A $U(1)_{X}$ extension to the MSSM with three families

We propose a supersymmetric extension of the anomaly-free and three families nonuniversal $U(1)$ model, with the inclusion of four Higgs doublets and two Higgs singlets. The quark sector is extended by adding three exotic quark singlets, while the lepton sector includes two exotic charged lepton singlets, three right-handed neutrinos and three sterile Majorana neutrinos to obtain the fermionic mass spectrum. By implementing an additional $\mathbb{Z}_2$ symmetry, the Yukawa coupling terms are suited in such a way that the fermion mass hierarchy is obtained without fine-tuning. The effective mass matrix for SM neutrinos is fitted to current neutrino oscillation data to check the consistency of the model with experimental evidence, obtaining that the normal-ordering scheme is preferred over the inverse ones. The electron and up, down and strange quarks are massless at tree level, but they get masses through radiative correction at one loop level coming from the squark and gluino or squark and Higgsinos contributions. We show that the model predicts a like-Higgs SM mass at electroweak scale by using the VEV according to the symmetry breaking and fermion masses.


I. INTRODUCTION
Regardless the success of the Standard Model of electroweak interactions (SM) [1] in explaining the experimental data, it is considered an incomplete model since some features remain satisfactorily unexplained. Among them, there is the fermion mass hierarchy problem as well as naturalness problem; both have motivated many extensions of the SM, or even complete new theories. In the case of supersymmetry, it is the model which best explains the Higgs mass naturalness thanks to the exact cancellation of the quadratic divergences between contributions of particles and superpartners in the Higgs mass radiative corrections, providing a finite mass value for the particle.
When considering the Minimal Supersymmetric Standard Model (MSSM) superpotential, there exists a mass-like parameter for the bilinear superfields coupling called µ, which is responsible of the Higgs and Higgsino masses. This parameter is expected to be at the order of SUSY breaking scale to provide Higgsino masses. However, the lightest Higgs mass is at the electroweak scale. Thus, it can not provide the correct neutralino masses and a phenomenological Higgs mass in accordance with the data presented by ATLAS and CMS experiments [2]. Additional to the above, the unexplained origin of this kind of coupling is what constitutes the µ-problem [3].
The Next to Minimal Supersymmetric Models (NMSSM) present an elegant solution to this problem [4] by introducing new scalar singlet field. Consequently, trilinear couplingsχφφ can be generated in such a way that a bilinear term, µφφ, arises when the singlet scalar field acquires a vacuum expectation value (VEV) at the SUSY breaking scale. On the other hand, when including new fields in the theory, the Higgs and Higgsinos mass matrices are changed through new coupling constants, allowing the explanation of these masses in accordance to experimental data or collider constraints.
Looking back to the MSSM, it is known that the lightest Higgs mass can be approximated to m 2 h ≈ m 2 Z cos 2 2β + ∆m 2 h , where ∆m 2 h comes from the 1-loop corrections due to the top quark and stops contributions [5]. Then, if we consider a big value for tan β, ∆m 2 h must be at the same order of the tree level contribution, and stop particles should have a big mass values in order to get a 125 GeV Higgs mass. Nevertheless, in extensions of the MSSM the radiative corrections due to stop particles wouldn't be necessary for explaining the ∆m 2 h term. This may come from a seesaw mechanism that creates an explicit dependence on the scalar singlet VEV [6]. Furthermore, if the scalar singlets come as part of a U (1) X extended gauge symmetry (USSM) [7], the respective D-term may give a new contribution to the lightest mass at tree level, causing the new mass eigenstate to be sharing the functional form of a SM-like Higgs boson.
The extensions of MSSM have other motivations. There are more scalar particles which can be searched in collider experiments, for instance a dark matter candidate [8]. Likewise, they can explain the small deviation of Higgs couplings to fermions, which turn out to be proportional to particle masses, as it has been found in experiments for tau lepton, top and bottom quarks.
The MSSM is a two Higgs doublets anomaly free theory, where the different hypercharge values allows to each Higgs doublet to couple with different kind of quarks, forbidding flavor changing neutral currents (FCNC). In order to extend the scalar doublets content without inducing any chiral anomaly, the minimun amount of them would be four. However, while a Yukawa linear combination can be diagonalized through a rotation, making it proportional to the particle mass, the other linear combination wouldn't be diagonal, generating then the FCNC [9]. On the other hand, from the LHC it is known the upper bound for the tch vertex [10] which can be explained, as new physics at tree level, from a model with multiple Higgs doublets. For this reason a SUSY theory with FNCN is still phenomenological relevant.
In the present work, it is done a SUSY extension of the three family U (1) X anomaly free model [11].
The non-supersymmetric model can explain the fermion mass hierarchy, as well as mixing angles for the CKM [12] and PMNS matrices [13] just by using two Higgs doublets and a scalar singlet field which breaks the U (1) X symmetry giving masses to exotic particles. A singlet scalar field without VEV is required for giving masses to light fermions. Then for the corresponding SUSY extension, the scalar sector has to be doubled with different X-charge in such a way that the anomaly induced by higgsinos are canceled and the model would be anomaly free. After symmetry breaking, the mass matrices for the scalar, vector and fermion sector are constructed. Furthermore, from the scalar CP-even mass matrix it is found that the theory is compatible with a 125 GeV mass for the lightest scalar particle, which we identify as the discovered Higgs boson in LHC. When considering the scalar CP-odd mass matrix, two would-be Goldstone bosons associated to the Z and Z particles are found. The rest of mass eigenstates are found to be above the electroweak scale. Likewise, from the charged scalar bosons another would-be Goldstone boson is found, associated to the W ± gauge bosons. In the present model, the masses of electron, quark up and quark down are zero. Then we consider the SUSY contributions to the self energies in order to generate the masses at one loop level.

II. GENERAL REMARKS OF THE MODEL
The non-supersymmetric model gives a scenario for solving the fermion mass hierarchy problem (FMH) with no need of unpleasant fine tunnings on the Yukawa coupling constants [14]. The way that such problem is addressed relies in having two Higgs doublets Φ 1 and Φ 2 ; the FMH is understood partially from the VEV hierarchy among the two doublets. Also, with the help of the set configuration of the U (1) X charges for all particles, the couplings allowed by the gauge symmetry give a natural scenario for exhibiting the FMH. The inclusion of a parity symmetry Z 2 helps in avoiding Yukawa terms in the Lagrangian that spoil the natural scenario wanted [15]. The new gauge symmetry extension comes in general with chiral anomalies, which have to be canceled in order to guarantee the renormalizability of the theory. In the model found in the literature [11], it was done by choosing the set configuration of U (1) X charges for all fermions [16], such that the following anomaly equations were canceled: where subscripts Q and l account for quarks and leptons, respectively. Moreover, subscripts L and R correspond to left-handed and right-handed chiralities, respectively. Exotic fermions were also included in the model for accomplishing a free anomaly model, in the way that new degrees of freedom enter into the equation (1). Thus, there is a bigger set of U (1) X charges than the SM particles for fulfilling both anomaly cancellation and FMH. In the quark sector, an up-like quark T and two down-like quarks, J 1 and J 2 , come into the bargain. The additional particles in the lepton sector are two charged leptons, E and E; three Dirac right handed neutrinos, ν e R , ν µ R and ν τ R ; and three Majorana neutrinos N 1,2,3 R .
The majorana particles do not contribute to the anomaly equations, but they were included for giving experiments which give information about squared mass differences and mixing angles. For breaking the new symmetry into the SM gauge symmetry , an scalar singlet χ was added with a VEV around the TeV scale. Therefore, the model contains the following spontaneous symmetry breaking chain: Because the lightest fermions, electron, down quark and up quark, did not acquire masses at tree level, another scalar singlet σ had to be included for giving masses to such particles through radiative corrections. However in the SUSY version of the model this scalar field is not necessary, because this radiative corrections can be done through superpartners into the loop. In the case of quarks, the contributions are coming from squarks and gluinos or squarks with Higgsinos.
For the minimal supersymmetric extension, all fields are upgraded to superfields; we denote a superfield with a hat symbol, as usual. The number of scalar particles has to be doubled in comparison of the non-SUSY version, otherwise the model would be anomalous due to Higgsinos. Therefore, new added fields areΦ 1 ,Φ 2 andχ . The introduced scalar fields have the same hypercharge and X charges as the non-primed partners, but with opposite sign to secure anomaly cancellation. For getting the right masses of the gauge bosons in the SM, the following condition must be imposed on the VEVs of the Higgs doublets, The bosonic and fermonic content of the model explored in this paper is shown in the tables I and II, respectively.

III. SCALAR AND GAUGE BOSON SECTOR
The Lagrangian for the scalar sector that describes the minimal supersymetric extension to the U (1) X model in the literature is given by the addition of a F-terms potential, a D-terms potential and a soft-supersymetry breaking potential. The F-terms were obtained from the following superpotential: which is obtained according to the symmetry properties of the scalar superfields given in Table I. Thus, the F-terms potential for scalar fields reads On the other hand, the D-terms potential, consequence of gauge invariance, turns out to be where the last term corresponds to the D-term associated to the U (1) X gauge symmetry and has a very important role for solving the mass problem of the lightest Higgs boson. This will be treated later.
Finally, the soft supersymmetry breaking potential turns out to be: where the last terms, proportional to the coupling constants named k 1 , k 2 , k 3 and k 4 , also break softly the parity symmetry. These trilinear terms avoid the massless feature of some scalar particles, as we will show later. It is important to mention that the soft supersymmetry breaking potential also includes bilinear terms for sfermions and gauginos. Nonetheless, since those terms are not required for our calculations we have decided that it is not required to present them in the current work. Then, by adding all contributions, the scalar potential for the Higgs bosons reads When considering the potential V F it can be seen that, before to include soft SUSY breaking, particles within the fields Φ i and Φ i , i=1,2, are expected to have the same mass µ i due to the absence of mixing terms. Then, with the inclusion of a soft breaking potential, m 2 i , m 2 i terms arise and the diagonal entries change according to the effective parameters m 2 Hk = m 2 k +µ 2 k and m 2 Hk = m 2 k +µ 2 k , ensuring that different Higgs doublets have now different mass eigenvalues. This is also exhibited by the scalar singlets, where diagonal entries are written in terms of the effective parameters M 2 χ = m 2 χ + µ 2 χ and M 2 χ = m 2 χ + µ 2 χ . The following minima conditions for the Higgs potential have to be fulfilled: where also we defined

A. CP-even masses
Taking the VEV for all scalar fields, we get the mass matrices for the different Higgs boson particles, that also respect the minima conditions, eq. (9). The 125 GeV Higgs boson is a CP-even scalar, and it must be obtained from the diagonalization of the following 6 × 6 mass matrix: In the (h 1 , h 1 , h 2 , h 2 , ξ χ , ξ χ ) basis, M φ is a 4 × 4 matrix accounting for the mixings among the CP-even fields of the doublets in the model, and it reads The mixings between scalar doublets and singlets are written in the 4 × 2 M φχ matrix and they are given by: Finally, the mixing matrix between Higgs singlets, M χχ , is written in the following equation Aimed in giving shorter expressions we have defined the coefficients f ng = g 2 +g 2 The rank of M h is 6. Therefore, all mass eigenstates are massive. Considering that a Higgs singlet has not been observed in the experiments, it would be expected for them to acquire mass at high energy scale. Thus, we can assume that the singlet mixing matrix M χχ has much greater entries than the other two matrices, leading us to the possibility of implementing a seesaw mechanism that decouples the 2 × 2 matrix M χχ . The rotated matrix takes the form: In this way it is possible to get an expression for the heaviest Higgs particles by the diagonalization of the 2 × 2 decoupled matrix. We are considering that the heaviest particle must depend on µ χχ , which is the biggest parameter in the theory; while for the lightest of these two depends then on v χ , v χ and k i with the condition that the sum must reconstruct the original trace. In this way we can get the tree level approximation of these eigenvalues which can be written as: As it is shown in eq. (14), the 4 × 4 matrixM φ receives contributions due to the seesaw mechanism.
While this matrix has rank 4, it decreases to 3 in the limit of small doublets' in comparison with µ 11 and µ 22 , which are considered as big parameters come from the SUSY breaking terms in eq. (7). In that way, it can be seen that the lightest Higgs particle depends exclusively on this VEVs since in this limit the determinant is equal to zero, and it becomes massless. When considering this limit we are able to get the next two heavy states by discarding the additive terms proportional to those VEV. As a result of this limit, the matrixM φ reduces to the following form: Therefore, we find two decoupled 2 × 2 matrices with determinant equal to zero, arising the two heaviest states. Then, in the tree level the eigenvalues are given by: At this point, there are only two CP-even particles for which one of them must be the like-Higgs SM.
Returning to the original mass matrix (14), it is known that eigenvalues are roots of the characteristic polynomial. In our case, the polynomial is a fourth degree one. It can be solved analytically by using a general solution given by Ferrari's method [19]. Therefore, taking again the small VEV limit mentioned before, v i , leads us to write the solution for one eigenvalue in an approximate form Finally we focus on the lightest CP-even Higgs particle, which so far in our approximation became massless, but must match the 125 GeV observed one. For starting, we considered the determinant of the 4 × 4 matrix given by eq. (14), taking into account just the dominant terms, e.g. in the small VEV limit. It reads: The lightest Higgs mass eigenvalue is found by dividing this determinant by the other three found mass eigenvalues given by equations (17)- (18), which can be written as Now, if we define the angles then the lightest scalar mass acquires the following form So in fact, when we consider the theory with additional scalar singlets and D-term, the correction term ∆m 2 h can be at the same order of tree level, and its experimental value can be explained with NMSSM and USSM. An interesting fact arises from the approximated expression for the lightest CPeven particle. Its tree level mass do not depend on the new physics energy scale, given by v χ and v χ .
Additionally the µ 11 and µ 22 factors canceled out due to the see saw mechanism, making an eigenvalue depend only on the electroweak scale VEV's, as it is expected.
As it will be shown later and has been already mentioned, the VEV should fulfill v 2 This was done for v 1 since it is not restricted directly by the FMH. The exploration in the parameter space was done by using the Montecarlo method for generating randomly the values of v 1 , v 1 , v 2 and g X . For addressing the fermion mass hierarchy, the domains of v 1 and v 2 were [170GeV, 200GeV] and [3GeV, 7GeV], respectively.
A similar plot is given in the figure (2), where the parameter space of v 2 vs g X is explored within the experimental constraints at 95% of confidence level. This was also done for this VEV, v 2 , since it is also not restricted directly by the FMH. The exploration in the parameter space was done by using the Montecarlo For addressing the fermion mass hierarchy, the domains of v 1 and v 2 were [170GeV, 200GeV] and [3GeV, 7GeV], respectively. Last by not least, the coupling parameter g X was explored in the domain [0, 1].
A clear dependence of the mass of the lightest CP-even particle with is found in the equation (21). In the figure (3), the parameter space cos 2β vs g X is explored. Negative values are found for cos 2β because non-primed VEV happened to be greater than the primed ones. VEVs. For obtaining considerable contributions to get the 125 GeV mass, those differences have to be relatively big. Thus v 1 , v 2 > v 1 , v 2 is preferred. As g X gets bigger, smaller differences on the VEVs are needed. Thus |cos 2β| approaches to smaller values.

B. CP-odd masses
The mass matrix for the CP-odd particles must contain the would-be Goldstone bosons to give masses to the Z µ and Z µ . Such matrix in the basis (η 1 , η 1 , η 2 , η 2 , ζ χ , ζ χ ) is given in the next equation: M ηη contains the mixings between the CP-odd part of doublets and it can be written as The matrix accounting for the mixings between the CP-odd parts of the doublets and the singlets is The mixings between the CP-odd part of the singlets, which are responsible of the would-be Goldstone boson due to the U (1) X symmetry breaking, are given by: The rank of the matrix M η turns out to be 4, so we can be sure that there are two null eigenvalues, corresponding to the would-be Goldstone bosons needed. It is important to notice that structure of this matrix is different from M h , but it preserves the same scale structure. That is to say, M ζζ ∼ µ χχ , v χ , v χ and M ηη ∼ µ ii , v i , v i which fulfill the conditions for a seesaw mechanism, M ζζ >> M ηη , M ηζ . When the rotation is done, the matrix reads:M It is worth noticing that this matrix has a big dependence on the parity breaking terms k i , i = 1, 2, 3, 4.
If we consider the limit in which these couplings go to zero, k i → 0 the mixing matrix M ηζ vanishes resulting in a 2×2 isolated singlet mixing matrix M ζζ (k i → 0) containing the U (1) X would-be Goldstone boson and the heaviest pseudoscalar particle predicted by the model. The two mass eigenstates are written as: In contrast, the 4 × 4 matrix has a rank of 3 ensuring the existence of the would-be Goldstone boson due to SU (2) × U (1) symmetry breaking. The scheme for getting an expression for the eigenvalues is the same we used for the CP-even mass matrix. Firstly, by considering the small VEV limit we can write the matrix as:M leading to 2 heavy eigenstates which at tree level can be written as: Finally for the lightest massive CP-odd particle it was used a small VEV limit for the general solution of the quartic equation given by Ferrari's method [19]. Due to have a massless state, allows us to reduce the characteristic polynomial to a third grade one. Ferrari's solutions implies the cubic general solution and, through this general expressions the lighest CP-odd Higgs particles can be written as and the other mass corresponds to the would-be Goldstone boson associated with Z µ m 2 η1 = 0. (32)

C. Charged scalar bosons
In the case of the charged components of the scalar fields, the corresponding mass matrix must contain a would-be Goldstone boson that gives mass to the W µ± gauge boson. The mass matrix in the The rank of the mass matrix for charged Higgs bosons is 3, so there is one would-be Goldstone boson that gives masses to the W µ± gauge boson. The procedure for obtaining the mass eigenvalues is straightforward. We perform a small VEV limit to get the heavy eigenvalues. The would-be Goldstone boson is ensured by the vanishing determinant and the lightest massive eigenvalue is found by taking a small VEV approximation in Ferrari's general solution for a cubic polynomial, giving the following expressions:

D. Gauge boson masses
As consequence of the inclusion of the symmetry U (1) X , there is a new gauge boson X µ , and it has to be included in the covariant derivate.
Therefore the gauge boson masses are determined by the interaction terms, which are present in the scalar field kinetic terms. On one hand, the charged bosons The neutral gauge bosons (W 3 µ , B µ , X µ ) make up a squared-mass matrix after SSB given by: Despite in this model we have four Higgs doublets, it recreates the same mass structure found in [11] when adopting the definitions eqs. (37)-(39). Nevertheless, it means that the neutral boson mass eigenvalues had been already determined, and they are giving by where tan θ W = g g , as it was defined in the Standard Model.

A. Quark Sector
According to the SU (2) L ⊗ U (1) Y ⊗ U (1) X ⊗ Z 2 symmetry, the most general superpotencial for the quark superfields is given by: where j = 1, 2, 3 label the down type singlet quarks, k = 1, 3 label the first and third quark doublets and a = 1, 2 is the index of the exotic J a L and J ca L quarks. It can be seen that this general superpotencial match the non supersymmetrical one given in ref [11] if we promote the conjugate Higgs fieldsφ i = iσ 2 φ * i to the independent ones superfield Φ i required for a suitable anomaly cancellation. As a consequence,

Non-SM Quarkŝ
taking the VEV of all scalar fields, the quark mass matrices at tree level have an identical structure to its non-SUSY counterpart, as it can be seen in the following equations. For up quark sector has and for the down quark, the matrices can be written as where It is worth to notice that up-like quarks acquire mass from Φ i Higgs fields while the down-like quarks do it from the Φ i ones. Thus, since the matrix structure is identical to its non-SUSY counterpart, the same analysis and eigenvalues gotten in [11] can be done, taking care now that the down-like eigenvalues are coupled to primed Higgs VEV. The mass expressions can be approximated to: The hierarchy between top and charm masses comes from the see-saw mechanism with the heavy T quark, which can be observed from the Yukawa coupling differences for the charm quark mass.
And for the the down-quarks their mass expressions can be approximated to: The m 2 u , m 2 d and m 2 s masses are equal to zero but they obtained by radiative corrections taking into account the SUSY contribution due to gluinos, Higgsinos and squarks into the self energies, as we will show later.

B. Lepton sector
Analogously, the lepton superpotencial corresponds to the non-SUSY Yukawa lagrangian; with the fields promoted to superfields and the conjugate Higgs fields promoted to the primed ones. Then the superpotenial reads as follows where p = e, µ , q = e, µ, τ , r = e, τ and i, j label the right handed and Majorana neutrinos. The superpotential presented in the equation (51) generates, taking the VEV for scalar fields, the same mass matrices structure as well for the neutral and charged leptons, compared to the non-SUSY model.

Charged leptons masses at tree level
The mass matrix for the charged leptons follows the same structure as the one obtained in the non-supersymmetrical model. It is shown right ahead: Just as it happened in the old model, the electron remains massless at tree level. Therefore radiative corrections must be employed to explain the mass feature of such particle. The mass eigenvalues at leading order are given by:

Neutrino masses at tree level
As for neutrinos, the mass matrix involves the Dirac and Majorana terms in order to provide a mass spectrum via the inverse seesaw mechanism (ISS). In the basis (ν q L , ν q L C , N i L C ), such matrix reads: where the block matrices constituting the neutrino mass matrix are given by: For the ISS to work, the assumption on small Majorana coupling constants is made, m D M M . Therefore, it can be shown that the matrix M ν can be approximately block diagonalized: where m light = m T D (M D ) −1 M M (M D ) −1 m D is the 3 × 3 mass matrix for the light neutrinos and it must explain the observed mixing parameters in the PMNS matrix. The rotation matrix V S can be calculated by: Lastly, the m heavy matrix involves the mixings of the exotic neutral leptons, and it is given by: Since the same structure as the non-SUSY model is followed also by the sector of neutral leptons, the same constraints are applied for the Yukawa parameters for explaining the quadratic difference of masses ∆m 2 and mixing angles in the neutrino oscillation [17]. The parameter values are then shown in [11].

C. Fermion masses at one loop level
As it was seen in the previous section, the electron and the up, down and strange quarks turned out to be massless at tree level. This was solved in [11] by including an scalar singlet σ with no VEV. Thus, the masses of these particles were generated from 1-loop radiative corrections. The supersymmetric extension is dotted now with a new advantage: there is no need of including an extra scalar singlet, because the superpartners of the existing particles provide one loop contributions to the fermions mass.
The radiative corrections to the masses of the up-like and down-like quarks due to the contribution of squarks and gluinons, running inside the loop, can be detailed in the figure (4). The corresponding expressions are summarized in the equation [20] Σ RL IJ,q (p 2 = 0) = 2 π α S i,a,β mgaT a † βα T a βα Z Ji q Z i(J+nq) qB 0 (p 2 = 0, mga, mq i ), Therefore, i = 1, ..., 10 and n d = 5. With I = 1, J = 1, 2, 3 and I = 2, J = 1, 2, 3, Σ RL IJ,q fills the 3 × 3 quark mass sub-matrices in the light sector in order to generate the q = u, d, s masses.
The self-energies of the quarks also receive contributions from sleptons and Higgsinos. However, we do not show them here because all what we want is to ensure the acquisition of the lighter quark masses through radiative corrections involving super partners.
In the case of the electron, we consider one loop correction via charginos and s-neutrinos. After performing the dimensional regularization, the lepton self-energies read where the divergence in B 0 is canceled thanks to unitarity of the matrix Z ν . With I, J = 1 the mass of the electron is obtained. In this case, the charginos with i = 1, ..., 4 are the superpartners of W µ and the three charged Higgs bosons. For the index J = 1, ..., 9, three of them account for the neutralinos associated with the gauge fields A µ 3 , B µ and X µ . The other six correspond to the neutralinos regarding to the neutral part of the four scalar doublets and two scalar singlets. The model studied here is the supersymmetric extension of the three families U (1) X model [11].
The SUSY extension requires four Higgs doublets and two scalar singlets in order to not induce chiral anomalies and giving mass to quarks, charged leptons and neutrinos. Additionally, the singlets generate the mass for exotic fermions and break the U (1) X gauge symmetry. An interesting prediction of this extension is that there are tree level flavor changing neutral currents.
In both versions of the model, SUSY and non SUSY, the lightest particles (electron and up, down and strange quarks) are massless at tree level. However, in the supersymmetric model they acquire a mass value via radiative corrections through superpartners into the loop, so an additional singlet with zero VEV is not required anymore in the model.
By implementing a seesaw mechanism among the singlet and doublet Higgs fields, together with the D-terms corresponding to the U (1) X , a ∆m 2 h is found at tree level and it turns out to be at the order of the MSSM contribution. The lightest scalar particle is identified as the Higgs boson and its mass is obtained at the order of 125 GeV. In fact, we show in figures (1) and (2) the region in the parameter space v 1 vs g X and v 2 vs g X which is compatible with a 125.3 GeV Higgs boson at 95% of C.L., where the VEVs v 1 and v 2 are fixed around the mass of the top quark and the bottom quark, respectively.
As a result, we found that the coupling constant g X , regarding to the U (1) X symmetry, takes values between 0.63 and 1 for 112 < v 2 (GeV) < 180 and 0 < v 1 (GeV) < 81. Thus, values below 0.63 for g X are excluded. Even more, if we take the LHC exclusion bound m Z > 3TeV, it implies from the expression for this gauge boson mass (eq. (42)) that the VEVs v χ ≈ v χ should be greater than 10 TeV.
The parameter space cos 2β vs g X was explored and negative values are found for cos 2β because non-primed VEV happened to be greater than the primed ones. This behavior lies in the fact that top quark mass (≈ v 1 ) is bigger than the bottom quark mass (≈ v 2 ). On the other hand, in the mass expression for m h , eq. 20, there are quadratic differences between non-primed and primed VEVs so v 1 , v 2 > v 1 , v 2 is preferred. Thus the allowed region is 0.38 < cosβ < 1.
Last but not less important, the model also predicts five CP-even, four CP-odd and three charged Higgs particles with a mass above the TeV scale. The would-be Goldstone bosons corresponding to Z µ , W ± µ and Z µ are also found.
Appendix A: Scheme for obtaining the scalar particle mass expressions For the CP-even particles an additional first step was made, which is to perform a seesaw like rotation, taking into account that the mixing in M φχ are small compared to the ones in M χχ . With that approximation, the matrix M h is transformed to a block diagonal form: with Θ h = M φχ (M χχ ) −1 . The heavy remaining block component, M χχ , is a 2 × 2 matrix, and therefore it's eigenvalues can be obtained trivially. On the other hand, the Feynman rule needed for obtaining the mass corrections for the electron corresponds to the diagram (7), and it is shown in the equation (B2) [21] − i e sin θ W Z IJ ν P L + Y e Z Ii P R (B2)