#### 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