Abstract
We develop a seesaw model for neutrino masses and mixing with an symmetry. It involves an interplay of TypeI and TypeII seesaw contributions of which the former is subdominant. The quantum numbers of the fermion and scalar fields are chosen such that the TypeII seesaw 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 TypeI seesaw 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. CPviolation is controlled by phases in the righthanded neutrino Majorana mass matrix, . In their absence, only normal ordering is admissible. When is complex the Dirac CPphase, , 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, seesaw, Leptonic CPviolation
A neutrino mass model with S3 symmetry and seesaw interplay
Soumita Pramanick^{*}^{*}*email: soumitapramanick5@gmail.com, Amitava Raychaudhuri^{†}^{†}†email: 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 CPviolation 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 objects^{1}^{1}1More 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 with^{2}^{2}2Note, however, there is no 3dimensional 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 331 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 twodimensional. 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 TypeI and TypeII seesaw 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 – .
(1) 
where and . The neutrino masses and mixings arise through a twostage mechanism. In the first step, from the TypeII seesaw 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 nonzero, and there are indications that is large but nonmaximal. Experiments have also set limits on . The TypeI seesaw addresses all the above issues and relates the masses and mixings to each other.
The starting form incorporates several wellstudied 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 feature^{3}^{3}3Such 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 ).
Model  TBM  BM  GR  NSM 

35.3  45.0  31.7  0.0 
In the following section we furnish a description of the model including the assignment of quantum numbers to the leptons and symmetrybreaking 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 seesaw features espoused earlier. All terms allowed by the symmetries of the model are included in the Lagrangian. No soft symmetrybreaking terms are required.
To begin it will be useful to formulate the conceptual structure behind the model. Neutrino masses arise from a combination of TypeI and TypeII seesaw contributions of which the latter dominates. In the neutrino mass basis, which is also the basis in which the Lagrangian will be presented, the TypeII seesaw yields a diagonal matrix in which two states are degenerate:
(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).
Fields  Notations  ()  

Lefthanded leptons  2 (1)  +1  
1 ()  
Righthanded charged leptons  1 (2)  +1  
Righthanded neutrinos  1 (0)  0  
1 () 
At this stage the mixing resides entirely in the charged lepton sector. We follow the convention
(3) 
for the fermions , so that the PMNS matrix, , is given by
(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 TypeI seesaw effects are included, the leptonic mixing matrix takes the form:
(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 righthanded charged leptons which are singlets.
In this basis, the matrices responsible for the TypeI seesaw have the forms:
(7) 
where and set the scale for the Dirac and righthanded 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 righthanded 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 generations^{4}^{4}4The scope of this model is restricted to the lepton sector. including three righthanded 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 righthanded 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.
Purpose  Notations  
()  ()  
1 (1)  2 (1)  0  
2 (1)  2 (1)  0  
Charged fermion mass  
2 ()  2 (1)  0  
1 ()  2 (1)  0  
Neutrino Dirac mass  1 (1)  2 (1)  1  
1 (1)  3 (2)  2  
TypeII seesaw mass  
1 ()  3 (2)  2  
1 ()  1 (0)  0  
Righthanded neutrino mass  
1 (0)  0 
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 TypeI and TypeII seesaw 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):
(8) 
The scalar multiplets are chosen such that the mass matrices appear with specific structures as discussed below^{5}^{5}5In 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 righthanded 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
(9)  
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 TypeI and TypeII seesaw 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 lefthanded 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 couplings^{6}^{6}6As 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 lefthanded and righthanded neutrino quantum numbers in Table 2 that only diagonal terms are allowed. Finally, the term is absent in the righthanded 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 nonvanishing 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:
(10) 
(11) 
and
(12) 
Notice that Eqs. (10) and (11) imply
(13) 
2. Lefthanded neutrino Majorana mass – Similarly, the mass matrix in Eq. (2) is obtained when
(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:
(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. Righthanded neutrino Majorana mass – Finally, the righthanded neutrino Majorana mass matrix follows from:
(16) 
We show in Appendix B how from a minimisation of the scalar potential the required scalar vevs may be obtained.
ii.1 TypeI seesaw contribution
In the previous section we have shown that the model results in a diagonal lefthanded neutrino mass matrix given in Eq. (2) through a TypeII seesaw. 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 TypeI seesaw contribution to the neutrino mass matrix as we discuss.
The TypeI seesaw arising from the Dirac and righthanded neutrino mass matrices in Eq. (7) is
(17) 
Iii Results
We have given above the contributions to the neutrino mass matrix from the TypeI and TypeII seesaw. Of these, the former is taken to be significantly smaller than the latter. As we have noted, in the absence of the TypeI seesaw 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 TypeII seesaw sets the solar mass splitting to be zero. The TypeI seesaw, 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 bestfit values of these masssplittings and angles.
iii.1 Data
From global fits the currently favoured 3 ranges of the neutrino mixing parameters are [12, 13]
(18) 
These data are from NuFIT2.1 of 2016 [12]. Here, , so that for normal (inverted) ordering. The data indicate two bestfit 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 ()
A limiting case, with less complications, corresponds to no CPviolation. 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
(19) 
The equality of two neutrino masses from the TypeII seesaw requires the use of degenerate perturbation theory to obtain corrections to the solar mixing parameters. The submatrix of relevant for this is:
(20) 
This results in:
(21) 
Model ()  TBM (35.3)  BM (45.0)  GR (31.7)  NSM (0.0) 

4.0 0.6  13.7 9.1  0.4 4.2  31.3 35.9  
4.0 0.6  14.5 9.3  0.4 4.2  44.0 56.7  
39.2 34.6  59.5 54.4  39.2 30.0  44.0 56.7 
A related quantity, , which is found useful later is given by
(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 TypeI seesaw is also obtained from Eq. (20).
(23) 
Furthermore, incorporating the leading order corrections to neutrino mixing from Eq. (19) one gets from Eq. (4):
(24) 
with
(25) 
The third column of the leptonic mixing matrix becomes:
(26) 
Since, as noted, is always positive, is positive (negative) for normal (inverted) ordering.
The righthandside of Eq. (26) has to be matched with the third column of eq. (1). This yields:
(27) 
and
(28) 
For ready reference, the ranges of allowed for the different mixing patterns are presented in Table 4. For normal ordering^{7}^{7}7We show in the following that inverted ordering is not consistent with real . the CPphase 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 CPconservation.
Combining Eqs. (23), (25), and (27) one can write:
(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:
(30) 
where is positive for both mass orderings. From Eq. (29) one has
(31) 
It is easy to verify from Eq. (30) that
(32) 
There is a onetoone correspondence of with the lightest neutrino mass . The quasidegeneracy 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
(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 bestfit value. The blue dotdashed 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:

Only the normal mass ordering is allowed.

can lie only in the first octant.

The TBM and GR alternatives are inconsistent with the allowed ranges of the neutrino mixing angles even after including the TypeI seesaw corrections.

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
Keeping real eliminates CPviolation. 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 TypeI seesaw 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
(35) 
The remaining calculation proceeds in much the same manner as for real while keeping the distinctive features of Eq. (34) in mind.
Mixing  Normal Ordering  Inverted Ordering  

Pattern  
quadrant  octant  quadrant  octant  
NSM  First/Fourth  First  Second/Third  Second 
BM, TBM, GR  Second/Third  First  First/Fourth  Second 
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 TypeI seesaw contribution. From Eq. (34) one finds:
(38) 
Eq. (26) is now replaced by:
(39) 
where
(40) 
Eq. (37) has been used, and the complex function is defined in Eq. (35).
is positive (negative) for normal (inverted) ordering. Comparing the righthandside of Eq. (39) with the third column of Eq. (1) we find
(41) 
(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 righthandside 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 TypeI seesaw contribution to can also be extracted from Eq. (39). One finds:
(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.