Modular $S_3$-invariant flavor model in SU(5) grand unified theory

Abstract

We present a flavor model with $S_3$ modular invariance in the framework of SU(5) grand unified theory (GUT). The $S_3$ modular forms of weights $2$ and $4$ give the quark and lepton mass matrices with a common complex parameter, the modulus $\tau$. The GUT relation of down-type quarks and charged leptons is imposed by the vacuum expectation value (VEV) of the adjoint 24-dimensional Higgs multiplet in addition to the VEVs of $5$ and $\bar 5$ Higgs multiplets of SU(5). The observed Cabibbo–Kobayashi–Maskawa and Pontecorvo–Maki–Nakagawa–Sakata mixing parameters as well as the mass eigenvalues are reproduced properly. We discuss the leptonic charge–parity phase and the effective mass of the neutrinoless double beta decay with the sum of neutrino masses.

Modular flavor symmetry on a magnetized torus

“…Γ N is called finite modular group. The experimental values corresponding to the lepton sectors, the masses of charged leptons, neutrino mass-square differences, three mixing angles, and the CP phase can be reproduced in models with modular symmetries of Γ 2 ≅ S 3 [10][11][12], Γ 3 ≅ A 4 [12][13][14][15][16], Γ 4 ≅ S 4 [17,18], and Γ 5 ≅ A 5 [19]. Modular symmetry is also applied to other physics beyond the standard model such as leptogenesis and inflation [20][21][22][23], and relationships between generalized CP symmetry [24,25] and the modular symmetry are also pointed out [26][27][28][29].…”

Modular invariant models of leptons at level 7

“…The finite modular groups Γ 2 ∼ = S 3 [3][4][5][6], Γ 3 ∼ = A 4 [1,3,4,[7][8][9][10][11][12][13][14], Γ 4 ∼ = S 4 [13,[15][16][17][18][19]] and Γ 5 ∼ = A 5 [18,20,21] have been considered. For example, simple A 4 modular models can reproduce the measured neutrino masses and mixing angles [1,8,12].…”

The minimal seesaw model with a modular S4 symmetry

“…In this case, the flavon fields are no longer needed. In the literature, there have been a great number of works on the model building based on the modular group Γ N , which for a given value of N is isomorphic to the well-known non-Abelian discrete symmetry groups, e.g., Γ 2 S 3 [26][27][28][29], Γ 3 A 4 [30][31][32][33][34][35][36][37][38][39][40], Γ 4 S 4 [41][42][43] and Γ 5 A 5 [44][45][46]. Moreover, other interesting aspects of modular symmetries have also been studied, such as the combination of modular symmetries and the generalized CP symmetry [47], multiple modular symmetries [48,49], the double covering of modular groups [50], the A 4 symmetry JHEP05(2020)017 from the modular S 4 symmetry [51,52], the modular residual symmetry [45,53] and the unification of quark and lepton flavors with modular invariance [55].…”