$S_3$ discrete group as a source of the quark mass and mixing pattern in $331$ models

We propose a model based on the $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}$ gauge symmetry with an extra $S_{3}\otimes Z_{2}\otimes Z_{4}\otimes Z_{12}$ discrete group, which successfully accounts for the SM quark mass and mixing pattern. The observed hierarchy of the SM quark masses and quark mixing matrix elements arises from the $Z_{4}$ and $Z_{12}$ symmetries, which are broken at very high scale by the $SU(3)_{L}$ scalar singlets ($\sigma$,$\zeta$) and $\tau $, charged under these symmetries, respectively. The Cabbibo mixing arises from the down type quark sector whereas the up quark sector generates the remaining quark mixing angles. The obtained magnitudes of the CKM matrix elements, the CP violating phase and the Jarlskog invariant are in agreement with the experimental data.


Introduction
The ATLAS and CMS experiments at the CERN Large Hadron Collider (LHC) have found signals consistent with a ∼ 126 GeV Higgs boson [1,2,3,4], increasing our knowledge of the Electroweak Symmetry Breaking (EWSB) sector and opening a new era in particle physics. This discovery offers the possibility to unveil the mechanism of Electroweak Symmetry Breaking (EWSB). Now the priority of the LHC experiments will be to measure precisely the couplings of the new particle to standard model fermions and gauge bosons. It also remains to look for further new states associated with the EWSB mechanism which will allow to discriminate among the different theoretical models.
One of the outstanding unresolved issues in Particle Physics is the origin of the masses of fundamental fermions. The current theory of strong and electroweak interactions, the Standard Model (SM), has proven to be remarkably successful in passing all experimental tests. Despite its great success, the Standard Model (SM) based on the SU (3) C ⊗ SU (2) L ⊗ U (1) Y gauge symmetry is unlikely to be a truly fundamental theory due to unexplained features [5,6]. Most of them are linked to the mechanism responsible for the stabilization of the weak scale, the origin of fermion masses and mixing and the existence of three generations of fermions. Because of these reasons, many people consider the Standard Model to be an effective framework of a more fundamental theory that one expects should have a dynamical explanation for the fermion masses and mixing. The lack of predictivity of the fermion masses and mixing in the SM has motivated many models based on extended symmetries, leading to specific textures for the Yukawa couplings. There are models with Multi-Higgs sectors, Grand Unification, Extradimensions and Superstrings, [7,8,9,10]. In particular, discrete flavour symmetries may play an important role in models of fermion mixing and many models based on flavour symmetries have been proposed in order to provide an explanation for the current pattern of fermion mixing, for recent reviews see Refs. [7,11,12]. One clear and outstanding feature in the pattern of quark masses is that they increase from one generation to the next spreading over a range of five orders of magnitude [6,13,14]. From the phenomenological point of view, it is possible to describe some features of the mass hierarchy by assuming zero-texture Yukawa matrices [15]. Models with spontaneously broken flavor symmetries may also produce hierarchical mass structures. These horizontal symmetries can be continuous and Abelian, as the original Froggatt-Nielsen model [16], or non-Abelian as for example SU (3) and SO(3) family models [17]. Recently, discrete groups such as A 4 , S 3 , S 4 , A 5 , ∆ (27) and T have been considered to explain the observed pattern of fermion masses and mixing [18,19,20,21,22]. Other models with horizontal symmetries have been proposed in the literature [23].
On the other hand, the origin of the structure of fermions can be addressed in family dependent models.
Alternatively, an explanation to this issue can also be provided by the models based on the gauge symmetry SU (3) c ⊗ SU (3) L ⊗ U (1) X , also called 3-3-1 models, which introduce a family non-universal U (1) X symmetry [24,25,26,27]. These models have a number of phenomenological advantages. First of all, the three family structure in the fermion sector can be understood in the 3-3-1 models from the cancellation of chiral anomalies [28] and asymptotic freedom in QCD. Secondly, the fact that the third family is treated under a different representation, can explain the large mass difference between the heaviest quark family and the two lighter ones [29]. Furthermore, these models contain a natural Peccei-Quinn symmetry, necessary to solve the strong-CP problem [30]. Finally, the 3-3-1 models with heavy sterile neutrinos, can provide cold dark matter candidates as weakly interacting massive particles (WIMPs) [31]. The 3-3-1 models extend the scalar sector of the SM into three SU (3) L scalar triplets: one heavy triplet field with a Vacuum Expectation Value (VEV) at high energy scale v χ , which breaks the symmetry SU (3) L ⊗ U (1) X into the SM electroweak group SU (2) L ⊗ U (1) Y , and two lighter triplets with VEVs at the electroweak scale v ρ and v η , which trigger Electroweak Symmetry Breaking. Besides that, as shown in Ref. [32], the 3-3-1 model can explain the number of events in the h → γγ decay, recently observed at the LHC, since the heavy exotic quarks, the charged Higges and the heavy charged gauge bosons contribute to this process. On the other hand, the 3-3-1 model reproduces a specialized Two Higgs Doublet Model type III (2HDM-III) in the low energy limit, where both triplets ρ and η are decomposed into two hypercharge-one SU (2) L doublets plus charged and neutral singlets.
One of the Yukawa couplings is for generating the quark masses, and the other one produces the flavor changing couplings at tree level. One way to remove both the huge FCNC and CP-violating effects, is by imposing discrete symmetries, obtaining two types of 3-3-1 models (type I and II models), which exhibit the same Yukawa interactions as the 2HDM type I and II at low energy. In the 3-3-1 model type I, one Higgs triplet (for example, ρ) provide masses to the phenomenological up-and down-type quarks, simultaneously. In the type II, one Higgs triplet (η) gives masses to the up-type quarks and the other triplet (ρ) to the down-type quarks. Recently, authors, respectively in Refs. [33] and [34] discuss the mass structures in the framework of the I-type and II-type 331 model. In this paper we obtain different mass structures for the S 3 type III-like model. We choose the S 3 discrete symmetry since it is the smallest non-Abelian discrete symmetry group that contains a two-dimensional irreducible representation which can connect two maximally mixed generations. Thereby we group the scalar fields into doublet and non trivial S 3 singlet representations. Regarding the quark sector, we group left handed quarks as well as the right handed bottom quark into S 3 singlet representations while the remaining right handed quarks are assigned to S 3 doublet representations. This paper is organized as follows. In Section 2 we briefly describe some theoretical aspects of the 3-3-1 model and its particle content, as well as the particle assignments under S 3 doublets and S 3 singlets, in particular in the fermionic and scalar sector in order to obtain the mass spectrum. Section 3 is devoted to discuss the resulting tree level quark mass textures. In Section 4 we employ six dimensional Yukawa operators for the up-type quark sector in order to generate the up and charm quark masses. In Section 5, we present our results in terms of quark mixing, which is followed by a numerical analysis. Finally in Section 6, we state our conclusions.

The model.
We consider the 3-3-1 model where the electric charge is defined by: (1, 1, −2). In order to avoid chiral anomalies, the model introduces in the fermionic sector the following (SU (3) C , SU (3) L , U (1) X ) left-and right-handed representations: where U i L and D i L for i = 1, 2, 3 are three up-and down-type quark components in the flavor basis, while ν i L and e i L are the neutral and charged lepton families. The right-handed sector transforms as singlets under SU (3) L with U (1) X quantum numbers equal to the electric charges. In addition, we see that the model introduces heavy fermions with the following properties: a single flavor quark T 1 with electric charge 2/3, two flavor quarks J 2,3 with charge −1/3, three neutral Majorana leptons (ν 1,2,3 ) c L and three right-handed Majorana leptons N 1,2,3 R (recently, a discussion about neutrino masses via double and inverse see-saw mechanism was perform in ref. [35]). On the other hand, the scalar sector introduces one triplet field with VEV υ χ , which provides the masses to the new heavy fermions, and two triplets with VEVs υ ρ and υ η , which give masses to the SM-fermions at the electroweak scale. The [SU (3) L , U (1) X ] group structure of the scalar fields is: The EWSB follows the scheme SU ( Furthermore, we impose the S 3 flavor symmetry for fermions and scalars so that the full symmetry of our model is extended to be SU ( The inclusion of the discrete S 3 symmetry will allow to reduce the number of parameters in the Yukawa and scalar sector of the SU (3) C ⊗ SU (3) L ⊗U (1) X model making it more predictive. Besides facilitating maximal mixing through its doublet representation, the S 3 discrete group provide two inequivalent singlet representations which play a crucial role in reproducing fermion masses and mixing [20].
The scalar fields are grouped into doublet and singlet representions of S 3 as follows: Regarding the quark sector, it is assumed that all quarks are assigned to S 3 doublets excepting the Q 1 L , Q 2 L , Q 3 L and D 3 R fields, which are assigned to S 3 singlets: With the above spectrum, we obtain the following SU ( Yukawa terms for the quark sector: where n = 1, 2 is the index that label the first and second quark triplet shown in Eq. (2.2), and h (f ) φij (f = U, D, T, J) are the i, j components of non-diagonal matrices in the flavor space associated with each scalar triplet φ : η, ρ, χ.
The interactions among the scalar fields are contained in the following most general potential that we can construct with three scalar triplets: Besides that, the scalar potential can be written in terms of the three scalar triplets as follows: After the symmetry breaking, as shown in detail in Appendix B, it is found that the mass eigenstates are related to the weak states in the scalar sector by: where the mixing angles are given by: 13) while the masses of the physical scalar fields are: where we have taken into account that f v χ v η , v ρ . Notice that after the spontaneous breaking of the gauge symmetry SU (3) L ⊗ U (1) X and rotations into mass eigenstates, the model contains 4 massive charged Higgs (H ± 1 , H ± 2 ), one CP-odd Higgs (A 0 ) and 4 neutral CP-even Higgs (h 0 , H 0 1 , H 0 2 , H 0 3 ) bosons. Here we identify the scalar h 0 with the SM-like 126 GeV Higgs boson observed at the LHC. We recall that the neutral Goldstone bosons G 0 1 , G 0 2 and G 0 3 correspond to the Z, K 0 and Z gauge bosons, respectively. Furthermore, the charged Goldstone bosons G ± 1 and G ± 2 correspond to the W ± and K ± gauge bosons, respectively [24,25,26,27].
Using the multiplication rules of the S 3 group given in Appendix A, it follows that the Yukawa terms for the quark sector take the following form: From the previous expression, it follows that the Yukawa mass terms for the quark sector are given by: ( 2.16) 3 Tree level quark mass textures.
From Eq. (2.16) it follows that the mass matrices for the up and down type quarks are given by: Without loss of generality we assume that the Yukawa couplings h Φ are real. These Yukawa couplings can be assumed to be complex but their phases can be removed by redefining the phases of the U 3 L and T 1 L fields. Then, the matrix M U satisfies the following relation: where: Then, it follows that the masses of the up-type quarks are given by: Therefore, the up and charm quarks are massless at tree level, while the top quark and exotic T quark get their masses from the η and χ triplets, respectively.
In the down type quark sector we assume that all Yukawa couplings are real. Then, the mass matrix M D for down-type quarks satisfies the following relation: where: The matrices S D and Y D can be diagonalized by a rotation matrix R D according to: 3.8) where: Furthermore, the masses of the down-type quarks, consistent with Eq. (3.6), are given by: From Eqs. (3.5) and (3.11), we obtain the following relation: Considering the fact that: we get: However, only some of the non vanishing entries of the corresponding mass matrix receive radiative corrections at one loop level. Figure 1 shows the form of generating a mass term at one loop level for the up type quark sector. In particular, theŪ 2 L U 1 R term is absent in the mass matrix of Eq. (2.16) and at one loop level it does not get value different from zero. From the Yukawa terms given in Eq. (2.15) we see that the Yukawa termQ 3 L ηU 1 R generates one of the interaction vertex in the loop diagram that includes the components ofQ 3 L as one internal particles in the loop. However, a mass insertion of the components ofQ 3 L with the fields U 1 R and D 3 R has to be done to obtain the fields with flipped quirality U 2 L and U 1 R , as seen from Eq. (2.16). A Yukawa term that connects the fields U 1 R and D 3 R with U 2,1 L does not exist. Therefore the Yukawa mass termsŪ 1 L U 1 R andŪ 2 L U 1 R cannot be generated at one loop level. The same argument can be used to justify that the remaining vanishing entries of the up type quark mass matrix do not receive one loop level corrections. Therefore, in our SU (3) model, radiative corrections at one loop level do not generate masses for the up and charm quarks. Due to this reason, we consider the following SU (3) (4.1) Using the multiplication rules of the S 3 group, the previous six dimensional terms can be rewritten as follows: After the scalars triplets get vacuum expectation values, the previous non renormalizable interaction yields the following contributions to the entries of the up type quark mass matrix: Therefore, the up type quark mass matrix takes the following form: Notice that the dimensionless couplings in Eq. (4.4) satisfy the following hierarchy: Here we consider the scenario where the dimensionless couplings of the aforementioned non renormalizable Yukawa terms, are complex. Then, the mass matrix for the up-type quarks satisfies the following relation: From Eqs. (4.7) and (4.8), it follows that the dimensionless couplings in Eq. (4.8) satisfy the following hierarchy: ω j << ζ l << σ ≈ κ, l = 1, 2, j = 1, 2, 3. (4.9) For the sake of simplicity we assume that the dimensionless couplings ε sn , with s = 1, 2 and n = 1, 2 are approximatelly equal and have the same complex phase. Then, f 1 ≈ f 2 ≈ f 3 ≈ f 4 . This assumption implies ζ l ζ, ω i ω with l = 1, 2 and i = 1, 2, 3, 4 as well as τ 1 = τ 2 = τ and τ 3 = 0. We define Notice that the parameter ϑ controls the breaking of universality in the dimensionless couplings of the six dimensional operators.
Then, after the aforementioned choice is made, the expression given by Eq. (4.8) becomes: As shown in detail in Appendix C, the matrix S U can be diagonalized by a rotation matrix R U , according to: (4.10) where: (4.11) with tan 2φ = 2 √ 2ζv ρ κv η , tan 2ψ = −ϑ tan 2φ. (4.12) And the masses of the SM up type quarks are given by: (4.13) Therefore, the top quark acquires a tree level mass, the charm quark gets its mass via six dimensional operators and the up quark mass arises from a seesaw type mechanism with the top quark and the six dimensional operators. Notice that the six dimensional operators are crucial to give masses the up and charm quarks. However if universality in the dimensionless couplings of the six dimensional operators were exact, the up quark would be massless. Thus, the smallness of the up quark mass can be explained by the suppression of universality violation in the dimensionless couplings of six dimensional operators.

Quark mixings.
With the rotation matrices for the up and down type quarks, we find the CKM mixing matrix: while the Jarlskog invariant which measures the amount of CP violation takes the following form [36]: The experimental values of CKM magnitudes and Jarlskog invariant are taken from Ref. [6]. These tables also show the obtained values of CKM magnitudes and Jarlskog invariant. The values of the parameters τ , θ, ψ and φ that successfully reproduce the CKM magnitudes and the values of the Jarlskog invariant and CP violating phase δ are given by: 3) The CP violating phase δ has been computed by using the expression [37]:   The obtained magnitudes of the CKM matrix elements and Jarlskog invariant are in agreement with the experimental data. Notice that we have assumed that the renormalizable Yukawa couplings for the up and down type quarks are real while the couplings of six dimensional operators are complex. The complex phase τ responsible for CP violation in the quark sector arises from six dimensional Yukawa terms for the up type quark sector.

Conclusions
In this work we proposed a model based on the extended group SU (3) C ⊗ SU (3) L ⊗ U (1) X ⊗ S 3 . By assuming specific particle assignments into S 3 doublets and singlets for scalars and quarks, we obtained a predictive model, where quark masses and mixing can be reproduced. Taking into account the multiplication rules of the S 3 discrete group, we obtained the complete mass terms of the scalar and quark sector. In particular, the down type quark masses are generated at tree level, where the light down and strange quarks acquire masses through one scalar triplet (η), where the bottom quark obtains its mass from the other triplet (ρ), while the two heavy quarks J A and J B get large masses from the third scalar triplet (χ). Furthermore, the hierarchy among the down type quark masses can be obtained from the down type quark Yukawa couplings and the VEVs of the scalar triplets (η) and (ρ). Regarding the up sector, the model predicts at tree level two massless quarks (the up and charm quarks), one massive quark at the electroweak scale (the top quark), and a very heavy T-quark. Due to the symmetries of the model, the massless quarks do not get masses from radiative corrections at one loop level. Then, these quarks get masses from non renormalizable six dimensional operators. In particular, the up quark mass arises from a seesaw type mechanism with the top quark and the six dimensional operators. Besides that, if universality of the non renormalizable couplings is assumed, the up quark remains massless. Only if the above universality is violated, the up quark gets mass. Thus, the smallness of the up quark mass can be understood as a consequence of a small deviation in the universality of non renormalizable effective couplings. We also have reproduced the magnitudes of the CKM matrix elements, the CP violating phase and the Jarlskog invariant. The complex phase responsible for CP violation in the quark sector has been assumed to come from six dimensional Yukawa terms for the up type quark sector. The S 3 symmetry is explicitly broken since v χ >> v η . Considering that the lightest exotic quark, i.e, J A has a mass of about A The product rules for S 3 .

CP violating phase Obtained Value Experimental Value
The S 3 group has three irreducible representations: 1, 1 and 2. Denoting (x 1 , x 2 ) T and (y 1 , y 2 ) T as the basis vectors for two S 3 doublets and y´a non trivial S 3 singlet, the multiplication rules of the S 3 group for the case of real representations are given by [12]: x 1 B Scalar potential and mass spectrum for the neutral and charged scalar fields.
The scalar potential of the model is constructed of the S 3 doublet Φ = (η, χ) and the non trivial S 3 singlet ρ fields, in the way invariant under the group SU ( This scalar potential is given by: 2,3 and all parameters of the scalar potential have to be real.
Using the multiplications rules of the S 3 group, the scalar potential can be written in terms of the three scalar triplets as follows: From the previous expressions and from the scalar potential minimization conditions, the following relations are obtained: Then, the following relation is obtained: The squared mass matrices for the neutral scalar fields in the basis ξ ρ , ξ η , ξ χ , η 0 3 , χ 0 1 , ζ ρ , ζ η , ζ χ are respectively given by: Here we assume that the mass parameter of the cubic term in the scalar potential satisfies f v χ . Since v χ v η , v ρ , the squared mass matrix M 2 N a can be block-diagonalized through the rotation matrix W , according to: where the relation λ 2 = −λ 4 given by Eq. (B.4) has been used.
From the condition of the vanishing of the off-diagonal submatrices in the previous expression, we obtain at leading order in B the following relations: where a and b are given by: (B.13) By using the method of recursive expansion of Ref. [38] taking into account the hierarchy m 2 H 0 3 << b n1 << a nm , (m, n = 1, 2) we find that the submatrix B is approximatelly given by: (B.14) The matrix M 2 N a is diagonalized by a rotation matrix is diagonalized by a rotation matrix: which satisfies: where we have used the Eq. (B.4) and the fact that Therefore, the physical neutral scalar mass eigenstates contained in the squared mass matrix M 2 N a are given by: which satisfies: (B.22) where we have used Eq. (B.4) and the fact that f v χ v η , v ρ .
so that the physical neutral scalar mass eigenstates contained in the matrix M 2 N b are given by: Therefore, the matrix M 2 N c is diagonalized by a rotation matrix P , according to: where we have taken into account that κ 0 since v χ v η , v ρ .
The masses of these physical neutral scalar mass eigenstates contained in the matrix M 2 N c are: The squared mass matrix for the charged scalar fields in the basis η + 2 , ρ + 1 , χ + 2 , ρ + 3 is given by: Hence, the physical charged scalar eigenstates are given by: 29) where: (B.31) and the masses of the physical charged scalars are: C Computation of the rotation matrix for SM up type quarks.
The mass matrix for up type quarks satisfies the following relation: The matrix S U can be diagonalized by a rotation matrix R U , according to: where: where the following relations are fullfilled: with the mixing angle ψ satisfying the following relation: tan 2ψ = −ϑ 2 √ 2ζv ρ σv η . (C.4) and we have taken into account that ϑ << 1 and ζ << σ.