A New S4 Flavor Symmetry in 3-3-1 Model with Neutral Fermions

Abstract

A new S4 flavor model based on SU(3[subscript])C[/subscript] [ecedil]7;SU(3[subscript])L[/subscript] [ecedil]7;U(1[subscript])X[/subscript] 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 S4. In this work, we assume the VEVs of the antisextet differ from each other under S4 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 S3 and our previous S4 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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

V. V. Vien 1 and H. N. Long 2

Academic Editor:Anastasios Petkou

1, Department of Physics, Tay Nguyen University, 567 Le Duan, Buon Ma Thuot, Vietnam
2, Hoang 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; 2 April 2014

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 .

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 [figure omitted; refer to PDF] 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 _A4 [14-29], _S3 [30-65], and _S4 [66-93].

An alternative to extend the standard model (SM) is the 3-3-1 models, in which the SM gauge group SU(2_)L [ecedil]7;U(1_)Y is extended to SU(3_)L [ecedil]7;U(1_)X 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 SU(3_)L . 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: [figure omitted; refer to PDF] where Q indicates the electric charge, _T3 and _T8 are two of the SU(3) generators, and X is the generator of U(1_)X . β 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 _S4 for 3-dimensional representations. In addition, due to the anomaly cancelation one family of quarks has to transform under SU(3_)L differently from the two others. _S4 can meet this requirement with the representations 1_ and 2_ .

There are two typical variants of the 3-3-1 models as far as lepton sectors are concerned. In the minimal version, three SU(3_)L lepton triplets are (_νL ,_lL ,^lRc ) , where _lR are ordinary right-handed charged leptons [94-98]. In the second version, the third components of lepton triplets are the right-handed neutrinos, (_νL ,_lL ,^νRc ) [99-105]. To have a model with the realistic neutrino mixing matrix, we should consider another variant of the form (_νL ,_lL ,^NRc ) where _NR 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 _S4 group, in which most of the Higgs multiplets are in triplets under _S4 except that χ is in a singlet, and the exact tribimaximal form [114-117] is obtained, in which _θ13 =0 . As we know, the recent considerations have implied _θ13 ...0;0 , but small as given in (1). This problem has been improved in [111] by adding a new triplet ρ and another antisextet ^{s[variant prime]} , in which ^{s[variant prime]} 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], _S4 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 2_ irreducible representation which can connect two maximally mixed generations. Besides the facilitating maximal mixing through 2_ , it provides two inequivalent singlet representations 1_ and ^{1_[variant prime]} 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 _S3 and our previous _S4 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 _S3 and _S4 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 _S4 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 _S4 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 SU(3_)C [ecedil]7;SU(3_)L [ecedil]7;U(1_)X , where the electroweak factor SU(3_)L [ecedil]7;U(1_)X 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 (_NR ) arranged under the SU(3_)L symmetry as a triplet (_νL ,_lL ,^NRc ) and a singlet _lR . The residual electric charge operator Q is therefore related to the generators of the gauge symmetry by [110-112] [figure omitted; refer to PDF] where _Ta (a=1,2,...,8) are SU(3_)L charges with Tr..._TaTb =(1/2)_δab and X is the U(1_)X 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 [Lagrangian (script capital L)] commuting with the gauge symmetry and related to the ordinary lepton number by diagonal matrices [110-112, 118] [figure omitted; refer to PDF] The lepton charge arranged in this way (i.e., L(_NR )=0 as assumed) is in order to prevent unwanted interactions due to U(1_{)[Lagrangian (script capital L)]} 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 U and D as well as new non-Hermitian gauge bosons ^X0 and ^Y± possess lepton charges as of the ordinary leptons: L(D)=-L(U)=L(^X0 )=L(^Y- )=1 . The lepton parity is introduced as follows: _Pl =(-^)L , which is a residual symmetry of L . The particles possess L=0 , ±2 such as _NR , ordinary quarks, and bileptons having _Pl =1 ; the particles with L=±1 such as ordinary leptons and exotic quarks having _Pl =-1 . Any nonzero VEV with odd parity, _Pl =-1 , will break this symmetry spontaneously [112]. For convenience in reading, the numbers L and _Pl of the component particles are given in Appendix C.

In this paper we work on a basis where 3_ and ^{3_[variant prime]} are real representations whereas the two-dimensional representation 2_ of _S4 is complex, ^2_* (^1* ,^2* )=2_(^2* ,^1* ) , and [figure omitted; refer to PDF]

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 3_ under _S4 , whereas in this model we put the first family of leptons in singlets 1_ of _S4 , while the two other families are in the doublets 2_ . In the quark content, the third family is put in a singlet 1_ and the two others in a doublet 2_ under _S4 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 [SU(3_)L ,U(1_)X ,U(1_{)[Lagrangian (script capital L)]} ,_{S_4} ] symmetries as proposed, the fermions of the model transform as follows: [figure omitted; refer to PDF] where the subscript numbers on field indicate respective families in order to define components of their _S4 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 _S4 . To generate masses for the charged leptons, we have introduced two SU(3_)L scalar triplets [varphi] and ^{[varphi][variant prime]} lying in 3_ and ^{3_[variant prime]} under _S4 , respectively, with VEVs Y9;[varphi]YA;=(v v v^)T and Y9;^{[varphi][variant prime]} YA;=(^{v[variant prime] v[variant prime] v[variant prime])T} . From the invariant Yukawa interactions for the charged leptons, we obtain the right-handed charged leptons mixing matrices which are diagonal ones, _UlR =1 , and the right-handed one given by [112] [figure omitted; refer to PDF]

Similar to the charged lepton sector, to generate the quark masses, we have additionally introduced the three scalar Higgs triplets χ , η , ^{η[variant prime]} lying in 1_ , 3_ , and ^{3_[variant prime]} under _S4 , respectively. Quark masses can be derived from the invariant Yukawa interactions for quarks with supposing that the VEVs of η , ^{η[variant prime]} , and χ are (u,u,u) , (^{u[variant prime]} ,^{u[variant prime]} ,^{u[variant prime]} ) , and w , where u=Y9;^η10 YA; , ^{u[variant prime]} =Y9;^{η1[variant prime]0} YA; , and w=Y9;^χ30 YA; . The other VEVs Y9;^η30 YA; , Y9;^{η3[variant prime]0} YA; , and Y9;^χ10 YA; vanish if the lepton parity is conserved. In addition, the VEV w also breaks the 3-3-1 gauge symmetry down to that of the standard model and provides the masses for the exotic quarks U and D as well as the new gauge bosons. The u , ^{u[variant prime]} as well as v , ^{v[variant prime]} 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 w is much larger than those of [varphi] and η [112]. The unitary matrices which couple the left-handed quarks _uL and _dL with those in the mass bases are unit ones (^ULu =1 , ^ULd =1 ), and the CKM quark mixing matrix at the tree level is then _UCKM =^UdL[dagger]_UuL =1 . 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 SU(3_)L antisextet lying in 1_ under _S4 and one SU(3_)L antisextet lying in 3_ under _S4 with the VEV of s being set as (Y9;_s1 YA;,0,0) under _S4 . The neutrino masses are explicitly separated and the lepton mixing matrix yields the exact tribimaximal form [112] where _θ13 =0 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: [figure omitted; refer to PDF] with VEVs Y9;[varphi]YA;=(0,v,0^)T and Y9;^{[varphi][variant prime]} YA;=(0,^{v[variant prime]} ,0^)T .

The Yukawa interactions are [figure omitted; refer to PDF]

The mass Lagrangian of the charged leptons reads [figure omitted; refer to PDF] It is then diagonalized, and [figure omitted; refer to PDF] This means that the charged leptons _l1,2,3 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 ^{[varphi][variant prime]} triplet. This is the reason why we introduce ^{[varphi][variant prime]} in addition to [varphi] .

The charged lepton Yukawa couplings _h1,2,3 relate to their masses as follows: [figure omitted; refer to PDF] The current mass values for the charged leptons at the weak scale are given by [7] [figure omitted; refer to PDF] Thus, we get [figure omitted; refer to PDF] It follows that if ^{v[variant prime]} and v are of the same order of magnitude, _h1 ...a;_h2 and _h2 ~_h3 . This result is similar to the case of the model based on _S3 group [111]. On the other hand, if we choose the VEV of [varphi] as v=100 GeV , then _h1 ~5 × 1^0-6 , _h3 ~_h2 ~1^0-4 .

4. Quark Mass

To generate the quark masses with a minimal Higgs content, we additionally introduce the following scalar multiplets: [figure omitted; refer to PDF] It is noticed that these scalars do not couple with the lepton sector due to the gauge invariance. The Yukawa interactions are then [figure omitted; refer to PDF]

Suppose that the VEVs of η , ^{η[variant prime]} , and χ are u , ^{u[variant prime]} , and w , where u=Y9;^η10 YA; , ^{u[variant prime]} =Y9;^{η1[variant prime]0} YA; , and w=Y9;^χ30 YA; . The other VEVs Y9;^η30 YA; , Y9;^{η3[variant prime]0} YA; , and Y9;^χ10 YA; vanish due to the lepton parity conservation [111]. The exotic quarks therefore get masses _mU =_f3 w and _mD1,2 =fw . In addition, w has to be much larger than those of [varphi] , ^{[varphi][variant prime]} , η , and ^{η[variant prime]} for a consistency with the effective theory. The mass matrices for ordinary up-quarks and down-quarks are, respectively, obtained as follows: [figure omitted; refer to PDF] Similar to the charged leptons, the masses of u-c and d-s quarks are in pair separated by the scalars ^{[varphi][variant prime]} and ^{η[variant prime]} , respectively. We see also that the introduction of ^{η[variant prime]} in addition to η is necessary to provide the different masses for u and c quarks as well as for d and s quarks.

The expressions (17) yield the relations: [figure omitted; refer to PDF] The current mass values for the quarks are given by [7] [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF] It is obvious that if u~v~^{v[variant prime]} ~^{u[variant prime]} , the Yukawa coupling hierarchies are ^hu ~^{h[variant prime]u} ...a;^h3u , ^hd ~^{h[variant prime]d} ...a;^h3d , and the couplings between up-quarks (^hu ,^{h[variant prime]u} ,^h3u ) and Higgs scalar multiplets are slightly heavier than those of down-quarks (^hd ,^{h[variant prime]d} ,^h3d ) , respectively.

The unitary matrices, which couple the left-handed up- and down-quarks with those in the mass bases, are ^ULu =1 and ^ULd =1 , respectively. Therefore we get the CKM matrix [figure omitted; refer to PDF] 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 Y9;^η30 YA; , Y9;^{η3[variant prime]0} YA; , and Y9;^χ10 YA; or a violation of [Lagrangian (script capital L)] and/nor _S4 symmetry due to unnormal Yukawa interactions, namely, _Q-3L χ_u3R , _Q-iL^χ*_diR , _Q-3L χ_uiR , _Q-iL^χ*_d3R , 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 ^ψ-αLc_ψαL , ^ψ-1Lc_ψ1L , and ^ψ-1c_ψαL to scalars, where ^ψ-αLc_ψαL transforms as ^3* [ecedil]5;6 under SU(3_)L and 1_[ecedil]5;2_[ecedil]5;3_[ecedil]5;^{3_[variant prime]} under _S4 , ^ψ-1Lc_ψ1L transforms as ^3* [ecedil]5;6 under SU(3_)L and 1_ under _S4 , and ^ψ-1Lc_ψαL transforms as ^3* [ecedil]5;6 under SU(3_)L and 2_ under _S4 . For the known scalar triplets ([varphi],^{[varphi][variant prime]} ,χ,η,^{η[variant prime]} ) , only available interactions are (^ψ-αLc_ψαL )[varphi] and (^ψ-αLc_ψαL )^{[varphi][variant prime]} but explicitly suppressed because of the [Lagrangian (script capital L)] -symmetry. We will therefore propose new _SU(3)L antisextets instead of coupling to ^ψ-Lc_ψL responsible for the neutrino masses which are lying in either 1_ , 2_ , 3_ , or ^{3_[variant prime]} under _S4 . In [112], we have introduced two SU(3_)L antisextets σ , s which are lying in 1_ and 3_ under _S4 , 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: [figure omitted; refer to PDF] where the numbered subscripts on the component scalars are the SU(3_)L indices, whereas i=1,2 is that of _S4 . The VEV of s is set as (Y9;_s1 YA;,Y9;_s2 YA;) under _S4 , in which [figure omitted; refer to PDF] Following the potential minimization conditions, we have several VEV alignments. The first is that Y9;_s1 YA;=Y9;_s2 YA; and then _S4 is broken into an eight-element subgroup, which is isomorphic to _D4 . The second is that Y9;_s1 YA;...0;0=Y9;_s2 YA; or Y9;_s1 YA;=0...0;Y9;_s2 YA; and then _S4 is broken into _A4 consisting of the identity and the even permutations of four objects. The third is that Y9;_s1 YA;...0;Y9;_s2 YA;...0;0 and then _S4 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 _K4 ). To obtain a realistic neutrino spectrum, we argue that both the breakings _S4 [arrow right]_D4 and _S4 [arrow right]_K4 must take place. We therefore assume that its VEVs are aligned as the former to derive the direction of the breaking _S4 [arrow right]_D4 , and this happens in any case bellow: [figure omitted; refer to PDF] The direction of the breaking _S4 [arrow right]_K4 is equivalent to the breaking _D4 [arrow right]{Identity} . In this direction, we set Y9;_s1 YA;=Y9;sYA;...0;Y9;_s2 YA;...0;0 . If _D4 is unbroken, we have Y9;_s1 YA;=Y9;_s2 YA;=Y9;sYA; as in (24), and on the contrary, if _D4 is unbroken, we have Y9;sYA;=Y9;_s2 YA;[approximate]Y9;_s1 YA; : [figure omitted; refer to PDF] The difference between Y9;_s1 YA; and Y9;_s2 YA; 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 _S3 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 _θ13 ...0;0 , while the old version [112] based on _S4 cannot provide nonvanishing _θ13 .

In general, the Yukawa interactions are [figure omitted; refer to PDF] With the alignments of VEVs as in (24) and (25), the mass Lagrangian for the neutrinos is determined by [figure omitted; refer to PDF] where ν=(_ν1 ,_ν2 ,_ν3^)T and N=(_N1 ,_N2 ,_N3^)T . The mass matrices are then obtained by [figure omitted; refer to PDF] with [figure omitted; refer to PDF] The VEVs _Λ1,2 break the 3-3-1 gauge symmetry down to that of the SM and provide the masses for the neutral fermions _NR and the new gauge bosons: the neutral ^{Z[variant prime]} and the charged ^Y± and ^X0,0* . The _λ1,2 and _v1,2 belong to the second stage of the symmetry breaking from the SM down to the SU(3_)C [ecedil]7;U(1_)Q symmetry and contribute the masses to the neutrinos. Hence, to keep a consistency we assume that _Λ1,s ...b;_v1,s ...b;_λ1,s [105].

Three active neutrinos therefore gain masses via a combination of type I and type II seesaw mechanisms derived from (27) and (28) as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] The following comments of _S4 breaking are in order.

(i) If _S4 is broken into _D4 (_D4 is unbroken), we have A=D=0 , _B1 =_B2 =B , and _C1 =_C2 =C , which is presented in Section 5.1.

(ii) If _S4 is broken into _K4 (_D4 is broken into {Identity}), we have A[approximate]0 , _B1 [approximate]_B2 , _C1 [approximate]_C2 , and D...0;0 but it is very small. In this case the disparity of two VEVs of Y9;sYA; 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 _S4 [arrow right]_D4 , and the second one is _S4 [arrow right]_K4 .

5.1. Experimental Constraints in the Case _S4 [arrow right]_D4

If _S4 is broken into _D4 , _λ1 =_λ2 ...1;_λs , _v1 =_v2 ...1;_vs , _Λ1 =_Λ2 ...1;_Λs , we have A=0 , _B1 =_B2 ...1;B , _C1 =_C2 ...1;C , and D=0 , and _Meff reduces to [figure omitted; refer to PDF] where [figure omitted; refer to PDF] We can diagonalize the matrix _Meff in (32) as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and the neutrino mixing matrix takes the form: [figure omitted; refer to PDF] Note that _m1m2 =-2^B2 . This matrix can be parameterized in three Euler's angles, which implies [figure omitted; refer to PDF] This case coincides with the data since ^sin2 (2_θ13 )<0.15 and ^sin2 (2_θ23 )>0.92 [119, 120]. For the remaining constraints, taking the central values from the data in [119] [figure omitted; refer to PDF] and we have a solution [figure omitted; refer to PDF] and B=-0.0202757i eV , C=0.0573631 eV , K=1.44667 , and |x/y|=0.707087 . It follows that tan_θ12 =0.977565 , (_θ12 ≈44.3⁵⁰ ) , and the neutrino mixing matrix form is very close to that of bimaximal mixing which takes the form: [figure omitted; refer to PDF] Now, it is natural to choose _λs , ^vs2 /_Λs in eV order, and suppose that _λs >^vs2 /_Λs . Let us assume _λs -^vs2 /_Λs =0.1 , and we have then x=0.399403i and y=-0.573631 .

This result is not obviously consistent with the recent data on neutrinos oscillation in which _θ13 ...0;0 , 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 _S4 [arrow right]_K4 takes place. This means that, for the model under consideration, both the breakings _S4 [arrow right]_D4 and _S4 [arrow right]_K4 (instead of _D4 [arrow right]{Identity} ) must take place in the neutrino sector.

5.2. Experimental Constraints in the Case _S4 [arrow right]_K4

In this case _S4 is broken into the Klein four group _K4 , _λ1 ...0;_λ2 , _v1 ...0;_v2 , and _Λ1 ...0;_Λ2 , and the direct consequence is A[approximate]0 , _B1 [approximate]_B2 , _C1 [approximate]_C2 , and D...0;0 . The general neutrino mass matrix in (30) can be rewritten in the form: [figure omitted; refer to PDF] where B and C are given by (33), accommodated in the first matrix, which is matched to the case of _S4 [arrow right]_D4 . The three last matrices in (41) are a deviation from the contribution due to the disparity of Y9;_s1 YA; and Y9;_s2 YA; , namely, A=^a2 r , _B1 -B=ap , _B2 -B=aq , q=_C1 -C , p=_C2 -C , and r=D , with the A , _B1,2 , _C1,2 , and D being defined in (31), which correspond to _S4 [arrow right]_K4 .

Substituting (29) into (31) we get [figure omitted; refer to PDF] Indeed, if _S4 [arrow right]_D4 , the deviations p , q , r will vanish, therefore the mass matrix _Meff in (30) reduces to its first term coinciding with (32). The first term of (41) provides bimaximal mixing pattern, in which _θ13 =0 as shown in Section 5.1. The other matrices proportional to p , q , r due to contribution from the disparity of Y9;_s1 YA; and Y9;_s2 YA; will take the role of perturbation for such a deviation of _θ13 . So, in this work we consider the disparity of Y9;_s1 YA; and Y9;_s2 YA; as a small perturbation and terminating the theory at the first order.

Without loss of generality, we consider the case of breaking _S4 [arrow right]_K4 , in which _λ1 ...0;_λs whereas _v1 =_vs and _Λ1 =_Λs . It is then p=r=0 , q=(_λ1 -_λs )y...1;...y with ...=_λ1 -_λs being a small parameter. In this case, the matrix _Meff in (41) reduces to [figure omitted; refer to PDF] At the first order of perturbation, the physical neutrino masses are obtained as [figure omitted; refer to PDF] where _m1,2,3 are the mass values as of the case _S4 [arrow right]_D4 given by (39). For the corresponding perturbed eigenstates, we put [figure omitted; refer to PDF] where U is defined by (36), and [figure omitted; refer to PDF] with [figure omitted; refer to PDF] The lepton mixing matrix in this case ^{U[variant prime]} can still be parameterized in three new Euler's angles ^{θij[variant prime]} , which are also a perturbation from the _θij in the case 1, defined by [figure omitted; refer to PDF] 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 x , y of the antisextet scalar s with the above mentioned VEVs. Indeed, taking the data in (1) we obtain ...≈0.0692 , x≈0.0728 , y≈-0.1562 , and B≈-0.0241 eV and C=0.022 eV , K=1.943 , and ^{t23[variant prime]} =0.9045 [^{θ23[variant prime]} ≈42.1^3o ,^sin2 (2^{θ23[variant prime]} )=0.98999 satisfying the condition ^sin2 (2^{θ23[variant prime]} )>0.95 ]. The neutrino masses are explicitly given as ^{m1[variant prime]} ≈-0.02737 eV , ^{m2[variant prime]} ≈-0.02870 eV , and ^{m3[variant prime]} ≈-0.05607 eV . The neutrino mixing matrix then takes the form: [figure omitted; refer to PDF]

6. Gauge Bosons

The covariant derivative of a triplet is given by [figure omitted; refer to PDF] where _λa (a=1,2,...,8) are Gell-Mann matrices, _λ9 =2/3diag...(1,1,1) , Tr..._λaλb =2_δab , Tr..._λ9λ9 =2 , and X is X -charge of Higgs triplets.

Let us denote the following combinations: [figure omitted; refer to PDF] and then _Pμ is rewritten in a convenient form as follows: [figure omitted; refer to PDF] with t=_gX /g . We note that _W4 and _W5 are pure real and imaginary parts of ^X0 and ^X0* , respectively. The covariant derivative for an antisextet with the VEV part is [121] [figure omitted; refer to PDF] The covariant derivative (53) acting on the antisextet VEVs is given by [figure omitted; refer to PDF] The masses of gauge bosons in this model are defined as follows: [figure omitted; refer to PDF] where ^{[Lagrangian (script capital L)]massGB} in (55) is different from the one in [122] by the difference of the components of the antisextet s . In [122], Y9;_s1 YA;=Y9;_s1 YA; , namely, _λ1 =_λ2 =_λs , _v1 =_v2 =_vs , and _Λ1 =_Λ2 =_Λs , 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 _λ1 ...0;_λ2 , _v1 ...0;_v2 , and _Λ1 ...0;_Λ2 . 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 [figure omitted; refer to PDF] We can separate ^{[Lagrangian (script capital L)]massGB} in (57) into [figure omitted; refer to PDF] where ^{[Lagrangian (script capital L)]mass_W5} is the Lagrangian part of the imaginary part _W5 . This boson is decoupled with mass given by [figure omitted; refer to PDF] In the limit _λ1 ,_λ2 ,_v1 ,_v2 [arrow right]0 we have [figure omitted; refer to PDF] ^{[Lagrangian (script capital L)]mixCGB} is the Lagrangian part of the charged gauge bosons W and Y : [figure omitted; refer to PDF] ^{[Lagrangian (script capital L)]mixCGB} in (60) can be rewritten in matrix form as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] The matrix ^MWY2 in (62) can be diagonalized as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] with [figure omitted; refer to PDF] With corresponding eigenstates, the charged gauge boson mixing matrix takes the form: [figure omitted; refer to PDF] The mixing angle θ is given by [figure omitted; refer to PDF] The physical charged gauge bosons are defined [figure omitted; refer to PDF] In our model, the following limit is often taken into account: [figure omitted; refer to PDF] With the help of (69), the Γ in (65) becomes [figure omitted; refer to PDF] It is then [figure omitted; refer to PDF] with [figure omitted; refer to PDF] In the limit _v1,2 [arrow right]0 the mixing angle θ tends to zero, Γ=2^Λ12 +2^Λ22 +^ω2 -^u2 -^{u[variant prime]2} , and one has [figure omitted; refer to PDF] With the help of (69), one can estimate [figure omitted; refer to PDF] In addition, from (73), it follows that ^MW2 is much smaller than ^MY2 . Note that, due to the above mixing, the new gauge boson Y will give a contribution to neutrinoless double beta decay (for details, see [123-125]).

^{[Lagrangian (script capital L)] mix NGB} is the Lagrangian that describes the mixing among the neutral gauge bosons _W3 , _W8 , B , _W4 . The mass Lagrangian in this case has the form [figure omitted; refer to PDF]

On the basis of (_Wμ3 ,_Wμ8 ,_Bμ ,_Wμ4 ) , the ^{[Lagrangian (script capital L)]mixNGB} in (75) can be rewritten in matrix form: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] The matrix ^M2 in (76) with elements in (77) has one exact eigenvalue, which is identified with the photon mass: [figure omitted; refer to PDF] The corresponding eigenvector of ^Mγ2 is [figure omitted; refer to PDF]

Note that in the limit _λ1,2 ,_v1,2 [arrow right]0 , ^M142 =^M242 =^M342 =0 , and _W4 does not mix with _W3μ , _W8μ , _Bμ . In the general case _λ1,2 ,_v1,2 ...0;0 , the mass matrix in (76) contains one exact eigenvalues as in (78) with the corresponding eigenstate given in (79).

The mass matrix ^M2 in (76) is diagonalized via two steps. In the first step, the basic (_Wμ3 ,_Wμ8 ,^{Bμ[variant prime]} ,_W4μ ) is transformed into the basic (_Aμ ,_Zμ ,^{Zμ[variant prime]} ,_W4μ ) by the matrix: [figure omitted; refer to PDF] The corresponding eigenstates are given by [figure omitted; refer to PDF] To obtain (80) and (81) we have used the continuation of the gauge coupling constant g of the SU(3_)L at the spontaneous symmetry breaking point, in which [figure omitted; refer to PDF] On this basis, the mass matrix ^M2 becomes [figure omitted; refer to PDF] where [figure omitted; refer to PDF] In the approximation ^λ1,22 ,^v1,22 ...a;^Λ1,22 ~^ω2 , we have [figure omitted; refer to PDF] with [figure omitted; refer to PDF] From (83), there exist mixings between _Zμ , ^{Zμ[variant prime]} and _Wμ4 . It is noteworthy that, in the limit _v1,2 =0 , the elements ^{M24[variant prime]2} and ^{M34[variant prime]2} vanish. In this case there is no mixing between _W4 and _Zμ , ^{Zμ[variant prime]} .

In the second step, three bosons gain masses via seesaw mechanism [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Combination of (87), (88), and (85) yields [figure omitted; refer to PDF] where [figure omitted; refer to PDF] with [figure omitted; refer to PDF] The ρ parameter in our model is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Let us assume the relations (A.17) and put _v2 ...1;_vs , ω=_Λ2 ...1;_Λs , and then [figure omitted; refer to PDF] From (92)-(94) we have [figure omitted; refer to PDF] The experimental value of the ρ parameter and _MW are, respectively, given in [7] [figure omitted; refer to PDF] It means [figure omitted; refer to PDF] From (95) one can make the relations between v , g , and k . Indeed, we have [figure omitted; refer to PDF] Figure 1 gives the relation between _vs and g , k provided that g=0.5 , and k∈(0.9,1.1) in which |_vs |∈(0,8.0) Gev .

Figure 1: The relation between _vs and g , k with g=0.5 and k∈(0.9,1.1) .

[figure omitted; refer to PDF]

Figure 2 gives the relation between g and _δtree , _vs provided that k=1 and _δtree ∈(0,0.0007) , _vs ∈(0,8.0) GeV in which |g|∈(0,2.0) GeV . The conditions (69) are satisfied. The Figure 3 gives the relation between k and g , _vs provided _δtree =0.0005 and g∈(0.4,0.6) , _vs ∈(0,8.0) GeV in which k∈(1,3) GeV (k is a real number, Figure 3(a)) or k=i_k1 , _k1 ∈(-1.2,-1.05) GeV (k is a pure complex number, Figure 3(b)). The conditions (69) are satisfied. From Figure 3 we see that a lot of values of k that is different from the unit but nearly it still can fit the recent experimental data [7]. It means that the difference of Y9;_s1 YA; and Y9;_s2 YA; as mentioned in this work is necessary.

Figure 2: The relation between g and _δtree , _vs with k=1 and _δtree ∈(0,0.0007) , _vs ∈(0,8.0) GeV .

[figure omitted; refer to PDF]

The relation between k and g , _vs provided that _δtree =0.0005 and g∈(0.4,0.6) , _vs ∈(0,8.0) GeV .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Diagonalizing the mass matrix ^{M2×2[variant prime]2} , we get two new physical gauge bosons [figure omitted; refer to PDF]

With the approximation as in (69), the mixing angle [varphi] is given by [figure omitted; refer to PDF] provided that _v1 ~_v2 , _Λ1 ~_Λ2 .

In the limit _λ1,2 ,_v1,2 [arrow right]0 the mixing angle [varphi] tends to zero, and the physical mass eigenvalues are defined by [figure omitted; refer to PDF] From (59) and (101) we see that the ^{Wμ4[variant prime]} and _W5 components have the same mass in the limit _λ1,2 ,_v1,2 [arrow right]0 . So we should identify the combination of ^{Wμ4[variant prime]} and _Wμ5 [figure omitted; refer to PDF] as physical neutral non-Hermitian gauge boson. The subscript "0" denotes neutrality of gauge boson X . Notice that the identification in (102) only can be acceptable with the limit _λ1,2 ,_v1,2 [arrow right]0 . In general, it is not true because of the difference in masses of ^{Wμ4[variant prime]} and _Wμ5 as in (58) and (99).

The expressions (74) and (100) show that, with the limit (69), the mixings between the charged gauge bosons W-Y and the neutral ones ^{Z[variant prime]} -_W4 are in the same order since they are proportional to _vi /_Λi (i=1,2 ). In addition, from (101), ^{MZμ[variant prime][variant prime] 2} ≈^g2 (4^Λ12 +4^Λ22 +^ω2 ) is little bigger than ^{MWμ4[variant prime] 2} ≈(^g2 /2)(^ω2 +2^Λ12 +2^Λ22 ) (or ^{MXμ0 2} ), and |^MY2 -^{MXμ0 2} |=(^g2 /2)(^u2 +^{u[variant prime]2} -^v2 -^{v[variant prime]2} ) is little smaller than ^MW2 =(^g2 /2)(^u2 +^{u[variant prime]2} +^v2 +^{v[variant prime]2} ) . In that limit, the masses of ^Xμ0 and Y degenerate.

7. Conclusions

In this paper, we have constructed a new _S4 model based on SU(3_)C [ecedil]7;SU(3_)L [ecedil]7;U(1_)X 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 _S4 . We argue how flavor mixing patterns and mass splitting are obtained with a perturbed _S4 symmetry by the difference of VEV components of the antisextet under _S4 . We have pointed out that this model is simpler than those of _S3 and _S4 [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 _θ13 ...0;0 can be obtained if the direction for breaking _S4 [arrow right]_K4 takes place. This corresponds to the requirement on the difference of VEV components of the antisextet under _S4 group. As a result, the value of _θ13 is a small perturbation by |_λ1 -_λ2 | . 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.

Acknowledgment

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

Conflict of Interests

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

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Appendix

Appendices

A. Vacuum Alignment

We can separate the general scalar potential into [figure omitted; refer to PDF] where _Vtri and _Vsext , respectively, consist of the SU(3_)L scalar triplets and sextets, whereas _Vtri-sext contains the terms connecting the two sectors. Moreover _{Vtri,sext,tri-sext} conserve [Lagrangian (script capital L)] -charge and _S4 symmetry, while V¯ 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 V- will be shown below.

The details on the potentials are given as follows. We first denote V(X[arrow right]_X1 ,Y[arrow right]_Y1 ,...)...1;V(X,Y,...)_{|X=X1 ,Y=Y1 ,...} Notice also that (Tr...A)(Tr...B)=Tr...(ATr...B) . _Vtri is a sum of [figure omitted; refer to PDF] _Vsext is only of V(s) , [figure omitted; refer to PDF] and _Vtri-sext is a sum of [figure omitted; refer to PDF] To provide the Majorana masses for the neutrinos, the lepton number must be broken. This can be achieved via the scalar potential violating U(1_{)[Lagrangian (script capital L)]} . However, the other symmetries should be conserved. The violating [Lagrangian (script capital L)] potential up to quartic interactions is given as [figure omitted; refer to PDF] We have not explicitly written, but there must additionally exist the terms in V- explicitly violating the only _S4 symmetry or both the _S4 and [Lagrangian (script capital L)] -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, 0...0;Y9;_s1 YA;...0;Y9;_s2 YA;...0;0 , _S4 , is broken into a subgroup including the elements {1,T^S2T2 ,^S2 ,^T2S2 T} which is isomorphic to the Klein four-group [75] [S=(1234) , T=(123) , obeying the relations ^S4 =^T3 =1 , S^T2 S=T , are generators of _S4 group given in [112]]. The second direction, Y9;_s1 YA;=Y9;_s2 YA;=Y9;sYA;...0;0 , _S4 , is broken into _D4 . The third direction, 0=Y9;_s1 YA;...0;Y9;_s2 YA; , or 0=Y9;_s2 YA;...0;Y9;_s1 YA; , _S4 , is broken into _A4 . 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 _Vtri . The flavons χ , [varphi] , ^{[varphi][variant prime]} , η , ^{η[variant prime]} with their VEVs aligned in the same direction (all of them are singlets) are an automatic solution from the minimization conditions of _Vtri . To explicitly see this, in the system of equations for minimization, let us put ^v* =v , ^{v[variant prime]*} =^{v[variant prime]} , ^u* =u , ^{u[variant prime]*} =^{u[variant prime]} , and ^vχ* =_vχ . Then the potential minimization conditions for triplets reduce to [figure omitted; refer to PDF] It is easily shown that the derivatives of _Vtri with respect to the variables u , ^{u[variant prime]} , v , ^{v[variant prime]} 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 (u , v , ^{u[variant prime]} , ^{v[variant prime]} ) 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 u=^{u[variant prime]} =v=^{v[variant prime]} . In this case, (A.7)-(A.10) reduce a unique equation, and system of (A.6)-(A.10) becomes [figure omitted; refer to PDF] This system has a solution as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Considering the potentials _Vsex and _Vtri-sex , we impose that [figure omitted; refer to PDF] and we obtain a system of equations of the potential minimization for antisextets: [figure omitted; refer to PDF] where _V1 is a sum of _Vsext and _Vtri-sext : [figure omitted; refer to PDF] It is easily shown that (A.15) takes the same form in couples. This system of equations yields the following relations: [figure omitted; refer to PDF] where κ 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 _S4 [arrow right]_D4 and _S4 [arrow right]_K4 as mentioned, in which κ=1 and κ...0;1 but approximates to the unit. In the case that κ=1 or _λ1 =_λ2 =_λs , _v1 =_v2 =_vs , and _Λ1 =_Λ2 =_Λs , the system of (A.15) reduces to system for minimal potential condition consisting of three equations as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] The system of (A.18) always has the solution (_λs , _vs , _Λs ) 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. _S4 Group and Clebsch-Gordan Coefficients

_{S 4} 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 1_ , ^{1_[variant prime]} , 2_ , 3_ , and ^{3_[variant prime]} as its 5 irreducible representations. Any element of _S4 can be formed by multiplication of the generators S and T obeying the relations ^S4 =^T3 =1 , S^T2 S=T . Without loss of generality, we could choose S=(1234) , T=(123) where the cycle (1234) denotes the permutation (1,2,3,4)[arrow right](2,3,4,1) , and (123) means (1,2,3,4)[arrow right](2,3,1,4) . The conjugacy classes generated from S and T are [figure omitted; refer to PDF]

The character table of _S4 is given as shown in Table 1, where n is the order of class and h is the order of elements within each class. Let us note that _C1,2,3 are even permutations, while _C4,5 are odd permutations. The two three-dimensional representations differ only in the signs of their _C4 and _C5 matrices. Similarly, the two one-dimensional representations behave the same.

Table 1: Class   n    h    _{χ 1 _    χ ^{1 _ [variant prime]}    χ 2 _    χ 3 _    χ ^{3 _ [variant prime]}}   

_{C 1}   1  1  1  1  2  3  3  

_{C 2}   3  2  1  1  2  -1  -1  

_{C 3}   8  3  1  1  -1  0  0  

_{C 4}   6  4  1  -1  0  -1  1  

_{C 5}   6  2  1  -1  0  1  -1  

We will work on a basis where 3_ and ^{3_[variant prime]} are real representations whereas 2_ is complex. One possible choice of generators is given as follows: [figure omitted; refer to PDF] where ω=^e2πi/3 =-1/2+i3/2 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 3_(1,2,3) which means some 3_ multiplet such as x=(_x1 ,_x2 ,_x3 )~3_ or y=(_y1 ,_y2 ,_y3 )~3_ , and similarly for the other representations. Moreover, the numbered multiplets such as (...,ij,...) mean (...,_xiyj ,...) where _xi and _yj are the multiplet components of different representations x and y , 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.: [figure omitted; refer to PDF] where the subscripts s and a , 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 _S3 and _A4 groups [14, 112].

In the text we usually use the following notations, for example, (x^{y[variant prime]}_{)3_} =[x^{y[variant prime]}_{]3_} ...1;(_x2^{y3[variant prime]} -_x3^{y2[variant prime]} ,_x3^{y1[variant prime]} -_x1^{y3[variant prime]} ,_x1^{y2[variant prime]} -_x2^{y1[variant prime]} ) which is the Clebsch-Gordan coefficients of _{3_a} in the decomposition of 3_[ecedil]7;^{3_[variant prime]} , whereas mentioned x=(_x1 ,_x2 ,_x3 )~3_ and ^{y[variant prime]} =(^{y1[variant prime]} ,^{y2[variant prime]} ,^{y3[variant prime]} )~^{3_[variant prime]} .

The rules to conjugate the representations 1, ^{1_[variant prime]} , 2, 3, and ^{3_[variant prime]} are given by [figure omitted; refer to PDF] where, for example, ^2_* (^1* ,^2* ) denotes some ^2_* multiplet of the form (^x1* ,^x2* )~^2_* .

C. The Numbers

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

Table 2: Particles   L    _{P l}   

_{N R} , u , d , ^[varphi]1+ , ^{[varphi]1[variant prime]+} , ^[varphi]20 , ^{[varphi]2[variant prime]0} , ^η10 , ^{η1[variant prime]0} , ^η2- , ^{η2[variant prime]-} , ^χ30 , ^σ330 , ^s330   0  1  

_{ν L} , l , U , ^D* , ^[varphi]3+ , ^{[varphi]3[variant prime]+} , ^η30 , ^{η3[variant prime]0} , ^χ10* , ^χ2+ , ^σ130 , ^σ23+ , ^s130 , ^s23+   -1  -1  

^{σ 11 0} , ^σ12+ , ^σ22++ , ^s110 , ^s12+ , ^s22++   -2  1