Higgs and gauge boson phenomenology of the 3-3-1 model with CKS mechanism

The gauge boson sector in the renormalizable 3-3-1 model for the SM fermion mass and mixing is explored. The data of rho parameter shows that a VEV of the first spontaneous symmetry breaking (SSB) ranges from 3.6 TeV to 6.1 TeV. Therefore the mass of new heavy neutral gauge boson Z' ranges from 1.42 TeV to 2.42 TeV, which is consistent with estimation from other 3-3-1 models. In that region of masses, we find that the total cross section for the production of the heavy neutral gauge boson Z' at the LHC via Drell-Yan mechanism ranges from 46.2 pb up to 2.89 pb. On the other hand, in a future 100 TeV proton-proton collider the total cross section for the Drell-Yan production of a heavy Z'neutral gauge boson gets significantly enhanced reaching values ranging from 1371 pb up to 235 pb. By the way, masses of new bilepton gauge bosons Y and X are around 800 GeV which are quite good. The Higgs sector of the model is explored. The Higgs potential with lepton number conserving was in details considered. The SM Higgs boson was derived and as expected, is most contained from \eta_1^0. For the total potential including lepton number violating, except CP-even sector, situation is similar. The potential consists of enough number of Goldstone bosons for massive gauge bosons. The potential contains a complex scalar candidate for Dark Matter $\va_2^0$ and a Mojoron but it is harmless since being a singlet. The constraints arising from the estimation of the Dark Matter (DM) relic density, set the mass of the scalar dark matter candidate in the range 300 GeV\ \lesssim m_{\va } \lesssim 600 GeV, for a quartic scalar coupling $\la _{h^{2}\va ^{2}}$ in the window $0.5\lesssim \la _{h^{2}\va ^{2}}\lesssim 1$.


Introduction
Despite its great successes, the Standard Model (SM) still puzzles over the hierarchies and structure of the fermion sector, which remain without compelling explanation. It is known that in the SM, masses of the matter fields are determined through Yukawa interactions. In addition, the CKM matrix is also constructed from the same Yukawa couplings. To solve this puzzles, some mechanisms have been suggested. To the best of our knowledge, the first attempt to explain the huge differences in fermion masses in the SM is the Froggatt -Nielsen (FN) mechanism [1]. According to the FM mechanism, the mass differences between generations follow from suppression factors depending on FN charges of particles. It has been noticed that in order to implement the aforementioned mechanism, the effective Yukawa interactions were introduced, thus making this theory non-renormalizable. From this point of view, the recent mechanism proposed by Cárcamo, Kovalenko and Schmidt [2] (called by CKS mechanism) based on sequential loop suppression mechanism, is more natural since its suppression factor is arisen from loop factor l ≈ (1/4π) 2 .
One of the main purposes of the models based on the gauge group SU (3) C × SU (3) L × U (1) X (for short, 3-3-1 model) [3][4][5][6][7][8][9] is concerned with the search of an explanation for the number of generations of fermions. Combining with the QCD asymptotic freedom, the 3-3-1 models provide an explanation for the number of generations to be three. Some other advantages of the 3-3-1 models are: i) they solve the electric charge quantization [10,11], ii) they contain several sources of CP violation [12,13], and iii) they have a natural Peccei-Quinn symmetry, which solves the strong-CP problem [14][15][16][17].
In the framework of the 3-3-1 models, most of research are focused on radiative seesaw mechanisms, and some but involving nonrenormalizable interactions (see references in Ref. [18]). However, most researches on the 3-3-1 models are not concerned with vast different masses among the generations.
The FN mechanism was implemented in the 3-3-1 models in Ref. [19]. It is interesting that the FN mechanism does not produce new scale, i.e., the scale of the flavour breaking is the same as the breaking scale of the symmetry of the model.
The CKS mechanism has been included in the 3-3-1 model without exotic electric charges (β = −1/ √ 3) in Ref. [18]. The implication of the CKS mechanism to the 3-3-1 model leads to the interesting 3-3-1 model in sense that the derived model is renormalizable, while it fits all current data on fermion masses and mixing [18]. It is worth mentioning that there exists a residual discrete Z (Lg) 2 lepton number symmetry arising from the breaking of the global U (1) Lg symmetry. Under this residual symmetry, the leptons are charged and other particles are neutral [18].
However, in the mentioned work, the authors have just focused on the data concerning fermions (both quarks and leptons including neutrino mass and mixing), but some questions are open for the future study.
The aim in this work is to consider, in more details, the phenomenology of the model such as gauge and Higgs sectors from which we can get a bound on the model scale v χ as well as on the mass of the new heavy Z boson. Due to the implemented symmetries, the Higgs sector is rather simple and can be completely solved. All Goldstone bosons and the SM like Higgs boson are defined.
The further content of this paper is as follows. In Sect. 2, we briefly present particle content and SSB of the model. Sect.3 is devoted to gauge boson mass and mixing. Taking into account of data on the ρ parameter, we get bounds on the VEV of the first step of the SSB and on the mass of the new heavy Z gauge boson. By the way, we also get a limit for masses of the bilepton charged/non-Hermitian bosons. The Higgs sector is considered in Sect. 4. The Higgs sector consists of two parts: the first part contains lepton number conserving terms and the second one is lepton number violating. We study in details the first part and show that the Higgs sector has all necessary ingredients. Sect. 5 is devoted to the production of the heavy Z and the heavy neutral scalar H 4 . In Sect. 6, we deal with the DM relic density. We make conclusions in Sect. 7.

Review of the model
To implement the CKS mechanism, only the heaviest particles such as the exotic fermions and the top quark get masses at tree level. The next -medium ones: bottom, charm quarks, tau and muon get masses at one-loop level. Finally, the lightest particles: up, down, strange quarks and the electron aquire masses at two-loop level. To forbid the usual Yukawa interactions, the discrete symmetries should be implemented. Hence, the full symmetry of the model under consideration is where L g is generalized lepton number defined in Refs. [18,20]. It is interesting to note that, in this model, the light active neutrinos also get their masses by a combination of linear and inverse seesaw mechanisms at two-loop level. As in the ordinary 3-3-1 model without exotic electric charges, the quark sector contains the following SU where ∼ denotes quantum numbers for the three above subgroups, respectively. Note that the SU (3) L singlet exotic up type quarks T L,R , down type quarks B L,R in the last line of Eq. (2.2) are newly introduced for implementation of the CKS mechanism.

Gauge boson masses and mixing
After SSB, the gauge bosons get masses arising from the kinetic terms for the η and χ SU (3) L scalar triplets, as follows: with the covariant derivative for triplet defined as where g and g X are the gauge coupling constants of the SU (3) L and U (1) X groups, respectively. Here, λ 9 = 2/3 diag(1, 1, 1) is defined such that Tr(λ 9 λ 9 ) = 2, similarly as the usual Gell-Mann matrix λ a , a = 1, 2, 3, · · · , 8. By matching gauge coupling constants at the SU (3) L × U (1) X breaking scale, the following relation is obtained [8] t Let us provide the definition of the Weinberg angle θ W . As in the SM, ones put g = g tan θ W , where g is gauge coupling of the U (1) Y subgroup satisfying the relation [8] (3.5) Denoting and substituting (3.2) and (3.6) into (3.1) ones get squared masses for charged/non-Hermitian gauge bosons as follows and v η = v = 246 GeV, as expected.

From (3.7) it follows a splitting of gauge boson masses
For neutral gauge bosons, the squared mass mixing matrix has the form where V T = (A µ3 , A µ8 , B µ ) and (3.10) The down-left entries in (3.10) are not written, due to the fact that the above matrix is symmetric. The matrix in (3.10) has vanishing determinant, thus giving rise to massless gauge boson, which corresponds to the photon. Diagonalization of matrix in (3.10) separates into two steps. In the first step, the massive fields are identified as where we have denoted s W = sin θ W , c W = cos θ W , t W = tan θ W . After the first step, matrix M 2 ngauge becomes block diagonal one, where the entry in the top is zero (due to the masslessness of the photon), while the 2 × 2 matrix for (Z µ , Z µ ) in the bottom has the form The matrix elements in (3.12) are given by Note that our formula of M 2 Z is consistent with that given in [21].
The last step of diagonalization is quite simple. The eigenstates are determined as where the mixing angle is given by It is very easy to prove that our definition of φ is consistent with that introduced in Ref. [22], which is needed to study the ρ parameter. The masses of physical neutral gauge bosons are determined as (3.16) Ones approximate In the limit v χ v η , the Z − Z mixing angle is tan φ (3.20) Before turning to the next section, we remind the usual relation e = gs W .

Limit on Z mass from the ρ parameter
The presence of the non SM particles modifies the oblique corrections of the SM, the values of which have been extracted from high precision experiments. Consequently, the validity of our model depends on the condition that the non SM particles do not contradict those experimental results. Let us note that one of the most important observables in the SM is the ρ parameter defined as For the model under consideration, the oblique correction leads to the following form of the ρ parameter [22] ρ − 1 where where α(m Z ) ≈ 1 128 [23]. Taking into account s 2 W = 0.23122 [23] and we have plotted ∆ρ as a function of v χ in Fig. 1    Substituting (3.26) into (3.19) and evaluating in figure 1(the right-panel) we get a bound on the Z mass as follows It is worth mentioning that the second term in (3.24) is much smaller the first one. Consequently, the limit deduced from the tree level is slightly different from the one with the oblique correction.
Then, the bilepton gauge boson mass is constrained to be in the range: Here we have used [23] m W = 80.379 GeV .
Note that the above limit is stronger than the one obtained from the wrong muon decay [29] M Y ≥ 230 GeV .
For conventional notation, hereafter we will call Z 1 and Z 2 by Z and Z , respectively. Now we turn into the main subject -the Higgs sector.

Higgs potential
The renormalizable potential contain three parts: the first one invariant under group G in (2.1) is given by The second part is a lepton number violating one (the subgroup U (1) Lg is violated) The last part which breaks softly Z 4 × Z 2 , is given by 3) The total potential is composed of three above mentioned parts (4.4) The scalar interactions needed for quark and charged lepton mass generation, read as follows For the neutrino mass generation, beside the first term in (4.5), the additional part is given as It is worth mentioning that for generation of masses for quark and charged lepton, only terms in conserving part V LN C are enough, while for the generation of the light active neutrino masses, one needs the lepton number violating scalar interactions of V LN V as well as the softly breaking part L scalars sof t [the last term in (4.6)] of the scalar potential.

Potential with lepton number conservation
Below we present lepton number conserving part V LN C . Expanding the Higgs potential around VEVs, ones get the constraint conditions at the three levels as follows The simplified form is Applying the constraint conditions in (4.8), the charged scalar sector contains two massless fields: η + 2 and χ + 2 which are Goldstone bosons eaten by the W + and Y + gauge bosons, respectively. The other massive fields are φ + 1 , φ + 2 and φ + 4 with respective masses In addition, in the basis (ρ + 1 , ρ + 3 , φ + 3 ), there is the mass mixing matrix where we have used the following notations From (4.11), it follows that in the limit v η v ξ , ρ + 1 is physical field with mass and two massive bilepton scalars ρ + 3 and φ + 3 mixing each other. Now we turn into CP-odd Higgs sector. There are three massless fields: I χ 0 3 , I η 0 1 and I ξ 0 . The field I ϕ 2 has the following squared mass where There are other two mass matrices, namely: 1. In the basis ( (4.16) The matrix in (4.16) provides two physical states where The field G 1 is massless while the field A 1 has mass as follows 2. In the basis (I ϕ 1 , I ρ ), the matrix is where we have denoted Generally, physical states of matrix (4.20) are where the mixing angle is given by and their squared masses as follows where Next, the CP-even scalar sector is our task. Ones have one massive field, namely R ϕ 2 with mass (4.26) As mentioned in Ref. [18], the lightest scalar ϕ 0 2 is possible DM candidate. Therefore from (4.26), the following condition is reasonable (4.27) In this case, the model contains the complex scalar DM ϕ 0 2 with mass Other three mass matrices are 1. In the basis (R χ 0 1 , R η 0 3 ), the matrix is (4.29) The above matrix is similar to that in (4.16) except the mixing angle has the opposite sign. Thus, two physical states are where R G 1 is massless while the field H 2 has mass as follows 2. In the basis (R ρ , R ϕ 1 ), the matrix is (4.32) The physical states of matrix (4.32) are where the mixing angle is given by and their squared masses identified by where 3. In the basis (R χ 0 3 , R η 0 1 , R ξ 0 ), the matrix is (4.37) Let us summarize the Higgs content: 1. In the charged scalar sector: there are two Goldstone bosons η − and χ − eaten by the gauge bosons W − and Y − . Three massive charged Higgs bosons are φ + 1 , φ + 2 and φ + 4 . The remaining fields ρ + 1 , φ + 3 and ρ + 3 are mixing.
2. In the CP-odd scalar sector: there is one massless Majoron scalar I ξ 0 which is denoted by G M . Fortunately, it is a gauge singlet, therefore, is phenomenologically harmless. Two massless scalars I η 0 1 and I χ 0 3 which are Goldstone bosons for the gauge bosons Z and Z , respectively. There exists another massless state denoted by G 1 , its role will be discussed below. Here we just mention that in the limit v η v χ , this field is I χ 0 1 . The massive CP-odd field are I ϕ 2 , A 1 and other two I ϕ 1 , I ρ are mixing. 3. In the CP-even scalar sector: There is one massless field: R G 2 , and in the limit v η v χ , it tends to R χ 0 1 . Combination of G 1 and R G 1 is Goldstone boson for neutral bilepton gauge boson X 0 . Hence The massive fields are: R ϕ 2 , H 1 , H 2 and three massive R χ , R η , R ξ 0 and the SM-like Higgs boson h. Note that there exists degeneracy when the contribution arising from Z 2 × Z 4 soft breaking scalar interactions is not considered Thus, the complex scalar ϕ 2 has mass given by This result is consistent with the prediction in Ref. [18]. To be a DM candidate, the first term in (4.40) is suggested to be eliminated the terms with large VEVs such as v χ and v ξ . Then the above DM candidate ϕ 2 gets a mass given by According [23], the WIMP candidate has mass around 10 GeV, therefore λ ηϕ 2 ≈ 0.04 (4.42) To get the second DM candidate, namely, ϕ 0 1 , we have to carefully choose conditions. Looking at Eqs. (4.17), (4.30) and (4.39) we come to the fact that there is a new complex scalar

Simplified solutions
To find solutions in Higgs sector, we should make some simplification.

The CP-odd Higgs bosons
Looking at the potential in (4.1), the relations below are completely reasonable The CP-odd Higgs sector contains three massless fields: I χ 0 3 , I η 0 1 and I ξ 0 . The field I ϕ 2 has the following squared mass (4.47) Next, in the basis (I χ 0 1 , I η 0 3 ), we have one Goldstone boson G 1 and a massive A 1 with mass Taking into account (4.46), the mass matrix in Eq. (4.20) for I ϕ 1 and I ρ becomes diagonal (4.49) Hence, according the above assumption, I ϕ 1 and I ρ are physical states with respective masses as follows In this case, we have A 2 = I ϕ 1 and A 3 = I ρ .
In summary, under assumption of (4.46), the CP-odd scalar sector consists of four massless fields: I χ 0 3 , I η , G M and G 1 . Four massive fields are A 1 , A 2 , A 3 and I ϕ 2 . The content of the CP-odd scalar sector is summarized in Table 3. Table 3. Squared mass of CP-odd scalars under condition in (4.46) and v χ v η . Fields

The CP-even and SM-like Higgs bosons
As mentioned before, one massive field R ϕ 2 has the same mass as of I ϕ 2 (4.52) As mentioned in Ref. [18], the lightest scalar ϕ 0 2 is possible DM candidate. Therefore from (4.26), the following condition is reasonable (4.53) In this case, the model contains the complex scalar DM ϕ 0 2 with mass Some other components are: In the basis (R χ 0 1 , R η 0 3 ), we have one massless R G 1 and one massive H 1 with mass equal to that of A 1 and the mixing angle is the same.
In the basis (R ρ , R ϕ 1 ), the matrix is .

(4.55)
The physical states of matrix (4.55) are H 2 and H 3 with mixing angle given by (4.56) Now we turn to the sector where the SM Higgs boson exists, i.e., -the matrix in the basis (R χ 0 3 , R η 0 1 , R ξ 0 ) is given by (4.57) Let us assume a simplified scenario worth to be considered is characterized by the following relations: In this scenario, the squared matrix (4.37) for the electrically neutral CP even scalars in the basis (R η 0 1 , R χ 0 3 , R ξ 0 ) takes the simple form: (4.59) Table 4. Squared mass of CP-even scalars under condition in (4.58) and v χ v η . Fields In this scenario, the squared mass matrix m 2 CP even3 given above can be perturbatively diagonalized as follows: (4.60) where we have taken into account that v χ v η = 246 GeV. Thus, we find the that the physical scalars included in the matrix m 2 CP even3 are: The content of the CP-even scalar sector is summarized in Table 4.

The charged Higgs bosons
The charged scalar sector contains two massless fields: G W + and G Y + which are Goldstone bosons eaten by the W + and Y + gauge bosons, respectively. The other massive fields are φ + 1 , φ + 2 and φ + 4 with respective masses Fields In the basis (ρ + 1 , ρ + 3 , φ + 3 ), the mass mixing matrix is given by Let us make effort to simplify the above matrix. Note that due to the conditions in (4.58) we have Therefore, it is reasonable to assume Imposing these conditions, we get From (4.69), we get two charged Higgs bosons with masses at electoweak scale and one massive with mass around TeV (∝ v χ ) as indicated in figures 2, 3 and 4. Figure 2 shows that the Higgs boson H + 1 which is mainly composed of ρ + 1 has mass around 100 GeV, while the H + 2 gets mass in the range of 200 GeV and the mass of H + 3 is around 3.5 TeV. In addition, the Higgs boson H + 1 almost does not carry lepton number, while two others do. The content of the charged scalar sector is summarized in Table 5. It is worth mentioning that the masses of three charged scalars φ + i , i = 1, 2, 4 are still not fixed. The potential including lepton number violations,i.e., V f ull = V LN C + V LN V is quite similar to the previous one. There are some differences:

Search for Z at LHC
In this section, we present two typical effects of the LHC, namely, production of single a particle in proton-proton collisions.

Phenomenology of Z gauge boson
In what follows we proceed to compute the total cross section for the production of a heavy Z gauge boson at the LHC via Drell-Yan mechanism. In our computation for the total cross section we consider the dominant contribution due to the parton distribution functions of the light up, down and strange quarks, so that the total cross section for the production of a Z via quark antiquark annihilation in proton proton collisions with center of mass energy √ S takes the form: [GeV] Figure 3. Correlation between the mass of the charged scalar H + 2 and the trilinear scalar coupling w 2 . Figure 5 displays the Z total production cross section at the LHC via Drell-Yan mechanism at the LHC for √ S = 13 TeV and as a function of the Z mass, which is taken to range from 1.42 TeV up to 2.42 TeV to fulfill the constraints imposed by the ρ parameter. In that region of masses for the heavy neutral gauge boson Z , we find that the total cross section for its production at the LHC via Drell-Yan mechanism ranges from 46.2 pb up to 2.89 pb. On the other hand, in a future 100 TeV proton-proton collider the total cross section for the Drell-Yan production of a heavy Z neutral gauge boson gets significantly enhanced reaching values ranging from 1371 pb up to 235 pb, as indicated in figure 6.
It is worth noting that the produced Z boson will decay to t andt, but in the model under consideration, there are no the decays of the top quark t to the SM h associated with c or u as well as cZ [18].

Phenomenology of H 4 Heavy Higgs boson
In what follows we proceed to compute the LHC production cross section of the singly heavy scalar H 4 . Let us note that the singly heavy scalar H 4 is mainly produced via gluon fusion mechanism mediated by a triangular loop of the heavy exotic quarks T , J 1 and J 2 . Thus, the total cross section for the production of the heavy scalar H 4 through gluon fusion mechanism in proton proton collisions with center of mass energy √ S takes the form:   Figure 7 displays the H 4 total production cross section at the LHC via gluon fusion mechanism for √ S = 13 TeV, as a function of the SU (3) L × U (1) X symmetry breaking scale v χ , which is taken to range from 3.57 TeV up to 6.1 TeV. The aforementioned range of values for the SU (3) L × U (1) X symmetry breaking scale v χ corresponds to a heavy scalar mass m H4 varying between 1.6 TeV and 2.7 TeV and was chosen to guarantee the consistency of our model with the constraints imposed by the ρ parameter. Here, for the sake of simplicity we have restricted to the simplified scenario described by Eq. (4.58) and we have chosen the exotic quark Yukawa couplings equal to unity, i.e, y (T ) = y (J1) = y (J2) = 1. In addition, the top quark mass has been taken to be equal to m t = 173 GeV. We find that the total cross section for the production of the H 4 scalar at the LHC takes a value close to about 0.5 fb for the lower bound of 3.57 TeV of the SU (3) L × U (1) X symmetry breaking scale v χ arising from the ρ parameter constraint and decreases when v χ takes larger values. We see that the total cross section at the LHC for the H 4 production via gluon fusion mechanism is small to give rise to a signal for the allowed values of the SU scale v χ , however in a future 100 TeV proton-proton collider, it can range from 134 fb up to 14 fb when v χ is varied between 3.57 TeV up to 6.1 as shown in figure 8.

Dark matter relic density
In this section we provide a discussion of the implications of our model for DM, assuming that the DM candidate is a scalar. Let us recall that our goal in this section is to provide an estimate of the DM relic density in our model, under some simplifying assumptions motivated by the large number of scalar fields of the model. We do not intend to provide a sophisticated analysis of the DM constraints of the model under consideration, which is beyond the scope of the present paper. We just intend to show that our model can accommodate the observed value of the DM relic density, by having a scalar DM candidate with a mass in the TeV range and a quartic scalar coupling of the order unity, within the perturbative regime. We start by surveying the possible scalar DM candidates in the model. Considering that the Z 4 symmetry is preserved and taking into account the scalar assignments under this symmetry, given by Eq. (1), we can assign this role to either any of the SU (3) L scalar singlets, i.e., Reϕ 0 n and Imϕ 0 n (n = 1, 2). In this work we assume that the ϕ I = Imϕ 0 1 is the lightest among the Reϕ 0 n and Imϕ 0 n (n = 1, 2) scalar fields and also lighter than the exotic charged fermions, as well as lighter than Ψ R , thus implying that its tree-level decays are kinematically forbidden. Consequently, in this mass range the Imϕ 0 1 scalar field is stable. The relic density is given by (c.f. Ref. [23,30]) 1) where σv is the thermally averaged annihilation cross-section, A is the total annihilation rate per unit volume at temperature T and n eq is the equilibrium value of the particle density. Furthermore, K 1 and K 2 are modified Bessel functions of the second kind and order 1 and 2, respectively [30] and m ϕ = m Im ϕ . Let us note that we assume that our scalar DM candidate is a stable weakly interacting particle (WIMP) with annihilation cross sections mediated by electroweak interactions mainly through the Higgs field. In addition we assume that the decoupling of the non-relativistic WIMP of our model is supposed to happen at a very low temperature. Because of this reason, for the computation of the relic density, we take T = m ϕ /20 as in Ref. [30], corresponding to a typical freeze-out temperature. We assume that our DM candidate ϕ annihilates mainly into W W , ZZ, tt, bb and hh, with annihilation cross sections given by the following relations [31]: . v rel σ (ϕ I ϕ I → ZZ) = λ 2 where √ s is the centre-of-mass energy, N c = 3 is the color factor, m h = 125.7 GeV and Γ h = 4.1 MeV are the SM Higgs boson h mass and its total decay width, respectively. Note that we have worked on the decoupling limit where the couplings of the 126 GeV Higgs boson to SM particles and its self-couplings correspond to the SM expectation. The vacuum stability and tree level unitarity constraints of the scalar potential are [32][33][34]: The dark matter relic density as a function of the mass m ϕ of the scalar field ϕ I is shown in Fig. 9, for several values of the quartic scalar coupling λ 2 h 2 ϕ 2 , set to be equal to 0.7, 0.8 and 0.9 (from top to bottom). The horizontal line corresponds to the experimental value Ωh 2 = 0.1198 for the relic density. We found that the DM relic density constraint gives rise to a linear correlation between the quartic scalar coupling λ h 2 ϕ 2 and the mass m ϕ of the scalar DM candidate ϕ I , as indicated in Fig. 10.
We find that we can reproduce the experimental value Ωh 2 = 0.1198 ± 0.0026 [35] of the DM relic density, when the mass m ϕ of the scalar field ϕ I is in the range 300 GeV m ϕ 600 GeV,for a quartic scalar coupling λ h 2 ϕ 2 in the window 0.5 λ h 2 ϕ 2 1, which is consistent with the vacuum stability and unitarity constraints shown in Eqs. (6.3) and (6.4). Note that our range of values chosen for the quartic scalar coupling λ h 2 ϕ 2 also allow the extrapolation of our model at high energy scales as well as the preservation of perturbativity at one loop level.

Conclusions
In this paper, we have studied the gauge and Higgs sectors of the renormalizable 3-3-1 model for the SM fermion masses and mixings. From the experimental data on the ρ parameter, it follows that the mass of new heavy Z ranges from 1.42 TeV to 2.42 TeV. This bound is available for the LHC search and also implies that the SU (3) L × U (1) X symmetry breaking scale v χ ranges from 3.57 TeV up to 6.1 TeV. In the aforementioned region of masses for the heavy neutral gauge boson Z , we find that the total cross section for the its production at the LHC via Drell-Yan mechanism ranges from 46.2 pb up to 2.89 pb. On the other hand, in a future 100 TeV proton-proton collider the total cross section for the Drell-Yan production of a heavy Z neutral gauge boson gets significantly  enhanced reaching values ranging from 1371 pb up to 235 pb. We find that the total cross section for the production of the H 4 scalar at the LHC with √ S = 13 TeV takes a value close to about 0.5 fb for the lower bound of 3.57 TeV of the SU (3) L × U (1) X symmetry breaking scale v χ required by the consistency of the ρ with the experimental data. Despite that small value, the H 4 production total cross section can be as large as 134 fb in a future 100 TeV proton-proton collider that can be useful to probe the scalar and gauge sector of the 3-3-1 model and shed light in the understanding of the underlying dynamics behind electroweak symmetry breaking.
By the way, the bilepton gauge bosons Y ± and X 0 have masses around 800 GeV. For the further studies, the neutral currents have been also presented.
The general Higgs sector is separated into two parts. The first part consists of lepton number conserving terms and the second one contains lepton number violating couplings. The first part of potential was in details considered and the SM Higgs boson was derived and as expected, is most contained from η 0 1 . We have showed that the total potential, except CP-even sector, has quite similar situation,i.e., the eigenmasses and eigenstates in the part with lepton number conservation and the total potential are similar. The potential consists of enough number of Goldstone bosons for massive gauge bosons. In the CP-odd scalar sector, there are four massive bosons and one of them is a DM candidate. The CP-even sector consists of seven massive fields including the SM Higgs boson and a DM candidate. The singly charged Higgs bosons sector contains six massive fields. Two of them have masses in the electroweak scale and the remaining has mass around 3.5 TeV. Masses of three charged bosons φ + i , i = 1, 2, 4 are not fixed. The scalar potential contains a Majoron but it is harmless, because it is a scalar singlet. There is complex scalar DM candidate ϕ 0 2 . To reproduce the Dark matter relic density, the mass of the scalar dark matter candidate has to be in the range 300 GeV m ϕ 600 GeV, for a quartic scalar coupling λ h 2 ϕ 2 in the window 0.5 λ h 2 ϕ 2 1. In addition, it has been shown in Ref. [18] that requiring that the DM candidate ϕ 0 lifetime be greater than the universe lifetime τ u ≈ 13.8 Gyr and assuming m ϕ 0 ∼ 1 TeV, we estimate the cutoff scale of our model Λ > 3 × 10 10 GeV. Thus we conclude that under the above specified conditions the model contains viable fermionic Ψ R and scalar ϕ 0 DM candidates. A sophisticated analysis of the DM constraints of our model is beyond the scope of the present paper and is left for future studies.