Fermion Mass Hierarchies, Large Lepton Mixing and Residual Modular Symmetries

Abstract

In modular-invariant models of flavour, hierarchical fermion mass matrices may arise solely due to the proximity of the modulus τ to a point of residual symmetry. This mechanism does not require flavon fields, and modular weights are not analogous to Froggatt-Nielsen charges. Instead, we show that hierarchies depend on the decomposition of field representations under the residual symmetry group. We systematically go through the possible fermion field representation choices which may yield hierarchical structures in the vicinity of symmetric points, for the four smallest finite modular groups, isomorphic to S 3 , A 4 , S 4 , and A 5 , as well as for their double covers. We find a restricted set of pairs of representations for which the discussed mechanism may produce viable fermion (charged-lepton and quark) mass hierarchies. We present two lepton flavour models in which the charged-lepton mass hierarchies are naturally obtained, while lepton mixing is somewhat fine-tuned. After formulating the conditions for obtaining a viable lepton mixing matrix in the symmetric limit, we construct a model in which both the charged-lepton and neutrino sectors are free from fine-tuning.

Flavor and CP from String Theory

“…refs. [27,28,29,30,31,32,33]). Model building includes free choices of representations and modular weights of the matter fields.…”

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

“…Other types of modular symmetries have also been proposed to understand masses, mixings, and phases of the standard model (SM) in refs. [79][80][81][82][83][84][85][86][87]110]. 3 Different applications to physics such as dark matter and origin of CP are found in refs.…”

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

“…The inhomogeneous finite modular group Γ N is the quotient group of the modular group P SL(2, Z) ∼ = Γ over the principal congruence subgroup Γ(N ). The phenomenologically viable modular invariant models have been widely discussed by using the inhomogeneous finite modular group Γ N in the literature, such as models based on the finite modular groups Γ 2 ∼ = S 3 [11][12][13][14][15], Γ 3 ∼ = A 4 [10][11][12], Γ 4 ∼ = S 4 [29,[44][45][46][47][48][49][50][51][52][53][54], Γ 5 ∼ = A 5 [49,55,56] and Γ 7 ∼ = P SL(2, Z 7 ) [57] have been studied. The modular forms of integral weights will be decomposed into irreducible representations of the homogeneous finite modular group Γ N which is the double covering of Γ N [58].…”

“…Recently the modular weight k has been extended to a rational number, and the corresponding finite modular group will be extended to its metaplectic covers [63,64]. It is remarkable that the modular symmetry also has the merit of conventional abelian flavor symmetry group, the structure of the modular form can produce texture zeros of fermion mass matrices exactly [30,59], 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 [15,41,65]. Furthermore, the modular invariance approach has been extended to the quark sector to explain the quark mixing and quark masses.…”