I Introduction
###### Abstract

We develop a see-saw model for neutrino masses and mixing with an symmetry. It involves an interplay of Type-I and Type-II see-saw contributions of which the former is subdominant. The quantum numbers of the fermion and scalar fields are chosen such that the Type-II see-saw generates a mass matrix which incorporates the atmospheric mass splitting and sets . The solar splitting and are absent, while the third mixing angle can achieve any value, . Specific choices of are of interest, e.g., (tribimaximal), (bimaximal), (golden ratio), and (no solar mixing). The role of the Type-I see-saw is to nudge all the above into the range indicated by the data. The model results in novel interrelationships between these quantities due to their common origin, making it readily falsifiable. For example, normal (inverted) ordering is associated with in the first (second) octant. CP-violation is controlled by phases in the right-handed neutrino Majorana mass matrix, . In their absence, only normal ordering is admissible. When is complex the Dirac CP-phase, , can be large, i.e., , and inverted ordering is also allowed. The preliminary results from T2K and NOVA which favour normal ordering and are indicative, in this model, of a lightest neutrino mass of 0.05 eV or more.

Key Words:  Neutrino mixing, , Solar splitting, S3, see-saw, Leptonic CP-violation

A neutrino mass model with S3 symmetry and see-saw interplay

Soumita Pramanick***email: soumitapramanick5@gmail.com, Amitava Raychaudhuriemail: palitprof@gmail.com

Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India

## I Introduction

Oscillation experiments over vastly different baselines and a range of neutrino energies have filled up a vast portion of the mass and mixing jigsaw of the neutrino sector. Yet, we still remain in the dark with regard to CP-violation in the lepton sector. Neither do we know the mass ordering – whether it is normal or inverted. Further open issues are the absolute mass scale of neutrinos and whether they are of Majorana or Dirac nature. While we await experimental guidance for each of the above unknowns, there have been many attempts to build models of lepton mass which capture much of what is known.

Here we propose a neutrino mass model based on the direct product group . The elements of correspond to the permutations of three objects111More details of can be found in Appendix A.. Needless to say, -based models of neutrino mass have been considered earlier [1, 2]. A popular point of view [3] has been to note that a permutation symmetry between the three neutrino states is consistent with222Note, however, there is no 3-dimensional irreducible representation of (see Appendix A). So these models entail fine tuning. (a) a democratic mass matrix, , all whose elements are equal, and (b) a mass matrix proportional to the identity matrix, . A general combination of these two forms, e.g., , where are complex numbers, provides a natural starting point. One of the eigenstates, namely, an equal weighted combination of the three states, is one column of the popular tribimaximal mixing matrix. Many models have been presented [3] which add perturbations to this structure to accomplish realistic neutrino masses and mixing. Variations on this theme [4] explore mass matrices with such a form in the context of Grand Unified Theories, in models of extra dimensions, and examine renormalisation group effects on such a pattern realised at a high energy. Other variants of the -based models, for example [5], rely on a 3-3-1 local gauge symmetry, tie it to a -extended model, or realise specific forms of mass matrices through soft symmetry breaking, etc. As discussed later, the irreducible representations of are one and two-dimensional. The latter provides a natural mechanism to get maximal mixing in the sector [6].

The present work, also based on symmetry, breaks new ground in the following directions. Firstly, it involves an interplay of Type-I and Type-II see-saw contributions. Secondly, it presents a general framework encompassing many popular mixing patterns such as tribimaximal mixing. Further, the model does not invoke any soft symmetry breaking terms. All the symmetries are broken spontaneously.

We briefly outline here the strategy of this work. We use the standard notation for the leptonic mixing matrix – the Pontecorvo, Maki, Nakagawa, Sakata (PMNS) matrix – .

 U=⎛⎜⎝c12c13s12c13s13e−iδ−c23s12+s23s13c12eiδc23c12+s23s13s12eiδs23c13s23s12+c23s13c12eiδ−s23c12+c23s13s12eiδc23c13⎞⎟⎠, (1)

where and . The neutrino masses and mixings arise through a two-stage mechanism. In the first step, from the Type-II see-saw the larger atmospheric mass splitting, , is generated while the solar splitting, , is absent. Also, , and the model parameters can be continuously varied to obtain any desired . Of course, in reality [7], the solar splitting is non-zero, and there are indications that is large but non-maximal. Experiments have also set limits on . The Type-I see-saw addresses all the above issues and relates the masses and mixings to each other.

The starting form incorporates several well-studied mixing patterns such as tribimaximal (TBM), bimaximal (BM), and golden ratio (GR) mixings within its fold. These alternatives all have and . They differ only in the value of the third mixing angle as displayed in Table 1. The fourth option in this Table, no solar mixing (NSM), exhibits the attractive feature333Such a mixing pattern was envisioned earlier in a model based on symmetry [8] which built on previous work along similar lines [9, 10]. that the mixing angles are either maximal, i.e., () or vanishing ( and ).

In the following section we furnish a description of the model including the assignment of quantum numbers to the leptons and symmetry-breaking scalar fields. The consequences of the model are described next where we also compare with the experimental data. A summary and conclusions follow. The scalar potential of this model has a rich structure. In two Appendices we present the essence of symmetry and discuss the invariant scalar potential, deriving the conditions which must be satisfied by the scalar coefficients to obtain the desired minimum.

## Ii The Model

In the model under discussion fermion and scalar multiplets are assigned quantum numbers in a manner such that spontaneous symmetry breaking naturally yields mass matrices which lead to the see-saw features espoused earlier. All terms allowed by the symmetries of the model are included in the Lagrangian. No soft symmetry-breaking terms are required.

To begin it will be useful to formulate the conceptual structure behind the model. Neutrino masses arise from a combination of Type-I and Type-II see-saw contributions of which the latter dominates. In the neutrino mass basis, which is also the basis in which the Lagrangian will be presented, the Type-II see-saw yields a diagonal matrix in which two states are degenerate:

 MνL=⎛⎜ ⎜ ⎜⎝m(0)1000m(0)1000m(0)3⎞⎟ ⎟ ⎟⎠. (2)

This mass matrix results in while . Later, we find the combinations useful. signals the mass ordering; it is positive for normal ordering (NO) and negative for inverted ordering (IO).

At this stage the mixing resides entirely in the charged lepton sector. We follow the convention

 Ψflavour=UΨΨmass, (3)

for the fermions , so that the PMNS matrix, , is given by

 U=U†lUν. (4)

As noted, at this level , where alternate choices of result in popular mixing patterns such as tribimaximal, bimaximal, and golden ratio with the common feature that and . is another interesting alternative [8] where initially the lepton mixing angles are either vanishing or maximal, i.e., (). Thus, till Type-I see-saw effects are included, the leptonic mixing matrix takes the form:

 U0=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝cosθ012sinθ0120−sinθ012√2cosθ012√21√2sinθ012√2−cosθ012√21√2⎞⎟ ⎟ ⎟ ⎟ ⎟⎠=U†lU0ν, (5)

where and the charged lepton mass matrix is:

 (6)

The identity matrix, , at the right in the first step above indicates that no transformation needs to be applied on the right-handed charged leptons which are singlets.

In this basis, the matrices responsible for the Type-I see-saw have the forms:

 MD=mD IandMνR=mR2xy⎛⎜ ⎜⎝0xe−iϕ1xe−iϕ1xe−iϕ1ye−iϕ2/√2−ye−iϕ2/√2xe−iϕ1−ye−iϕ2/√2ye−iϕ2/√2⎞⎟ ⎟⎠, (7)

where and set the scale for the Dirac and right-handed Majorana masses while and are dimensionless real quantities of (1). We take the Dirac mass matrix proportional to the identity for ease of presentation. We have checked that the same results can be reproduced so long as is diagonal. The right-handed neutrino Majorana mass matrix, , has a discrete symmetry. This choice too can be relaxed without jeopardising the final outcome.

We will show later how the mass matrices in eqs. (2) - (7) lead to a good fit to the neutrino data and yield testable predictions. But before this we must ensure that the above matrices can arise from the symmetric Lagrangian.

The behaviour of the fermions, i.e., the three lepton generations444The scope of this model is restricted to the lepton sector. including three right-handed neutrinos, is summarised in Table 2. The gauge interactions of the leptons are universal and diagonal in this basis. A feature worth noting is that the right-handed neutrinos have lepton number . We discuss later how this leads to a diagonal neutrino Dirac mass matrix. The lepton mass matrices arise from the Yukawa couplings allowed by the symmetry.

The structure of the lepton sector is matched by a rich scalar sector which we have presented in Table 3. The requirement of charged lepton masses and Type-I and Type-II see-saw neutrino masses dictates the inclusion of singlet, doublet, and triplet scalar fields. The properties of the scalars are chosen bearing in mind the and combination rules. In particular, for the former the representations are , , and which satisfy the multiplication rules (see Appendix A):

 1×1′=1′,1′×1′=1, and2×2=2+1+1′. (8)

The scalar multiplets are chosen such that the mass matrices appear with specific structures as discussed below555In general the multiple scalar fields in models based on discrete symmetries also result in flavour changing neutral currents induced by the neutral scalars. Discussions of this aspect in the context of can be found, for example, in [11].. It can be seen from Table 3 that all neutral scalars pick up a vev. The vev of the singlets, namely, and , can be much higher than the electroweak scale, , and determine the masses of the right-handed neutrinos. The other vev break . We take .

Charged lepton and neutrino masses are obtained from the Yukawa terms in a Lagrangian constructed out of the fields in Tables 2 and 3. Including all terms which respect the gauge symmetry and the flavour symmetry so long as lepton number, , is also conserved one is led to the Lagrangian mass terms

 Lmass = f1 ¯eL(μRϕ02−τRϕ01)+f2 ¯μL(μRϕ04−τRϕ03)+f3 ¯τL(μRϕ04+τRϕ03) (9) + f4 ¯μLeRα0+f5 ¯eLeRη0  (charged lepton mass) + (h1¯νeLN1R+h2¯νμLN2R+h3¯ντLN3R)β0  (neutrino Dirac mass) + 12g1 νTeLC−1νeLΔ0L+12(g2 νTμLC−1νμL+g3 νTτLC−1ντL)ρL  (neutrino Type−II see−saw mass) + 12([k1NT2RC−1N2R+k2NT3RC−1N3R]χ+k3NT2RC−1N3Rγ) + 12(k4NT1RC−1N2R~χ+k5NT1RC−1N3R~γ)  (rh neutrino mass)+h.c..

Here, and are charge conjugated fields which transform under as . For each term in the Lagrangian the fermion masses which arise therefrom have been indicated. Both Type-I and Type-II see-saw contributions for neutrino masses are present.

The above Lagrangian gives rise to the mass matrices in Eqs. (2) - (7) through the Yukawa couplings in Eq. (9) and the vevs in Table 3. Before turning to these let us note how the quantum number assignments of the fermion and scalar fields force certain entries in the mass matrices to be vanishing. For example, the mass term is zero in Eq. (6) because there is no doublet field which transforms as a under . Similarly the diagonal nature of the left-handed neutrino Majorana mass matrix in Eq. (2) is ensured by the absence of an triplet field which transforms either as (i) a under or (ii) as under . The neutrino Dirac mass matrix in Eq. (7) arises from the Yukawa couplings666As the carry , conservation of lepton number forbids any contribution to the Dirac mass from the scalar doublets which generate the charged lepton masses. of the doublet scalar . Since it transforms as 1 under both and it can be seen from the left-handed and right-handed neutrino quantum numbers in Table 2 that only diagonal terms are allowed. Finally, the term is absent in the right-handed neutrino Majorana mass matrix in Eq. (7) since there is no singlet among the singlet scalars.

Before proceeding further it may be useful to comment on the sizes of the various vacuum expectation values in Table 3. The doublets acquire vevs which are while the triplet vevs are several orders of magnitude smaller. This is in consonance with the smallness of the neutrino masses as also the parameter of electroweak symmetry breaking. Needless to say, the singlet fields and can acquire vevs well above the electroweak scale.

The non-vanishing entries in the mass matrices in Eqs. (2) - (7) which arise from the Yukawa couplings entail the following relationships:

1. Charged lepton masses – On matching the Lagrangian in Eq. (9), the scalar doublet vevs in Table 3 and the charged lepton mass matrix in Eq. (6) one gets:

 f1⟨ϕ01⟩=−mτ√2sinθ012,f1⟨ϕ02⟩=−mμ√2sinθ012, (10)
 f2⟨ϕ03⟩=mτ√2cosθ012,f2⟨ϕ04⟩=mμ√2cosθ012,f3⟨ϕ03⟩=mτ√2,f3⟨ϕ04⟩=mμ√2, (11)

and

 f4⟨α0⟩=mesinθ012,f5⟨η0⟩=mecosθ012. (12)

Notice that Eqs. (10) and (11) imply

 w2w1=w4w3=mμmτ. (13)

2. Left-handed neutrino Majorana mass – Similarly, the mass matrix in Eq. (2) is obtained when

 g1⟨Δ0L⟩=m01=g2⟨ρ0L⟩,g3⟨ρ0L⟩=m03. (14)

The first equation above requires a matching between two sets of Yukawa couplings and vevs. This is to ensure degeneracy of two neutrino states, implying the vanishing of the solar mass splitting at this stage. Notice that the relatively large size of the atmospheric mass splitting requires and to be of different order.

3. Neutrino Dirac mass – The Dirac mass matrix in Eq. (7) is due to the relations:

 h1=h2=h3=handh⟨β0⟩=mD. (15)

The equality of the three Yukawa couplings, , above is only a simplified choice. We have checked that deviations from this relation, i.e., a diagonal Dirac mass matrix but not proportional to the identity, can also readily lead to the results which we discuss in this paper.

4. Right-handed neutrino Majorana mass – Finally, the right-handed neutrino Majorana mass matrix follows from:

 k1⟨χ0⟩=mRe−iϕ22√2x=k2⟨χ0⟩,k3⟨γ0⟩=−mRe−iϕ22√2x,k4⟨~χ0⟩=mRe−iϕ12y=k5⟨~γ0⟩. (16)

We show in Appendix B how from a minimisation of the scalar potential the required scalar vevs may be obtained.

### ii.1 Type-I see-saw contribution

In the previous section we have shown that the model results in a diagonal left-handed neutrino mass matrix given in Eq. (2) through a Type-II see-saw. The charged lepton mass matrix as given in Eq. (6) is not diagonal and induces a mixing in the lepton sector. This mixing, Eq. (5), receives further corrections from a smaller Type-I see-saw contribution to the neutrino mass matrix as we discuss.

The Type-I see-saw arising from the Dirac and right-handed neutrino mass matrices in Eq. (7) is

 M′=[MTD(MνR)−1MD]=m2DmR⎛⎜ ⎜ ⎜ ⎜ ⎜⎝0y eiϕ1y eiϕ1y eiϕ1x eiϕ2√2−x eiϕ2√2y eiϕ1−x eiϕ2√2x eiϕ2√2⎞⎟ ⎟ ⎟ ⎟ ⎟⎠. (17)

## Iii Results

We have given above the contributions to the neutrino mass matrix from the Type-I and Type-II see-saw. Of these, the former is taken to be significantly smaller than the latter. As we have noted, in the absence of the Type-I see-saw the leptonic mixing matrix in this model is determined entirely by the charged lepton mass matrix. It has , , and arbitrary. We will be considering four mixing patterns which fall within this scheme and in each of which the value of is specified, namely, the TBM, BM, GR, and NSM cases. In addition, in this model the Type-II see-saw sets the solar mass splitting to be zero. The Type-I see-saw, whose effect we incorporate perturbatively, brings all the above leptonic parameters into agreement with their values preferred by the data. Before we proceed further with this discussion it will be useful to summarise the global best-fit values of these mass-splittings and angles.

### iii.1 Data

From global fits the currently favoured 3 ranges of the neutrino mixing parameters are [12, 13]

 Δm221 = (7.02−8.08)×10−5eV2,θ12=(31.52−36.18)∘, |Δm231| = (2.351−2.618)×10−3eV2,θ23=(38.6−53.1)∘, θ13 = (7.86−9.11)∘,δ=(0−360)∘. (18)

These data are from NuFIT2.1 of 2016 [12]. Here, , so that for normal (inverted) ordering. The data indicate two best-fit points for in the first and second octants. Later, we also remark about the compatibility of this model with the recent T2K and NOVA hints [14, 15] of being near -.

### iii.2 Real MνR (ϕ1=0 or π,ϕ2=0 or π)

A limiting case, with less complications, corresponds to no CP-violation. This happens when is real, i.e., the phases in Eq. (17) are 0 or . These cases can be compactly considered by keeping and real but allowing them to be of either sign, i.e., four alternatives. We show below how the experimental data picks out one or the other out of these.

Without the phases , i.e., for real , one gets

 M′=m2DmR⎛⎜ ⎜ ⎜⎝0yyyx√2−x√2y−x√2x√2⎞⎟ ⎟ ⎟⎠. (19)

The equality of two neutrino masses from the Type-II see-saw requires the use of degenerate perturbation theory to obtain corrections to the solar mixing parameters. The submatrix of relevant for this is:

 M′2×2=m2DmR(0yyx/√2). (20)

This results in:

 θ12=θ012+ζ,tan2ζ=2√2(yx). (21)

A related quantity, , which is found useful later is given by

 sinϵ=y√y2+x2/2and cosϵ=x/√2√y2+x2/2,i.e., tanϵ=12tan2ζ. (22)

Once a mixing pattern is chosen, i.e., fixed, the experimental limits on as given in Eq. (18) set bounds on the range of and also from Eq. (22) on . These are displayed for the four mixing patterns in Table 4. If is positive (negative) then the ratio will also be positive (negative). In addition, from Eq. (22) the sign of is fixed by the value of . Taking these points into account one can conclude that is always positive, i.e., has to be 0, while must be positive, (negative, ) for NSM (BM). For the other mixing patterns, i.e., TBM and GR, both signs of are possible.

The solar mass splitting arising from the Type-I see-saw is also obtained from Eq. (20).

 Δm2solar=√2m2DmR m(0)1√x2+8y2=√2m2DmR m(0)1xcos2ζ. (23)

Furthermore, incorporating the leading order corrections to neutrino mixing from Eq. (19) one gets from Eq. (4):

 U=U0Uνwith Uν=⎛⎜⎝cosζ−sinζκrsinϵsinζcosζ−κrcosϵκrsin(ζ−ϵ)κrcos(ζ−ϵ)1⎞⎟⎠, (24)

with

 κr≡m2DmRm−√y2+x2/2=m2DmRm−x√2cosϵ. (25)

The third column of the leptonic mixing matrix becomes:

 |ψ3⟩=⎛⎜ ⎜ ⎜ ⎜⎝κrsin(ϵ−θ012)1√2[1−κrcos(ϵ−θ012)]1√2[1+κrcos(ϵ−θ012)]⎞⎟ ⎟ ⎟ ⎟⎠   . (26)

Since, as noted, is always positive, is positive (negative) for normal (inverted) ordering.

The right-hand-side of Eq. (26) has to be matched with the third column of eq. (1). This yields:

 sinθ13cosδ=κrsin(ϵ−θ012), (27)

and

 tan(π/4−θ23)≡tanω=κrcos(ϵ−θ012). (28)

For ready reference, the ranges of allowed for the different mixing patterns are presented in Table 4. For normal ordering777We show in the following that inverted ordering is not consistent with real . the CP-phase is 0 () when is positive (negative). From Table 4 one can then observe that for the NSM mixing pattern and is for the three other cases. Needless to say, both correspond to CP-conservation.

Combining Eqs. (23), (25), and (27) one can write:

 Δm2solar=2 m−m(0)1 sinθ13cosδcosϵcos2ζ sin(ϵ−θ012). (29)

Eq. (29) leads to the conclusion that inverted ordering is not allowed for this case of real . To establish this property one can define:

 z≡m−m(0)1/Δm2atmosandtanξ≡m0/√|Δm2atmos|, (30)

where is positive for both mass orderings. From Eq. (29) one has

 z=(Δm2solar|Δm2atmos|)(cos2ζ sin(ϵ−θ012)2sinθ13|cosδ|cosϵ). (31)

It is easy to verify from Eq. (30) that

 z = sinξ/(1+sinξ) i.e.,0≤z≤12(for normal ordering), z = 1/(1+sinξ) i.e.,12≤z≤1(for inverted ordering). (32)

There is a one-to-one correspondence of with the lightest neutrino mass . The quasi-degeneracy limit, i.e., large, is approached as for both mass orderings.

In Eq. (31) for real . Using the global fit mass splittings and mixing angles given in Sec. III.1 and Table 4 one finds or smaller for all four mixing patterns. This excludes the inverted mass ordering option for real .

From Eqs. (27) and (28) one has

 tanω=sinθ13cosδtan(ϵ−θ012). (33)

The noteworthy point is that for normal ordering Eq. (28) implies that is always positive irrespective of the mixing pattern. So, in this model is restricted to the first octant only for real .

Eqs. (21) and (22) can be used to express in terms of and thereby put in Eq. (33) as a function of and only. In Fig. 1, is shown as a function of for the NSM (thick green lines) and BM (thin pink lines) mixing patterns. The ranges of and have been kept within their 3 allowed limits from global fits as given in Sec. III.1. The TBM and GR cases are excluded because for the allowed values of they predict beyond the 3 range. The solid lines in the figure correspond to the 3 limiting values of and the dashed line is for its best-fit value. The blue dot-dashed horizontal and vertical lines display the 3 experimental bounds on and .

Using Eq. (31) any allowed point in the plane and the associated can be translated to a value of or equivalently , provided the solar and atmospheric mass splittings are given. We find that for both the allowed mixing patterns the range of variation of is very small. For the NSM (BM) case this range is meV meV ( meV meV) when both neutrino mass splittings and all mixing angles are varied over their full 3 ranges.

To summarise the real case:

1. Only the normal mass ordering is allowed.

2. can lie only in the first octant.

3. The TBM and GR alternatives are inconsistent with the allowed ranges of the neutrino mixing angles even after including the Type-I see-saw corrections.

4. For the NSM and BM mixing patterns real can give consistent solutions for the neutrino masses and mixings. The ranges of allowed lightest neutrino masses are very tiny.

### iii.3 Complex MνR

Keeping real eliminates CP-violation. Further, inverted ordering is disallowed. Also, the TBM and GR mixing patterns cannot be accommodated. These restrictions can be ameliorated by taking in its general complex form giving rise to the Type-I see-saw contribution as given in Eq. (17). Recall that this introduces the phases and and take only positive values.

With its complex entries, is now not hermitian any more. To address this we consider the combination , and treat as the leading term with acting as a perturbation at the lowest order, both hermitian by construction. The unperturbed eigenvalues are thus . The perturbation matrix is

 (34)

In the above

 f(φ)=m+cosφ−im−sinφ. (35)

The remaining calculation proceeds in much the same manner as for real while keeping the distinctive features of Eq. (34) in mind.

In place of Eqs. (21) and (22) for the real case, we get from (34)

 θ12=θ012+ζ,tan2ζ=2√2 yx cosϕ1cosϕ2, (36)

and

 sinϵ=ycosϕ1√y2cos2ϕ1+x2cos2ϕ2/2,cosϵ=xcosϕ2/√2√y2cos2ϕ1+x2cos2ϕ2/2,tanϵ=12tan2ζ. (37)

The allowed ranges of and depend on the mixing pattern and are given in Table 4. It is seen that for all patterns is positive. Therefore, from Eq. (37) we can immediately conclude that must be always in the first or fourth quadrants. The possible quadrants of are also determined from the range of for the different mixing patterns. From the first relation in Eq. (37) we find that has to be in the first or fourth (second or third) quadrants if is positive (negative). Using the results in Table 4 we conclude that the first (second) option is valid for the NSM (BM) patterns. For TBM and GR cases spans a range over positive and negative values and so both options are included.

The solar mass splitting is induced entirely through the Type-I see-saw contribution. From Eq. (34) one finds:

 Δm2solar=√2m(0)1 m2DmR√x2cos2ϕ2+8y2cos2ϕ1=√2m(0)1 m2DmR xcosϕ2cos2ζ=√2m(0)1 m2DmR2√2ycosϕ1sin2ζ. (38)

Eq. (26) is now replaced by:

 |ψ3⟩=⎛⎜ ⎜ ⎜ ⎜⎝κc[sinϵcosϕ1f(ϕ1)cosθ012−cosϵcosϕ2f(ϕ2)sinθ012]/m+1√2{1−κc[sinϵcosϕ1f(ϕ1)sinθ012+cosϵcosϕ2f(ϕ2)cosθ012]/m+}1√2{1+κc[sinϵcosϕ1f(ϕ1)sinθ012+cosϵcosϕ2f(ϕ2)cosθ012]/m+}⎞⎟ ⎟ ⎟ ⎟⎠, (39)

where

 κc=m2DmRm− √y2cos2ϕ1+x2cos2ϕ2/2, (40)

Eq. (37) has been used, and the complex function is defined in Eq. (35).

is positive (negative) for normal (inverted) ordering. Comparing the right-hand-side of Eq. (39) with the third column of Eq. (1) we find

 sinθ13cosδ=κc sin(ϵ−θ012), (41)
 sinθ13sinδ=κc m−m+cosϕ1cosϕ2[sinϵsinϕ1cosϕ2cosθ012−cosϵcosϕ1sinϕ2sinθ012]. (42)

As indicated in Table 4, always remains in the first (fourth) quadrant for the NSM (BM, TBM, and GR) mixing pattern. For normal ordering Eq. (41) then implies that for the NSM (BM, TBM, and GR) case(s) lies in the first or fourth (second or third) quadrants. For inverted ordering of masses, changes sign and so the quadrants are accordingly modified. The different possibilities are indicated in Table 5. For any mixing pattern and mass ordering there are two allowed quadrants of which have of opposite sign. Which of these is chosen is determined by the phases through the sign of the right-hand-side of Eq. (42). As noted above, can be in either the first or fourth quadrants and the quadrant of is determined by the mixing pattern in such a way that can be of either sign. Thus the the phases and can always be chosen such that can be of any particular sign. Therefore the two alternate quadrants of for every case in Table 5 are equally viable in this model.

The perturbative Type-I see-saw contribution to can also be extracted from Eq. (39). One finds:

 tanω=sinθ13cosδtan(ϵ−θ012). (43)

Recalling that Eq. (41) correlates and through one can readily conclude that for all mixing patterns always lies in the first (second) octant for normal (inverted) ordering. This important conclusion from these models is shown in Table 5.

In the expression for the solar mass splitting in Eq. (38) one can trade the factor   in terms of and use Eq. (41) to get

 Δm2solar=2m−m(0)1sinθ13cosδcosϵsin(ϵ−θ012) cos2ζ. (44)

The strategy that we have followed to extract the predictions of this model relies on utilising Eqs. (43) and (44). We take the three mixing angles , , and as inputs. With these at hand Eq. (