1 Introduction


Tri-Bimaximal Neutrino Mixing and the Flavor Symmetry

Gui-Jun Ding 111Email: dinggj@ustc.edu.cn

Department of Modern Physics,

University of Science and Technology of China, Hefei, Anhui 230026, China Department of Physics, University of Wisconsin-Madison,

1150 University Avenue, Madison, WI 53706, USA

We present a supersymmetric model for the tri-bimaximal neutrino mixing, and the complete flavor group is . At leading order, the residual symmetry of the charged lepton sector is , and the symmetry is broken completely in the neutrino sector. The charged lepton mass hierarchies are determined by the spontaneous breaking of the flavor symmetry, both the type I see-saw mechanism and the Weinberg operator contribute to generating the light neutrino masses. Tri-bimaximal mixing is exact at leading order while subleading contributions introduce corrections of order to the three lepton mixing angles. The vacuum alignment and subleading corrections are studied in detail, a moderate hierarchy of order between the vacuum expectation values of the flavon fields in the charged lepton and neutrino sectors can be accommodated.

1 Introduction

The presence of two large and one small mixing angles in the lepton sector [1, 2, 3],


suggests that the observed neutrino mixing matrix is remarkably compatible with the so called tri-bimaximal (TB) structure [4] within measurement errors. The simple form of the TB mixing matrix implies an underlying family symmetry between the three generations of leptons. It has been realized that the TB mixing can naturally arise as the result of a particular vacuum alignment of scalars that break spontaneously certain flavor symmetries. In the past years, much effort has been devoted to produce TB mixing based on some family symmetry. It has been shown that TB mixing can be understood with the help of discrete flavor symmetries, such as [5, 6, 7, 8, 9, 10], [11] , [12], [13, 14] and [15], or continuous flavor symmetry [16] and [17]. Discrete non-abelian groups appear to be particularly suitable to reproduce the TB mixing pattern, some higher order discrete groups such as [18], [19], [20] and [21] are also considered for neutrino mixing besides the above mentioned simple groups, the extension of the discrete flavor symmetry to the quark sector and grand unification theory (GUT) have been investigated as well [7, 8], please see Refs.[22, 23] for a review. In this work, we shall study another 39 element simple discrete group in flavor model building, which has gained much less attention.

Recently 76 discrete groups with 3-dimensional representation were scanned, it is suggested that is the group with the largest fraction of TB mixing models [24]. But the authors set all the couplings to be equal to 1, the vacuum expectation values are chosen to be 0 or 1, and the vacuum alignment is not considered dynamically in Ref.[24]. It is very interesting to investigate the possible consistent realizations of TB mixing based on group from this point of view. As far as we know, the group as a discrete flavor symmetry has not been discussed extensively. We note that a flavor model was put forward in Ref. [25], and its implication in the indirect detection of dark matter was studied. However, the motivation is not to produce the TB mixing 222The vacuum alignment and the next leading order correction are not discussed in Ref.[25], a set of numerical values are chosen by hand for the model parameters so that the resulting lepton masses and flavor mixing are consistent with experimental data.. We have tried many possible assignments for the involved fields, we find the TB mixing can be produced exactly at leading order (LO) in some scenarios, but meanwhile we face the difficulties that the first and third light neutrino are degenerate or the corresponding vacuum alignment is very difficult to be realized or some other problems. In particularly, the realizations of TB mixing based on symmetry are drastically constrained after taking into account the vacuum alignment issue. After lots of trial and error, we construct a flavor model described in this work, where TB mixing is obtained exactly at LO. It is well-known that discrete group or continuous one like are usually introduced to eliminate unwanted couplings, to ensure the need vacuum alignment and to reproduce the observed charged charged lepton mass hierarchies. In the present work, the auxiliary symmetry is introduced for this purpose. It is notable that the charged lepton mass hierarchies are determined by the flavor symmetry itself without invoking a Froggatt-Nielsen symmetry.

This paper is organized as follows. In section 2, we discuss the relevant features of group. In section 3, the structure of the model is described, the LO results for neutrino as well as charged lepton mass matrices are presented. In section 4, we show how to get in a natural way the required vacuum alignment used throughout the paper. In section 5, we present the study on the corrections introduced by the higher order terms, which is responsible for the deviation from TB mixing. Finally section 6 is devoted to our conclusion. We give the explicit representation matrices and the Clebsch-Gordan coefficients of group in Appendix A. The analysis of the subleading corrections to the vacuum alignment is presented in Appendix B.

2 The discrete group

The discrete group is a subgroup of , and it is smallest discrete group with two complex irreducible three-dimensional representations. is isomorphic to [26, 27], consequently it has 39 group elements. can be generated by two elements and obeying the relations


The 39 elements of the group belong to 7 conjugate classes and are generated from and as follows,


The group has 7 inequivalent irreducible representations , , , , , and . It is easy to see that the one-dimensional representations are given by


where . The three-dimensional representations are given by


where , the and representations can be obtained by performing the complex conjugation of and respectively. We can straightforwardly calculate the character table of , which is shown in Table 1. Then the multiplication rules between various representations follow immediately,


1 13 13 3 3 3 3

1 3 3 13 13 13 13

1 1 1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

3 0 0

3 0 0

3 0 0

3 0 0

Table 1: Character table of the group, where , , and . denotes the number of the elements contained in the class , and is the order of the elements of .

where the indices , indicates any irreducible representation, and the subscript and denote symmetric and anti-symmetric products respectively. The explicit representation matrices of the group elements for the three dimensional irreducible representations are listed in Appendix A. From these representation matrices, one can directly calculate the Clebsch-Gordan coefficients for the decomposition of the product representations, which are given in Appendix A as well.

3 The structure of the model

The model is supersymmetric and based on the discrete symmetry . Supersymmetry (SUSY) is introduced in order to simplify the discussion of the vacuum alignment. All the fields of the model, together with their transformation properties under the flavor group, are listed in Table 2. We assign the 3 generation of left-handed lepton doublets to be the representation, while the right-handed charged lepton , and transform as , and respectively. It is notable that the three right-handed neutrinos , and are assigned as , and as well, they transform in the same way as the right-handed charged lepton fields. This is an interesting feature of the model. We note that in popular and models, the right-handed neutrinos are frequently treated to be a triplet [5, 13]. Lepton masses and mixing arise from the spontaneous breaking of the flavor symmetry by means of the flavon fields, they are neutral under the standard model gauge group and are divided into two sets and . We note that all the flavon fields are triplets under in this work, is responsible for the charged lepton masses and for the neutrino masses at LO. In the following, we shall discuss the LO predictions for fermion masses and flavor mixings. For the time being we assume that the scalar components of the flavon fields acquire vacuum expectation values (VEV) according to the following scheme,


1 i -1 -i 1 1 1 1 i i 1 1 -1 -1 -1 1 -i

1 1 1 1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 -1

1 1 1 1 1 1 1 0 0 0 0 0 2 2 2 2 2
Table 2: The transformation properties of the matter fields, the electroweak Higgs doublets, the flavon fields and the driving fields under the flavor symmetry .

In section 4 we shall show that the above alignment is indeed naturally realized at LO from the most general superpotential allowed by the symmetry of the model.

3.1 Charged leptons

The charged lepton masses are described by the following superpotential




We note that the subscripts , , , etc denote the contractions. In the above superpotential , for each charged lepton, only the lowest order operators in the expansion in powers of are displayed explicitly. Dots stand for higher dimensional operators which will be discussed later. It is remarkable that the symmetry imposes different powers of and for the electron, muon and tauon terms, i.e., only the tau mass is generated at LO, the muon and the electron masses are generated by high order contributions. After electroweak and flavor symmetry breaking, we have


where , the parameters and are parameterized as and . As a result, the charged lepton mass matrix has the form


Obviously the charged lepton mass matrix is diagonalized by performing the transformation , where is


and the charged lepton masses are given by


We see that the charged lepton mass hierarchies are generate by the spontaneous breaking of the flavor symmetry. To estimate the order of magnitudes of and , we can use the experimental data on the ratios of charged lepton masses. Assuming that the coefficients , and are of , we have


These relations are satisfied for


we see that the amplitudes of both and are roughly of the same order about , where is the Cabibbo angle. It is interesting to investigate the flavor symmetry breaking pattern in the charged lepton sector, it is induced by the VEVs of and at LO. Given the explicit representation matrices listed in Appendix A, it is obvious that the VEVs of and are invariant under the action of and . Furthermore, we can check that the hermitian matrix is invariant under both and as well. Therefore we conclude that the flavor symmetry is broken down to the subgroup generated by the element in the charged lepton sector at LO.

3.2 Neutrinos

The superpotential for the neutrino sector can be written as


where and are constants with dimension of mass, they are naturally of the same order as the cutoff scale , and the factor of is a normalization factor for convenience. We note that denotes the lagrangian of the type I see-saw mechanism, and is the collection of higher dimensional Weinberg operators. Taking into account the vacuum alignment shown in Eq.(7), we can read the Dirac and Majorana mass matrices immediately from as follows


where is the vacuum expectation value of the Higgs field . It is remarkable that the eigenvalues of the Majorana mass matrix are , and , two of the right handed neutrinos are degenerate at LO. This is a distinguished feature of our model from the previous flavor models in which the right-handed neutrinos are usually treated to form a triplet. It is very interesting to discuss the assignment of right-handed neutrinos as singlets and the corresponding phenomenological implications in flavor models based on , , and so on. The light neutrino mass matrix from see-saw mechanism is given by the well-known see-saw formula




The superpotential leads to the following effective light neutrino mass matrix




Therefore in the flavor basis where the charged lepton mass matrix is diagonal, the light neutrino mass matrices read


Both the light neutrino mass matrices and are invariant, and they satisfy the magic symmetry . Therefore they are exactly diagonalized by the TB mixing matrix


We note that the contribution from the see-saw mechanism is of the same order as coming from the Weinberg operators, consequently both contributions should be included. The light neutrino mass matrix is the sum of and


Obviously is still diagonalized by the TB mixing matrix, and the light neutrino masses are given by


where is the well-known TB mixing matrix


We note that the contributions proportional to and can be absorbed into and by redefinition and , therefore the light neutrino masses depend on three unrelated complex parameters. There are more freedoms to tune the mass differences and then satisfy the constraints associated to neutrino oscillation, the neutrino mass spectrum can be normal hierarchy or inverted hierarchy. In contrast with some ”constrained” flavor models, no neutrino mass sum-rules [28] can be found in this model. We could certainly remove the right-handed neutrinos from our model, then the neutrino masses are described by the Weinberg operators , the above conclusions remain invariant. However, if we only concentrate on the see-saw realization , the second neutrino would be massless although the lepton mixing is of TB form, this scenario is ruled out by the experimental observations.

It is notable that the VEVs of and are always changed under the action of any group element except unit element, consequently the flavor symmetry is broken down to nothing in the neutrino sector at LO. Reminding that ones usually break the flavor symmetry into the low energy neutrino symmetry group Klein four [29, 30, 31] or [32, 33] to guarantee TB mixing for neutrinos, it is really amazing we can still obtain TB mixing even if the flavor symmetry is broken completely in the neutrino sector at LO.

In short summary, at the LO the flavor symmetry is broken down to subgroup and nothing in the charged lepton and neutrino sectors respectively. This breaking chain lets us to find the TB scheme at LO as the lepton mixing matrix. However, the mixing angles generally deviate from the TB values after the corrections of the higher order terms are included. It is remarkable that this symmetry breaking pattern of our model has not been studied, as far as we know. It is attractive to investigate whether we can still reproduce TB mixing in models with or symmetry, if the flavor symmetry is broken completely in the neutrino sector at LO.

4 Vacuum alignment

The vacuum alignment problem of the model can be solved by the supersymmetric driving fields method introduced in Ref.[33]. This approach exploits a continuous symmetry under which matter fields have , while Higgses and flavon fields have . Such a symmetry will be eventually broken down to the R-parity by small SUSY breaking effects which can be neglected in the first approximation in our analysis. The spontaneous breaking of can be employed by introducing the so-called driving fields with , which enter linearly into the superpotential. Five driving fields , , , and are introduced in our model, their transformation rules under the flavor symmetry are shown in Table 2. We note that the driving fields and are necessary to stabilize the vacuum alignment under subleading corrections. At LO, the most general superpotential dependent on the driving fields, which is invariant under the flavor symmetry group , is given by


In the SUSY limit, the vacuum configuration is determined by the vanishing of the derivative of with respect to each component of the driving fields


The above equations are satisfied by the alignment


with the relation


Without assuming any fine-tuning among the parameters and , the VEVs and are expected to be of the same order of magnitude, this is consistent with the conclusion drew from the charged lepton mass hierarchies. We note that if one component of or has vanishing VEV, e.g. , Eqs.(28a)-(28c) imply . This means that the VEV of any component of the flavons or should be non-zero in order to obtain a non-trivial vacuum configuration. As has been shown in the previous section, at LO the flavor symmetry is spontaneously broken by the VEVs of and in the neutrino sector, their vacuum configurations are determined by


The first three equations Eq.(33a)-(33c) lead to two un-equivalent vacuum configurations 333We note that the equations can be satisfied by two additional solutions as well. One is (35) Another one is (36) where , , and are undetermined. However, the above two solutions can be obtained by acting on the vacuum Eq.(39) with the elements and respectively. Therefore these two solutions are equivalent to the configuration in Eq.(39). , the first is




The second solution is


where , and are constrained. Using the alignment of in Eq.(31), for the first solution shown in Eq.(37), we can immediately infer from Eq.(34)


We are led to the trivial solutions or , which can be removed by the interplay of radiative corrections to the scalar potential and soft SUSY breaking terms for the flavon fields. Therefore we choose the second solution in this work, this vacuum configuration can produce the results in the previous section. Then the minimization equation Eq.(34) implies


This indicates that and have to be equal up to a relative sign, thus is fully aligned as


Starting from the vacuum configurations given in Eq.(31), Eq.(39) and Eq.(42) and acting on them with the elements of the flavor symmetry group , we can generate other minima of the scalar potential. However, these new minima are physically equivalent to the original one, it is not restrictive to analyze the model by choosing the vacuum in Eqs.(31,39,42) as local minimum. It is important to check the stability of the LO vacuum configuration, if we introduce small perturbations to the VEVs of the flavon fields as follows,


After some straightforward algebra, we find that the only solution to the minimization equations is


Therefore the LO vacuum alignment is stable, then we turn to consider the magnitudes of flavon VEVs. Since the VEVs of and are closely related with each other through the equations Eqs.(33a)-(33c), and they have the same charges under the auxiliary symmetry , we expect a common order of magnitude for the VEVs and . However, the VEVs of and can be in principle different and they are subject to phenomenological constraints. As we have shown in section 3.1, is responsible for the charged lepton mass hierarchies, and it is required to satisfy


Among the three neutrino mixing angles, the solar neutrino mixing angle is measured most precisely so far, the experimentally allowed departures of from its TB value are at most of order [1, 2, 3]. It is well-known that the superpotentials , , and are corrected by higher dimensional operators in the expansion (please see section 5 and Appendix B for detail), which mostly can be constructed by including the combination on top of each LO term, thus all the three mixing angles receive corrections of order (please see section 5 for detail). Requiring that the mixing angles particular lie in the ranges allowed by neutrino oscillation data, we obtain the condition