A variant of 3-3-1 model for the generation of the SM fermion mass and mixing pattern

We propose an extension of the 3-3-1 model with an additional symmetry group $Z_{2}\times Z_{4} \times U(1)_{L_g}$ and an extended scalar sector. To our best knowledge this is the first example of a renormalizable 3-3-1 model, which allows explanation of the SM fermion mass hierarchy by a sequential loop suppression: tree-level top and exotic fermion masses, 1-loop bottom, charm, tau and muon masses; 2-loop masses for the light up, down, strange quarks as well as for the electron. The light active neutrino masses are generated from a combination of linear and inverse seesaw mechanisms at two loop level. The model also has viable fermionic and scalar dark matter candidates.


Introduction
Despite the great consistency of the Standard Model (SM) with experimental data, recently confirmed by the LHC discovery of the 126 GeV Higgs boson [1,2], it has several unexplained issues [3]. Among the most pressings ones are the smallness of neutrino masses, the fermion mass and mixing hierarchy, and the existence of three fermion families.
In the SM the flavor structure of the Yukawa interactions is not restricted by gauge invariance. Consequently, fermion masses and mixings are left unfixed, and the SM does not provide an explanation for their large hierarchy, which spreads over a range of five orders of magnitude in the quark sector, and a dramatically broader range of about 11 orders of magnitude, if we include the neutrinos. Even though in the SM these parameters appear only through Yukawa interaction terms and not in explicit mass terms, this mechanism does not provide an explanation for their values, but only translates the problem to fitting different Yukawa couplings, one for each mass and with disparate values for some of them. The origin of quark mixing and the size of CP violation in this sector is also a related issue. A fundamental theory is expected to provide a dynamical explanation for the masses and mixings.
While the mixing angles in the quark sector are very small, in the lepton sector two of the mixing angles are large, and one mixing angle is small. This suggests a different kind of New Physics for the neutrino sector from the one present in the quark mass and mixing pattern. Experiments with solar, atmospheric and reactor neutrinos have brought clear evidence of neutrino oscillations from the measured non vanishing neutrino mass squared splittings. This brings compelling and indubitable evidence that at least two of the neutrinos have non vanishing masses, much smaller, by many orders of magnitude, than the SM charged fermion masses, and that the three neutrino flavors mix with each other.
The flavor puzzle of the SM indicates that New Physics has to be advocated in order to explain the prevailing pattern of fermion masses and mixings. To tackle the limitations of the SM, various extensions, including larger scalar and/or fermion sectors, as well as extended gauge groups with additional flavor symmetries, have been proposed in the literature . Recent reviews on flavor symmetries are provided in Refs. [57][58][59][60][61][62]. Another approach to describe the fermion mass and mixing pattern consists in postulating particular mass matrix textures (see Refs  for works which consider textures). In addition, the hierarchy of SM charged fermion masses can also be explained by considering the charged fermion Yukawa matrices as products of a few random matrices, which typically feature strong hierarchies in their eigenvalue spectrum, even though the individual entries are of order unity, as was recently observed in Ref. [94].
Concerning models with an extended gauge symmetry, those based on the gauge symmetry SU (3) c × SU (3) L × U (1) X , also called 3-3-1 models, which introduce a family nonuniversal U (1) X symmetry [95][96][97][98][99][100][101][102][103][104][105][106], can provide an explanation for the origin of the family structure of the fermions. These models have the following phenomenological advantages: (i) The three family structure in the fermion sector can be understood in the 3-3-1 models from the cancellation of chiral anomalies and asymptotic freedom in QCD. (ii) 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. (iii) The 3-3-1 models allow for the quantization of electric charge [107,108]. (iv) These models have several sources of CP violation [109,110]. (v) These models explain why the Weinberg mixing angle satisfies sin 2 θ W < 1 4 . (vi) These models contain a natural Peccei-Quinn symmetry, which solves the strong-CP problem [111][112][113][114]. (vii) The 3-3-1 models with heavy sterile neutrinos include cold dark matter candidates as weakly interacting massive particles (WIMPs) [115][116][117][118]. A concise review of WIMPs in 3-3-1 Electroweak Gauge Models is provided in Ref. [119].
In most versions of 3-3-1 models, one heavy triplet field with a Vacuum Expectation Value (VEV) at a high energy scale breaks the symmetry SU (3) L × U (1) X into the SM electroweak group SU (2) L ×U (1) Y , thus generating masses for the non SM fermions and non SM gauge bosons, while other two lighter triplets with VEVs at the electroweak scale, trigger the Electroweak Symmetry Breaking [80] and provide the masses for the SM particles. To provide an explanation for the observed pattern of SM fermion masses and mixings, various 3-3-1 models with flavor symmetries [33][34][35][36][37][38][39][40][41][42][43][44][45][120][121][122][123] and radiative seesaw mechanisms [80,120,[124][125][126][127][128][129][130][131][132][133] have been proposed in the literature. However, some of them involve non renormalizable interactions [39,40,42,43,45], others are renormalizable but do not address the observed pattern of fermion masses and mixings due to the unexplained huge hierarchy among the Yukawa couplings [33, 35-38, 122, 123, 134] and others are only focused either in the quark mass hierarchy [34,128,130], or in the study of the neutrino sector [120, 124-127, 129, 132, 133, 135, 136], or only include the description of SM fermion mass hierarchy, without addressing the mixings in the fermion sector [131]. It is interesting to find an alternative explanation for the observed SM fermion mass and mixing pattern, in the framework of 3-3-1 models, by considering that it arises by a sequential loop suppression, so that the masses are generated according to: three level top quark mass, one loop level bottom, charm, tau and muon masses and two loop level masses for the light up, down and strange quarks as well as for the electron and neutrinos. This way of generating the SM fermion mass hierarchy was proposed for the first time in Ref. [137]. However, the proposed model includes non-renormalizable Yukawa terms with a quite low cutoff scale. In this paper we propose the first renormalizable extension of the 3-3-1 model with the electric charge constructed from the SU (3) L generators as Q = T 3 + βT 8 + XI with β = − 1 √ 3 . The model explains the SM fermion mass and mixing pattern by a sequential loop suppression mechanism.
The paper is organized as follows. In section 2 we present the theoretical setup of the proposed model. In section 3 we discuss the quark masses and mixings within the model, while the discussion of the lepton masses and mixings is given in section 4.

The model
The 3 and right-handed Majorana neutrinos in the SU (3) L lepton triplet was proposed for the first time in [138]. However, the observed pattern of fermion masses and mixings was not addressed at that time due to the unexplained huge hierarchy among the Yukawa couplings [122,129,134]. Here we propose the first renormalizable extension of the 3-3-1 model with the parameter β = − 1 √ 3 , which includes a loop suppression mechanism to generate the observed pattern of the SM fermion masses and mixings. In our model only the top quark and the charged exotic fermions acquire tree level masses, whereas the remaining SM fermions get their masses via radiative corrections: 1 loop bottom, charm, tau and muon masses; 2-loop masses for the light up, down, strange quarks as well as for the electron. Light active neutrinos acquire their masses from a combination of linear and inverse seesaw mechanisms at two loop level, and the quark mixings arise from a combination of one and two loop level effects.
In order to realize this scenario we extend the SU (3) C × SU (3) L × U (1) X group with an extra Z 4 × Z 2 discrete group, where the Z 4 symmetry is softly broken and the remaining Z 2 symmetry is broken both spontaneously and softly. We also introduce a global U (1) Lg of the generalized lepton number L g [127], which is spontaneously broken down to a residual discrete Z (Lg) 2 lepton number symmetry by a VEV of a gauge-singlet scalar ξ 0 to be introduced below. The corresponding massless Goldstone boson, Majoron, is phenomenologically harmless being a gauge-singlet. The full symmetry G of the model experiences a two-step spontaneous breaking, as follows: where the different symmetry breaking scales satisfy the following hierarchy which corresponds in our model to the VEVs of the scalar fields to be introduced below. In the 3-3-1 model under consideration, the electric charge is defined as [96,122,138]: where T 3 and T 8 are the SU (3) L diagonal generators, I is the 3 × 3 identity matrix and X is the U (1) X charge. Different versions of the 3-3-1 models are determined by the choice of the β parameter, which is related to the different possible fermion assignments. The most studied versions of 3-3-1 models have β = ± 1 √ 3 [95,138] and β = ± √ 3 [97,99,139], and if we want to avoid exotic charges we are led to only two different models: contain non SM fermions with non-exotic electric charges, i.e., equal to the electric charge of some SM fermions [140][141][142]. Those with β = ± √ 3 have non SM fermions with large exotic electric charges and require a departure from the perturbative regime at a scale of several TeV, in order to successfully account for the measured value of the weak mixing angle at low energies, as shown in detail in Ref. [143]. Other versions of 3-3-1 models have β = 0, ± 2 √ 3 and contain non SM particles with fractional electric charges [144]. For instance, 3-3-1 models with β = 0 contains exotic quarks and exotic charged leptons with electric charges 1 6 and − 1 2 , respectively [144]. Since electric charge conservation implies that the lightest exotic particles of the 3-3-1 models with β = 0, ±2/ √ 3 should be stable, the phenomenological viability of such models requires a detailed analysis of the abundance of such stable exotic charged particles in cosmology.
For these reasons, 3-3-1 models with β = − 1 √ 3 have advantages over those with β = 0, ± 2 √ 3 , ± √ 3. In addition, choosing β = − 1 √ 3 implies that the third component of the weak lepton triplet is a neutral field ν C R , which allows building the Dirac matrix with the usual field ν L of the weak doublet. If one introduces a sterile neutrino N R in the model, the light neutrino masses can be generated via low scale seesaw mechanisms, which could be inverse or linear. The 3-3-1 models with β = − 1 √ 3 can also provide an alternative framework to generate neutrino masses, where the neutrino spectrum includes the light active sub-eV scale neutrinos, as well as sterile neutrinos, which could be dark matter candidates, if they are light enough, or candidates for detection at the LHC, if their masses are at the TeV scale. Therefore, pair production of TeV scale sterile neutrinos via the Drell-Yan mechanism at the LHC could be a signal supporting models with extended gauge symmetries such as the 3-3-1 models. In addition, Drell-Yan heavy vector pair production processes at the LHC may help to distinguish the 3-3-1 models from other models with extended gauge symmetry. With respect to the quark spectrum, we assign each of the first two families of quarks to an SU (3) L antitriplet 3 * , whereas the third family is assigned to a SU (3) L triplet 3, as required by the SU (3) L anomaly cancellation condition. Therefore, considering that there are 3 quark colors, we have six 3 * irreducible representations. In addition, there are six SU (3) L triplets 3 of fermionic fields, considering the three lepton families. Thus, the SU (3) L representations are vector like and anomaly free. The quantum numbers for the fermion families are assigned in such a way that the combination of the U (1) X representations with other gauge sectors is anomaly free. As a consequence, one finds that the number of chiral fermion generations is an integer multiple of the number of colors, which provides an explanation for the existence of three generations of quarks and leptons in terms of the 3 colors. The U (1) X -charge assignments of the fermionic fields are obtained from Eq. (2.3) and the requirement of reproducing the electric charges of the SM quarks and leptons. Then the U (1) X charge of the first two families of quark antitriplets is where U iL and D iL (i = 1, 2, 3) are the left handed up and down type quarks fields in the flavor basis, respectively. The right handed SM quarks, i.e., U iR and D iR (i = 1, 2, 3) and right handed exotic quarks, i.e., T R and J nR (n = 1, 2) are assigned to be SU (3) L singlets with U (1) X quantum numbers equal to their electric charges. Furthermore, the requirement of chiral anomaly cancellation constrains the leptons to the following SU (3) C × SU (3) L × U (1) X left-and right-handed representations [96,101,145]: where ν iL , ν c ≡ ν c R and e iL (e L , µ L , τ L ) are the neutral and charged lepton families, respectively. Let us note that we assign the right-handed leptons to SU (3) L singlets, which implies that their U (1) X quantum numbers correspond to their electric charges.   To implement the radiative seesaw mechanisms that generate the observed hierarchy of the SM charged fermion masses and mixing angles by a sequential loop suppression and the light active neutrino masses from a combination of linear and inverse seesaw mechanisms at two loop level, we extend both the fermion and the scalar sectors of the 3-3-1 models with β = − 1 √ 3 previously considered in the literature. We introduce SU (3) L singlet exotic up type quarks T L,R , down type quarks B L,R and charged leptons E L,R as well as four gauge group Eq.
The U (1) Lg ×Z 4 ×Z 2 assignments for all the fermions of the model are shown in Tables 1, 2.
The spontaneous symmetry breaking (2.1) in our model is triggered by the VEVs (2.2) of the scalar fields χ, η and ξ 0 , neutral under the Z 4 discrete symmetry. As seen from (2.1), the first stage of the breaking is done by a TeV scale VEV v χ of an SU (3) L triplet χ handing masses to the non-SM fermions and gauge bosons as well as by the TeV scale VEV v ξ of the gauge-singlet scalar ξ 0 , which spontaneously breaks the generalized lepton number symmetry U (1) Lg . The corresponding Majoron is a gauge-singlet and, therefore, unobservable. Note that Lepton Number (LN) is broken together with Generalized Lepton Number (GLN) by the VEV of ξ 0 , which has both LN and GLN equal to −2. Since the gauge singlet scalar ξ 0 breaks U (1) Lg in a way that respects the condition |∆L g | = |∆L| = 2, there survives a residual discrete Z Let us explain the VEV pattern of the SU (3) L scalar triplets χ and η. Since the χ triggers the SU (3) L × U (1) X → SU (2) L × U (1) Y breaking, the following conditions have to be fulfilled: whereas the remaining generators do not leave the vacuum χ invariant. From the first three conditions for χ given in Eq. (2.9), it follows that: The last condition in Eq. (2.9) for χ , i.e, (βT 8 + XI) χ = 0, yields the following relation between the U (1) X charge of the SU (3) L scalar triplet χ and the β parameter: Requiring that all the SU (3) L generators are broken, with the exception of the electric charge generator Q, we arrive at the following VEV pattern From the requirement of the U (1) EM invariance we have thus producing the following relation for the U (1) X charge X η of the η field: as indicated by Eq. (2.8). Note that, the difference between the η and χ Higgs triplets can be explained using the generalized lepton number L g , discussed in Appendix A. Its values for the fields of the model are specified in Tables 1-3.
The choice of the VEV structure in (2.10) and (2.12) shows that only the neutral Higgs field without lepton number is allowed to have the VEV. In addition, the patterns of the SU (3) L scalar triplets χ and η shown in Eq. (2.10) and (2.12) are consistent with a global minimum of the scalar potential of our model for all the region of parameter space. We adopt X ρ = 2/3 for another SU (3) L scalar triplet ρ in Eq. (2.8) from the simplified versions of the 3-3-1 model [101,102,138], where both η and ρ scalars participate in the electroweak symmetry breaking. The extra SU (3) L scalar triplet ρ is introduced in simplified versions of the 3-3-1 models to give masses to charged leptons, as well as to the bottom, up and charm quarks. In our model the SU (3) L scalar triplet ρ is crucial to give one loop level masses for the bottom and charm quarks, to the tau and muon leptons as well as two loop level masses for the up, down and strange quarks as well to the electron, as shown in Figs. 1, 2. The SU (3) L scalar triplet ρ also contributes to some entries of the neutrino mass matrix as indicated in Fig. 3. On the other hand, the conditions similar to (2.12), (2.13) are applied to ρ as well and lead to X ρ = 2/3. In our model we have ρ = 0 due to the Z 4 conservation (2.1), and the above symmetry breaking conditions do not restrict X ρ . We choose X ρ = 2/3 in order to maintain resemblance with the previous versions of the 3-3-1 model. Another motivation for the choice X ρ = 2/3 is the U (1) X invariance of the SU (3) L invariant trilinear scalar interaction χηρ. Let us note that our choice β = − 1 √ 3 yields X η = X χ = − 1 3 , which in turn leads to X ρ = 2/3. With the above particle content, the relevant quark and lepton Yukawa terms invariant under the symmetry group (2.1) of our model take the form: where the dimensionless parameters in Eqs.
(2.17) After the spontaneous breaking of the electroweak symmetry, the above-given Yukawa interactions generate the observed hierarchy of SM fermion masses and mixing angles by a sequential loop suppression, provided that one introduces the Z 4 × Z 2 soft breaking mass terms for the electroweak singlet fermions: as well as soft Z 4 × Z 2 breaking in the electroweak singlet scalar sector: Let us note that in the simplified version of the 3-3-1 model with β = − 1 √ 3 , whose scalar sector contains three SU (3) L scalar triplets, the flavor constraints can be fulfilled by considering the scale of breaking of the SU (3) L × U (1) X gauge symmetry much larger than the electroweak symmetry breaking scale v = 246 GeV, which corresponds to the alignment limit of the mass matrix for the CP-even Higgs bosons [147]. Our model has a more extended scalar sector since it is composed of three SU (3) L scalar triplets (from which one is inert SU (3) L triplet) and six SU (3) L scalar singlets. Consequently, following Ref. [147], we expect that the FCNC effects as well as the constraints arising from K 0 −K 0 , B 0 −B 0 and D 0 −D 0 mixings will be fulfilled in our model, by considering the scale of breaking of the SU (3) L × U (1) X gauge symmetry much larger than the scale of breaking of the electroweak symmetry. The scalar sector of our model is not predictive as its corresponding scalar potential has many free uncorrelated parameters that can be adjusted to get the required pattern of scalar masses. Therefore, the loop effects of the heavy scalars contributing to certain observables can be suppressed by the appropriate choice of the free parameters in the scalar potential. Fortunately, all these adjustments do not affect the charged fermion and neutrino sector, which is completely controlled by the fermion-Higgs Yukawa couplings.
Despite the fact that the scalar and fermion sectors of our model are considerably larger than the corresponding to the simplified version of the 3-3-1 model with β = − 1 √ 3 , and the fields assignments under the discrete group Z 4 × Z 2 look rather sophisticated, each introduced element plays its own role in the implementation of the radiative seesaw mechanisms that allow us to explain the SM fermion mass hierarchy by a sequential loop suppression. In what follows we provide a justification and summary of our above presented model setup: 1. The spontaneously and softly broken Z 2 symmetry is crucial to generate a two loop level electron mass as it distinguishes the first generation left leptonic triplet, i.e., L 1L , neutral under Z 2 from the second and third generation ones i.e., L 2L and L 3L which are Z 2 even. This symmetry also separates the SM right handed charged leptonic field, i.e, e 1R , which is Z 2 odd from the remaining SM right handed charged leptonic fields, i.e, e 2R and e 3R , neutral under the Z 2 symmetry. This results in one loop level tau and muon lepton masses and a two loop level mass for the electron.
2. The softly broken Z 4 symmetry separates the third generation left handed quark fields from the first and second generation ones, giving rise to a tree level top and exotic quark masses and to radiatively generated masses for the remaining quarks. Besides that, the Z 4 symmetry differentiates the second generation right handed SM up quark fields, i.e., U 2R , charged under this symmetry, from the first generation SM one, i.e., U 1R , which is Z 4 neutral, thus giving rise to a one loop level charm quark mass and two loop level up quark mass.
3. The scalar sector of our model is composed of three SU (3) L scalar triplets, i.e., χ, η and ρ, seven SU (3) L scalar singlets, from which three are electrically neutral, i.e., ϕ 0 1 , ϕ 0 2 and ξ 0 and four electrically charged, i.e., φ + 1 , φ + 2 , φ + 3 and φ + 4 . The inclusion of the spontaneously and softly broken Z 2 symmetry requires the introduction of a SU (3) L scalar singlet φ + 3 , which is odd under this symmetry. The presence of the SU (3) L scalar singlet φ + 3 , is needed in order to build the Z 2 invariant trilinear scalar interactions required to generate two loop level down and strange quark masses, as shown in Fig. 1. Besides that, in order to implement a two loop level radiative seesaw mechanism for the generation of the up, down and strange quark masses as well as the electron mass, the Z 4 charged SU (3) L scalar singlets ϕ 0 1 , ϕ 0 2 , φ + 1 , φ + 2 (which do not acquire a vacuum expectation value) are also required in the scalar sector. The Z 4 charged SU (3) L scalar singlet ϕ 0 1 is also needed for the implementation of the one loop level radiative seesaw mechanism that generates the charm, the bottom quark masses as well as the tau and muon lepton masses, as shown in Fig. 2. The Z 4 charged SU (3) L scalar singlets ϕ 0 2 and φ + 3 as well as the SU (3) L scalar singlet φ + 4 , neutral under Z 4 are also crucial for the implementation of two loop level linear and inverse seesaw mechanisms that give rise to the light active neutrino masses. The SU (3) L scalar singlet ξ 0 is introduced to spontaneously break the U (1) Lg generalized lepton number symmetry and thus giving rise to a tree-level mass for the right handed Majorana neutrino Ψ R . Lets us note that ξ 0 is the only electrically neutral SU (3) L scalar singlet that has a non-vanishing generalized Lepton Number L g . It is crucial for generating two loop-level masses for the down and strange quarks. This is due to the fact that the electrically charged SU (3) L scalar singlets φ + 2 and φ + 3 appearing in the two loop level diagrams that give rise to the down and strange quark masses carry non-vanishing generalized Lepton Numbers thus implying that the quartic scalar interaction λ 3 φ − 3 ρη † ξ 0 is crucial to generate the masses for the down and strange quarks, as shown in Fig. 1. Note that we assign non-vanishing generalized Lepton Numbers for φ + 2 and φ + 3 because φ + 3 mix with φ + 4 as well as with φ + 2 via the soft breaking mass terms of Eq.  1, 2, 3), Ψ R . The SU (3) L singlet exotic down type quarks, i.e. B, is crucial for the implementation of the one loop level radiative seesaw mechanism that generate the bottom quark mass. The SU (3) L singlet exotic up type quarks, i.e., T 1 and T 2 , are needed to generate a one loop level charm quark mass as well as two loop level down and strange quark masses. The three SU (3) L singlet exotic charged leptons, i.e., E j (j = 1, 2, 3), are required in order to provide the radiative seesaw mechanisms that generate one loop level tau and muon masses and two loop level electron mass. The four right handed Majorana neutrinos, i.e., N jR (j = 1, 2, 3), Ψ R , are crucial for the implementation of the two loop level linear and inverse seesaw mechanisms that give rise to the light active neutrino masses. It is worth mentioning that out of these four right handed Majorana neutrinos, only Ψ R acquires a tree level mass, whereas the three remaining right handed Majorana neutrinos, i.e., N jR (j = 1, 2, 3), get their masses via a one loop level radiative seesaw mechanism mediated by Ψ R and ϕ 0 2 , as shown in Fig. 3.
In what follows we briefly comment on some phenomenological aspects of our model concerning LHC signals of non-SM fermions. From the quark Yukawa interactions it follows that the heavy exotic SU (3) L singlet down (up) type quark(s), i.e., B ( T n (n = 1, 2)) will decay predominantly into a SM down (up) type quark and the Reϕ 0 1 or Imϕ 0 1 neutral scalar, which is identified as missing energy, due to the preserved Z 4 symmetry. Furthermore, from the lepton Yukawa interactions it follows that the heavy SU (3) L singlet exotic charged leptons, i.e. E j (j = 1, 2, 3), will have a dominant decay mode into a SM charged lepton and a neutral CP even ξ ρ or CP odd ζ ρ scalar state, which can also be identified as missing energy, due to the preserved Z 4 symmetry. The exotic SU (3) L singlet up type quarks, i.e. T 1 and T 2 and down type quark, i.e., B, are produced in pairs at the LHC via gluon fusion and the Drell-Yan mechanism, and the charged exotic leptons E j (j = 1, 2, 3) are also produced in pairs but only via the Drell-Yan mechanism. Thus, observing an excess of events with respect to the SM background in the dijet and opposite sign dileptons final states at the LHC, can be a signal in support of this model. With respect to the exotic T , J 1 or J 2 quarks, they mainly decay into a top quark and either neutral or charged scalar. The precise signature of the decays of the exotic quarks depends on details of the spectrum and other parameters of the model. The present lower limits on the Z gauge boson mass in 3-3-1 models arising from LHC searches, reach around 2.5 TeV [148]. These bounds can be translated into limits of about 6.3 TeV on the SU (3) C × SU (3) L × U (1) X gauge symmetry breaking scale v χ . Furthermore, electroweak data from the decays B s,d → µ + µ − and B d → K * (K)µ + µ − set lower bounds on the Z gauge boson mass ranging from 1 TeV up to 3 TeV [146,[149][150][151][152]. The exotic quarks can be pair produced at the LHC via Drell-Yan and gluon fusion processes mediated by charged gauge bosons and gluons, respectively. A detailed study of the exotic quark production at the LHC and the exotic quark decay modes is beyond the scope of this work and is deferred for a future publication.
Furthermore, from the quark Yukawa terms of Eq. (2.16), it follows that the flavor changing top quark decays t → hc, t → hu and t → cZ are absent in our model. Besides that, the decays of charged Higgses into a SM up-type and SM down-type quarks, namely, s, b), are forbidden at tree level in our model. Out of the charged Higgs decays into SM quarks, only the decays H + 1 → u 1dn , H − 1 →ū 1 d n (n = 1, 2) appear at one loop level whereas the decays H + 1 → u 2dn , H − 1 →ū 2 d n (n = 1, 2) are allowed at two loop level. In addition, the dominant SM leptonic decay modes of the charged Higgses H ± 1 and H ± 2 only appear at one loop level and correspond to the processes H ± 1 → ν 1 e ± and H ± 2 → ν c 1 e ± . The remaining decay modes H ± 1 → ν 1 µ ± , H ± 2 → ν c 1 µ ± , H ± 1 → ν 1 τ ± , H ± 2 → ν c 1 τ ± are very tiny with respect to the decays H ± 1 → ν 1 e ± and H ± 2 → ν c 1 e ± , due to the very small mixing angles in the rotation matrix that connects the SM right handed charged leptonic fields in the interaction eigenstates with the physical SM right handed charged leptonic fields. Consequently, a measurement of the branching fraction for the t → hc, t → hu, t → cZ, τ ± decays at the LHC will be crucial for ruling out this model.

Tadpole cancellation mechanisms
Notice that after Z 2 × Z 4 is softly broken, the terms E nL ϕ 0 1 E mR and (m E ) nm E nL E mR (m, n = 2, 3) will generate a tadpole for ϕ 0 1 . Since this contribution is known to give an infinite value, in order to make the theory renormalizable without giving a VEV to the ϕ 0 1 , one has to consider also the contribution to the ϕ 0 1 tadpole arising from the scalar interaction ω 1 ϕ 0 1 (ϕ 0 2 ) 2 with the virtual ϕ 0 2 in the loop. We require that these two tadpoles cancel so that ϕ 0 1 = 0 be guaranteed at one-loop level. This requirement of tadpole cancellation is an ad hoc condition of viability of our model. It implies fine-tuning of the model parameters (m E ) nm and ϕ 0 1 (ϕ 0 2 ) 2 , which is unstable under the renormalization flow. In our model we do not have a symmetry to stabilize the required tadpole cancellation. Moreover, it is not possible to introduce such a symmetry without a radical modification of the model structure with all its nice features. The solution to this problem can be expected from the appropriate imbedding of our model into a more fundamental setup with additional symmetries protecting the tadpole cancelation. Given that this condition relates the parameters of the fermionic and scalar sector one may think of imbedding our model into a supersymmetric or warped five-dimensional framework (see Refs. [153,154] for recent reviews on extra-dimensions). Thinking of a supersymmetric (SUSY) version of our model (for some examples of SUSY 3-3-1 models see Refs. [121,131,[155][156][157][158][159][160][161][162][163][164][165][166][167][168][169][170]) we hope that even in the case of softly-broken SUSY the tadpole cancelation would be technically natural. More conservatively we may expect a violation of this cancelation not stronger than logarithms of the high-scale cutoff. In this case ϕ 0 1 = 0, but due to the logarithmic sensitivity to the cutoff, it would be around the electroweak scale. This is phenomenologically safe, giving rise to tree level mixingF L f R between an exotic, F , and a SM, f , charged fermions. Despite the presence of this mixing terms, the first and second rows of the up type quark mass matrix as well as the first three rows of the down type quark and charged lepton mass matrices will still be vanishing at tree level, which is a consequence of the symmetries of the model as well as from the fact that the SU (3) L scalar triplet ρ is inert. This implies that only the top quark and exotic fermions do acquire tree level masses, whereas the remaining SM fermions will be massless at tree level. The masses for the remaining SM fermions will still appear via the radiative seesaw mechanisms described in the previous subsection. The implementation of supersymmetry or embedding our model in a warped extra-dimensional setup, requires careful studies, which are beyond the scope of the present paper and will be addressed elsewhere.

Quark masses and mixings
From the quark Yukawa interactions (2.16) it follows that the SM quark mass matrices are given by: where y 1 is generated at tree level from the renormalizable Yukawa interaction Q 3L ηU 3R , thus giving rise to a tree level top quark mass. Furthermore, ε  Fig. 1. In what follows we will show that the SM quark mass matrices given above are consistent with the low energy quark flavor data. To this end, and considering that the ε where l ≈ (1/4π) 2 ≈ 2.0×λ 4 is the loop suppression factor and λ = 0.225 is the Wolfenstein parameter. Then we expect in the model that a nj (n, m = 1, 2 and j = 1, 2, 3) be O(1) parameters.
Let us note that the large amount of independent model parameters in the fermion and scalar sectors of our model, entering in the Feynman diagrams contributing to the entries of the SM fermion mass matrices, can be absorbed in the effective parameters ε being t, u, c, d, s, b quark masses, θ 12 , θ 23 , θ 13 mixing angles and the Jarlskog parameter. Therefore, the model in its present form does not predict these observables. However, as we already commented, we only pretend to reproduce the hierarchy of the quark masses via the loop suppression predicted by the model and expressed by Eq.  2)). Thus the model reproduces fairly well the hierarchical structure of the observable quark mass spectrum as a result of the sequential loop suppression mechanism, where the top quark mass is generated at tree level, the masses for the bottom and charm quarks arise at one loop level and the light up, down and strange quarks get their masses at two loop level.
the form: where ε (l) jn (n = 2, 3 and j = 1, 2, 3) are dimensionless parameters generated at one loop level, whereas the dimensionless parameter ε (l) 11 arises at two loop level. In order to express the loop order suppression explicitly we define new parameters 11 · l 2 , j = 1, 2, 3.  All the values are not unnaturally small compared to the loop hierarchy (2-loop level)/(1loop level)∼ l ≈ 6.3 × 10 −3 . Thus, as in the quark sector, the model proves to be able to reproduce the observed charged lepton mass hierarchy by a sequential loop suppression. From the neutral lepton Yukawa interactions in Eq. (2.16) we find the neutral lepton mass terms: where the neutrino mass matrix M ν is generated by the loop diagrams shown in Fig. 3.  where the matrix elements d l (l = 1, 2, · · · , 6) arise at tree level, ij , ε j and M ij (i, j = 1, 2, 3) at one loop level, whereas a nm , a 1n , b 1n and b nm (n, m = 2, 3) arise at two loop level. Let us note that a 1n and b 1n are generated by the h (L) ρ11 and h (L) ρnm terms in Eq. (2.16). These terms give rise to the four Feynman diagrams shown in the last two lines of Fig. 3. Consequently, the light active neutrino masses are generated by a combination of linear and inverse seesaw mechanisms at two loop level.
By performing the perturbative block diagonalization of the 9 × 9 neutrino mass matrix M ν of Eq. (4.5), which is shown in Appendix B, we find that the light active neutrino mass matrix has the form: Let us analyze Eq. (4.8) and see what are the typical mass scales of the model, which allow us to reproduce the neutrino mass scale of m ν ∼50 meV. The non-zero matrix elements of M 1 are determined by the 2-loop diagrams in Fig. 3. For the benchmark region where (4.9) This 2-loop-level contribution has typical inverse or linear seesaw structure proportional to the soft symmetry breaking parameter µ 2 1 . This parameter is stable against radiative corrections due to the model symmetries and, therefore, any of its possible values is technically natural. Then we can choose it arbitrarily small to adjust the observable neutrino mass scale m ν ∼ 50 meV. Note that a ij → 0 in the limit m Ψ → 0, since m Ψ is the only Lepton Number Violating parameter in our model.
The second term in Eq. vχ with respect to the ones of M 3ν , as indicated by Eq. (4.8). These heavy quasi Dirac neutrinos can be produced in pairs at the LHC, via a Drell-Yan mechanism, mediated by a heavy non Standard Model neutral gauge boson Z . The heavy quasi Dirac neutrinos can decay into a Standard Model charged lepton and W gauge boson, due to their mixings with the light active neutrinos. Thus, the observation of an excess of events in the dilepton final states with respect to the SM background at the LHC would be a signal supporting this model. Studies of inverse seesaw neutrino signatures at the LHC and ILC as well as the production of Heavy neutrinos at the LHC are performed in Refs. [174,175]. A detailed study of the collider phenomenology of this model is beyond the scope of the present paper and is left for future studies.

Discussions and conclusions
We have built the first renormalizable extension of the 3-3-1 model with β = − 1 √ 3 , which explains the SM fermion mass hierarchy by a sequential loop suppression. Our model, based Re ϕ 0 2 , Im ϕ 0 on the 3-3-1 symmetry extended with the U (1) Lg × Z 4 × Z 2 group is consistent with the low energy fermion flavor data. In the model only the top quark and the charged exotic fermions acquire tree level masses, whereas the remaining SM fermions get their masses via radiative corrections: 1 loop bottom, charm, tau and muon masses; 2-loop masses for the light up, down, strange quarks as well as for the electron. Furthermore, the light active neutrinos acquire their masses from a combination of linear and inverse seesaw mechanisms at two loop level. In our model the quark and lepton mixings arise from radiative effects. At tree level there is no quark mixing, the mixing angles in the quark sector are generated from a combination of one and two loop level radiative seesaw mechanisms. In the lepton sector, the contribution to the leptonic mixing angles coming from the charged leptons arise at one loop level, whereas the mixings in the light active neutrino sector are generated from a two loop level radiative seesaw mechanism.
Furthermore our model predicts the absence of the decays t → hc, t → hu, t → cZ, which implies that a measurement of the branching fraction for these decays at the LHC will be crucial for ruling out the model. Consequently, charged Higgses can be searched at the LHC through the their decay into a SM up-type (downtype) and a exotic SM down-type B ( T k (k = 1, 2) up-type) quarks, as well as into a exotic charged lepton and neutrino. Since the heavy exotic SU (3) L singlet down (up) type quark(s), i.e., B ( T n (n = 1, 2)) will decay predominantly into a SM down (up) type quark and the Reϕ 0 1 or Imϕ 0 1 neutral scalar (which is identified as missing energy, due to the preserved Z 4 symmetry), it follows that the observation of an excess of events with respect to the SM background in the dijet final states at the LHC can be a signal of charged Higgs decays of this model. Finally, it is worth mentioning that since charged exotic fermions are produced in pairs, and they predominantly decay into a SM charged fermion and a electrically neutral Z 4 charged scalar (identified as a missing energy), observing an excess of events with respect to the SM background in the dijet and opposite sign dilepton final states at the LHC can be a signal in support of this model. A detailed study of the exotic charged fermion production at the LHC and the exotic charged fermion decay modes is beyond the scope of this work and is deferred for a future publication.
The final remark deals with the possible DM candidates in our model, which could be either the right handed Majorana neutrinos N iR (i = 1, 2, 3), Ψ R , or the lightest scalars ϕ R 1 ≡ Reϕ 0 1 , ϕ I 1 ≡ Imϕ 0 1 as well as ϕ R 2 ≡ Reϕ 0 2 , ϕ I 2 ≡ Imϕ 0 2 . Let us note that the masses m R,I 1 , m R 2 of the scalars ϕ R,I 1 , ϕ R 2 and the fermion Ψ R mass m Ψ are arbitrary parameters, since the corresponding mass terms are compatible with all the symmetries of the model, while ϕ I 2 squared mass is (m R 2 ) 2 − 4µ 2 1 . The mass splitting parameter µ 2 1 is the soft Z 4 breaking mass (2.19), which already showed up in the light neutrino sector (4.9). Since the light neutrino mass scale should be small, then the ϕ R 2 − ϕ I 2 mass splitting should not be very large. A superficial survey shows that N iR could be a DM candidate only in a rather restricted domain of the model parameter space, due to the presence of N R − ν L -mixing U νN at least at one-loop level (fifth diagram in Fig. 3). As a result, there is the SM charged current decay N R → e − L ν 1 e + R . The requirement that the DM lifetime be greater than the universe lifetime sets stringent constraints on U νN . We do not analyze the impact of this constraint on the model parameter space and the possible correlations of the DM and the light neutrino sectors. Instead we consider the other more viable DM candidates. The gauge group singlet Ψ R is one of them, if its mass satisfies the condition m Ψ < m R 2 , and then, as follows from Eq. (2.16), it does not decay at tree level. Assuming that our model be valid only up to some high-energy scale Λ v χ , we have to consider the possible nonrenormalizable operators induced by the physics beyond this scale. It is easy to check that all such operators compatible with the symmetry group G of our model involve the exotic scalar ϕ 0 2 . The lowest dimensional operator is Therefore, with the condition m Ψ < m Ψ 2 the non-renormalizable operators do not lead to kinematically allowed decays of Φ R , making it a stable DM particle. There is also a viable scalar DM candidate ϕ 0 in our model. This is the lightest of the exotic scalars Reϕ 0 1,2 , Imϕ 0 1,2 , which is also lighter than the exotic charged fermions, as well as lighter than Ψ R , and then, as follows from Eqs. (2.16), (2.16), its tree-level decays are kinematically forbidden. However, as before, we check the possible non-renormalizable operators originating from the scales Λ, above the theoretical validity of our model. In the case of the DM candidate ϕ 0 ≡ Reϕ 0 1 or Imϕ 0 1 , we find the dominant operator compatible with all the symmetries of our model. This operator induces the decays ϕ 0 → e + 1 e − 2,3 ξ η , ϕ 0 → e + 1 e − 2,3 ζ η , ϕ 0 → e + 1 e − 2,3 ξ χ , ϕ 0 → e + 1 e − 2,3 ζ χ , ϕ 0 → e + 1 e − 2,3 , respectively. Here ξ η = cos αh 0 + sin αH 0 1 , ζ η G 0 1 , with tan θ ∼ O( v vχ ) and α a mixing angle, which depends on the scalar potential parameters. Furthermore, h is the 126 GeV SM Higgs boson, H 0 1 is one of the physical heavy neutral Higges, whereas G 0 1 is the Goldstone boson associated with the longitudinal component of the Z gauge boson. For the scenario where the scalar ϕ 0 is heavier than the 126 GeV Higgs, the partial decay rates of the kinematically allowed processes can be estimated as Requiring that the DM candidate ϕ 0 lifetime be greater than the universe lifetime τ u ≈ 13.8 Gyr, taking into account the limit (4.11) and assuming m ϕ 0 ∼ 1 TeV, we estimate the cutoff scale of our model Λ > 3 × 10 10 GeV. (5.4) Thus we conclude that under the above specified conditions the model contains viable fermionic Ψ R and scalar ϕ 0 DM candidates. A detailed study of the dark matter constraints in our model is beyond the scope of the present paper and will be considered elsewhere.
Since the lepton and anti-lepton lie in the triplet, the lepton number operator L does not commute with the SU (3) C × SU (3) L × U (1) X gauge symmetry and has the form [127] where the upper and lower signs correspond to triplet and antitriplet of SU (3) L , respectively. Here L g is a conserved charge corresponding to the U (1) Lg global symmetry, which commutes with the gauge symmetry. According to the analysis done in Ref. [127], the SU (3) L Higgs triplets χ and η have different L g charges, which are given by:

B Perturbative diagonalization of the neutrino mass matrix
In this appendix we show explicitly the perturbative diagonalization of the 9 × 9 neutrino mass matrix M ν of our model, which is given by Eqs. (4.4)-(4.7). The elements of the submatrices M 1,2,3,4 obey the following hierarchy: with i, j = 1, 2, 3. We first apply the following orthogonal transformation to the matrix M ν : where I is the 3 × 3 identity matrix.
Then, a second orthogonal transformation is applied under the matrix M ν , as follows: where the rotation matrix R 1ν is given by: The partial diagonalization condition: Now, a third orthogonal transformation is applied under the matrix M ν : and the rotation matrix R 2ν is given by: The resulting partial diagonalization condition: yields the following relation: