Trimaximal neutrino mixing from modular A4 invariance with residual symmetries

Abstract

We construct phenomenologically viable models of lepton masses and mixing based on modular A 4 invariance broken to residual symmetries Z T 3 or Z ST 3 and Z S 2 respectively in the charged lepton and neutrino sectors. In these models the neutrino mixing matrix is of trimaximal mixing form. In addition to successfully describing the charged lepton masses, neutrino mass-squared differences and the atmospheric and reactor neutrino mixing angles θ 23 and θ 13 , 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 the existence of specific correlations between i) the values of the solar neutrino mixing angle θ 12 and the angle θ 13 (which determines θ 12 ), ii) the values of the Dirac CPV phase δ and of the angles θ 23 and θ 13 , iii) the sum of the neutrino masses and θ 23 , iv) the neutrinoless double beta decay effective Majorana mass and θ 23 , and v) between the two Majorana phases. *

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.…”

“…In order to accommodate the observed lepton mixing angles, it is assumed the complex modulus τ in the charged lepton and neutrino mass matrices takes two different values τ f,e and τ f,ν which break the modular symmetry into the residual subgroups generated by γ f,e and γ f,ν respectively in the charged lepton and neutrino sectors. In this approach, the trimaximal TM2 lepton mixing pattern can be obtained from the A 4 modular group [21] and the models giving TM1 mixing are constructed with multiple modular S 4 groups [38,39]. Inspired by the tri-direct CP approach [43,44,48], in the following we shall present the tri-direct modular model which has three fixed moduli.…”

“…The modular group Γ(3) has been extensively studied in the literature [12,14,15,[18][19][20][21][22]. The finite modular group Γ 3 is isomorphic to A 4 .…”

“…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].…”