The $SU(3)_C\times SU(3)_L\times U(1)_X$ Model from $SU(6)$

We propose the $SU(3)_C\times SU(3)_L\times U(1)_X$ model arising from $SU(6)$ breaking. One family of the Standard Model (SM) fermions arises from two $\bar{6}$ representations and one $15$ representation of $SU(6)$ gauge symmetry. To break the $SU(3)_C\times SU(3)_L\times U(1)_X$ gauge symmetry down to the SM, we introduce three $SU(3)_L$ triplet Higgs fields, where two of them comes from $\bar{6}$ representation while the other one from $15$ representation. We study the gauge boson masses and Higgs boson masses in details, and find that the Vacuum Expectation Value (VEV) of the Higgs field for $SU(3)_L\times U(1)_X$ gauge symmetry breaking is around 10 TeV. The neutrino masses and mixing can be generated via the littlest inverse seesaw mechanism. In particular, we have normal hiearchy for neutrino masses and the lightest active neutrino is massless. Also, we consider constraints from the charged lepton flavor changing decays as well. Furthermore, introducing two $SU(3)_L$ adjoint fermions, one $SU(3)_C$ adjoint scalar, and one $SU(3)_L$ triplet scalar, we can achieve gauge coupling unification within 1\%. These extra particles can provide a dark matter candidate as well.


I. INTRODUCTION
The Standard Model (SM) has made a great achievement in explaining the experimental result. However, many significant problems remain to be answered. One of the most import issues is the fermion generation and the U(1) Y hypercharge. Since the SM did not explain the origin of the hypercharge, one may expect that the quantum number comes from a bigger group, for example, the Grand Unified Theory (GUT). In the traditional SU(3) C × SU(3) L × U(1) X (331) model, it successfully explained why there are three generations by tactfully eliminating SU(3) L gauge anomalies. However, the U(1) X number of is given by hand just like U(1) Y in the SM, which is not satisfying and inspires us to embed the 331 model into a bigger group to understand the U(1) X number more naturally. In this paper, we shall propose a 331 model generated from a SU(6) model, where the U(1) X charge is determined from the SU(6) breaking.
For models with β = √ 3 [9][10][11], it is obvious that all the three scalar triplets are all in different representations, because Q = ±diag[1 + X, X, −1 + X] for particles in (anti)fundamental representation. Moreover, to generate all charged fermion masses in tree level, we need three scalar triplets and one scalar sextet. Such models also contain exotic charged particles such as double charged Higgs and quarks with charge ± 5 3 and ± 4 3 . In particular, there exists the Landau pole problem for U(1) X not far from the TeV scale.
We for the first time propose the SU(3) C × SU(3) L × U(1) X model, which can be obtained from the SU(6) breaking. One family of the SM fermions arises from twō 6 representations and one 15 representation of SU(6) gauge symmetry. To break the SU(3) C × SU(3) L × U(1) X gauge symmetry down to the SM gauge symmetry, we introduce three SU(3) L triplet Higgs fields, where two of them arises from6 representation while the other one from 15 representation. We discuss the gauge boson masses and Higgs boson masses in details, and show that the Vacuum Expectation Value (VEV) of the Higgs field for SU(3) L × U(1) X gauge symmetry breaking is around 10 TeV. We explain the neutrino masses and mixing via the littlest inverse seesaw mechanism. Especially, the normal hiearchy for neutrino masses is realized and the lightest active neutrino is massless. Moreover, we study constraints from the charged lepton flavor changing decays as well. Furthermore, introducing two SU(3) L adjoint fermions, one SU(3) C adjoint scalar, and one SU(3) L triplet scalar, we can achieve gauge coupling unification within 1%. These extra particles can give us a dark matter candidate as well.
The paper is organized as follows. In section II, we present the models and Yukawa terms. The gauge sector and Higgs sector are studied in section III and section IV, respectively. We discuss the neutrino masses and mixing, as well as the charged lepton flavor changing decays in section V. In section VI, we consider gauge coupling unification and dark matter candidate. Our conclusion is in section VII.
In our 3-3-1 model, the SU(3) C × SU(3) L × U(1) X gauge group arises from a large SU(6) gauge group. With U(1) X charge operator for the 6 representation of the SU(6) group being The following representions of the SU(6) group can be decomposed into representations of the SU(3) C × SU(3) L × U(1) X group as below One family of the SM fermions and extra fermions in our model is In SU(6) model, two6 anti-fundamental representations and one 15 anti-symmetric representation of the fermions are anomaly free. Thus, our model is anomaly free. To be concrete, we can verify it is easily as well. According to [25,26], first, for U(1) X , we have which makes U(1) X gauge structure anomaly free. For gauge structure of SU(3) L /SU(3) C , since the number of fermion multiplets in 3 representation equals to the number of fermion multiplets in3 representation for every generation, it is also anomaly free.
Our model has 3 scalar multiplets coming from two6 and one 15 representations of the SU(6) group, which are (2.14) We use to parameterize the 3 VEVs, which break the SU(3) L × U(1) X gauge group down to the U(1) EM gauge group. We write the U(1) EM charge operator as Then the conditon, which only neutral states of the scalar multiplets can get VEVs, gives To make SM particles have the same electric charges as in the SM, we find leading to The Yukawa terms and Majorana mass terms of our model are where M s , M s and M ss are 3 × 3 matrix. For simplicity, we do not include all the gauge invariant terms in Eq (2.21).

III. GAUGE BOSONS
We write W a (a = 1, 2, . . . , 8), which is in the adjoint representation of SU(3) L in the form of For the adjoint representation of the SU(3) L group, the electric charge operator is we get And we get To make W ± , which is the familiar W ± gauge boson in the SM, have the right mass, we have where A, Z and Z are the eigenstates of the mixing of B, W 3 and W 8 . A and Z are the photon and the Z gauge boson in the SM, respectively. We also find With the condition that |k| 1, we have and where θ W is the Weinberg angle. According to [27] M Z larger than 4.5 TeV, |v t | needs to be larger than 10 TeV.

IV. HIGGS SECTOR
The most general Higgs potential in our model is . . , 5), we get 4 independent relations, which are From the Higgs potential V Higgs , we get leading to the following eigenstates (4.10) Apparently η ± 1 and η ± 2 are Goldstone bosons.

B. Mixing of σ i
We have where The eigenstates are 14) and their masses satisfy a 1 , a 2 and a 3 are Goldstone bosons.

C. Mixing of ρ i
From the Higgs potential V Higgs , we have The lightest eigenstate, is massless, which is a Goldsten boson. The next to the lightest eigenstate is the SM Higgs boson, whose mass M H should be 125 GeV. The independent parameters in the Higgs potential affecting M H are tan θ, k, l 1 , l 2 , l 3 , l 12 , l 13 , l 23 , l 12 , y 1 , y 2 , y 3 , y 12 and A. All these parameters except A are dimensionless. For simplicity, in FIG. 1,

V. NEUTRINO MASS, MIXING AND FCNC
From Eq. (2.21), the neutrino mass matrix in the basis (ν L , ν L , ν c R , N, N s , N s , N ) is Every element in M is a 3 × 3 matrix. Because y ν − y νT is an antisymmetric matrix, we have which means the lightest neutrino eigenstate is massless. We choose M ss to be 0 for simplicity. In the limits of tan θ 1 and |k| 1, we approximately get that (ν L , N, N s ) are only mixing with themselves and the mass matrix is . Notice that the situation here looks very similar to the littlest inverse seesaw (LIS) model [28,29], in which the elements of M s is very small to generate the very small neutrino masses. Since det[M D ] is zero, the lightest eigenstate of the mixing of ν L , N and N s is massless.
The three light eigenvalues of M † M forms the SM neutrino mass squares, which are constrained by neutrino oscillation experiments. According to [28], in the case that M D , M s M N , the three light neutrino mass squares are eigenvalues of M † ν M ν with For simplicity, we set M N and M s to be diagonal, which are

6)
Since M D is antisymmetric, it can be written as So we have Suppose eigenvalues of M † ν M ν are m 2 1 = 0, m 2 2 , and m 2 3 . However, we can always rescale d i (i = 1, 2, 3) and k j (j = 1, 2, 3) to 10 −R D d i and R s k j without changing the neutrino mixing pattern and m 3 m 2 . But the masses will be changed to 10 −2R D R s m i (i =, 1, 2, 3). Because the lightest neutrino in our model is massless, we should choose appropriate values of a i , k j , c N , tan θ and k to give where U ν is parameterized by θ 12 , θ 13 , θ 23 ,and δ, i.e., the Normal Hierarchy (NH) for neutrino masses. We choose where R s is determined by tan θ, k, c N , and R D to give the right neutrino masses. For example, when tan θ = 6, k = −60, c N = −1 and R D = 1, R s needs to be 1.8 × 10 −4 GeV, giving us the three mixing angles, CP violating phase δ and neutrino masses in TABLE I.

Observable
Model bpf ± 1σ bpf ± 1σ  Next, we shall disscuss the implication of the 3-3-1 model in the charged lepton flavor changing decays. There are in total three processes, which are µ → eγ, τ → µγ and τ → eγ. The branch ratio of lepton e i decaying to lepton e j is . Experiment results ask us that the branch ratio of charged lepton decay should satisfy Independent parameters influencing BR(e i → e j γ) are tan θ, k, R D , and c N , while R s is determined by other parameters to give the right neutrino masses. In FIG. 2, we show the dependence of BR(µ → eγ) on these parameters. We find that BR(µ → eγ) mainly depends on R D . To make that BR(µ → eγ) ≤ 4.2 × 10 −13 , R D needs to be larger than 2.5, which means that (d 1 , d 2 , d 3 ) < (1.55, 0.92, 2.81) × 10 −3 . In the case that R D ∼ 2.5, BR(τ → µγ), and BR(τ → eγ) are around 10 −14 and 10 −13 respectively.

VI. UNIFICATION OF GAUGE COUPLINGS
The Renormalization Group Equation (RGE) for gauge coupling is where i stands for the i-loop correction in RGE running. In this section, we consider two-loop correction. Equations of 1-loop and 2-loop corrections are To make gauge couplings unify at the GUT scale, we add two fermion multiplets, FA -1. Rj T/M * . Thus, we have two cases. First, SA can be a dark matter candidate if Z 2 symmetry is imposed to forbid SA decaying to quarks. We will leave this part of work in the future. For simplicity, we make all the particles beyond the SM take part in the RGE running at the energy scale of 2 TeV, then the gauge coupling unification can be satisfied with accuracy of 0.65% at the energy scale of 5.2 × 10 16 GeV, which is shown in FIG. 3.
Alternatively, to make SA deday, we can add two fermion multiplets in 6 and6 representation of the SU(6) gauge group respectively, then the gauge coupling unification can be satisfied with accuracy of 0.68% at the energy scale of 6.2 × 10 16 GeV, which is shown in Fig. 4. Also, we make all the particles beyond the SM take part in the RGE running at the energy scale of 2 TeV.

VII. CONCLUSIONS
We have proposed a new SU(3) C × SU(3) L × U(1) X model, in which gauge symmetry can be realized from SU(6) breaking. The SM fermions in each of the three generations come from two6 representations and one 15 representation of the SU(6) gauge group besides two singlets of the SU(3) C × SU(3) L × U(1) X gauge group. There are three scalar multiplets, where two come from6 representations of SU(6) and one from 15 representation. And their VEVs are v u , v d and v t , respectively. There are additional gauge bosons, W ± , Z and V/V * , in our model besides the SM gauge bosons. v t needs to be larger than 10 TeV to make the mass of Z larger than 4 TeV. It is easy to give the 125 GeV Higgs boson mass when we set all the dimentionless parameters in the Higgs potential to be ∼ 0.1 and A to be ∼ 1 TeV. When M ss are set to be a zero matrix and in the limits of tan θ 1 and |k| 1, the mixing of ν L , N and N s is the same as in the littlest inverse seesaw model. The lightest neutrino in our model is massless. With parameters in y ν , y N and M s set to be appropriate values, we obtained the light active neutrino masses, leptonic mixing angles, and CP violating phase highly consistent with the experimental datas for the scenario of NH neutrino mass. To make BR(µ → eγ) ≤ 4.2 × 10 −13 , parameters in y ν needs to be smaller than ∼ 10 −3 , and in this case BR(τ → µγ) and BR(τ → eγ) are around 10 −14 and 10 −13 respectively. With additional two fermion multiplets, FA and FA , as well as two scalar multiplets, SA and T , the gauge coupling unification can be realized with accuracy of 0.68% at the energy scale of 6.2 × 10 16 GeV. SA can be a dark matter candidate if Z 2 symmetry is imposed. Alternatively, we can add two fermionic multiplets in 6 and6 representations of the SU(6) gauge group to make SA deday, then the gauge coupling unification can be satisfied with accuracy of 0.65% at the energy scale of 5.2 × 10 16 GeV.