Ameliorating the popular lepton mixings with A4 symmetry: A see-saw model for realistic neutrino masses and mixing

A model for neutrino masses and mixing is devised appointing the see-saw mechanism. The proffered model is fabricated with a combination of 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 conducive to obtain the mass matrices viable for the scheme. The Type -II see-saw mass matrix accommodates atmospheric mass splitting and maximal mixing in the atmospheric sector ($\theta_{23}=\pi/4$). It is characterized by vanishing solar mass splitting and $\theta_{13}$ whereas the third neutrino mixing angle is free to acquire any value of $\theta_{12}^0$. Particular alternatives of $\theta_{12}^0$ corresponding to the popular lepton mixings viz. $\theta_{12}^0=35.3^\circ$ (tribimaximal), $45.0^\circ$ (bimaximal), $31.7^\circ$ (golden ratio) are accounted for. Another choice of $\theta_{12}^0=0^\circ$ (no solar mixing) is reckoned. The subdominant Type-I see-saw constituent of the model propels all the neutrino oscillation parameters into the ranges allowed by the data which in its turn get interrelated owing to their common origin. This makes the model testable in the light of future experimental data. As an example, $\theta_{23}$ emerges in the first (second) octant for normal (inverted) ordering. CP-violation is governed by phases present in the right-handed Majorana neutrino mass matrix, $M_{\nu R}$. Only normal ordering is allowed if these phases are absent. If $M_{\nu R}$ is complex the Dirac CP-violating phase $\delta$, is capable of being large, i.e., $\sim \pm \pi/2$, and inverted ordering of neutrino masses is also permitted. T2K and NOVA preliminary data favouring normal ordering and $\delta \sim -\pi/2$ predicts lightest neutrino mass to be 0.05 eV or more within the framework of this model.


I Introduction
Intensive experimental investigations worldwide have determined neutrino masses and mixing to a great extent. In spite of these neutrinos retain certain mysteries including the ordering of their masses, their absolute mass scale, their Dirac or Majorana nature, the octant of the atmospheric mixing angle θ 23 and CP-violation in lepton sector. While future experiments address these riddles, here a model of neutrino masses and mixing in concord with the experimental observations is proposed. The two small quantities θ 13 and the ratio, R ≡ ∆m 2 solar /∆m 2 atmos can get interrelated when both are derived from a single perturbation [1]. In [2] larger mixing parameters like ∆m 2 atmos and θ 23 = π/4 were ascribed to the dominant fundamental structure of neutrino masses and mixing whereas the other oscillation parameters i.e., θ 13 , θ 12 , the deviation of θ 23 from π/4, and ∆m 2 solar originated from a smaller see-saw [3]  can give rise to vanishing θ 13 rather easily and new models based on perturbations of such structures are also common in literature [5,6].
Here, a schematic outline of the current exercise is given. The following standard parametrization of the lepton mixing matrix -the Pontecorvo, Maki, Nakagawa, Sakata (PMNS) matrix -U has been used where c ij = cos θ ij and s ij = sin θ ij . Neutrino masses and mixing are generated by a two-component Lagrangian, one of the dominant Type-II see-saw kind while the subdominant contribution originates from Type-I see-saw. The larger atmospheric mass splitting, ∆m 2 atmos and maximal atomspheric mixing (θ 23 = π/4) is embedded within the Type-II see-saw structure whereas the solar splitting, ∆m 2 solar and θ 13 are kept to be zero. The solar mixing angle can vary continuously and acquire any desired value of θ 0 12 . Needless to mention that neither ∆m 2 solar nor θ 13 are vanishing [7]. Evidences of non-maximal yet large θ 23 exist. The solar mixing angle θ 12 is also constrained by experiments. The Type-I see-saw alleviates all these issues. Since the solar splitting is vanishing in the Type-II see-saw scenario, the first two mass eigenstates are degenerate. In order to lift this degeneracy with the help of Type-I see-saw contribution one has to use degenerate perturbation theory. As a consequence of this, corrections to the solar mixing angle can be large.
The starting structure can be of tribimaximal (TBM), bimaximal (BM), and golden ratio (GR) mixings. All of these have θ 13 = 0 and θ 23 = π/4, θ 0 12 being the only discriminating factor as specified in Table 1. In this Table, the fourth option corresponds to no solar mixing (NSM) i.e., θ 0 12 = 0 which has the virtue of the mixing angles to be either maximal, i.e., π/4 (θ 23 ) or vanishing (θ 13 and θ 0 12 ). An A4-based model with identical objectives only for the NSM case was studied in [8]. This attempt along with [8] differ from the other earlier works on A4 [9,10,11] as in most of them neutrino mass matrix was derived as an outcome of a Type-II see-saw mechanism and obtaining TBM was of chief importance. Recent activities directed towards more realistic mixing patterns [12] often leading to breaking of A4 symmetry can be found in [13].
A few distinctive aspects of this model are worth noting at this point. Firstly, a combination of Type-I and Type-II see-saw is considered. Secondly, the model is constructed to accommodate many popular mixing patterns. This is the first attempt of this kind using A4 flavour symmetry that amends several popular lepton mixing patterns in a single stroke in which Type-II see-saw is the dominant contribution whereas Type-I see-saw is the subdominant component. The symmetries are broken spontaneously. Further, soft symmetry breaking terms are prohibited. All symmetry conserving terms are included in the Lagrangian. Scalars and leptons involved in the model are assigned suitable A4 charges to implement this feature. An analogous pursuit based on S3 × Z3 resulted in [14]. Right-handed charged leptons l 2R 1 ′ 1 (-2) 1 l 3R 1 ′′ Right-handed neutrinos N iR 3 1 (0) -1 Table 2: The lepton catalogue of the model. The A4 quantum numbers assignments of the fields are featured together with their SU (2) L properties. The hypercharge, Y , and lepton number, L, are displayed.
All the three neutrino mixing angles and the solar mass splitting receives first order corrections from a single source -the Type -I see-saw in this model. Owing to the common origin, they all get interrelated. These correlations are characteristic features of this particular model. Indeed the model has a large number of parameters, but it must noted that only the region of the parameter space allowed by the neutrino mass and mixing data obeying these correlations is considered.
An analysis of the model initiates the discussion. In the next section, the operational strategy is described. The results so obtained are compared to the experimental data in the following section, succeeded by the conclusions and inferences of this work. Some essential ideas of the of the discrete symmetry A4 are presented in Appendix A. A detailed study of the rich scalar sector to the extent of local minimization of the scalar potential is furnished in Appendix B. In Appendix C algebraic details of the mass matrix calculations while going to the flavour basis of the neutrinos from the Lagrangian basis can be found.

II The Mass Model
The model comprises of scalars and leptons with specific A4 charges. All terms allowed by the symmetries under consideration are included in the Lagrangian. No soft symmetry-breaking term is included.
The right-handed charged leptons transform as 1(e R ), 1 ′ (µ R ), and 1 ′′ (τ R ) under A4. The left-handed lepton doublets of three flavours constitute an A4 triplet, so does the right-handed neutrinos 2 . Table  2 shows the lepton constituents of the model together with their transformation properties under A4 and SU (2) L . The hypercharge and lepton number assignments are also shown 3 . The choices of A4 properties of the fields are not unique. A list of all possible options can be found in [15] of which this Masses of all leptons originate from A4-invariant Yukawa couplings. Several scalar fields have to be included 5 that acquire suitable vacuum expectation values (vevs). The strategy of choosing the scalar field multiplets requires some elaboration. An idea of the mass matrices of the left-and right-handed neutrinos in the flavour basis (charged lepton mass matrix diagonal) that are suitable for our avowed goal can be acquired from our previous work [14]. The Lagrangian is written down in a basis which is unitarily related to the flavour basis. Consequently, the mass matrices in this defining basis have somewhat complicated structures for which the motivation is not initially obvious. These forms of the mass matrices (below) arise from a rather large set of scalars and their vevs.
The charged leptons acquire their masses through the SU (2) L doublet scalar fields Φ i (i = 1, 2, 3) forming an A4 triplet. The neutrino Dirac mass matrix is generated by an A4 invariant SU (2) L doublet η, having lepton number 2. SU (2) L triplet scalars are required for the Type-II see-saw for left-handed neutrino mass matrix that include A4 triplet fields∆ L a and∆ L b along with ∆ L ζ , ζ = 1, 2, 3 transforming as 1, 1 ′ , 1 ′′ of A4. These are used to construct the dominant Type-II see-saw neutrino mass matrix. Effects of the subdominant Type-I see-saw contribution is included perturbatively. A4 conserving Yukawa couplings produce the right-handed neutrino mass matrix as well. Several SU (2) L singlet scalars are involved in generation of the Majorana masses for the right-handed neutrinos viz. ∆ R p (p = a, b, c) transforming as A4 triplets and ∆ R γ (γ = 1, 2, 3) transforming as 1, 1 ′ and 1 ′′ under A4. Table 3 evinces transformation properties of the model scalars under A4 and SU (2) L together with their hypercharge, lepton number and vev configurations. The vevs of the SU (2) L doublet scalars are of O(M W ) while that of the SU (2) L triplets are several orders of magnitude smaller than the doublet vevs in concord with the small neutrino masses as well as the ρ parameter of electroweak symmetry breaking. As expected, the vevs of the SU (2) L singlets responsible for right-handed neutrino mass lies much above the electroweak scale. The mass terms of the neutrinos (both Type-I and Type-II see-saw) and that of the charged leptons are generated by a SU (2) L × U (1) Y conserving Lagrangian that preserves A4 as well 6 : The scalars acquire the following vevs (SU (2) L part is suppressed): An elaborate study of the A4 conserving scalar potential involving the fields listed in Table 3 is presented in Appendix B of this paper. Local minimization is performed and the conditions corresponding to the particular vev structures as indicated in Eqs. (3)(4)(5) are obtained.
The mass matrix for the charged leptons and the left-handed Majorana neutrinos so obtained are: where the choice of Y L 2 = Y L 3 is made. The Yukawa couplings involved in the charged lepton mass matrix satisfies y 1 v = m e , y 2 v = m µ , y 3 v = m τ . The neutrino mass matrix of Dirac nature and the right-handed neutrino mass matrix of Majorana kind acquires the following structures: m D sets the scale of Dirac masses of the neutrinos where one can identify f u = m D . The scale of the Type-II see-saw neutrino masses is much smaller than that of the charged leptons i.e., Such a possibility that the triplet vev is much smaller than the doublet vev can be obtained as shown in [18], albeit in a model with fewer scalars. The scale of the right-handed Majorana neutrino masses is set by m R and χ i in Eq. (7) are dimensionless quantities 7 of O(1).
The mass matrices in Eq. (6) could be expressed in a more convenient form by applying a couple of transformations. The non-hermitian charged lepton mass matrix can be diagonalised by applying a transformation U L (below) on the left-handed lepton doublets and no transformation on the righthanded charged leptons. The transformation matrices are expressed as: This basis in which the charged lepton mass matrix is diagonal and the entire lepton mixing is governed by the neutrino sector is termed as the flavour basis in which the mass matrices acquire the following forms: Here m ± ≡ m 1 . Therefore, m − is positive (negative) for normal (inverted) ordering. As noted earlier, M f lavour νL , which arises from the Type-II see-saw, is the dominant contribution to the neutrino mass.
Demanding that the neutrino Dirac mass matrix, which couples the left-and right-handed neutrinos, preserves its proportionality to the identity matrix necessitates that the transformation applied on the right-handed neutrino fields must be V R = U L . Thus we get,   r 11 r 12 r 13 r 12 r 22 r 23 r 13 r 23 r 33   .
7 See Appendix C for exact expressions of χi in Eq. (7).
The matrices in Eq. (10) will take part in the Type-I see-saw mechanism 8 . Various identification of the products of the Yukawa couplings and the vevs with the neutrino mass and mixing parameters are necessary for the mass matrices to be expressed in the forms as presented in Eqs. (9) and (10). Appendix C comprises of these algebraic details.

III Modus Operandi
The four mass matrices in the flavour basis obtained from the model are given in Eq. (9) and (10).
In this basis the entire lepton mixing and CP-violation is controlled solely by the neutrino sector to which we restrict our discussion now onwards. The Type-II see-saw derived M f lavour νL is the dominant component to which the subdominant contribution attributed by the Type-I see-saw is incorporated by perturbation theory. The flavour basis mass matrices have to undergo one more basis transformations for successful implementation of this scheme. More precisely they ought to be expressed in the mass basis of the neutrinos which by definition has the left-handed neutrino mass matrix diagonal in it. Thus, where, The left-handed neutrino fields in the mass basis (|ν mass L ) are connected to the ones in the flavour basis (|ν f lavour L ) by this U 0 furnished in Eq. (12). One can obtain the |ν mass L by applying U 0 † on |ν f lavour L i.e., |ν mass L = U 0 † |ν f lavour L . It immediately follows from Eqs. (11), (1) and (12) that in the Type-II see-saw component solar splitting is absent, θ 13 = 0 and θ 23 = π/4. The columns of U 0 are the unperturbed flavour basis.
Once again we demand that in the mass basis the neutrino Dirac mass matrix remains proportional to identity. In order to satisfy this the same transformation (U 0 † ) has to be applied on the righthanded neutrino fields. This leads to changes in form of right-handed neutrino mass matrix given by M mass νR = (U 0 † M f lavour νR U 0 ). The matrices contributing in Type-I see-saw are as follows: Here a and b are dimensionless quantities 9 of O(1). It is imperative to note that a and b can in general be complex. One can in principle trade off a and b in terms of complex numbers ye −iφ 2 and xe −iφ 1 respectively, where x and y are dimensionless real quantities of O(1). The Type-I see-saw contribution so obtained is given by: Here the Dirac mass matrix is proportional to identity. It was checked that the same results can follow as long as M D is diagonal. M mass νR exhibits a N 2R ↔ N 3R discrete symmetry. The results remain intact even if that choice is relaxed. Now onwards the entire procedure is carried on in the mass basis of the neutrinos using the mass matrices expressed in Eqs. (11) and (14).
The method followed below essentially consists of the following steps. Form the Type-II see-saw a lepton mixing of the form of Eq. (12) is generated, with θ 0 12 of any preferred value. At this stage, only the atmospheric mass splitting is non-zero and atmospheric mixing is maximal. Next, the Type-I see-saw is included using degenerate perturbation theory. The solar mass splitting and the desired θ 12 are first obtained. Then the third column of the mixing matrix is calculated and compared with Eq.

IV Results
The neutrino mass matrices derived from Type-I and Type-II see-saw mechanism have been discussed in the previous section, of which the former is significantly smaller than the latter. In absence of the Type-I see-saw contribution the leptonic mixing matrix is characterized by θ 13 = 0, θ 23 = π/4, and θ 0 12 is free to vary. Consequences for four choices of the value of θ 0 12 corresponding to TBM, BM, GR, and NSM cases together with vanishing solar splitting are examined. This along with the atmospheric mass splitting allowed by the data depict the Type-II see-saw structure. Inclusion of Type-I see-saw corrections perturbatively up to first order modulates the neutrino oscillation parameters into the ranges preferred by data. Owing to the vanishing solar splitting in the Type-II see-saw contribution the first two mass eigenstates are degenerate. Thus in the solar sector degenerate perturbation theory has to be applied. Hence the first order corrections to the solar mixing angle can be large. The global best-fit of the oscillation parameters are displayed in the next section.

IV.1 Data
The current 3σ global fits of the neutrino oscillation parameters are: [19,20] These numbers are taken from NuFIT2.1 of 2016 [19]. Needless to mention, ∆m 2 ij ≡ m 2 i − m 2 j , such that ∆m 2 31 > 0 for normal ordering (NO) and ∆m 2 31 < 0 for inverted ordering (IO). Two best-fit points of θ 23 are evinced by the data in the first and in the second octants. Towards the end of the paper it is discussed how the model can accommodate the recent T2K and NOVA hints [21,22] of δ close to -π/2. Table 4: Data allowed 3σ ranges of ζ (Eq. (18)), ǫ (Eq. (19)), and (ǫ − θ 0 12 ) for different popular mixing patterns are shown.
As a warm-up exercise let us consider the simpler case of M νR real. In such a scenario there is no CP-violation as the phases φ 1,2 of Eq. (14) are 0 or π. This leads to four different alternatives available for choosing φ 1 and φ 2 . These are captured compactly by taking x and y real and allowing them to assume both signs for notational convenience. It will be soon clear how the experimental observations prefer one or the other of these four alternatives. Thus for real M νR the Type -I see-saw contribution appears like: The degeneracy of the two neutrino masses in the Type-II see-saw ensuring the vanishing solar splitting necessitates the application of degenerate perturbation theory to obtain the corrections for the solar sector mixing parameters 10 . The entire dynamics of this sector is dictated by the upper 2×2 submatrix of M ′ given by: This gives rise to: For functional ease it is useful to define a quantity, ǫ as: Once a mixing pattern is selected, the corresponding θ 0 12 gets fixed and the experimental bounds of θ 12 determines the 3σ ranges of ζ and ǫ by means of Eq. (15) and Eq. (19) as featured in Table 4. The ratio (y/x) is positive (negative) when ζ is positive (negative). From Eq. (19) it is evident that the sign of y is regulated by the value of ǫ. Putting all these facts together it is easy to infer that x is positive always, or in other words φ 2 must be 0, while y has to be positive, φ 1 = 0 (negative, φ 1 = π) for NSM (BM). In case of TBM and GR, both signs of y are admissible. The solar splitting provided by the Type-I see-saw as extracted from Eq. (17) is: For the mass basis form of the mass matrix in Eq. (11), the mixing in the leptonic sector is completely given by the U 0 given in Eq. (12). After including the Type-I see-saw correction to the mass matrices there is a further contribution to the mixing matrix as well, now given by: with The third column of the lepton mixing matrix is: As already pointed out, x is always positive, κ r is positive (negative) for NO (IO).
Eq. (23) when mapped to the third column of Eq. (1) leads to: and The allowed ranges of (ǫ − θ 0 12 ) for the different mixing patterns is given in Table 4 . The CP-phase δ is 0 (π) when sin(ǫ − θ 0 12 ) is positive (negative) in case of normal ordering 11 . It can be immediately concluded that δ = 0 for the NSM from Table 4 and δ = π for the rest of the options under study. CP is conserved for both the values of δ.
Using Eqs. (20), (22), and (24) it can be found: For real M νR inverted ordering is forbidden as can be seen from Eq. (26). In order to justify this one can define: where z is positive for both the orderings of neutrino masses. With the help of Eq. (26) it can be written as: From Eq. (27) it is straightforward to show that: z = sin ξ/(1 + sin ξ) i.e., 0 ≤ z ≤ 1 2 (for normal ordering), 11 Inverted ordering is prohibited for real MνR. : ω = (π/4 − θ 23 ) -vs-θ 12 plot for normal ordering. The 3σ allowed range of sin θ 13 is marked by the solid lines whereas the dashed line indicates the best-fit value. Thin pink (thick green) lines denote the BM (NSM) case. The horizontal and vertical lines represent the data allowed 3σ range. The first octant of θ 23 is preferred since ω is positive always. Although ω is positive for TBM and GR mixing patterns its value lies beyond the 3σ range. Best-fit values of atmospheric and solar mass splittings are taken. Inverted ordering is disallowed for M νR real.
The lightest neutrino mass m 0 has a one-to-one correspondence with z. In the quasi-degenerate limit, i.e., m 0 → large, z → 1 2 for both orderings. For real M νR , | cos δ| = 1 in Eq. (28). It simply follows from the global fit mass splittings and mixing angles in Sec. IV.1 and Table 4 that z ∼ 10 −2 or smaller for all four popular mixing alternatives. Thus inverted ordering is forbidden for real M νR .
Eq. (25) implies that ω is positive always for normal ordering irrespective of the mixing pattern. Thus θ 23 is confined only to the first octant for real M νR . ǫ can be expressed in terms of θ 12 using Eqs. (18) and (19). Thus ω in Eq. (30) can be expressed as a function of θ 13 and θ 12 only. Fig. 1 exhibits ω as a function of θ 12 for BM (thin pink lines) and NSM (thick green lines) alternatives. θ 12 and ω varied within 3σ allowed ranges as shown in Sec. IV.1. The TBM and GR cases are excluded owing as for the allowed values of θ 12 they predict θ 23 beyond the 3σ range. The 3σ limiting values of θ 13 are marked by the solid lines whereas the dashed lines indicate its best-fit value. The vertical and horizontal blue dot-dashed lines denote the 3σ experimental limits of θ 12 and θ 23 .
With the help of Eq. (28), one can translate any allowed point in the ω − θ 12 plane and the θ 13 associated with it to a value of z or equivalently m 0 , when the solar and the atmospheric mass splittings are provided. For both the allowed mixing patterns m 0 varies over a very small range. This range is found to be 2.13 meV ≤ m 0 ≤ 3.10 meV (3.20 meV ≤ m 0 ≤ 4.42 meV) for NSM (BM) when both mass splittings and all the three mixing angles are allowed to vary over their entire 3σ ranges.
The salient features of real M νR case are: 1. Only the normal ordering of neutrino masses is allowed.
2. Only the first octant of θ 23 is admissible.
3. Type-I see-saw corrections is unable to make the TBM and GR mixing patterns consistent with the allowed ranges of the mixing angles.
4. NSM and BM alternatives can produce solutions in agreement with the observed neutrino masses and mixing. The allowed ranges of lightest neutrino mass is very narrow.

IV.3 Complex M νR
Real M νR has several limitations viz. inverted ordering and CP-violation is forbidden. Moreover TBM and GR mixing patterns cannot be included within the ambit of the model when M νR is real. In order to overcome these constraints the general complex form of M νR leading to Type-I see-saw contribution M ′ furnished in Eq. (14) has to be considered. It is worth reminding ourselves that this choice introduces the complex phases φ 1,2 while x and y can only be positive.
Thus, M ′ is no longer hermitian. To retain the hermitian nature the combination are treated as the leading term and the perturbation at the lowest order respectively. The unperturbed eigenvalues are given by (m i ) 2 and perturbation matrix is: where, The rest of the procedure is analogous to what was done in case of real M νR keeping in mind the discriminating factors of Eq. (31). Now, instead of Eqs. (18) and (19) of the real M νR case, the solar mixing obtained from Eq. (31) is given by and sin ǫ = y cos φ 1 Table 4 shows the allowed ranges of ζ and ǫ which depend on the mixing patterns. For all mixing alternatives cos ǫ is found to be positive. Thus from Eq. (34) φ 2 must always lie in the first or fourth quadrants. For the different mixing patterns the ranges of φ 1 are also given by that of ǫ. When ǫ is positive (negative) then from the first relation contained in Eq. (34), it is evident that φ 1 has to be in the first or fourth (second or third) quadrants. Using the results displayed in Table 4   Applying degenerate perturbation theory the solar mass splitting attributed completely to the Type-I see-saw contribution can be obtained from Eq. (31): (35) In place of Eq. (23) one gets: where, Here Eq. (34) sin θ 13 sin δ = κ c m − m + cos φ 1 cos φ 2 sin ǫ sin φ 1 cos φ 2 cos θ 0 12 − cos ǫ cos φ 1 sin φ 2 sin θ 0 12 .
From Table 4, it is obvious that (ǫ − θ 0 12 ) exists in the first (fourth) quadrant for the NSM (BM, TBM, and GR) mixing pattern. From Eq. (38) one can immediately conclude that for NSM (BM, TBM, and GR) case(s) δ remains in the first or fourth (second or third) quadrants in case of normal ordering. κ c changes sign for inverted ordering. Thus the quadrants get modified accordingly. The different alternatives are furnished in Table 5. There are two allowed quadrants of δ having sin δ of opposite sign for any mixing option and ordering of neutrino masses. The sign of the right-hand-side of Eq. (39) governs the phases φ 1,2 which in its turn decides the quadrants CP-phase δ out of the two allowed options. As already discussed, φ 2 can be in either the first or fourth quadrants. The quadrant of φ 1 depends on the mixing pattern in such a manner that sin φ 1 can be of either sign. Therefore, the phases φ 1 and φ 2 can be chosen in a way such that sin δ can acquire any particular sign. Thus the two alternate quadrants of δ for every case in Table 5 are equally allowed in the model.
The Type-I see-saw perturbative contribution to the atmospheric mixing angle can be obtained from Eq. (36) as: Let us recall, Eq. (38) relates δ and (ǫ − θ 0 12 ) through κ c . Thus for all mixing alternatives θ 23 always remains in first (second) octant for NO (IO). This is one of the most important results of the model as shown in Table 5.
In the solar splitting expressed in Eq. (35), the factor of m 2 D /m R can be replaced in terms of κ c . This together with Eq. (38) gives, Predictions of the model can be extracted from Eqs. (40) and (41). The three mixing angles θ 13 , θ 12 , and θ 23 are taken as inputs. Eq. (40) determines a value of the CP-violating phase δ. With the help of these and the experimentally observed solar splitting the combination m  . Normal and inverted orderings are always associated with the first and second octants of the atmospheric mixing angle θ 23 respectively. For NSM case δ lies in the first (second) quadrant for normal (inverted) ordering, while for the rest of the mixing options it is in the second (first) quadrant. For inverted neutrino mass ordering |δ| remains close to π/2 for the complete range of m 0 . The CP-phase δ lies near π/2 for normal ordering for m 0 larger than around 0.05 eV.
From Table 5 it is evident that if δ is a solution for some m 0 then by properly choosing alternate values of the phases φ 1,2 appearing in M νR one can also obtain a second solution with the phase −δ. This mirror set of solutions are not shown in Fig. 2. The preliminary data presented by the T2K [21] and NOVA [22] collaborations can be considered as primary hint of normal ordering associated with δ ∼ −π/2. The consistency of this model with these observations is clearly visible from Fig. 2 with δ ∼ −π/2 favouring m 0 in the quasi-degenerate regime, i.e., m 0 ≥ O(0.05 eV), for normal ordering. If this result is determined with better accuracy in the future analysis then the model will predict neutrino masses to be in a range that ongoing experiments are capable of probing [23,24].
These interrelationships between the octant of θ 23 , the quadrant of the CP-violating phase δ, and the neutrino mass ordering provide a clear set of correlations characteristic of this A4 based model. In the model the corrections to the three neutrino mixing angles and ∆m 2 solar all have a common originthe Type-I see-saw. As a result these parameters get correlated. Such interrelationships are specific to this model. Although the model has a large number of parameters, only this correlated region of the parameter space allowed by neutrino mass and mixing data leads to testable predictions in Table 5.

V Conclusions
In this paper an A4 based see-saw model for neutrino masses and mixing has been proposed. The flavour quantum numbers suitable for the model are assigned to the leptons and the scalars. The Lagrangian is inclusive of all the symmetry conserving terms. No soft breaking of symmetry is entertained. The Yukawa couplings induce the charged lepton masses, Dirac and Majorana masses for the left-and right-handed neutrinos after the symmetry is broken spontaneously. Neutrino masses are produced by a combined effect of both Type-I and Type-II see-saw terms present in the Lagrangian of which the former can be thought of to be a small correction. The Type-II see-saw dominant contribution is associated with the atmospheric mass splitting, no solar splitting, keeps θ 23 = π/4, and θ 13 = 0 and θ 12 can be given any preferred value. In particular, this model is scrutinized in context of tribimaximal, bimaximal, golden ratio, and 'no solar mixing' patterns. The contribution of Type-I see-saw can be treated as a perturbation that generates the solar splitting and tunes the mixing angles to values in agreement with the global fits. As a corollary a correlation between the octants of θ 23 and neutrino mass ordering followed -first (second) octant is allowed for normal (inverted) ordering of neutrino mass. The model has several testable predictions including that of the CP-phase δ, relationships between mixing angles and mass splittings. Moreover, inverted ordering got associated with near-maximal CP-phase δ and arbitrarily small neutrino masses are allowed. In case of normal ordering δ can vary over a larger range and maximality is accomplished in the quasi-degenerate regime. The lightest neutrino mass has to be at least a few meV for this case.

A Appendix: The group A4
A4 is the even permutation group of four objects having 12 elements and two generators S and T satisfying the property S 2 = T 3 = (ST ) 3 = I. It has four inequivalent irreducible representations viz. one 3 dimensional representation and three 1 dimensional representations namely, 1, 1 ′ and 1 ′′ . These three dimension-1 representations are singlets under S whereas they transform as 1, ω, and ω 2 respectively under the action of T , ω being a cube root of unity. Therefore it is apparent that 1 ′ ×1 ′′ = 1. The pertinent form of the generators S and T acting on the 3 dimensional representations are given by 12 , It is imperative to note the product rule for the three dimensional representation is: When two triplets of A4 given by 3 a ≡ a i and 3 b ≡ b i , with i = 1, 2, 3; are combined according to Eq. (A.2), then the resultant triplets can be represented by 3 c ≡ c i and 3 d ≡ d i where, and the 1, 1 ′ and 1 ′′ so obtained can be scripted as: The group is studied in extensive details in [9,10].

B Appendix: Minimization of the scalar potential
Some detailed analysis of the nature of the scalar potential is presented in this Appendix. The conditions that have to be satisfied by the parameters of the potential so that the vevs acquire the values considered in the model are extracted. The conditions so obtained guarantee the potential is locally minimized by those choices. To confirm if those choices are in concurrence with the global minimum is beyond the scope of this work 13 .
12 This choice of basis has the generator S diagonal. One can equivalently perform an analogous analysis in a basis in which the generator T is diagonal. Needless to mention that the two bases are related by some unitary basis transformation. 13 As an example one can take a look at [25] where a comparatively simpler scenario consisting of an A4 triplet composed of three SU (2)L doublet scalar or in other words an A4 symmetric three Higgs doublet model (3HDM) was analyzed in terms of the global minimization of the scalar potential. In [26], it is shown that alignment follows as a natural consequence when the vevs acquire the configurations corresponding to those global minima. Three Higgs doublets symmetric under A4 group has been vividly discussed in [27]. A model for leptons using an A4 symmetric 3HDM can be found in [28]. Table 3 comprise of scalars having lepton numbers as well as A4, SU (2) L , and U (1) Y charges. The scalar potential must be of the most general quartic nature conserving all the symmetries under consideration. Thus all the terms allowed by the symmetries are included in the discussion below. Verification of SU (2) L , U (1) Y and lepton number are familiar exercises. A4 invariance requires elaborate discussion as presented in the following section.

B.1 A4 conserving terms: Notations and general principles
Let us summarize a few salient features of this model to fix the notations to be followed for the A4invariant terms. As already noted, the scalar spectrum has fields transforming as 1, 1 ′ , 1 ′′ , and 3 under A4. One has to consider all the combinations of these fields up to quartics that can yield A4 invariants. The product rules for 1, 1 ′ and 1 ′′ are easy, but that for the triplets of A4 needs to be emphasized.
properties that are not considered for the time being in the immediate course of discussion. As furnished in Eq. (A.2), one can combine A and B to obtain The scalar potential can be formulated implementing this notation and keeping in mind that the scalar sector of this model is devoid of any field which is invariant under all the symmetries under consideration. Therefore the scalar potential will contain terms of the following kind (only A4 properties are exhibited): Here W is any field, X, X ′ , and X ′′ represent generic fields transforming as 1, 1 ′ , and 1 ′′ under A4 while Y happens to be generic A4 triplet field. The invariants constructed by using X † , X ′ † , X ′′ † , and Y † are not listed separately.
Owing to the large number of scalars in the model -e.g., SU (2) L singlets, doublets, and tripletsthe scalar potential consists of many terms. In order to simplify the discussion, cubic terms in the fields are excluded and all the couplings are taken to be real. The antisymmetric triplet arising from the combination of two A4 triplets i.e., the terms denoted by T a in Eq. (B.3) are not included in the potential throughout for ease of calculation. The potential is studied piece-wise: (a) consisting of terms that arise from combination of fields belonging to same SU (2) L sector, and (b) comprising of terms obtained by combining scalars of different SU (2) L sectors. The vev of the SU (2) L singlets giving rise to the right-handed neutrino mass are larger than the vev of the other scalar fields. Thus in the latter category the combinations of SU (2) L singlets with the doublets and triplets of SU (2) L are considered, whereas, doublet-triplet inter-sector terms are dropped owing to the smallness of the triplet vev responsible for the left-handed Majorana neutrino mass. Also the electroweak precision measurements put a stringent bound on the triplet vev compelling it to be very small.

B.2 SU(2) L Singlet Sector:
The SU (2) L singlet scalar sector consists of three A4 triplets∆ R p with p = a, b, c denoting each one of them. These three triplets possess identical quantum numbers, their vev being the only discriminating criterion. Also there are three more fields viz. ∆ R 1 , ∆ R 2 and ∆ R 3 transforming as 1, 1 ′ and 1 ′′ under A4. From Eq. (B.1) we can see that two same∆ R p triplets can combine to produce several A4 irreducible representations. For notational simplicity let us define: Using two different triplets∆ R p and∆ R q where p = q analogous combinations can be defined: Generically, it is convenient to use O ip or T sp if the second triplet in the argument is replaced by its hermitian conjugate. As an example, One can also consider: Also the following combinations are required: The A4 singlets ∆ R i (i = 1, 2, 3) can be combined to yield Needless to mention that such terms are singlets of all the symmetries under consideration.
Having devised the essential notations one can write the most general scalar potential for the SU (2) L singlet sector of this model as: .

(B.10)
Here λ s 3p , λ s 3pq and λ s 3pq are taken as the common coefficient of the different A4 invariants generated by combining two∆ R and two (∆ R ) † fields. Similar policy will be adopted for the fields with other SU (2) L properties.

B.3 SU(2) L Doublet Sector:
The SU (2) L doublet scalar precinct consists of the two fields Φ and η transforming as 3 and 1 of A4 respectively. Opposite hypercharges are assigned to Φ and η. The A4 triplet Φ combinations are denoted as: (B.11) and that of the A4 singlet η are: The potential for the SU (2) L doublet sector is given by:

B.4 SU(2) L Triplet Sector:
The SU (2) L triplet sector comprises of five fields. There are two A4 triplets∆ L a and∆ L b together with the fields the ∆ L 1 , ∆ L 2 and ∆ L 3 transforming as 1, 1 ′ , 1 ′′ of A4 respectively.
It is useful to define: The scalar potential for this sector: .

(B.19)
B.5 Inter-sector terms in the scalar potential: The terms in the scalar potential involving scalar fields of identical SU (2) L behavior are already taken into account. Apart from them, the scalar potential will also receive contributions from terms generated by combining scalars of two different SU (2) L sectors that constitute the main objective of the following discussion. In this category the combinations of the SU (2) L singlet scalars with that belonging to either of the doublet or the triplet sector. The other variety of inter-sector terms -doublet-triplet type -are not included. This seems to be a reasonable approximation as the vevs of the singlet fields are the largest.

B.5.1 Singlet-Doublet inter-sector terms:
Let us consider the combinations: Using this notations: (B.22) In the last two terms a simplifying assumption of using a common couplings λ sd 7p and λ sd 7pq for the terms in the scalar potential that are generated from various combinations of (Φ † Φ)(∆ R †∆R ), all four of the fields involved being triplets of A4.
Following the convention introduced already: The inter-sector potential for this case is given by: It must be noted that while writing the last λ ts 7−12 terms the different couplings corresponding to the combinations of O ts inp with (∆ L † i ∆ R j ) and O ts inp with (∆ L † i ∆ R † j ) are set to be equal.

B.6 The conditions for minimization:
With the scalar potential in hand it is necessary to derive the conditions for which the particular vev configurations used in this model -see Eqs. (3), (4) and (5) and Table 3 -corresponds to the local minimum. For immediate reference the vevs are: (B.28) where the SU (2) L nature of the scalars has been suppressed.
Eq. (B.26) shows that the A4 triplet fields -∆ L,R and Φ -have vev configurations that have been verified to be the global minima in [25]. This result was for a single A4 triplet considered in isolation.
In the current scenario since many other fields are involved, it is not straight-forward to directly adopt the conclusions of [25].
The conditions for which the vev configurations shown in Eqs. (3), (4) and (5) correspond to minimum are shown sector by sector.
For minima of the scalar potential, the first derivatives of the scalar potential with respect to the vevs have to vanish and the second derivatives have to satisfy some conditions. Since the scalar sector is very rich, the expressions look very complicated. The conditions arising by setting the first derivatives to be zero have been discussed for each of the SU (2) L sectors. As a sample, constraints coming from the second derivatives have been shown only for the SU (2) L singlet sector. Similar exercise can be carried out for the other SU (2) L sectors but are not presented here.
B.6.1 SU (2) L singlet sector: The SU (2) L singlet vevs are much larger than those of the doublet and triplet scalars. Thus it is safe to neglect the contributions to the minimization equations from the inter-sector terms.
Let us remind ourselves that v Rp (p = a, b, c) are real and define: v For ease of presentation, let us set the following masses and couplings equal: With the help of the singlet sector potential in Eq. (B.10), the equalities in Eq. (B.30) and the vev in Eqs. (3), (4) and (5) one can obtain: Besides the first derivatives discussed above, second derivatives are also needed to established minimality. For example, and Further mixed derivatives such as: are also necessary to establish minimality in the most general case. The results presented for the first and second derivatives are calculated using the most general expression of the scalar potential in terms of the vevs and putting (v Needless to mention that v * Rp i =ṽ Rp i for (p = a, i = 1, 2, 3) and (p = b, c and i = 1). Similar equations can be obtained by minimizing the potential wrt u 2R , u 3R ,ṽ Ra2 ,ṽ Ra3 andṽ Rpi for (p = b, c) and (i = 1, 2, 3). For the sake of brevity those are not mentioned. Similar strategy will be adopted for the SU (2) L doublet and SU (2) L triplet sector. It is worth noting that this exercise for all the three sectors are for illustrative purpose only and the minimization equations are achieved by setting the different couplings equal.
B.6.2 SU (2) L doublet sector: For this sector contributions from both the doublet sector itself -Eq. (B.13) -together with the singlet-doublet inter-sector are considered. Let us define The following couplings are set to be equal: For the vevs in Eqs. (B.26), (B.27) and (B.28) correspond to the minimum of the scalar potential it is necessary to satisfy the following conditions: and In order to satisfy Eqs. (B.37) and (B.38) some degree of fine-tuning is necessary that involve both SU (2) L doublet and singlet vev of varying magnitudes. Similar equations can be obtained by minimizing the potential wrt v * 2 and v * 3 .

(B.41)
It is worth noticing that certain fine-tuning is essential to satisfy Eqs. (B.40) -(B.41). Also similar equations can be obtained by minimizing the potential wrt u * Lj where (j = 2, 3), v * Lni where for n = b one has (i = 1, 2, 3) and for n = a we have (i = 2, 3). Those are not mentioned here. This exercise is performed to illustrate the scenario in a simplified limit achieved by setting several masses and couplings to be equal.

C Appendix: Flavour basis form of the mass matrices
Mass matrices expressed in the Lagrangian basis in Eqs. (6) and (7) can be transformed to simpler forms in the flavour basis as in Eqs. (9) and (10) with the help of a unitary transformation written in Eq. (8). Certain straight-forward algebraic calculations related to this derivation of the forms the mass matrices in the flavour basis is furnished in this Appendix.
The Lagrangian in Eq. (2) produces the following mass matrix for the charged leptons and the lefthanded Majorana neutrinos: (C.1) where, the Yukawa coupling Y L 2 is chosen to be equal to Y L 3 . Also, y 1 v = m e , y 2 v = m µ , y 3 v = m τ is satisfied. The dominant Type-II see-saw component of the neutrino mass matrix, M νL , gives rise to the atmospheric splitting and maximal atmospheric mixing but is devoid of solar splitting and is therefore characterized by two masses m where, Here m R is the right-handed Majorana neutrino mass scale and χ i are dimensionless O(1) quantities. In order to achieve the right-handed Majorana neutrino mass matrix of the form expressed in Eq. (10), the vev and Yukawa couplings products have to obey: Y R 1 u 1R = m R (r 11 + 2r 23 ), Y R 2 u 2R = m R (r 22 + 2r 13 ), Y R 3 u 3R = m R (r 33 + 2r 12 ) Y R a v Ra = 2m R (r 11 − r 23 ),Ŷ R b v Rb = 2m R (r 22 − r 13 ) andŶ R c v Rc = 2m R (r 33 − r 12 ). (C.4) The r ij in Eq. (C.4) are given by : r 11 ≡ √ 2b sin 2θ 0 12 + a sin 2 θ 0 12 ,