Modular symmetry at level 6 and a new route towards finite modular groups

Abstract

We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as Γ(N )/Γ(N ), and the modular group SL(2, Z) is extended to a principal congruence subgroup Γ(N ). The original modular invariant theory is reproduced when N = 1. We perform a comprehensive study of Γ 6 modular symmetry corresponding to N = 1 and N = 6, five types of models for lepton masses and mixing with Γ 6 modular symmetry are discussed and some example models are studied numerically. The case of N = 2 and N = 6 is considered, the finite modular group is Γ(2)/Γ(6) ∼ = T , and a benchmark model is constructed.

Radiative neutrino masses from modular $A_4$ symmetry and supersymmetry breaking

“…[60,68,69], double covering of A 4 in refs. [70][71][72], double covering of S 4 in refs. [73,74], and double covering of A 5 in refs.…”

A radiative seesaw model in a supersymmetric modular $A_4$ group

“…[59,67,68], double covering of A 4 in refs. [69][70][71], double covering of S 4 in refs. [72,73], and double covering of A 5 in refs.…”

Dark matter stability at fixed points in a modular $A_4$ symmetry

“…S 4 [48][49][50][51][52][53][54][55][56][57][58][59][60], A 5 [53,61,62], double covering of A 4 [63][64][65], S 4 [66,67], A 5 [68][69][70][71], multiple modular symmetries [72] and other types of groups [73][74][75][76][77][78][79], which have nicely predicted masses, mixings, and CP phases for the SM fermions 1 . Interesting ideas on DM have been proposed [6, 10, 13,52], in which modular weights assure the stability of DM.…”