Modular S4 and A4 symmetries and their fixed points: new predictive examples of lepton mixing

Abstract

In the modular symmetry approach to neutrino models, the flavour symmetry emerges as a finite subgroup Γ N of the modular symmetry, broken by the vacuum expectation value (VEV) of a modulus field τ . If the VEV of the modulus τ takes some special value, a residual subgroup of Γ N would be preserved. We derive the fixed points τ S = i, τ ST = (−1 + i √ 3)/2, τ T S = (1 + i √ 3)/2, τ T = i∞ in the fundamental domain which are invariant under the modular transformations indicated. We then generalise these fixed points to τ f = γτ S , γτ ST , γτ T S and γτ T in the upper half complex plane, and show that it is sufficient to consider γ ∈ Γ N . Focussing on level N = 4, corresponding to the flavour group S 4 , we consider all the resulting triplet modular forms at these fixed points up to weight 6. We then apply the results to lepton mixing, with different residual subgroups in the charged lepton sector and each of the right-handed neutrinos sectors. In the minimal case of two right-handed neutrinos, we find three phenomenologically viable cases in which the light neutrino mass matrix only depends on three free parameters, and the lepton mixing takes the trimaximal TM1 pattern for two examples. One of these cases corresponds to a new Littlest Modular Seesaw based on CSD(n) with n = 1 + √ 6 ≈ 3.45, intermediate between CSD (3) and CSD(4). Finally, we generalize the results to examples with three right-handed neutrinos, also considering the level N = 3 case, corresponding to A 4 flavour symmetry. *

Fermion mass hierarchies from modular symmetry

“…The modular group Γ(3) has been extensively studied in the literature [16,18,19,[22][23][24][25][26]28].…”

“…In the present work we shall adopt the same convention as [16,27,28]. The finite modular group Γ 3 is isomorphic to A 4 which is the symmetry group of the tetrahedron.…”

“…i∞ in the fundamental domain are invariant under modular transformations, and there are many other examples in the upper half complex plane [28]. For example, τ T = i∞ implies Y…”

“…The finite modular groups Γ 2 ∼ = S 3 [18][19][20][21], Γ 3 ∼ = A 4 [16,18,19,[22][23][24][25][26][27][28], Γ 4 ∼ = S 4 [28][29][30][31][32] and Γ 5 ∼ = A 5 [32][33][34] have been considered. For example, simple A 4 modular models can reproduce the measured neutrino masses and mixing angles [16,23,27].…”

“…It has been realised that, if the VEV of the modulus τ takes some special value, a residual subgroup of the finite modular symmetry group Γ N would be preserved. The phenomenological implications of the residual modular symmetry have been discussed in the context of modular A 4 [25,28], S 4 [28,30] and A 5 [33] symmetries. If the modular symmetry is broken down to a residual Z 3 (or Z 5 ) subgroup in charged lepton sector and to a Z 2 subgroup in the neutrino sector, the trimaximal TM1 and TM2 mixing patterns can be obtained [25,30].…”