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 ΓN of the modular symmetry, broken by the vacuum expec- tation value (VEV) of a modulus field τ. If the VEV of the modulus τ takes some special value, a residual subgroup of ΓN would be preserved. We derive the fixed points τS = i, τST = (−1 + i sqrt{3} )/2, τTS = (1 + i sqrt{3} )/2, τT = i∞ in the fundamental domain which are in-variant under the modular transformations indicated. We then generalise these fixed points to τf = γτS, γτST, γτTS and γτT in the upper half complex plane, and show that it is suffi-cient to consider γ ∈ ΓN. Focussing on level N = 4, corresponding to the flavour group S4, 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 + sqrt{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 A4 flavour symmetry.