Hierarchical Fermions and Detectable $Z^\prime$ from Effective Two-Higgs-Triplet 3-3-1 Model

We develop a ${\rm SU}(3)_C\otimes {\rm SU}(3)_L\otimes {\rm U}(1)_X$ model where the number of fermion generations is fixed by cancellation of gauge anomalies, being a type of 3-3-1 model with new charged leptons. Similarly to the economical 3-3-1 models, symmetry breaking is achieved effectively with two scalar triplets so that the spectrum of scalar particles at the TeV scale contains just two CP even scalars, one of which is the recently discovered Higgs boson, plus a charged scalar. Such a scalar sector is simpler than the one in the Two Higgs Doublet Model, hence more attractive for phenomenological studies, and has no FCNC mediated by scalars except for the ones induced by the mixing of SM fermions with heavy fermions. We identify a global residual symmetry of the model which guarantees mass degeneracies and some massless fermions whose masses need to be generated by the introduction of effective operators. The fermion masses so generated require less fine-tuning for most of the SM fermions and FCNC are naturally suppressed by the small mixing between the third family of quarks and the rest. The effective setting is justified by an ultraviolet completion of the model from which the effective operators emerge naturally. A detailed particle mass spectrum is presented, and an analysis of the $Z^\prime$ production at the LHC run II is performed to show that it could be easily detected by considering the invariant mass and transverse momentum distributions in the dimuon channel.


I. INTRODUCTION
The measurements of the Higgs boson properties and their actual agreement with the Standard Model predictions [1][2][3][4] have corroborated the simplest implementation of the Higgs mechanism as the source of electroweak symmetry breaking. Although the CERN Large Hadron Collider (LHC) has not yet provided clear evidence for new physics, the Standard Model (SM) consolidation has helped us to put in a firmer footing a series of its theoretical problems such as the severe hierarchy of the Yukawa couplings (the flavor problem), including the neutrino masses and mixing problem; the number of fermion generations; the chiral nature of the electroweak interaction; matter-antimatter asymmetry of the Universe; the strong CP problem; the dark matter content of the Universe; and the vacuum stability. The seeking of solutions for one or more of these problems has often guided the development of new models by extending the field content of the SM or, sometimes simultaneously, enlarging its symmetries.
Concerning the empirical observation of just three generations of fermions, 3-3-1 models offer a plausible explanation [5][6][7][8][9][10][11]. In these theoretical constructions the SU(2) L ⊗ U(1) Y symmetry group of the electroweak interactions is extended to SU(3) L ⊗U(1) X , in such a way that cancellation of all gauge anomalies involves necessarily all the three fermion generations. As it happens, there are different types of 3-3-1 models depending on the matter content fixed by a parameter β in the electric charge operator where T 3 and T 8 are the diagonal generators of SU(3) L built as T a = λa 2 from the Gell-Mann matrices λ a , with a = 1, ..., 8; and X refers to the U(1) X charge. Standard Model left-handed lepton fields take part in SU(3) L triplets,  [5,9,11,12]. Other well developed constructions have β = − √ 3 and, in this case, E iL carry one unit of electric charge so that it could be identified with the charged anti-leptons, i.e., E iL ≡ (l − iR ) c [6,7], or even represent new charged leptons [10]. For other possibilities, see [13].
For all models, the cancellation of gauge anomalies requires two anti-triplets (triplets) and one triplet (anti-triplet) of left-handed quarks when taking into account triplets (anti-triplets) of leptons. This feature leads to flavor changing neutral currents (FCNC) mainly through a vector boson Z whose mass is related to the energy scale in which the SU(3) L ⊗ U(1) X symmetry is broken down spontaneously to SU(2) L ⊗ U(1) Y . Bounds on the Z boson mass have been obtained from the LHC data for the versions with β = ±1/ √ 3, − √ 3 in Ref. [14], and recent analyses on FCNC have been performed in Refs. [15][16][17].
Many works have been published exploring the theoretical and phenomenological benefits of these models, showing that they are good candidates for describing new physics. For example, it is possible to include supersymmetry in such a context [18][19][20][21], as well as to construct left-right extensions [22], which have recently been subject of some studies [23][24][25][26].
To implement spontaneous symmetry breaking, three or more scalar multiplets getting vacuum expectation value (vev) at the GeV-TeV scale have been considered in 3-3-1 models. As a consequence, the scalar potential has many free parameters, being more complex than e.g. the Two Higgs Doublet Model (see Ref. [27] for a review). However, it has been shown that it is possible to break the symmetries down in some 3-3-1 models by taking into account two scalar triplets only [17,[28][29][30][31]. These constructions are phenomenologically attractive once they have a simpler scalar potential, predicting only three Higgs bosons. With the introduction of effective operators, masses for all fermions can be generated [17,28].
Our aim in this work is to develop a version of the 3-3-1 model with β = 1 √ 3 distinct from its first proposals [32,33] and from other similar models focused on neutrino masses and mixing [34,35].
We focus on this model because the versions with β = ± √ 3 [6,7,10] become strongly interacting at an energy of few TeV [7,[36][37][38][39]. We comment on the case β = −1/ √ 3 when relevant. We show that a consistent symmetry breaking pattern is obtained for this model with only two scalar triplets getting vev. Also, we present a simple mechanism where the low energy effective operators required to generate mass for some fermions arise after the integration of a supposedly heavy scalar triplet. These effective operators share similarities with those in the Froggatt-Nielsen mechanism in the sense that they generate more natural, less fine-tuned, masses for most of the fermions of the model, when compared to the mass generation in the Standard Model. The scalar particle spectrum of the model is composed of just two neutral CP even scalars, with one of them directly identified with the discovered 125 GeV Higgs boson, plus a charged one and its antiparticle. Also, five new vector bosons, V + , V − , V 0 , V 0 † , and Z , are predicted by the model. A study of the production signals at the LHC of the Z boson is performed.
The paper is organized as follows: in Sec. II, the essential aspects of the model are presented, including symmetry breakdown, residual symmetries, and the particle spectra of scalar and vector bosons; Fermion masses are treated in Sec. III; flavor changing interactions are analyzed in Sec.
IV; in Sec. V, a simple UV-completion able to generate the needed effective operators for fermion masses is discussed; the Z boson phenomenology is presented in Sec. VI; and our conclusions are given in Sec. VII.

II. THE MODEL
We focus on the model with β = 1/ √ 3 in Eq. (1). Therefore, the left-handed lepton fields form SU(3) L triplets, with the right-handed lepton fields in SU(3) L singlets, as follows where i = 1, 2, 3, is the generation index, and s = 1, ..., 6, with e − sR ≡ (e − iR , E − iR ). The numbers in parentheses refer to the field transformation properties under SU(3) C , SU(3) L , and U(1) X , respectively. We consider the right-handed neutrino fields, ν iR , in order to generate small masses to the left-handed neutrinos through the usual seesaw mechanism. The fields E − iL , required to complete the SU(3) L representation, along with the right-handed components, give rise to three heavy leptons.
Given the above lepton multiplets, as first observed long ago, gauge anomalies are canceled when the three families of quarks are included non-universally into two anti-triplets and one triplet of SU(3) L for the left-handed parts, and the corresponding right-handed fields assigned to singlets: where a = 1, 2, m = 1, . . . , 5, n = 1, . . . , 4, with u mR ≡ (u iR , U aR ) and d nR ≡ (d iR , D R ). Besides the quark fields of the SM, this model has two extra up-type quark fields, U a , and one downtype field, D. Such fields, as well as E i , get their masses at the energy scale w, in which the SU(3) L ⊗ U(1) X is supposedly broken down to SU(2) L ⊗ U(1) Y . Once that energy scale must be higher than the electroweak scale, i.e., w > v = 246 GeV, it is natural for the new elementary fermions associated with those fields to be heavier than the standard ones.
As we have already mentioned, the set of fields in Eqs. (2) and (3) is such that the cancellation of gauge anomalies involves the three fermion generations. This contrasts with the SM where the cancellation of anomalies occurs in each family, independently.
In principle, the choice of which generation of left-handed quark is assigned to a triplet is arbitrary. But, the fact that not all left-handed quark multiplets have the same transformation properties leads to new sources of FCNC. This has been explored in various works considering different versions of 3-3-1 models. Constructions with the third generation transforming differently from the first two are less restricted by bounds of processes involving FCNC. We show in Sec. IV that FCNC interactions are naturally suppressed in our model due to its peculiar mass generation mechanism for the fermions.
The following two scalar triplets realize the spontaneous breaking of the SU(3) L ⊗ U(1) X symmetry down to U(1) Q of the electromagnetic interactions: This is the minimal set of scalar fields that can perform the required symmetry breakdown.
From the fermionic and the scalar multiplets in Eqs. (2), (3), and (4), we write down the following Yukawa Lagrangian where the complex coupling constants are such that: h E is is a 3 × 6 matrix; h ν ij , is a 3 × 3 matrix; m ij = m i δ ij is a 3 × 3 diagonal matrix; h U am and h d an are 2 × 5 and 2 × 4 matrices, respectively; f u m and f D n are 1 × 5 and 1 × 4 matrices, respectively. With only the two scalar triplets in Eq. (4), the most general renormalizable scalar potential is simply given by We assume that the quadratic mass parameters, µ 2 1,2 < 0, and the self-interaction coupling constants, λ i , i = 1, ...4, are such that the scalar fields will develop non-vanishing vevs, χ , ρ = 0. The latter symmetry is the continuous version of the center of SU(3) L in which every triplet, 3, carries unit charge while every anti-triplet, 3 * , carries the opposite charge. This is possible in our case due to the absence of any trilinear couplings of the form 3 × 3 × 3. Therefore, when the scalar fields get vev, the U(1) P Q symmetry will be broken, but its charges will be part of another remaining global symmetry as we show below. In Table I the quantum numbers associated with the U(1) symmetries are presented for all the matter fields.
where U can be an arbitrary SU(2) transformation in the 2-3 sub-block or any diagonal transformation. 2 It is the former group which is special to this case, and we denote this group as SU (2)  Therefore, without loss of generality we can consider that the minimum of the potential in Eq. (6) is attained at the vevs If we had considered the more general vev χ = (0, v , w ) T / √ 2, the reparametrization symmetry in (7) would allow us to rotate χ to the original form in Eq. (8) without affecting ρ in Eq. (9).
Consequently, the fields are also transformed so that Eqs. (8) and (9) can be taken from the start.
A direct consequence is that the vector bosons V ± and W ± do not mix at tree level. It has to be pointed out that the reparametrization symmetry applies to other models, like the 3-3-1 model defined by β = −1/ √ 3, having a scalar triplet with two neutral components that could acquire vevs, so that we can make a rotation in order to have just one component with vev.
Given that we are considering w > v, the spontaneous symmetry breaking induced by the vevs in Eqs. (8) and (9) happens in two stages: first with χ realizing the breakdown with the hypercharge given by Y = T 8 / √ 3 + X, and the charges of the global symmetry P Q = 3X − P Q; and second with ρ realizing the breakdown with Q the electric charge operator of Eq. (1) and charges of a global symmetry given by 3 It is easy to see that this operator is unbroken by the vevs of Eqs. (8) and (9)  at the perturbative level, as pointed out in [42]. In order to overcome this problem we introduce in Sec. III dimension-5 operators which explicitly break the U(1) P Q and, consequently, the U(1) G symmetry. As we will see, such operators can be generated at low energies in an UV-completed model with a heavy scalar triplet which is integrated out. In such a setting the G charge is only broken by the soft breaking of PQ, and thus it remains approximately conserved. 3 If both neutral components of χ acquire a vev, i.e. √ 2 χ = (0, u, w), there still remains a conserved symmetry generated by G θ = 2(1 − 2 sin 2 θ)T3 + sin (2θ)T6 + (1 − 3 sin 2 θ)X − 1 3 P Q , with tan θ = u/w for the case that u, w are real and positive. If they were complex a more general expression can be written reparametrized by SU(2)rep. 4 It is also clear that U(1)G would be generally broken if there is an additional scalar triplet that can acquire a vev in its second component.

B. Scalar bosons
In order to find the scalar field masses and corresponding physical states, let us first write the scalar triplets as where we have decomposed the neutral fields which acquire a non-vanishing vev into scalar and pseudo-scalar contributions, S i and A i , respectively. In the approximation that the global charge G is exactly conserved, we can expect from its conservation that both χ 0 2 and ρ − 2 already have definite masses (they are would-be Goldstone bosons) and the possible pairs that can mix are ( The minimum condition for the potential leads to the constraint equations from which the quadratic mass parameters µ 2 1,2 can be eliminated. The mass matrix, derived from the scalar potential, for the CP even scalars in the basis (S 1 , S 3 ) is This leads to the quadratic mass eigenvalues corresponding, respectively, to the mass eigenstates where tan 2θ = λ 3 vw/(λ 2 w 2 − λ 1 v 2 ). We identify h as the state corresponding to the observed Higgs boson with mass of 125 GeV. In the limit θ → 0, the tree level couplings of h to the electroweak vector bosons W and Z are the same as the Standard Model Higgs boson.
The particle spectrum of the model does not contain CP odd neutral scalar fields. The pseudoscalar fields A 1 and A 3 are absorbed in the massive vector bosons Z and Z . In particular, the complex field χ 0 2 = (S 2 + iA 2 )/ √ 2 in the triplet χ does not get a mass term and plays the role of the Goldstone boson absorbed in the non-Hermitian neutral vector boson V 0 (both have G charge −1). This contrasts with the Two Higgs Doublet Models, which necessarily contain a neutral pseudoscalar in the particle spectrum.
For the charged scalar fields, it can be seen that only ρ + 3 and χ + 1 mix with each other so that their mass matrix, in the basis (ρ ± 3 , χ ± 1 ), is whose the nonzero eigenvalue corresponds to the squared mass of a charged scalar state given by The orthogonal eigenstates are Goldstone bosons which are absorbed to form the longitudinal components of the vector bosons V ± and W ± .
Thus, we see that four, from the initial twelve, degrees of freedom contained in the two scalar triplets remain as the physical scalars h, H, ϕ ± . The other eight degrees of freedom become the Goldstone modes needed to give mass to the vector bosons W ± , Z, V ± , V 0 , V 0 † , and Z . So ρ contains predominantly the SM Higgs h within the SU(2) L doublet and the heavy charged Higgs ϕ − in its third component while χ contains predominantly the heavy Higgs H in the third component and a small admixture of the ϕ + within the SU(2) L .

C. Vector bosons
As usual, to determine the physical gauge bosons and their masses, we look at the covariant derivative terms for the scalar fields: in which the covariant derivative is defined as where T a , with a = 1, ..., 8, are the SU(3) L generators as defined in Eq. (1), and X denotes the U(1) X charge of the field on which D µ acts; g, g X are the gauge coupling constants related to SU(3) L and U(1) X , respectively. The gauge coupling constant g is the same as in the Standard Model, since in 3-3-1 models the gauge group SU(2) L is totally embedded in SU(3) L . Additionally, the gauge coupling constants are related to the Standard Model electroweak mixing angle θ W according to Then, the P µ matrix for 3 can be written as where we have defined the following fields The G charge carried by these gauge fields is given by 2T 3 and yields The vector boson masses arise when the scalar fields in Eq. (23) acquire vevs as in Eqs. (8) and (9). The vector boson fields W ± , V ± , V 0 , and V 0 † get the following squared masses A direct consequence of breaking down the symmetries with just two scalar triplets is the tree Another peculiarity of the model is that a novel sort of neutral current might occur involving V 0 , V 0 † , since these vector bosons intermediate transitions between standard and new leptons with the same electric charge.
The gauge boson fields W 3 , W 8 , and B of the symmetry basis mix with each other leading to the mass matrix, in the basis (W 3 , W 8 , B), The mass eigenstates from this matrix give the photon field, A µ , and two massive fields, Z 1µ and Z 2µ , where the normalization constants are The masses of the neutral vector bosons, Z 1µ and Z 2µ , can be written as in such a way that the dominant contributions are And there is also the prediction, at tree level, that where we have used sin 2 θ W ≈ 0.231.
It is convenient for the discussion on the FCNC in the model to express the mass eigenstates Z 2µ and Z 1µ as linear combinations of the fields Z µ and Z µ , which result from the sequential symmetry breakdown SU(3) L ⊗U(1) X and SU(2) L ⊗U(1) Y , respectively, The unitary matrix in Eq. (41) diagonalizes the Z µ -Z µ mass matrix where In terms of these elements, the angle θ is For example, taking M Z ≈ 4 TeV, the angle is then θ Due to the fact that M 2 Z 1 also depends on the energy scale w related to the breakdown of SU(3) L ⊗ U(1) X to SU(2) L ⊗ U(1) Y , the model presents a deviation from the Standard Model ρ 0 parameter prediction ρ 0 = M 2 W / cos 2 θ W M 2 Z 0 = 1, with M Z 0 standing for the Standard Model Z 0 boson mass at tree level. Such a deviation, up to order (v/w) 2 1 in M 2 Z 1 , is given by where The actual experimental data furnishes ∆ρ 0 ≡ ρ 0 − 1 0.0006 [43]. Thus, if the tree level contribution is dominant over the radiative corrections we obtain the lowerbound w 6.5 TeV, taking into account the value v = 246 GeV. For definiteness, we take w = 10 TeV which furnishes M Z 2 ≈ 4 TeV, and M V ± ≈ M V 0 ≈ 3.2 TeV, although in our phenomenological analysis in Sec. VI a lower value for the Z boson mass is also considered.

III. FERMION MASSES
With the vevs χ and ρ the Yukawa Lagrangian in Eq. (5) leads to the mass matrix for neutrinos, in the basis (ν iL , ν iR c ), where h ν = (h ν ij ). We analogously denote by boldface the various Yukawa matrices appearing in Eq.
The mass matrix (46) has the usual seesaw texture which generates masses at the sub-eV scale for the left-handed neutrinos assuming, for example, h ν ij of order one and m ∼ 10 14 GeV.
As we have already pointed out in Sec. II A, due to the residual U(1) G symmetry, the Yukawa Lagrangian in Eq. (5) is not sufficient for giving mass to all charged fermion fields. In order to overcome this problem, we consider the following dimension-5 effective operators, 5 which can emerge from a simple ultraviolet completion shown in Sec. V, The large mass scale is Λ w and the matrices of coefficients y e is , y u am and y d n have sizes 3 × 6, 2 × 5 and 1 × 4 respectively; they are denoted by y e , y u and y d when the indices are suppressed.
where TeV and ∼ m τ ∼ 1 GeV, we have Λ ∼ 10 3 TeV. Although there is still an unexplained fine 5 Effective operators like these have also been considered recently in a similar model in Ref. [17].
tuning of order 10 −3 for the electron mass relative to the scale , this situation contrasts with the Standard Model where a tuning of order 10 −5 relative to the electroweak scale is required.
For the up-type quark mass matrix, in the basis (u 1 , u 2 , u 3 , U 1 , U 2 ) T L,R , we obtain similarly of /v ∼ 10 −2 between the first two rows and the third. By conveniently rotating the right-handed components, we can write the mass matrix for the SM up-type quarks, where m t is the top mass of order v andM u ,M ut are of order or smaller. The mass matrix M u is naturally of order ∼ 1 GeV and gives masses for (u, c). We could have chosenM u to be diagonal instead of M U from the rotation on Q aL . The mixing between t L and (u L , c L ) is naturally For the down-type quarks, in the basis (d 1 , d 2 , d 3 , D) T L,R , we have the mass matrix withh d an of order one and ∼ m s ∼ 0.1 GeV. This suppression further decreases the mixing between D L and b L to /w ∼ 10 −4 and one order of magnitude smaller for the mixing with (d L , s L ). We show in the following a possible mechanism responsible for this further suppression.
Assuming this hierarchy for the moment, we obtain the mass matrix for (d, s, b): Thus, we would have effectively that h d an =h d an ϕ /Λ , where ϕ /Λ ∼ /v ∼ 6 × 10 −4 1. The absence of the tree-level term Q aL ρ * d nR can be arranged by introducing a Z 2 symmetry under which only ϕ, Q aL , u aR , U aR are odd. One of the up-type quarks, u 3R , is kept even so that the top mass is still generated correctly by Eq. (5). The direct interaction terms involving the bilinear forms Q aL u 3R , Q 3L u aR and Q 3L U aR will be absent but effectively induced by the replacements Q aL → ϕ/Λ Q aL , u aR → u aR ϕ/Λ , and U aR → U aR ϕ/Λ , so that the mixing between t L and the heavy U aL or the lighter (u L , c L ) will be further suppressed by /v ∼ 10 −3 compared to the estimates discussed above. This property renders the top quark essentially unmixed with the rest.
Therefore, our minimal mechanism of breaking the 3-3-1 symmetry by the use of just two triplets, together with the Z 2 symmetry above, correctly displays the qualitative hierarchy between the masses for the third family quarks and those of the first two families. Comparing the mass matrices in Eqs. (50) and (53), it is clear that we have enough freedom to obtain the correct masses for the SM quarks and the correct Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix in a quantitative way. We can write already discarding the unobservable rotation matrices for the right-handed quarks. The CKM matrix is fixed by so that V d L can be considered free, while V u L is fixed by V CKM and V d L . From the discussion above, V u L is essentially block-diagonal with the third family practically decoupled from the rest. Another possibility to naturally suppress the coupling h d an would be to implement the Froggatt-Nielsen mechanism [45] (see also [46] for a proposal along this line in a 3-3-1 model). We leave this question for future investigations.
We briefly comment on the model with β = −1/ √ 3 where the heavy quark content is inverted -we would have two heavy quarks D a of charge −1/3 and just one quark U of charge 2/3-and the heavy charged leptons E i are replaced by neutral leptons N i that could participate in the mass generation mechanism for light neutrinos. The natural hierarchy in the up-type and down-type quark sectors would be very similar and the implementation of the Z 2 symmetry is analogous.

IV. SUPPRESSED FLAVOR CHANGING CURRENTS
We consider first the FCNC interactions mediated by scalars. To that end it is instructive to consider first the breaking of SU (3) L ⊗ U (1) X to the SM gauge group by χ and rewrite where φ 1 , φ 2 are SU (2) L doublets of Y = 1/2. At this stage, only √ 2 Re χ 0 3 acquires a vev w ∼ 10 TeV and √ 2 Re(χ 0 3 ) and ρ − 3 will be heavy SM singlet scalars of Y = 0 and Y = −1 respectively. The scalar √ 2 Im χ 0 3 and the doublet φ 1 will be absent in the unitary gauge because they will be absorbed by the gauge bosons Z µ and (V + µ , V 0 µ ) T . 6 See Sec. II B for their composition after EWSB. When taking into account the effective operators in (47), it is convenient to write where is a SM effective Higgs doublet. The dominant contribution coming from (47) at this stage will be We also separate the quark triplets in Eq. (3) and lepton triplets in Eq. (2) into SM doublets and singlets as where q aL is the usual quark doublet of family a = 1, 2, σ 2 is one of the Pauli matrices, and l iL are the usual lepton doublets.
At this stage of breaking, the model is equivalent to the SM with additional heavy singlet vectorlike quarks (VLQ) D, U 1,2 , singlet vector-like leptons E i and heavy singlet scalars √ 2 Re(χ 0 3 ), ρ ± 3 , with the additional heavy gauge bosons. The Yukawa interactions in (5) and (47) that only involve See appendix A for other interactions involving heavy particles. Note that d 3R ≈ b R , u 3R ≈ t R and q 3L ≈ (t L , b L ) T are almost the mass eigenstates except for suppressed mixing with the lighter quarks or the heavier D, U a . With only one effective Higgs doublet, there is natural flavor conservation and no flavor changing neutral interactions mediated by scalars [47], except for the ones induced by the small mixing between SM and heavy quarks [44]. This contrasts with the usual mechanism for breaking the 3-3-1 symmetry involving three SU(3) L triplets: usually there are two light Higgs doublets which induce suppressed but nonvanishing neutral flavor changing interactions at tree level [48].
We can consider now the currents coupling with the gauge bosons. There are two types of FCNC interactions for SM fermions: (i) the ones coming from the mixing between the third family of quarks and the first two families and (ii) the ones coming from the mixing between heavy fermions and the SM fermions. The first type (i) inevitably appears in all 3-3-1 models because one of the quark families is treated differently by the SU(3) L . Treating the third family differently is the only option if we want to avoid unrealistic flavor changing contributions in the mass differences of the K 0 , D 0 and B 0 systems for a Z µ 2 of mass no larger than a few TeV [48]. Sufficiently suppressed mixing of the third family with the lighter ones thus leads to experimentally allowed flavor changing contributions. In our model, however, because of the different mass generation mechanism, such a mixing is already naturally suppressed at least at the level of /v ∼ 10 −2 for the up-type quarks (further suppressed with the Z 2 implementation) and at least m s /m b ∼ 10 −2 for the down-type quarks. The flavor changing interactions of type (ii) are well known to be naturally suppressed by the hierarchy of SM fermions and heavy fermions [44]. We discuss these suppressed interactions in more detail in the following.
The fermionic currents couple with a vector boson V µ as For the neutral gauge boson V = Z , we have and Ψ is the collection of all the fermions in the symmetry basis. The flavor changing piece can be extracted as where c 2W ≡ cos 2θ W , and we have subtracted the flavor universal part common to the first two families of quarks and three family of leptons. Small mixing terms appear when we go to the basis of physical fields of definite masses. We can clearly see in (65) the two types of FCNC discussed above: the first line induces interactions of type (i) while the rest induces the type (ii) currents.
For type (i), only the mixing between the third family and the first two families will be observable.
This mixing is naturally suppressed by /v ∼ m s /m b ∼ 10 −2 in our model which is typically below the current limits coming from meson mixing [49].
As the interaction with the Z boson is identical to the ones in the SM for the usual quarks and leptons, FCNCs are only of type (ii): where g Z ≡ g/c W . We have again subtracted the family universal contributions from the first families of quarks and leptons. The coupling with W is the same as in the SM in the symmetry basis and small non-unitary effects appear through mixing between heavy and SM fermions. Flavor changing interactions are constrained by the search for singlet VLQs at the LHC [50] and by indirect constraints coming from precision electroweak observables and from the Large Electron-Positron Collider (LEP) [51]. The former constrains the masses to be above around 1 TeV and the latter constrains the mixing angle between the heavy quarks and the third family to be less than 0.04 for the down-type quarks and 0.14 for the up-type quarks for heavy quarks of 1 TeV. So our heavy quarks of masses at the scale w ∼ 10 TeV and mixing angle of less than 10 −2 are not currently observable. If the heavy quark masses are lowered to few TeV, and the Z 2 that decouples the top is present, the dominant channel for U a will be U a → W b + X, as U a → ht + X and U a → Zt + X will be negligible. For D, the channels D → hb + X and D → W t + X are similarly important.
The constraints for singlet vector-like leptons are much more relaxed.
For completeness, we also collect the interactions with the heavy gauge bosons V 0 , V + : where the heavy gauge bosons (V + µ , V 0 µ ) T have the same gauge quantum numbers as the SM Higgs doublet. These gauge bosons lie at the scale w and interactions with two SM fermions are suppressed by the heavy-light mixing.

V. AN ULTRAVIOLET COMPLETION OF THE MODEL
We show in this section a simple ultraviolet completion of the model allowing for the generation of the effective operators in Eq. (47). In order to achieve that, we add a scalar field η ∼ (1, 3, 1/3) which transforms in the same way as χ under SU(3) C ⊗ SU(3) L ⊗ U(1) X . We assume that η has a mass M w, v much larger than the rest of the other fields in the model. Thus, as we describe below, η can be integrated out so that the remaining effective theory is exactly the model studied above with the dimension-5 effective operators given in (47).
With the introduction of η, the total scalar potential is where V (χ, ρ) is given in Eq. (6) and where M 2 > 0 is the quadratic mass for η; with the coupling constant f , which we take as being real, having dimension of mass; and the λ's (< 4π) are the usual scalar field perturbative selfinteraction coupling constants. Also, we consider a basis (χ, η) in which the bilinear terms in these fields are diagonal. In fact, bilinear terms, such as µ 2 3 η † χ, can be eliminated through a rotation to the diagonal basis, implying effectively a change on the original quadratic mass parameters along with a redefinition of the quartic coupling constants.
We can see that the approximate conservation of the global charge G in this UV completion is guaranteed by the fact that the breaking is induced solely by a soft breaking of the PQ symmetry through the f term in (69). Therefore the breaking effects are all proportional to the breaking parameter f even if we consider radiative corrections, and this fact justifies the approximate conservation of G at low energies.
The Yukawa interactions involving η are similar to those for χ in Eq. (5), adding to such an equation the terms Assuming M |f | w, at low energies η is effectively given by where the ellipsis stands for operators which are even more suppressed by M . consequently, on the new gauge boson masses [17]. For other versions, a lower-bound of 3 TeV for the Z mass has been found [52].
In addition, previous studies concerning the Z branching ratios for the β = ± √ 3 versions have identified a leptophobic character of such a neutral gauge boson [53]. In our case, within a scenario where the exotic masses are just above 1 TeV, the Z branching ratios are divided into  to the observable width of a peak structure appearing over a SM background. In this approach, theoretical cross section predictions for specific models are usually calculated in the narrow width approximation. Obviously, when the width is wide, the resonance appears as a broad shape and can be almost flat around the Z pole.
Thus, within this narrow width approximation, we show below the invariant mass and transverse momentum p T distributions of the emerging leptons in the processes p + p −→ µ + + µ − + X at 14 TeV, involving the Z of this new 3-3-1 version. We leave for a future work the study of the effects of a Z with wide width, like the one predicted in the so-called minimal version.
To carry out our analysis, we consider the general Lagrangian for the neutral currents involving Z and Z contributions, where f stands for leptons and quarks, g is the weak coupling constant, and g V , g A , g V and g A , are the SM and 3-3-1 couplings which are presented in the  By following previous studies on Z concerning strategies for the identification of this particle on the muon channel [54][55][56], as well as the last ATLAS report [57], we have applied some cuts in order to obtain clear distributions for the invariant masses and transverse momentum of the final muons.
In agreement with the ATLAS detector performance, the cuts adopted for the pseudorapidity and the transverse momentum of the muons are: |η| < 2.5 and p T > 30 GeV. For the invariant mass of the muon pair, we have used a strong cut (M µ µ > 1000 GeV) in order to suppress the SM background.
In our simulations, we have made use of the CompHep [58] and the MadAnalysis [59] packages and adopted the CTEQ6L [60] parton distribution functions set, evaluated at the √ŝ factorization/renormalization scale, i.e., the center-of-mass energy at the parton level.
Upon assuming 3, 4 and 5 TeV for the Z mass, we observe the resonance peaks around the respective masses in the invariant mass distribution. If we consider two values for the projected LHC integrated luminosity (L = 100 fb −1 and L = 300 fb −1 ) at √ s = 14 TeV, we obtain the number of events as shown in Fig. 1. As the width of the heavy boson satisfies the relation Γ Z ∼ 5%M Z , our results are in accordance with Ref. [56].
In order to identify the new gauge boson, we also consider the muon p T distribution, where two peaks are expected to appear, corresponding to one half of the resonance masses (M Z , M Z ) but, in this case, due to the invariant mass cut adopted, the first peak moves justly to one half of this cut (∼ 500GeV ). By applying an additional cut in the muon transverse momentum, p T > 500 GeV, we obtain the distributions shown in the Fig. 2. We can identify the peak around the M Z /2 values, with M Z = 3, 4 and 5 TeV, respectively. It is clear that, for M Z = 3 and 4 TeV, the peak is well defined, and for greater masses, the peak is smooth like the SM background. Then, with a stronger cut on the invariant mass or on the muon p T , we might obtain a clearer peak for higher masses in the p T distribution.
Therefore, as claimed in [54,56], we can use the transverse momentum distribution of the final muons as an additional tool to distinguish the signal coming from the Z from the SM background.
Thus, by a simple analysis of the invariant masses and transverse momentum of the the final µ, we get clear signals for the Z . Moreover, by considering the projected luminosities for the LHC run-II, we obtain a considerable number of events revealing the existence of the new gauge boson. In a recent work [61], a phenomenological analysis of the Z has been performed for the version considered in this paper. The authors have suggested that the possibility of detecting the new particle can be achieved by considering just the production of a pair of leptons. From their analysis using, for example, the Forward-Backward asymmetry, they have concluded that at the It is beyond the scope of our work to make a detailed analysis of the final states, including Z − Z interferences, detector efficiencies, hadronization, etc. However, based on our results, it is not hard to establish the existence or to exclude the Z predicted by this model. Finally, a complete analysis involving the Z predicted by different versions of the 3-3-1 model within the next stage of the LHC energy is mandatory, but we postpone this study to a future work. The model has three extra charged leptons, E i , two up-type quarks U a , and one down-type quark D, beyond the Standard Model fermion content. Although the model with just two scalar triplets has a consistent pattern of gauge symmetry breakdown to the electromagnetic factor U(1) Q , some of the standard fermionic fields remain massless due to a residual global U(1) G symmetry which, as we observed in Eq. (12), involves diagonal generators of spontaneously broken gauge symmetries plus a sort of Peccei-Quinn symmetry. This U(1) G symmetry seems to be a common feature of 3-3-1 models with just two scalar triplets and has also been identified in another version of the model [42]. To overcome this problem, we have introduced a heavy scalar triplet with mass M w, which is integrated out from the low energy theory leaving it with effective operators breaking U(1) G explicitly, completing the mass generation mechanism for the fermions. As we have shown, the effective operators furnish a less fine-tuned mass generation for leptons and up-type quarks compared to the Standard Model. Such a mechanism, however, does not work as naturally for the standard down-quark mass hierarchy and a solution based on an additional Z 2 symmetry has also been provided. Natural hierarchies between the heavy quarks and the third family quarks, and between the third family and the lighter two families, arise, and this feature naturally suppresses the mixing between them leading to suppressed FCNC interactions.
From the phenomenological standpoint, we have explored the possibility to discover the pre- (A2) Note that there are also effective Yukawa interactions involving ρ ± 3 in (62) coming from the terms within φ 3 .