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 ex- tended to a principal congruence subgroup Γ(N′). The original modular invariant theory is reproduced when N′ = 1. We perform a comprehensive study of {Gamma}_6^{prime } modular symmetry corresponding to N′ = 1 and N″ = 6, five types of models for lepton masses and mixing with {Gamma}_6^{prime } 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.
Highlights
Two Majorana CP violation phases α21 and α31
It is remarkable that the modular symmetry has the merit of conventional abelian flavor symmetry group, the structure of the modular form can produce texture zeros of fermion mass matrices exactly [30, 60], the modular weights can play the role of Froggatt-Nielsen charges [34, 49], and the hierarchical charged lepton masses can arise solely due to the proximity of the modulus τ to a residual symmetry conserved point [15, 41, 66]
We generalized the modular invariance approach [10] by replacing the modular group SL(2, Z) with the principal congruence subgroup Γ(N ), the original modular invariant theory is the special case of N = 1
Summary
The finite modular groups and their cover groups have been widely studied as the flavor symmetry groups. Modulo its normal subgroups Γ(N ), where SL(2, Z) ≡ Γ called the full modular group is defined as ab SL(2, Z) = c d ad − bc = 1 , a, b, c, d ∈ Z. Γ2 = Γ(1)/Γ(2) ∼= Γ(3)/Γ(6) ∼= Γ(5)/Γ(10) , Γ3 = Γ(1)/Γ(3) ∼= Γ(2)/Γ(6) =∼ Γ(4)/Γ(12) , Γ4 = Γ(1)/Γ(4) ∼= Γ(3)/Γ(12) ∼= Γ(5)/Γ(20) , Γ5 = Γ(1)/Γ(5) ∼= Γ(2)/Γ(10) ∼= Γ(3)/Γ(15) This implies that quotient of the principal congruence subgroups can give rise to the homogeneous finite modular groups ΓN. If one considers the more general congruent subgroups of SL(2, Z), other finite modular groups can be constructed by the quotient procedure. Where ρ(γ) is the irreducible representation of quotient group Γ(N )/Γ(N )
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