following

Transformation of integral weight modular forms under ΓN

Modular invariant supersymmetic theory

Conclusion

A Group theory of T