A model with three scalar doublets can be conveniently accommodated within an A4 symmetric framework. The A4 symmetry permits only a restricted form for the scalar potential. We show that for the global minima of this potential alignment follows as a natural consequence. We also verify that in every case positivity and unitarity constraints are satisfactorily met.

We present an A4-based model where neutrino masses arise from a combination of see-saw mechanisms. The model is motivated by several small mixing and mass parameters indicated by the data. These are θ 13 , the solar mass splitting, and the small deviation of θ 23 from maximal mixing (= π/4). We take the above as indications that at some level the small quantities are well-approximated by zero. In particular the mixing angles, to a zero order, should be either 0 or π/4. Accordingly, in this model the Type-II see-saw dominates and generates the larger atmospheric mass splitting and sets θ 23 = π/4. The other mixing angles are vanishing as is the solar splitting. We show how the A4 assignment for the lepton doublets leads to this form. We also specify the A4 properties of the right-handed neutrinos which result in a smaller Type-I see-saw contribution that acts as a perturbation and shifts the angles θ 12 and θ 13 into the correct range and the desired value of ∆m 2 solar is produced. The A4 symmetry results in relationships between these quantities as well as with a small deviation of θ 23 from π/4. If the right-handed neutrino mass matrix, M R , is chosen real then there is no leptonic CP-violation and only Normal Ordering is admissible. If M R is complex then Inverted Ordering is also allowed with the proviso that the CP-phase, δ, is large, i.e., ∼ π/2 or −π/2. The preliminary results from NOνA favouring Normal Ordering and δ near −π/2 imply quasi-degenerate neutrino masses in this model. PACS No:14.60.Pq Key Words: Neutrino mixing, θ 13 , Solar splitting, A4, see-saw I IntroductionMany neutrino oscillation experiments have established that neutrinos are massive and non-degenerate and that the flavour eigenstates are not identical with the mass eigenstates. For the three neutrino paradigm the two independent mass square splittings are the solar (∆m 2 solar ) and the atmospheric (∆m 2 atmos ). The mass and flavour bases are related through the Pontecorvo, Maki, Nakagawa, Sakata -PMNS -matrix usually parametrized as: where c ij = cos θ ij and s ij = sin θ ij .The recent measurement of a non-zero value for θ 13 [1], which is small compared to the other mixing angles, has led to a flurry of activity in developing neutrino mass models which incorporate this feature. Earlier we had demonstrated [2] that a direction which bears exploration is whether two small quantities * email: soumitapramanick5@gmail.com † email: palitprof@gmail.com 1 in the neutrino sector, namely, θ 13 and the ratio, R ≡ ∆m 2 solar /∆m 2 atmos , could in fact be related to each other, both resulting from a small perturbation. Subsequently we had shown [3] that it is possible to envisage scenarios where only the larger ∆m 2 atmos and θ 23 = π/4 are present in a basic structure of neutrino mass and mixing and the rest of the quantities, namely, θ 13 , θ 12 , the deviation of θ 23 from π/4, and ∆m 2 solar all have their origin in a smaller see-saw induced perturbation 1 . Obviously, this gets reflected in constraints on the measured quantitie...

A model for neutrino masses and mixing is presented using the see-saw mechanism. The model combines Type -I and Type-II see-saw contributions of which the latter dominates. The scalars and the leptons in the model are assigned A4 charges suitable to obtain the mass matrices required for the scheme. The Type -II see-saw accommodates atmospheric mass splitting and maximal mixing in the atmospheric sector (θ 23 = π/4). It is characterized by vanishing solar mass splitting and θ 13 whereas the third neutrino mixing angle can acquire any value, θ 0 12 . Particular alternatives of θ 0 12 viz. θ 0 12 = 35.3 • (tribimaximal), 45.0 • (bimaximal), 31.7 • (golden ratio) are accounted for. Another choice of θ 0 12 = 0 • (no solar mixing) is also considered. Incorporating the corrections provided by the subdominant Type-I see-saw involves degenerate perturbation theory due to vanishing solar splitting in the Type -II see-saw enabling the solar mixing angle to receive substantial corrections. Apart from amending the solar sector the Type-I see-saw also tunes all the neutrino oscillation parameters into the allowed ranges thus interrelating them all. Thus the model is testable in the light of future experimental data. As an example, θ 23 emerges in the first (second) octant for normal (inverted) ordering. CP-violation is controlled by phases present in the right-handed Majorana neutrino mass matrix, M νR . Only normal ordering is allowed if these phases are absent. If M νR is complex the Dirac CP-violating phase δ, can be large, i.e., ∼ ±π/2, and inverted ordering is also allowed. T2K and NOVA preliminary data favouring normal ordering and δ ∼ −π/2 predicts lightest neutrino mass to be 0.05 eV or more within the model framework. 1 Earlier attempts on neutrino mass models with some oscillation parameters much smaller than the others can be located in [4].

Realistic neutrino mixing is achieved at one-loop level radiatively using S3 × Z 2 symmetry. The model comprises of two right-handed neutrinos, maximally mixed to produce the structure of the lefthanded Majorana neutrino mass matrix characterized by θ 13 = 0, θ 23 = π/4 and any value of θ 0 12 particular to the Tribimaximal (TBM), Bimaximal (BM) and Golden Ratio (GR) or other mixings. A small deviation from this maximal mixing between the two right-handed neutrinos could generate non-zero θ 13 , shifts of the atmospheric mixing angle θ 23 from π/4 and also could correct the solar mixing angle θ 12 by a small amount altogether in a single step. In this scotogenic mechanism of generating non-zero θ 13 by shifting from maximal mixing in the right-handed neutrino sector, two Z 2 odd inert scalar SU (2) L doublets were used, the lightest of which can serve as a dark matter candidate. *