Modular S4 models of lepton masses and mixing

Abstract

We investigate models of charged lepton and neutrino masses and lepton mixing based on broken modular symmetry. The matter fields in these models are assumed to transform in irreducible representations of the finite modular group Γ 4 S 4 . We analyse the minimal scenario in which the only source of symmetry breaking is the vacuum expectation value of the modulus field. In this scenario there is no need to introduce flavon fields. Using the basis for the lowest weight modular forms found earlier, we build minimal phenomenologically viable models in which the neutrino masses are generated via the type I seesaw mechanism. While successfully accommodating charged lepton masses, neutrino mixing angles and mass-squared differences, these models predict the values of the lightest neutrino mass (i.e., the absolute neutrino mass scale), of the Dirac and Majorana CP violation (CPV) phases, as well as specific correlations between the values of the atmospheric neutrino mixing parameter sin 2 θ 23 and i) the Dirac CPV phase δ, ii) the sum of the neutrino masses, and iii) the effective Majorana mass in neutrinoless double beta decay. We consider also the case of residual symmetries Z ST 3 and Z S 2 respectively in the charged lepton and neutrino sectors, corresponding to specific vacuum expectation values of the modulus.

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

“…(72). As a result, the lepton mixing matrix would have six zeros for non-degenerate ρ L (γ f ) or four zeros for partially degenerate ρ L (γ f ) [21,26], and this is not consistent with the experimental data. Therefore there are no phenomenologically viable models with one common fixed point τ f in both neutrino and charged lepton sectors.…”

“…The modular forms of level 4 has been constructed in [25,26] in terms of η (τ )/η(τ ), where η(τ ) and η (τ ) are the Dedekind eta function and its derivative. In this section, we shall construct the modular forms of level 4 from the products of η(τ ).…”

“…Higher weight modular forms can be constructed in the same fashion, see Refs. [25,26] for modular forms of weight 8 and weight 10. Note that in our working basis the representation matrices are different from those of [25,26], and they are related to the choices of [25,26] by unitary transformations.…”

“…The finite modular groups Γ 2 ∼ = S 3 [14][15][16][17], Γ 3 ∼ = A 4 [12,14,15,[18][19][20][21][22][23], Γ 4 ∼ = S 4 [25][26][27] and Γ 5 ∼ = A 5 [30,31] have been studied and some simple modular models have been constructed. It is remarkable that even the A 4 modular models can reproduce the measured neutrino masses and mixing angles [12,19,23].…”

“…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 [21], S 4 [26] and A 5 [30] 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 [21,26].…”