Advances in High Energy Physics

Volume 2014 (2014), Article ID 192536, 24 pages

http://dx.doi.org/10.1155/2014/192536

Research Article

A New Flavor Symmetry in 3-3-1 Model with Neutral Fermions

V. V. Vien¹ and H. N. Long²

¹Department of Physics, Tay Nguyen University, 567 Le Duan, Buon Ma Thuot, Vietnam

²Hoang Ngoc Long, Institute of Physics, VAST, P.O. Box 429, Bo Ho, Hanoi 10000, Vietnam

Received 25 September 2013; Revised 29 January 2014; Accepted 30 January 2014; Published 2 April 2014

Academic Editor: Anastasios Petkou

Copyright © 2014 V. V. Vien and H. N. Long. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

Abstract.

A new S ₄ flavor model based on gauge symmetry responsible for fermion masses and mixings is constructed. The neutrinos get small masses from only an antisextet of SU(3)L which is in a doublet under S ₄. In this work, we assume the VEVs of the antisextet differ from each other under S ₄ and the difference of these VEVs is regarded as a small perturbation, and then the model can fit the experimental data on neutrino masses and mixings. Our results show that the neutrino masses are naturally small and a deviation from the tribimaximal neutrino mixing form can be realized. The quark masses and mixing matrix are also discussed. The number of required Higgs multiplets is less and the scalar potential of the model is simpler than those of the model based on S ₃ and our previous S ₄ model. The assignation of VEVs to antisextet leads to the mixing of the new gauge bosons and those in the standard model. The mixing in the charged gauge bosons as well as the neutral gauge bosons is considered.

1. Introduction

The experiments on neutrino oscillation confirm that neutrinos are massive particles [1–6]. The parameters of neutrino oscillations such as the squared mass differences and mixing angles are now well constrained. The data in PDG2012 [7–11] imply

These large neutrino mixing angles are completely different from the quark mixing ones defined by the CKM matrix [12, 13]. This has stimulated work on flavor symmetries and non-Abelian discrete symmetries are considered to be the most attractive candidate to formulate dynamical principles that can lead to the flavor mixing patterns for quarks and lepton. There are many recent models based on the non-Abelian discrete symmetries, such as [14–29], [30–65], and [66–93].

An alternative to extend the standard model (SM) is the 3-3-1 models, in which the SM gauge group is extended to which is investigated in [94–109]. The extension of the gauge group with respect to SM leads to interesting consequences. The first one is that the requirement of anomaly cancelation together with that of asymptotic freedom of QCD implies that the number of generations must necessarily be equal to the number of colors, hence giving an explanation for the existence of three generations. Furthermore, quark generations should transform differently under the action of . In particular, two quark generations should transform as triplets, one as an antitriplet.

A fundamental relation holds among some of the generators of the group:

where indicates the electric charge, and are two of the generators, and is the generator of . is a key parameter that defines a specific variant of the model. The model thus provides a partial explanation for the family number, as also required by flavor symmetries such as for 3-dimensional representations. In addition, due to the anomaly cancelation one family of quarks has to transform under differently from the two others. can meet this requirement with the representations and .

There are two typical variants of the 3-3-1 models as far as lepton sectors are concerned. In the minimal version, three lepton triplets are , where are ordinary right-handed charged leptons [94–98]. In the second version, the third components of lepton triplets are the right-handed neutrinos, [99–105]. To have a model with the realistic neutrino mixing matrix, we should consider another variant of the form where are three new fermion singlets under SM symmetry with vanishing lepton numbers [110–113].

In our previous works we have also extended the above application to the 3-3-1 models [110–113]. In [112] we have studied the 3-3-1 model with neutral fermions based on group, in which most of the Higgs multiplets are in triplets under except that is in a singlet, and the exact tribimaximal form [114–117] is obtained, in which . As we know, the recent considerations have implied , but small as given in (1). This problem has been improved in [111] by adding a new triplet and another antisextet , in which is regarded as a small perturbation. Therefore the model contains up to eight Higgs multiplets, and the scalar potential of the model is quite complicated.

In this paper, we propose another choice of fermion content and Higgs sector. As a consequence, the number of required Higgs is fewer and the scalar potential of the model is much simpler. The resulting model is near that of our previous work in [111] and includes those given in [112] as a special case and the physics is also different from the former. With the similar analysis as in [111], contains two triplets irreducible representation, one doublet and two singlets. This feature is useful to separate the third family of fermions from the others which contains a irreducible representation which can connect two maximally mixed generations. Besides the facilitating maximal mixing through , it provides two inequivalent singlet representations and which play a crucial role in consistently reproducing fermion masses and mixing as a perturbation. We have pointed out that this model is simpler than that of and our previous model, since fewer Higgs multiplets are needed in order to allow the fermions to gain masses and to break the gauge symmetry. Indeed, the model contains only six Higgs multiplets. On the other hand, the neutrino sector is simpler than those of and models [111, 112].

The rest of this work is organized as follows. In Sections 2 and 3 we present the necessary elements of the 3-3-1 model with flavor symmetry as in the above choice, as well as introducing necessary Higgs fields responsible for the charged-lepton masses. In Section 4, we discuss on quark sector. Section 5 is devoted to the neutrino mass and mixing. In Section 6 we discuss the gauge boson pattern of the model. We summarize our results and make conclusions in Section 7. Appendix A is devoted to the Higgs potential and minimization conditions. Appendix B is devoted to group with its Clebsch-Gordan coefficients. Appendix C presents the lepton numbers and lepton parities of model particles.

2. Fermion Content

The gauge symmetry is based on , where the electroweak factor is extended from those of the SM whereas the strong interaction sector is retained. Each lepton family includes a new fermion singlet carrying no lepton number arranged under the symmetry as a triplet and a singlet . The residual electric charge operator is therefore related to the generators of the gauge symmetry by [110–112]

where are charges with and is the charge. This means that the model under consideration does not contain exotic electric charges in the fundamental fermion, scalar, and adjoint gauge boson representations.

The particles in the lepton triplet have different lepton numbers (1 and 0), so the lepton number in the model does not commute with the gauge symmetry unlike the SM. Therefore, it is better to work with a new conserved charge commuting with the gauge symmetry and related to the ordinary lepton number by diagonal matrices [110–112, 118]

The lepton charge arranged in this way (i.e., as assumed) is in order to prevent unwanted interactions due to symmetry and breaking (due to the lepton parity as shown below), such as the SM and exotic quarks, and to obtain the consistent neutrino mixing.

By this embedding, exotic quarks and as well as new non-Hermitian gauge bosons and possess lepton charges as of the ordinary leptons: . The lepton parity is introduced as follows: , which is a residual symmetry of . The particles possess , such as , ordinary quarks, and bileptons having ; the particles with such as ordinary leptons and exotic quarks having . Any nonzero VEV with odd parity, , will break this symmetry spontaneously [112]. For convenience in reading, the numbers and of the component particles are given in Appendix C.

In this paper we work on a basis where and are real representations whereas the two-dimensional representation of is complex, , and

The lepton content of this model is similar to that of [111] but is different from the one in [112]; namely, in [112] three left-handed leptons are put in one triplet under , whereas in this model we put the first family of leptons in singlets of , while the two other families are in the doublets . In the quark content, the third family is put in a singlet and the two others in a doublet under which satisfy the anomaly cancelation in 3-3-1 models. The difference in fermion content leads to the difference between this work and our previous work [112] in physical phenomenon as seen bellow. Under the symmetries as proposed, the fermions of the model transform as follows:

where the subscript numbers on field indicate respective families in order to define components of their multiplets. In the following, we consider possibilities of generating masses for the fermions. The scalar multiplets needed for this purpose would be introduced accordingly.

3. Charged Lepton Mass

In [112], both three families of left-handed fermions and three right-handed quarks are put in a triplet under . To generate masses for the charged leptons, we have introduced two scalar triplets and lying in and under , respectively, with VEVs and . From the invariant Yukawa interactions for the charged leptons, we obtain the right-handed charged leptons mixing matrices which are diagonal ones, , and the right-handed one given by [112]

Similar to the charged lepton sector, to generate the quark masses, we have additionally introduced the three scalar Higgs triplets , , lying in , , and under , respectively. Quark masses can be derived from the invariant Yukawa interactions for quarks with supposing that the VEVs of , , and are , , and , where , , and . The other VEVs , , and vanish if the lepton parity is conserved. In addition, the VEV also breaks the 3-3-1 gauge symmetry down to that of the standard model and provides the masses for the exotic quarks and as well as the new gauge bosons. The , as well as , break the SM symmetry and give the masses for the ordinary quarks, charged leptons, and gauge bosons. To keep consistency with the effective theory, we assume that is much larger than those of and [112]. The unitary matrices which couple the left-handed quarks and with those in the mass bases are unit ones (, ), and the CKM quark mixing matrix at the tree level is then . For a detailed study on charged lepton and quark mass the reader can see [112].

In [112], to generate masses for neutrinos, we have introduced one antisextet lying in under and one antisextet lying in under with the VEV of being set as under . The neutrino masses are explicitly separated and the lepton mixing matrix yields the exact tribimaximal form [112] where which is a small deviation from recent neutrino oscillation data [7]. However, this problem will be improved in this work.

Because the fermion content of the model, as given in (6), is the same as that of one in [111] under all symmetries, so the charged-lepton mass is also similar to the one in [111]. Indeed, to generate masses for the charged leptons, we need two scalar triplets:

with VEVs and .

The Yukawa interactions are

The mass Lagrangian of the charged leptons reads

It is then diagonalized, and

This means that the charged leptons by themselves are the physical mass eigenstates, and the lepton mixing matrix depends on only that of the neutrinos that will be studied in Section 5.

We see that the masses of muon and tauon are separated by the triplet. This is the reason why we introduce in addition to .

The charged lepton Yukawa couplings relate to their masses as follows:

The current mass values for the charged leptons at the weak scale are given by [7]

Thus, we get

It follows that if and are of the same order of magnitude, and . This result is similar to the case of the model based on group [111]. On the other hand, if we choose the VEV of as , then , .

4. Quark Mass

To generate the quark masses with a minimal Higgs content, we additionally introduce the following scalar multiplets:

It is noticed that these scalars do not couple with the lepton sector due to the gauge invariance. The Yukawa interactions are then

Suppose that the VEVs of , , and are , , and , where , , and . The other VEVs , , and vanish due to the lepton parity conservation [111]. The exotic quarks therefore get masses and . In addition, has to be much larger than those of , , , and for a consistency with the effective theory. The mass matrices for ordinary up-quarks and down-quarks are, respectively, obtained as follows:

Similar to the charged leptons, the masses of and quarks are in pair separated by the scalars and , respectively. We see also that the introduction of in addition to is necessary to provide the different masses for and quarks as well as for and quarks.

The expressions (17) yield the relations:

The current mass values for the quarks are given by [7]

Hence

It is obvious that if , the Yukawa coupling hierarchies are , , and the couplings between up-quarks and Higgs scalar multiplets are slightly heavier than those of down-quarks , respectively.

The unitary matrices, which couple the left-handed up- and down-quarks with those in the mass bases, are and , respectively. Therefore we get the CKM matrix

This is a good approximation for the realistic quark mixing matrix, which implies that the mixings among the quarks are dynamically small. The small permutations such as a breaking of the lepton parity due to the VEVs , , and or a violation of and/nor symmetry due to unnormal Yukawa interactions, namely, , , , , and so forth, will disturb the tree level matrix resulting in mixing between ordinary and exotic quarks and possibly providing the desirable quark mixing pattern. A detailed study on these problems is out of the scope of this work and should be skipped.

5. Neutrino Mass and Mixing

The neutrino masses arise from the couplings of , , and to scalars, where transforms as under and under , transforms as under and under , and transforms as under and under . For the known scalar triplets , only available interactions are and but explicitly suppressed because of the -symmetry. We will therefore propose new antisextets instead of coupling to responsible for the neutrino masses which are lying in either , , , or under . In [112], we have introduced two antisextets , which are lying in and under , respectively. Contrastingly, in this work, with fermion content as proposed, to obtain a realistic neutrino spectrum, the model needs only one antisextet which transforms as follows:

where the numbered subscripts on the component scalars are the indices, whereas is that of . The VEV of is set as under , in which

Following the potential minimization conditions, we have several VEV alignments. The first is that and then is broken into an eight-element subgroup, which is isomorphic to . The second is that or and then is broken into consisting of the identity and the even permutations of four objects. The third is that and then is broken into a four-element subgroup consisting of the identity and three double transitions, which is isomorphic to Klein four group [75] (in this paper we denote this group by ). To obtain a realistic neutrino spectrum, we argue that both the breakings and must take place. We therefore assume that its VEVs are aligned as the former to derive the direction of the breaking , and this happens in any case bellow:

The direction of the breaking is equivalent to the breaking . In this direction, we set . If is unbroken, we have as in (24), and on the contrary, if is unbroken, we have :

The difference between and is very small which is regarded as a small perturbation as considered bellow. It is noteworthy that the derivation in this paper contains a fewer, in comparison with the model based on the group [111], number of Higgs triplets; consequently the Higgs sector and the minimization condition of the potential are much simpler. Moreover, the obtained model, despite the compact in Higgs sector, can fit the current data with , while the old version [112] based on cannot provide nonvanishing .

In general, the Yukawa interactions are

With the alignments of VEVs as in (24) and (25), the mass Lagrangian for the neutrinos is determined by

where and . The mass matrices are then obtained by

with

The VEVs break the 3-3-1 gauge symmetry down to that of the SM and provide the masses for the neutral fermions and the new gauge bosons: the neutral and the charged and . The and belong to the second stage of the symmetry breaking from the SM down to the symmetry and contribute the masses to the neutrinos. Hence, to keep a consistency we assume that [105].

Three active neutrinos therefore gain masses via a combination of type I and type II seesaw mechanisms derived from (27) and (28) as

where

The following comments of breaking are in order.

(i)If is broken into ( is unbroken), we have , , and , which is presented in Section 5.1.(ii)If is broken into ( is broken into {Identity}), we have , , , and but it is very small. In this case the disparity of two VEVs of is regarded as a small perturbation as shown in Section 5.2.

We next divide our considerations into two cases to fit the data: the first case is , and the second one is .

5.1. Experimental Constraints in the Case

If is broken into , , , , we have , , , and , and reduces to

where

We can diagonalize the matrix in (32) as follows:

where

and the neutrino mixing matrix takes the form:

Note that . This matrix can be parameterized in three Euler’s angles, which implies

This case coincides with the data since and [119, 120]. For the remaining constraints, taking the central values from the data in [119]

and we have a solution

and , , , and . It follows that , , and the neutrino mixing matrix form is very close to that of bimaximal mixing which takes the form:

Now, it is natural to choose , in eV order, and suppose that . Let us assume , and we have then and .

This result is not obviously consistent with the recent data on neutrinos oscillation in which , but small as given in [7]. However, as we will see in Section 5.2, this situation will be improved if the direction of the breaking takes place. This means that, for the model under consideration, both the breakings and (instead of ) must take place in the neutrino sector.

5.2. Experimental Constraints in the Case

In this case is broken into the Klein four group , , , and , and the direct consequence is , , , and . The general neutrino mass matrix in (30) can be rewritten in the form:

where and are given by (33), accommodated in the first matrix, which is matched to the case of . The three last matrices in (41) are a deviation from the contribution due to the disparity of and , namely, , , , , , and , with the , , , and being defined in (31), which correspond to .

Substituting (29) into (31) we get

Indeed, if , the deviations , , will vanish, therefore the mass matrix in (30) reduces to its first term coinciding with (32). The first term of (41) provides bimaximal mixing pattern, in which as shown in Section 5.1. The other matrices proportional to , , due to contribution from the disparity of and will take the role of perturbation for such a deviation of . So, in this work we consider the disparity of and as a small perturbation and terminating the theory at the first order.

Without loss of generality, we consider the case of breaking , in which whereas and . It is then , with being a small parameter. In this case, the matrix in (41) reduces to

At the first order of perturbation, the physical neutrino masses are obtained as

where are the mass values as of the case given by (39). For the corresponding perturbed eigenstates, we put

where is defined by (36), and

with

The lepton mixing matrix in this case can still be parameterized in three new Euler’s angles , which are also a perturbation from the in the case 1, defined by

It is easily to show that our model is consistent since the five experimental constraints on the mixing angles and squared neutrino mass differences can be, respectively, fitted with two Yukawa coupling parameters , of the antisextet scalar with the above mentioned VEVs. Indeed, taking the data in (1) we obtain , , , and and , , and [ satisfying the condition ]. The neutrino masses are explicitly given as , , and . The neutrino mixing matrix then takes the form:

6. Gauge Bosons

The covariant derivative of a triplet is given by

where are Gell-Mann matrices, , , , and is -charge of Higgs triplets.

Let us denote the following combinations:

and then is rewritten in a convenient form as follows:

with . We note that and are pure real and imaginary parts of and , respectively. The covariant derivative for an antisextet with the VEV part is [121]

The covariant derivative (53) acting on the antisextet VEVs is given by

The masses of gauge bosons in this model are defined as follows:

where in (55) is different from the one in [122] by the difference of the components of the antisextet . In [122], , namely, , , and , are taken into account, and the contribution of perturbation has been skipped at the first order. In the following, we consider the general case in which , , and . As a consequence, the fewer number of Higgs multiplets is needed in order to allow the fermions to gain masses and with the simpler scalar Higgs potential as mentioned above.

Substitution of the VEVs of Higgs multiplets into (55) yields

We can separate in (57) into

where is the Lagrangian part of the imaginary part . This boson is decoupled with mass given by

In the limit we have

is the Lagrangian part of the charged gauge bosons and :

in (60) can be rewritten in matrix form as follows:

where

The matrix in (62) can be diagonalized as follows:

where

with

With corresponding eigenstates, the charged gauge boson mixing matrix takes the form:

The mixing angle is given by

The physical charged gauge bosons are defined

In our model, the following limit is often taken into account:

With the help of (69), the in (65) becomes

It is then

with

In the limit the mixing angle tends to zero, , and one has

With the help of (69), one can estimate

In addition, from (73), it follows that is much smaller than . Note that, due to the above mixing, the new gauge boson will give a contribution to neutrinoless double beta decay (for details, see [123–125]).

is the Lagrangian that describes the mixing among the neutral gauge bosons , , , . The mass Lagrangian in this case has the form

On the basis of , the in (75) can be rewritten in matrix form:

where

The matrix in (76) with elements in (77) has one exact eigenvalue, which is identified with the photon mass:

The corresponding eigenvector of is

Note that in the limit , , and does not mix with , , . In the general case , the mass matrix in (76) contains one exact eigenvalues as in (78) with the corresponding eigenstate given in (79).

The mass matrix in (76) is diagonalized via two steps. In the first step, the basic is transformed into the basic by the matrix:

The corresponding eigenstates are given by

To obtain (80) and (81) we have used the continuation of the gauge coupling constant of the at the spontaneous symmetry breaking point, in which

On this basis, the mass matrix becomes

where

In the approximation , we have

with

From (83), there exist mixings between , and . It is noteworthy that, in the limit , the elements and vanish. In this case there is no mixing between and , .

In the second step, three bosons gain masses via seesaw mechanism

where

Combination of (87), (88), and (85) yields

where

with

The parameter in our model is given by

where

Let us assume the relations and put , , and then

From (92)–(94) we have

The experimental value of the parameter and are, respectively, given in [7]

It means

From (95) one can make the relations between , , and . Indeed, we have

Figure 1 gives the relation between and , provided that , and in which .

Figure 1: The relation between and , with and .

Figure 2 gives the relation between and , provided that and , in which . The conditions (69) are satisfied. The Figure 3 gives the relation between and , provided and , in which ( is a real number, Figure 3(a)) or , ( is a pure complex number, Figure 3(b)). The conditions (69) are satisfied. From Figure 3 we see that a lot of values of that is different from the unit but nearly it still can fit the recent experimental data [7]. It means that the difference of and as mentioned in this work is necessary.

Figure 2: The relation between and , with and , .

(a)

(b)

Figure 3: The relation between and , provided that and , .

Diagonalizing the mass matrix , we get two new physical gauge bosons

With the approximation as in (69), the mixing angle is given by

provided that , .

In the limit the mixing angle tends to zero, and the physical mass eigenvalues are defined by

From (59) and (101) we see that the and components have the same mass in the limit . So we should identify the combination of and

as physical neutral non-Hermitian gauge boson. The subscript “0” denotes neutrality of gauge boson . Notice that the identification in (102) only can be acceptable with the limit . In general, it is not true because of the difference in masses of and as in (58) and (99).

The expressions (74) and (100) show that, with the limit (69), the mixings between the charged gauge bosons and the neutral ones are in the same order since they are proportional to   (). In addition, from (101), is little bigger than   (or ), and is little smaller than . In that limit, the masses of and degenerate.

7. Conclusions

In this paper, we have constructed a new model based on gauge symmetry responsible for fermion masses and mixing which is different from our previous work in [112]. Neutrinos get masses from only an antisextet which is in a doublet under . We argue how flavor mixing patterns and mass splitting are obtained with a perturbed symmetry by the difference of VEV components of the antisextet under . We have pointed out that this model is simpler than those of and [111, 112] with the fewer number of Higgs multiplets needed in order to allow the fermions to gain masses but with the simple scalar Higgs potential. Quark mixing matrix is unity at the tree level. The realistic neutrino mixing in which can be obtained if the direction for breaking takes place. This corresponds to the requirement on the difference of VEV components of the antisextet under group. As a result, the value of is a small perturbation by . The assignation of VEVs to antisextet leads to the mixing of the new gauge bosons and those in the SM. The mixing in the charged gauge bosons as well as the neutral gauge boson was considered.

Appendices

A. Vacuum Alignment

We can separate the general scalar potential into

where and , respectively, consist of the scalar triplets and sextets, whereas contains the terms connecting the two sectors. Moreover conserve -charge and symmetry, while includes possible soft terms explicitly violating these charges. Here the soft terms as we meant include the trilinear and quartic ones as well. The reason for imposing will be shown below.

The details on the potentials are given as follows. We first denote Notice also that . is a sum of

is only of ,

and is a sum of

To provide the Majorana masses for the neutrinos, the lepton number must be broken. This can be achieved via the scalar potential violating . However, the other symmetries should be conserved. The violating potential up to quartic interactions is given as

We have not explicitly written, but there must additionally exist the terms in explicitly violating the only symmetry or both the and -charge too. In the following, most of them will be omitted, and only the terms of the kind of interest will be provided.

There are the several scalar sectors corresponding to the expected VEV directions. The first direction, , , is broken into a subgroup including the elements which is isomorphic to the Klein four-group [75] [, , obeying the relations , , are generators of group given in [112]]. The second direction, , , is broken into . The third direction, , or , , is broken into . As mentioned before, to obtain a realistic neutrino spectrum, we have thus imposed both of the first and the second directions to be performed.

Let us now consider the potential . The flavons , , , , with their VEVs aligned in the same direction (all of them are singlets) are an automatic solution from the minimization conditions of . To explicitly see this, in the system of equations for minimization, let us put , , , , and . Then the potential minimization conditions for triplets reduce to

It is easily shown that the derivatives of with respect to the variables , , , shown in (A.7), (A.8), (A.9), and (A.10) are symmetric to each other. The system of (A.6)–(A.10) always has the solution (, , , ) as expected, even though it is complicated. It is also noted that the above alignment is only one of the solutions to be imposed to have the desirable results. We have evaluated that (A.7)–(A.10) have the same structure solution. Consequently, to have a simple solution, we can assume that . In this case, (A.7)–(A.10) reduce a unique equation, and system of (A.6)–(A.10) becomes

This system has a solution as follows:

where

Considering the potentials and , we impose that

and we obtain a system of equations of the potential minimization for antisextets:

where is a sum of and :

It is easily shown that (A.15) takes the same form in couples. This system of equations yields the following relations:

where is a constant. It means that there are several alignments for VEVs. In this work, to have the desirable results, we have imposed the two directions for breaking and as mentioned, in which and but approximates to the unit. In the case that or , , and , the system of (A.15) reduces to system for minimal potential condition consisting of three equations as follows:

where

The system of (A.18) always has the solution (, , ) as expected, even though it is complicated. It is also noted that the above alignment is only one of the solutions to be imposed to have the desirable results.

B. Group and Clebsch-Gordan Coefficients

is the permutation group of four objects, which is also the symmetry group of a cube. It has 24 elements divided into 5 conjugacy classes, with , , , , and as its 5 irreducible representations. Any element of can be formed by multiplication of the generators and obeying the relations , . Without loss of generality, we could choose , where the cycle denotes the permutation , and means . The conjugacy classes generated from and are

The character table of is given as shown in Table 1, where is the order of class and is the order of elements within each class. Let us note that are even permutations, while are odd permutations. The two three-dimensional representations differ only in the signs of their and matrices. Similarly, the two one-dimensional representations behave the same.

Table 1

Class

1
1
1
1
2
3
3

3
2
1
1
2
−1
−1

8
3
1
1
−1
0
0

6
4
1
−1
0
−1
1

6
2
1
−1
0
1
−1

We will work on a basis where and are real representations whereas is complex. One possible choice of generators is given as follows:

where is the cube root of unity. Using them we calculate the Clebsch-Gordan coefficients for all the tensor products as given below.

First, let us put which means some multiplet such as or , and similarly for the other representations. Moreover, the numbered multiplets such as mean where and are the multiplet components of different representations and , respectively. In the following the components of representations in l.h.s will be omitted and should be understood, but they always exist in order in the components of decompositions in r.h.s.:

where the subscripts and , respectively, refer to their symmetric and antisymmetric product combinations as explicitly pointed out. We also notice that many group multiplication rules above have similar forms as those of and groups [14, 112].

In the text we usually use the following notations, for example, which is the Clebsch-Gordan coefficients of in the decomposition of , whereas mentioned and .

The rules to conjugate the representations 1, , 2, 3, and are given by

where, for example, denotes some multiplet of the form .

C. The Numbers

In Table 2 we will explicitly point out the lepton number () and lepton parity () of the model particles (notice that the family indices are suppressed).

Table 2

Particles

, , , , , , , , , , , , ,
0
1

, , , , , , , , , , , , ,
−1
−1

, , , , ,
−2
1

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no. 103.01-2011.63.

References

Super-Kamiokande Collaboration, Y. Fukuda, T. Hayakawa, et al., “Evidence for oscillation of atmospheric neutrinos,” Physical Review Letters, vol. 81, pp. 1562–1567, 1998.

Super-Kamiokande Collaboration, S. Fukuda, Y. Fukuda et al., “Solar ⁸B and hep neutrino measurements from 1258 days of Super-Kamiokande data,” Physical Review Letters, vol. 86, no. 25, pp. 5651–5655, 2001.

S. Fukuda, Y. Fukuda, M. Ishitsuka et al., “Constraints on neutrino oscillations using 1258 days of Super-Kamiokande solar neutrino data,” Physical Review Letters, vol. 86, no. 25, pp. 5656–5660, 2001.

S. Fukuda, Y. Fukuda, M. Ishitsuka, et al., “Determination of solar neutrino oscillation parameters using 1496 days of Super-Kamiokande-I data,” Physics Letters B, vol. 539, pp. 179–187, 2002.

Super-Kamiokande Collaboration, Y. Ashie, J. Hosaka, et al., “Evidence for an oscillatory signature in atmospheric neutrino oscillation,” Physical Review Letters, vol. 93, Article ID 101801, 2004.

Super-Kamiokande Collaboration, Y. Ashie, J. Hosaka, et al., “A measurement of atmo-spheric neutrino oscillation parameters by Super-Kamiokande I,” Physical Review D, vol. 71, Article ID 112005, 35 pages, 2005.

J. Beringer, J.-F. Arguin, R. M. Barnett, et al., “Review of particle physics,” Physical Review D, vol. 86, no. 1, Article ID 010001, 1528 pages, 2012.

T. Schwetz, M. Tórtola, and J. W. F. Valle, “Where we are on θ ₁₃: addendum to ‘global neutrino data and recent reactor fluxes: status of three-flavor oscillation parameters’,” New Journal of Physics, vol. 13, Article ID 109401, 2011.

K. Abe, N. Abgrall, Y. Ajima, et al., “Indication of electron neutrino appearance from an accelerator-produced off-axis muon neutrino beam,” Physical Review Letters, vol. 107, Article ID 041801, 8 pages, 2011.

P. Adamson, D. J. Auty, D. S. Ayres, et al., “Improved search for muon-neutrino to electron- neutrino oscillations in MINOS,” Physics Letters B, vol. 107, no. 18, Article ID 181802, 2011.

G. L. Fogli, E. Lisi, A. Marrone, A. Palazzo, and A. M. Rotunno, “Evidence of θ ₁₃> 0 from global neutrino data analysis,” Physical Review D, vol. 84, no. 5, Article ID 053007, 2011.

N. Cabibbo, “Unitary symmetry and leptonic decays,” Physical Review Letters, vol. 10, no. 12, pp. 531–533, 1963.

M. Kobayashi and T. Maskawa, “CP-violation in the renormalizable theory of weak interaction,” Progress of Theoretical Physics, vol. 49, pp. 652–657, 1973.

E. Ma and G. Rajasekaran, “Softly broken A ₄ symmetry for nearly degenerate neutrino masses,” Physical Review D, vol. 64, no. 11, Article ID 113012, 2001.

K. S. Babu, E. Ma, and J. W. F. Valle, “Underlying A ₄ symmetry for the neutrino mass matrix and the quark mixing matrix,” Physics Letters B, vol. 552, no. 3-4, pp. 207–213, 2003.

G. Altarelli and F. Feruglio, “Tri-bimaximal neutrino mixing, A ₄ and the modular symmetry,” Nuclear Physics B, vol. 741, no. 1-2, pp. 215–235, 2006.

E. Ma, “Tribimaximal neutrino mixing from a supersymmetric model with A ₄ family symmetry,” Physical Review D, vol. 73, no. 5, Article ID 057304, 2006.

X.-G. He, Y.-Y. Keum, and R. R. Volkas, “A ₄ flavour symmetry breaking scheme for understanding quark and neutrino mixing angles,” Journal of High Energy Physics, vol. 2006, no. 4, article 039, 2006.

S. Morisi, M. Picariello, and E. Torrente-Lujan, “Model for fermion masses and lepton mixing in SO(10) × A ₄,” Physical Review D, vol. 75, no. 7, Article ID 075015, 2007.

C. S. Lam, “Symmetry of lepton mixing,” Physics Letters B, vol. 656, no. 4-5, pp. 193–198, 2007.

F. Bazzocchi, S. Kaneko, and S. Morisi, “A SUSY A ₄ model for fermion masses and mixings,” Journal of High Energy Physics, vol. 2008, no. 3, article 063, 2008.

F. Bazzocchi, M. Frigerio, and S. Morisi, “Fermion masses and mixing in models with SO(10) × A ₄ symmetry,” Physical Review D, vol. 78, no. 11, Article ID 116018, 2008.

G. Altarelli, F. Feruglio, and C. Hagedorn, “A SUSY SU(5) grand unified model of tri-bimaximal mixing from A ₄,” Journal of High Energy Physics, vol. 2008, no. 3, article 052, 2008.

M. Hirsch, S. Morisi, and J. W. F. Valle, “Tribimaximal neutrino mixing and neutrinoless double beta decay,” Physical Review D, vol. 78, no. 9, Article ID 093007, 2008.

E. Ma, “Dark scalar doublets and neutrino tribimaximal mixing from A ₄ symmetry,” Physics Letters B, vol. 671, no. 3, pp. 366–368, 2009.

Y. Lin, “A predictive A ₄ model, charged lepton hierarchy and tri-bimaximal sum rule,” Nuclear Physics B, vol. 813, no. 1-2, pp. 91–105, 2009.

Y. H. Ahn and C.-S. Chen, “Nonzero U e3 and TeV leptogenesis through A ₄ symmetry breaking,” Physical Review D, vol. 81, no. 10, Article ID 105013, 2010.

J. Barry and W. Rodejohanny, “Deviations from tribimaximal mixing due to the vacuum expectation value misalignment in A ₄ models,” Physical Review D, vol. 81, Article ID 093002, 2010.

G.-J. Ding and D. Meloni, “A model for tri-bimaximal mixing from a completely broken A ₄,” Nuclear Physics B, vol. 855, no. 1, pp. 21–45, 2012.

L. Wolfenstein, “Oscillations among three neutrino types and CP violation,” Physical Review D, vol. 18, no. 3, pp. 958–960, 1978.

S. Pakvasa and H. Sugawara, “Discrete symmetry and Cabibbo angle,” Physics Letters B, vol. 73, no. 1, pp. 61–64, 1978.

E. Derman and H.-S. Tsao, “[] Sn flavor dynamics and a bound on the number of flavors,” Physical Review D, vol. 20, no. 5, pp. 1207–1215, 1979.

Y. Yamanaka, H. Sugawara, and S. Pakvasa, “Permutation symmetries and the fermion mass matrix,” Physical Review D, vol. 25, no. 7, pp. 1895–1903, 1982.

K. Kang, J. E. Kim, and P. Ko, “A simple modification of the maximal mixing scenario for three light neutrinos,” Zeitschrift für Physik C, vol. 72, no. 4, pp. 671–675, 1996.

H. Fritzsch and Z.-Z. Xing, “Lepton mass hierarchy and neutrino oscillations,” Physics Letters B, vol. 372, no. 3-4, pp. 265–270, 1996.

K. Kang, S. K. Kang, J. E. Kim, and P. Ko, “Almost maximally broken permutation symmetry for neutrino mass matrix,” Modern Physics Letters A, vol. 12, no. 16, pp. 1175–1184, 1997.

M. Fukugita, M. Tanimoto, and T. Yanagida, “Atmospheric neutrino oscillation and a phenomenological lepton mass matrix,” Physical Review D, vol. 57, no. 7, pp. 4429–4432, 1998.

H. Fritzsch and Z.-Z. Xing, “Large leptonic flavour mixing and the mass spectrum of leptons,” Physics Letters B, vol. 440, no. 3-4, pp. 313–318, 1998.

Y. Koide, “Universal seesaw mass matrix model with an S ₃ symmetry,” Physical Review D, vol. 60, no. 7, Article ID 077301, 4 pages, 1999.

H. Fritzsch and Z.-Z. Xing, “Maximal neutrino mixing and maximal CP violation,” Physical Review D, vol. 61, no. 7, Article ID 073016, 7 pages, 2000.

M. Tanimoto, “Large mixing angle MSW solution in S ₃ flavor symmetry,” Physics Letters B, vol. 483, no. 4, pp. 417–424, 2000.

G. C. Branco and J. I. Silva-Marcos, “The symmetry behind extended flavour democracy and large leptonic mixing,” Physics Letters B, vol. 526, no. 1-2, pp. 104–110, 2002.

M. Fujii, K. Hamaguchi, and T. Yanagida, “Leptogenesis with almost degenerate Majorana neutrinos,” Physical Review D, vol. 65, no. 11, Article ID 115012, 2002.

J. Kubo, A. Mondragón, M. Mondragón, and E. Rodríguez-Jáuregui, “The flavor symmetry,” Progress of Theoretical Physics, vol. 109, no. 5, pp. 795–807, 2003.

J. Kubo, “Majorana phase in minimal S ₃ invariant extension of the standard model,” Physics Letters B, vol. 578, no. 1-2, pp. 156–164, 2004.

P. F. Harrison and W. G. Scott, “Permutation symmetry, tri-bimaximal neutrino mixing and the S ₃ group characters,” Physics Letters B, vol. 557, no. 1-2, pp. 76–86, 2003.

S. L. Chen, M. Frigerio, and E. Ma, “Large neutrino mixing and normal mass hierarchy: a discrete understanding,” Physical Review D, vol. 70, Article ID 073008, 2004.

H. Fritzsch and Z.-Z. Xing, “Democratic neutrino mixing reexamined,” Physics Letters B, vol. 598, no. 3-4, pp. 237–242, 2004.

W. Grimus and L. Lavoura, “S ₃ × ℤ ₂ model for neutrino mass matrices,” Journal of High Energy Physics, vol. 2005, article 013, 2005.

R. N. Mohapatra, S. Nasri, and H.-B. Yu, “S ₃ symmetry and tri-bimaximal mixing,” Physics Letters B, vol. 639, no. 3-4, pp. 318–321, 2006.

R. Jora, S. Nasri, and J. Schechter, “An approach to permutation symmetry for the electroweak theory,” International Journal of Modern Physics A, vol. 21, no. 28-29, pp. 5875–5894, 2006.

J. E. Kim and J. C. Park, “Quantum numbers of heavy neutrinos, tri-bi-maximal mixing through double seesaw with permutation symmetry, and comment on + θ cπ/4,” Journal of High Energy Physics, vol. 2006, article 017, 2006.

Y. Koide, “S ₃ symmetry and neutrino masses and mixings,” The European Physical Journal C, vol. 50, no. 4, pp. 809–816, 2007.

A. Mondragón, M. Mondragón, and E. Peinado, “Lepton masses, mixings, and flavor-changing neutral currents in a minimal S ₃-invariant extension of the standard model,” Physical Review D, vol. 76, no. 7, Article ID 076003, 2007.

M. Picariello, “Neutrino CP violating parameters from nontrivial quark lepton correlation: a S ₃ × GUT model,” International Journal of Modern Physics A, vol. 23, no. 27-28, pp. 4435–4448, 2008.

C.-Y. Chen and L. Wolfenstein, “Consequences of approximate S ₃ symmetry of the neutrino mass matrix,” Physical Review D, vol. 77, no. 9, Article ID 093009, 2008.

R. Jora, J. Schechter, and M. N. Shahid, “Perturbed S ₃ neutrinos,” Physical Review D, vol. 80, Article ID 093007, 2009.

D. A. Dicus, S.-F. Ge, and W. W. Repko, “Neutrino mixing with broken S ₃ symmetry,” Physical Review D, vol. 82, no. 3, Article ID 033005, 2010.

Z.-Z. Xing, D. Yang, and S. Zhou, “Broken S ₃ flavor symmetry of leptons and quarks: mass spectra and flavor mixing patterns,” Physics Letters B, vol. 690, no. 3, pp. 304–310, 2010.

R. Jora, J. Schechter, and M. N. Shahid, “Doubly perturbed S ₃ neutrinos and the s ₁₃ mixing parameter,” Physical Review D, vol. 82, no. 5, Article ID 053006, 2010.

S. Dev, S. Gupta, and R. R. Gautam, “Broken S ₃ symmetry in the neutrino mass matrix,” Physics Letters B, vol. 702, no. 1, pp. 28–33, 2011.

D. Meloni, S. Morisi, and E. Peinado, “Fritzsch neutrino mass matrix from S ₃ symmetry,” Journal of Physics G, vol. 38, no. 1, Article ID 015003, 2011.

G. Bhattacharyya, P. Leser, and H. Päs, “Exotic Higgs boson decay modes as a harbinger of S ₃ flavor symmetry,” Physical Review D, vol. 83, no. 1, Article ID 011701, 2011.

T. Kaneko and H. Sugawara, “Broken S ₃ symmetry in flavor physics,” Physics Letters B, vol. 697, no. 4, pp. 329–332, 2011.

S. Zhou, “Relatively large θ ₁₃ and nearly maximal θ ₂₃ from the approximate S ₃ symmetry of lepton mass matrices,” Physics Letters B, vol. 704, no. 4, pp. 291–295, 2011.

R. N. Mohapatra, M. K. Parida, and G. Rajasekaran, “High scale mixing unification and large neutrino mixing angles,” Physical Review D, vol. 69, no. 5, Article ID 053007, 2004.

C. Hagedorn, M. Lindner, and R. N. Mohapatra, “S ₄ flavor symmetry and fermion masses: towards a grand unified theory of flavor,” Journal of High Energy Physics, vol. 2006, no. 6, article 042, 2006.

E. Ma, “Neutrino mass matrix from S ₄ symmetry,” Physics Letters B, vol. 632, no. 2-3, pp. 352–356, 2006.

H. Zhang, “Flavor S ₄ ⊗ Z₂ symmetry and neutrino mixing,” Physics Letters B, vol. 655, no. 3-4, pp. 132–140, 2007.

Y. Koide, “S ₄ flavor symmetry embedded into SU(3) and lepton masses and mixing,” Journal of High Energy Physics, vol. 2007, no. 8, article 086, 2007.

H. Ishimori, Y. Shimizu, and M. Tanimoto, “S ₄ flavor symmetry of quarks and leptons in SU(5) GUT,” Progress of Theoretical Physics, vol. 121, no. 4, pp. 769–787, 2009.

F. Bazzocchi, L. Merlo, and S. Morisi, “Fermion masses and mixings in a S ₄ based model,” Nuclear Physics B, vol. 816, no. 1-2, pp. 204–226, 2009.

F. Bazzocchi and S. Morisi, “S ₄ as a natural flavor symmetry for lepton mixing,” Physical Review D, vol. 80, no. 9, Article ID 096005, 2009.

G. Altarelli and F. Feruglio, “Discrete flavor symmetries and models of neutrino mixing,” Reviews of Modern Physics, vol. 82, no. 3, pp. 2701–2729, 2010.

G.-J. Ding, “Fermion masses and flavor mixings in a model with S ₄ flavor symmetry,” Nuclear Physics B, vol. 827, no. 1-2, pp. 82–111, 2010.

Y. H. Ahn, S. K. Kang, C. S. Kim, and T. P. Nguyen, “A direct link between neutrinoless double beta decay and leptogenesis in a seesaw model with S ₄ symmetry,” Physical Review D, vol. 82, no. 9, Article ID 093005, 2010.

H. Ishimori, Y. Shimizu, M. Tanimoto, and A. Watanabe, “Neutrino masses and mixing from S ₄ flavor twisting,” Physical Review D, vol. 83, no. 3, Article ID 033004, 2011.

H. Ishimori and M. Tanimoto, “Slepton mass matrices μ → eγ decay and EDM in SUSY S ₄ flavor model,” Progress of Theoretical Physics, vol. 125, no. 4, pp. 653–675, 2011.

R.-Z. Yang and H. Zhang, “Minimal seesaw model with S ₄ flavor symmetry,” Physics Letters B, vol. 700, no. 5, pp. 316–321, 2011.

S. Morisi and E. Peinado, “Admixture of quasi-Dirac and Majorana neutrinos with tri-bimaximal mixing,” Physics Letters B, vol. 701, no. 4, pp. 451–457, 2011.

T. Phong Nguyen and P. V. Dong, “Radiatively generated leptogenesis in S ₄ flavor symmetry models,” Advances in High Energy Physics, vol. 2012, Article ID 254093, 21 pages, 2012.

L. Dorame, S. Morisi, E. Peinado, J. W. F. Valle, and A. D. Rojas, “A new neutrino mass sum rule from inverse seesaw,” Physical Review D, vol. 86, Article ID 056001, 2012.

D. Hernandez and A. Yu. Smirnov, “Lepton mixing and discrete symmetries,” Physical Review D, vol. 86, Article ID 053014, 2012.

Z. Zhao, “Understanding for flavor physics in the lepton sector,” Physical Review D, vol. 86, Article ID 096010, 2012.

R. Krishnan, P. F. Harrison, and W. G. Scott, “Simplest neutrino mixing from symmetry,” Journal of High Energy Physics, vol. 2013, no. 4, article 087, 2013.

R. Krishnan, “A model for large θ ₁₃ constructed using the eigenvectors of the S ₄ rotation matrices,” Journal of Physics: Conference Series, vol. 447, Article ID 012043, 2013.

I. M. Varzielas and L. Lavoura, “Flavour models for TM₁ lepton mixing,” Journal of Physics G, vol. 40, no. 8, Article ID 085002, 2013.

W. Grimus, “Discrete symmetries, roots of unity, and lepton mixing,” Journal of Physics G, vol. 40, no. 7, Article ID 075008, 2013.

S. F. King and C. Luhn, “Neutrino mass and mixing with discrete symmetry,” Reports on Progress in Physics, vol. 76, no. 5, Article ID 056201, 2013.

R. G. Felipe, H. Serodio, and J. P. Silva, “Models with three Higgs doublets in the triplet repre- sentations of A ₄ or S ₄,” Physical Review D, vol. 87, Article ID 055010, 2013.

Y. Daikoku and H. Okada, “Phenomenology of S ₄ flavor symmetric extra U(1) model,” Physical Review D, vol. 88, no. 1, Article ID 015034, 27 pages, 2013.

F. Feruglio, C. Hagedorn, and R. Ziegler, “A realistic pattern of lepton mixing and masses from S ₄ and CP,” The European Physical Journal C, vol. 74, article 2753, 2014.

C. Luhn, “Trimaximal TM₁ neutrino mixing in S ₄ with spontaneous CP violation,” Nuclear Physics B, vol. 875, no. 1, pp. 80–100, 2013.

J. W. F. Valle and M. Singer, “Lepton-number violation with quasi-Dirac neutrinos,” Physical Review D, vol. 28, no. 3, pp. 540–545, 1983.

F. Pisano and V. Pleitez, “SU(3) ⊗ U(1) model for electroweak interactions,” Physical Review D, vol. 46, no. 1, pp. 410–417, 1992.

P. H. Frampton, “Chiral dilepton model and the flavor question,” Physical Review Letters, vol. 69, no. 20, pp. 2889–2891, 1992.

R. Foot, O. F. Hernández, F. Pisano, and V. Pleitez, “Lepton masses in an SU(3)L ⊗ U(1)N gauge model,” Physical Review D, vol. 47, no. 9, pp. 4158–4161, 1993.

J. C. Montero, F. Pisano, and V. Pleitez, “Neutral currents and Glashow-Iliopoulos-Maiani mechanism in SU(3)L ⊗ U(1)N models for electroweak interactions,” Physical Review D, vol. 47, no. 7, pp. 2918–2929, 1993.

M. Singer, J. W. F. Valle, and J. Schechter, “Canonical neutral-current predictions from the weak-electromagnetic gauge group SU(3) × U(1),” Physical Review D, vol. 22, no. 3, pp. 738–743, 1980.

R. Foot, H. N. Long, and T. A. Tran, “SU(3)L ⊗ U(1)N and SU(4)L ⊗U(1)N gauge models with right-handed neutrinos,” Physical Review D, vol. 50, no. 1, pp. R34–R38, 1994.

J. C. Montero, F. Pisano, and V. Pleitez, “Neutral currents and Glashow-Iliopoulos-Maiani mechanism in SU(3)L ⊗ U(1)N models for electroweak interactions,” Physical Review D, vol. 47, no. 7, pp. 2918–2929, 1993.

H. N. Long, “SU(3)L ⊗ U(1)N model for right-handed neutrino neutral currents,” Physical Review D, vol. 54, no. 7, pp. 4691–4693, 1996.

H. N. Long, “SU(3)C ⊗ SU(3)L ⊗ U(1)N model with right-handed neutrinos,” Physical Review D, vol. 53, no. 1, pp. 437–445, 1996.

H. N. Long, “Scalar sector of the 3-3-1 model with three Higgs triplets,” Modern Physics Letters A, vol. 13, no. 23, pp. 1865–1873, 1998.

P. V. Dong and H. N. Long, “Neutrino masses and lepton flavor violation in the 3-3-1 model with right-handed neutrinos,” Physical Review D, vol. 77, no. 5, Article ID 057302, 2008.

W. A. Ponce, Y. Giraldo, and L. A. Sánchez, “Minimal scalar sector of 3-3-1 models without exotic electric charges,” Physical Review D, vol. 67, no. 7, Article ID 075001, 2003.

P. V. Dong, H. N. Long, D. T. Nhung, and D. V. Soa, “SU(3)C ⊗ SU(3)L ⊗ U(1)X model with two Higgs triplets,” Physical Review D, vol. 73, no. 3, Article ID 035004, 2006.

P. V. Dong and H. N. Long, “The economical SU(3)C ⊗ SU(3)L ⊗ U(1)X model,” Advances in High Energy Physics, vol. 2008, Article ID 739492, 75 pages, 2008.

P. V. Dong, H. T. Hung, and H. N. Long, “Question of Peccei-Quinn symmetry and quark masses in the economical 3-3-1 model,” Physical Review D, vol. 86, no. 3, Article ID 033002, 13 pages, 2012.

P. V. Dong, L. T. Hue, H. N. Long, and D. V. Soa, “The 3-3-1 model with A ₄ flavor symmetry,” Physical Review D, vol. 81, no. 5, Article ID 053004, 2010.

P. V. Dong, H. N. Long, C. H. Nam, and V. V. Vien, “S ₃ flavor symmetry in 3-3-1 models,” Physical Review D, vol. 85, no. 5, Article ID 053001, 2012.

P. V. Dong, H. N. Long, D. V. Soa, and V. V. Vien, “The 3-3-1 model with S ₄ flavor symmetry,” The European Physical Journal C, vol. 71, no. 2, article 1544, 2011.

V. V. Vien and H. N. Long, “The D₄ flavor symmetry in 3-3-1 model with neutral leptons,” International Journal of Modern Physics A, vol. 28, no. 32, Article ID 1350159, 48 pages, 2013.

P. F. Harrison, D. H. Perkins, and W. G. Scott, “Tri-bimaximal mixing and the neutrino oscillation data,” Physics Letters B, vol. 530, no. 1–4, pp. 167–173, 2002.

Z.-Z. Xing, “Nearly tri-bimaximal neutrino mixing and CP violation,” Physics Letters B, vol. 533, no. 1-2, pp. 85–93, 2002.

X.-G. He and A. Zee, “Some simple mixing and mass matrices for neutrinos,” Physics Letters B, vol. 560, no. 1-2, pp. 87–90, 2003.

X.-G. He and A. Zee, “Neutrino masses with a “zero sum” condition: ,” Physical Review D, vol. 68, no. 3, Article ID 037302, 2003.

D. Chang and H. N. Long, “Interesting radiative patterns of neutrino mass in an SU(3)C ⊗ SU(3)L ⊗ U(1)X model with right-handed neutrinos,” Physical Review D, vol. 73, no. 5, Article ID 053006, 2006.

K. Nakamura, K. Hagiwara, K. Hikasa, et al., “Review of particle physics,” Journal of Physics G, vol. 37, Article ID 075021, 2010.

M. C. Gonzalez-Garcia, M. Maltonic, and J. Salvadod, “Updated global fit to three neutrino mixing: status of the hints of θ ₁₃ > 0,” Journal of High Energy Physics, vol. 2010, no. 4, article 056, 2010.

P. V. Dong and H. N. Long, “U(1)Q invariance and SU(3)C ⊗ SU(3)L ⊗ U(1)X models with β arbitrary,” The European Physical Journal C, vol. 42, no. 3, pp. 325–329, 2005.

P. V. Dong, V. T. N. Huyen, H. N. Long, and H. V. Thuy, “Gauge boson mixing in the 3-3-1 models with discrete symmetries,” Advances in High Energy Physics, vol. 2012, Article ID 715038, 18 pages, 2012.

D. V. Soa, P. V. Dong, T. T. Huong, and H. N. Long, “Bilepton contributions to the neutrinoless double beta decay in the economical 3-3-1 model,” Journal of Experimental and Theoretical Physics, vol. 108, no. 5, pp. 757–763, 2009.

J. C. Montero, C. A. D. S. Pires, and V. Pleitez, “Neutrinoless double beta decay with and without Majoron-like boson emission in a 3-3-1 model,” Physical Review D, vol. 64, no. 9, Article ID 096001, 2001.

A. G. Dias, A. Doff, C. A. S. Pires, and P. S. Rodrigues da Silva, “Neutrino decay and neutrinoless double beta decay in a 3-3-1 model,” Physical Review D, vol. 72, no. 3, Article ID 035006, 8 pages, 2005.