The scalar sector of the SU(3) c SU(3) L U(1) X model

Transcript

1 arxiv:hep-ph/98v Mar 4 The scalar sector of the SU c SU U X model Rodolfo A. Diaz, R. Martínez, and F. Ochoa Departamento de Física, Universidad Nacional, Bogotá-Colombia February, 8 Abstract A complete study of the Higgs sector of the SU c SU U X model is carried out, obtaining all possible cases of vacuum expectation values that permit the spontaneous symmetry breaking pattern SU U X SU U Y U Q. We find the most general Higgs potentials that contain three triplets of Higgs and one sextet. A detailed study of the scalar sector for different models with three Higgs triplets is done. The models end up in an electroweak two Higgs doublet model after the first symmetry breakdown; we find that the low energy limit depends on a trilinear parameter of the Higgs potential, and that the decoupling limit from the electroweak two Higgs doublet model to the minimal standard model can be obtained quite naturally. PACS numbers:..rd,.5.ex,.6.cn,.6.fr Keywords: models, scalar sector, three Higgs triplets, decoupling limit. Introduction Though the Standard Model SM [ is a good phenomenological theory and coincides very well with all experimental results [, it leaves several unanswer questions which suggest that SM should be an effective model at low energies, originated from a more fundamental theory. Some of the unexplained aspects in SM are: the existence of three families [, [4, the mass hierarchy problem[, [4, the quantization of the electric charge [5, the large number of free parameters to fit the model, the absence of an explanation for the matter anti-matter asymmetry [, the fact that radiaz@ciencias.unal.edu.co romart@ciencias.unal.edu.co frecho@hotmail.com

2 R. A. Diaz, R. Martinez, and F. Ochoa the SM says nothing about the stability of the proton, gravity cannot be incorporated as a gauge theory [4, [6, it cannot account on the neutrino deficit problem [7, [8, etc. Some of these problems can be understood introducing a larger particle content or enlarging the group of symmetry where the SM is embedded. The SU5 grand unification model of Georgi and Glashow [9 unifies the interactions and predicts the electric charge quantization; the group E 6 unifies the electroweak and strong interactions and might explain the masses of the neutrinos [. Nevertheless, such models are for only one fermion family. Models with larger symmetries that may allow to understand the origin of the families have been proposed[. In some models, it is also possible to understand the number of families from the cancelation of quiral anomalies, necessary to preserve the renormalizability of the theory[. This is the case of the SU c SU U X, or models, which are an inmediate extension of the SM [4, [5, [6, [7. There is a great variety of such models, which have generated new expectatives and possibilities to solve several problems of the SM. Refs. [8 have studied the different models based on the criterium of cancelation of anomalies. Cancelation of chiral anomalies demands some conditions, for instance, the number of fermionic triplets must be equal to the number of antitriplets[9; notwithstanding, an infinite number of models with exotic charges are found. However, only two models with no exotic charges exist, and they are classified according to the fermion assignment. In the present work, we find those possible models from a different point of view. As well as the cancelation of anomalies, the scalar sector should be taken into account. This sector couples to the fermions by means of the Yukawa agrangian generating additional constraints for the quantum numbers. From the Yukawa agrangian, the scalar fields neccesary to endow fermions with masses are selected. The vacuum is aligned in such a way that it respects the symmetry breaking scheme SU U X SU U Y U Q. where the different solutions to align the vacuum fit quantum numbers which are important in the determination of the electric charge and to determine the different models. In section, we examine the structure of the representations for fermions and vector bosons under the quiral group SU c SU U X. In section, we analize the representations in the Higgs sector, as well as the possible vacuum alignments that generates the scheme of spontaneous symmetry breaking SSB. Along with these possibilities, section 4 shows the most general Higgs potentials for any type of models containing three Higgs triplets and one Higgs sextet, necessary for all the fermions to acquire their masses. In section 5., we apply our general results about the scalar sector for a model of type β = / with three Higgs triplets, obtaining the complete scalar spectrum. Section 7 shows the behavior of the β = / model in the low energy regime. Finally, section 8 is regarded

3 The scalar sector of the model for our conclusions. Fermion and vector representations. Fermion representations The fermions present the following general structure of transformations under the quiral group SU c SU U X { q :,,Xq ψ =,,X = q,,x q, l :,,Xl =,,X l,,x l, { q ψ = :,, Xq =,, Xq,, X q, l :,, Xl =,, Xl,, X l, { qr :,,Xq ψ R, R = l R :,,Xl R., where the quarks q are color triplets and the leptons l are color singlets. The second equality corresponds to the branching rules SU SU. Both possibilities and are included in the flavor sector SU, since the same number of fermion triplets and antitriplets must be present in order to cancel anomalies [9. The generator of U X conmute with the matrices of SU, therefore it should be a matrix proportional to the identity, where the factor of proporcionality X p takes a value according to the representations of SU and the anomalies cancelation. The electric charge is defined in general as a linear combination of the diagonal generators of the group Q = a T + β T 8 + XÎ,. with T = diag,, and T 8 = diag,,, where the normalization chosen is TrT α T β = δ αβ, and I = diag,, is the identity matrix. In this work we shall show that these quantum numbers can also be fixed by choicing the scalar sector and vacuum alignment to break the symmetry of the model.. Representation of gauge bosons The gauge bosons associated to the group SU transform according to the adjoint representation and are written in the form W µ = Wµ α G α = Wµ + Wµ 8 W + µ K Q µ W µ Wµ + Wµ 8 K Q µ.. K Q µ K Q µ Wµ 8

4 4 R. A. Diaz, R. Martinez, and F. Ochoa Therefore, the electric charge takes the general form Q W + β + β β β..4 From Eq..4 we see that the cases β = ± lead to exotic charges and only the cases β = ±/ does not. As for the gauge field associated to U X, it is represented as B µ = I B µ,.5 which is a singlet under SU, and its electric charge is given by Q Bµ..6 From the previous expressions we see that three gauge fields with charges equal to zero are obtained, and in the basis of mass eigenstates they correspond to the photon, Z and Z. Two fields with charges ± associated to W ± and four fields with charges that depend on the choice of β. Demanding that the model contains no exotic charges is equivalent to settle β = / [6, and β = /. The Higgs potential of the model with β = / has been discussed recently by the third of Refs. [8, however we consider the most general Higgs potential with three Higgs triplets and we also show that it reduces to the HDM in a natural way when the decoupling limit is taken. In both of the cases β = ±/, the fields K ±Q µ and K ±Q µ posess charges ± and. It is important to take into account the scalar sector and the symmetry breakings to fix this quantum number, which in turn determine the would-be Goldstone bosons associated to the gauge fields, with the same electric charge of the gauge fields that are acquiring mass in the different scales of breakdown. When β takes a value different from the previous ones we get fields with exotic electric charges.. Fermionic anomalies The coefficient of triangular anomalies is defined as } A αβγ = Tr [{G α T ψ, G β T ψ G γ T ψ {G α T ψ R, G β T ψ R }G γ T ψ R.7 where G α Tψ h are the matrix representations for each group generator acting on the basis ψ with helicity left or right. In the particular case of the gauge models,

5 The scalar sector of the model 5 such conditions become a [SU c A αβγ =, b c [SU c SU A αβγ =, [SU c U X A αα = ±Xq sing XR q =, d SU c [SU A αβγ =, e SU c [U X A γ =, f SU c SU U X A αα =, g [SU A αβγ =, h [SU U X A αα = m i SU [U X A γ =, j [U X A 4 = m sing ±X l ± 9 X q ±X l m ± X q =, X R l sing X R q =..8 When we impose for the fermionic representations of SU c SU to be vectorlike, then the number of color triplets should be equal to the number of color antitriplets, and same for the left triplets. In this case, all the conditions of Eqs..8 are satisfied, except the ones that involve the quantum numbers asociated to U X. Considering a particle content as the one in table ; the non-trivial conditions are reduced to A αα = ±Xq Xq R = A αα = m 4 A = m singlets singlets ±X l m ± X q = ±X l m ± 9 X q X R l singlets X R q = [Grav U X A grav = ±X l ± 9X m q m X R l X R q = singlets singlets where we have included the gravitational anomaly condition as well. These conditions still permit an infinite number of solutions for the quantum numbers, that can be characterized by the values of a and β in the definition of the electromagnetic charge Eq... However, a = is required to obtain isospin doublets to embed properly the SU U Y model into SU U X. Now, in order to restrict β, we could demand that the exotic component of the triplet has ordinary electromagnetic charge, in whose case β = ±/. On the other hand, we can obtain

6 6 R. A. Diaz, R. Martinez, and F. Ochoa relations between β and the quantum numbers associated to U X by imposing gauge invariance in the Yukawa sector, and by taking into account the symmetry breaking pattern. q = l m = q q q l m l m l m q R q δ = q l δm = l m Q ψ ± ± X q ± X q ± X q ± ± X l m ± X l m ± X l m Y ψ ± 6 ± X q ± 6 ± X q ± X q ± 6 ± X l m ± 6 ± X l m ± X l m qr qr qr ±Xq R ±Xq R l n R Xl R n Xl R n Table : Particle content for the fermionic sector of the model for a one family scenario. q refers to triplets of color, m =,, and n is arbitrary. Scalar representations. SSB scheme From the phenomenological point of view, the SM is an effective theory at the electroweak scale, hence, the SSB follows the scheme Φ SU U Φ X SU U Y UQ. In the first breakdown there are five gauge fields that acquire mass proportional to < Φ > and in the second breakdown electroweak breakdown, three gauge fields acquire mass of the order of < Φ >, leaving a massless field associated to the unbroken generator Q, the photon... SU U X SU U Y transition In the first transition the VEV s of Φ break the symmetry SU U X /SU U Y where the hypercharge is defined as Ŷ = β T 8 + XÎ. The conditions that

7 The scalar sector of the model 7 should be satisfied in this breaking step are [ T, Φ = [ T, Φ = [ T, Φ = [β T 8 + XÎ, Φ =, [ T 4, Φ [ T 5, Φ [ T 6, Φ [ T 7, Φ [β T 8 XÎ, Φ. These conditions reflect the fact that five bosons acquire mass and in order to preserve the number of degrees of freedom, at least five components in Φ are needed to represent the would be Goldstone bosons... SU U Y U Q transition In the second transition the VEV s of Φ breaks the symmetry SU U Y /U Q. The conditions that should be satisfied in this breakdown read [Q, Φ =, [ T, Φ [ T, Φ [ T β T 8 + XÎ, Φ,. where three gauge bosons acquire mass, from which Φ needs three components associated to the would be Goldstone bosons.. Yukawa terms Other conditions fixed by the scalar sector are related to the fermion-fermion-scalar couplings that generate the fermion masses. The scalar fields have to be coupled to fermions by Yukawa terms invariant under SU U X ; these couplings can be written as ψ i ψ RΦ : Φ = Φ =, ψ i ψ j c Φ : Φ = Φ = = 6, ψ R ψ R c Φ : Φ = Φ =, ψ R c ψ i c Φ : Φ = Φ =..4 Hence, in order to generate the masses of the fermions, the Higgs bosons should lie either in the singlet, triplet, antitriplet or sextet representations of SU.

8 8 R. A. Diaz, R. Martinez, and F. Ochoa. Scalar representations for the first transition Φ The choice of the scalar sector should fulfill the following basic conditions. The VEV s should accomplish the conditions. and. in the first and second transition respectively.. When the VEV s are replaced in Eq..4, the fermions should acquire the appropiate masses.. The number of components of Φ should be at least the number of would be Goldstone bosons for each transition of SSB... Triplet representation One of the possible solutions of the Eq..4 is the fundamental representation where the VEV s can be written in the form ν χ χ = ν χ..5 ν χ By demanding the conditions of Eq.. for the first breaking of symmetry, it is found that the vacuum should be aligned in the following way χ = ν χ.6 with the condition X χ β =, β ν χ,.7 The choice of vacuum above generates the following mass terms Y = Γ χ ψi ψ R χ i + Γ χ ψ R c ψ i c χ i + h.c..8 = Γ χ ν χ ψ ψ R + Γ χ ν χ ψ R c ψ c + h.c..9.. Antisymmetric representation Other possible representation to generate masses from Eq..4 is the antisymmetric one. If we use this representation for the first breaking of the symmetry, then the conditions of Eq.. should be satisfied, implying for the vacuum to be aligned as χ = ν χ.

9 The scalar sector of the model 9 with the condition X χ β =, β ; ν χ.. The anti-triplet antisymmetric representation only admits mass Yukawa terms of the form Y = Γ Φ ǫ ijk ψ i ψ j c χ k + h.c = Γ χ ν χ ǫ ij ψ i ψ j c + h.c.... Symmetric representation 6 As for the possibility of Φ = S symmetric, we have: S ij = ν ν ν ν ν 4 ν 5 ν ν 5 ν 6 and requiring the conditions of Eq.., it is obtained S ij = ν 6,..4 with X S ij β =, β ν 6..5 The mass terms generated from the Yukawa agrangian are given by Y = Γ Φ ψ i ψ j c S ij + h.c = Γ χ ν 6 ψ ψ c + h.c..6.4 Scalar representations for the second transition Φ The first breaking align the vacuum according to the conditions. and impose a requirement on the quantum numbers of the representations X[Φ β/ =. This condition gives freedom to choice the electric charge of the fields of the scalar representations in different directions. To align the vacuum in the second breakdown according to the conditions. we shall write explicitly the electric charge of the components of the scalar fields. [ Q, Φ = [ T, Φ [ + β [ T8, Φ + X, Φ.

10 R. A. Diaz, R. Martinez, and F. Ochoa The charges for the representations, and 6 are given respectively by Q = Q = Q 6 = + β + X Φ + β + X Φ β + X Φ β β β X Φ X Φ X Φ,.7,.8 + β β + X Φ ij + X Φ ij + X Φ ij β + X Φ ij + β + X Φ ij β + X Φ ij β + X Φ ij β + X β Φij + X Φ ij β..9 We should notice that the choice of a particular vacuum alignment fixes the value of the quantum number X Φ, and it in turn fixes the β parameter according to the solutions.7,. and.5 respectively. The quantum numbers X for the fermions can be determined by means of the Yukawa sector, according to the couplings of fermions to the scalar sector and same for the gauge fields which acquire mass through the covariant derivative of the scalar fields. Therefore, it should exist a relation amongst both the fermion and gauge sectors with the scalar one. Owing to it, the quantum numbers of the scalar sector were settled free for the different symmetry breakings to fix them and find the different models. For this transition the vacuum should be chosen in the direction in which the scalar fields have null charge. From Eqs..7.9 we observe that the components depend on the value assigned to β and X Φ. We will align the vacuum arbitrarily and choice the quantum numbers β and X Φ that define the electric charge operator according to the conditions...4. Fundamental representation ike in section.., the possibility for a triplet is considered but for the second transition. Thus, the conditions. should be fulfilled. In table the possible vacuum alignments for a representation are shown..4. Antisymmetric representation For the anti-triplet antisymmetric representation, we get the same type of solutions as in table, and they are shown in table.

11 The scalar sector of the model ρ ν ρ ν ρ X ρ β = β = β ± ν ρ ν ρ ν ρ ν ρ ν ρ β ν ρ Table : Solutions for the second SSB with Higgs triplets in the fundamental representation. β ρ X ρ β = β = β ± ν ρ ν ρ ν ρ ν ρ ν ρ ν ρ ν ρ β ν ρ β Table : Solutions for the second SSB with Higgs triplets in the antisymmetric representation..4. Symmetric representation 6 For the representation of dimension 6 we have the solutions to the conditions. shown in table 4. ρ ij X ρ ij β = β = ν ν ν ν ν ν ν 4 ν 5 ν ν 5 ν ν 5 ν 6 ν ν 6 ν 5 ν ν 4 6 6

12 R. A. Diaz, R. Martinez, and F. Ochoa ρ ij X ρ ij β = β = ν ν ν ν ν 4 ν 5 ν ν 5 ν ν 5 ν 4 ν 5 ρ ij ν X ρ ij β ν ν 4 + β 4 β ± ; ± ν 4 β ν 5 ν 5 + β 4 4 Table 4: Vacuum alignments for the second SSB with Higgs sextets, and for all values of β. In this way, we have found the Higgs bosons necessary to break the symmetry according to the scheme SU U X SU U Y U Q, and generate the masses of the fermions and gauge fields. This choice fixes the values of the quantum numbers X and the value of β needed to define the electric charge. Among the possible solutions we have β =,, / and /. They correspond to the models in Refs. [4, [5, [6, and [8 respectively; the Higgs sector of the latter will be developed in detailed in this paper. For β = ±/ the models does not contain exotic electric charges. For values of β different from ±/, exotic charges arise i.e. electric charges different from ± and. 4 Higgs Potential 4. Three Higgs triplets The triplet field χ and the antitriplet χ only introduce VEV on the third component for the first transition, as it is indicated by the solutions.7 and. respectively, it induces the masses of the exotic fermionic components. In the second transition, it is observed in the tables and that pairs of solutions are obtained according to the

13 The scalar sector of the model value of β. A careful analysis of such solutions shows that the pair of multiplets is necessary to be able to give masses to the quarks of type up and down respectively. Therefore, in the second transition is necessary to introduce two triplets ρ and η associated to each pair of solutions in the tables and. In table 5, it is shown the minimal contents of Higgs bosons necessary to break the symmetry for the different values of β and X. st SSB χ β = β = β ± nd SSB ν χ ν χ X χ ρ ν ρ ν ρ X ρ η ν ρ ν χ β ν ρ β X η β Table 5: Higgs triplets neccesary for the SSB scheme: for a fix value of β. In order to study the most general form of the Higgs potential, we take all possible linear combinations among the three triplets forming quadratic, cubic, and quartic products invariant under SU U X. The different potentials, renormalizable and SU U X invariant are. For β = V higgs = µ χ i χ i + µ ρ i ρ i + µ η i η i + µ 4 χ i ρ i + h.c + f χ i ρ j η k ε ijk + h.c + λ χ i χ i + λ ρ i ρ i + λ η i η i + λ 4 χ i χ i ρ j ρ j + λ 5 χ i χ i η j η j + λ 6 ρ i ρ i η j η j + λ 7 χ i η i η j χ j + λ 8 χ i ρ i ρ j χ j + λ 9 η i ρ i ρ j η j + λ χ i χ i χ j ρ j + h.c. + λ ρ i ρ i ρ j χ j + h.c. + λ η i η i χ j ρ j + h.c. + λ χ i ρ i χ j ρ j + h.c. + λ 4 η i χ i ρ j η j + h.c.. 4. After the first breaking, the model is reduced to the SM with two Higgs doublets[,. The choice of three triplets does not ensure necessarily the masses of all leptons, an example is the model of Pisano and Pleitez, where an additional sextet is needed [4.

14 4 R. A. Diaz, R. Martinez, and F. Ochoa. For β = V higgs = µ χ i χ i + µ ρ i ρ i + µ η i η i + µ 4 χ i η i + h.c + f χ i ρ j η k ε ijk + h.c + λ χ i χ i + λ ρ i ρ i + λ η i η i + λ 4 χ i χ i ρ j ρ j + λ 5 χ i χ i η j η j + λ 6 ρ i ρ i η j η j + λ 7 χ i η i η j χ j + λ 8 χ i ρ i ρ j χ j + λ 9 η i ρ i ρ j η j + λ χ i χ i χ j η j + h.c. + λ η i η i η j χ j + h.c. + λ ρ i ρ i χ j η j + h.c. + λ χ i η i χ j η j + h.c. + λ 4 ρ i χ i η j ρ j + h.c For β = V higgs = µ χi χ i + µ ρi ρ i + µ ηi η i + f χ i ρ j η k ε ijk + h.c + λ χ i χ i + λ ρ i ρ i + λ η i η i + λ 4 χ i χ i ρ j ρ j + λ 5 χ i χ i η j η j + λ 6 ρ i ρ i η j η j + λ 7 χ i η i η j χ j + λ 8 χ i ρ i ρ j χ j + λ 9 η i ρ i ρ j η j + λ ρ i χ i ρ j η j + h.c For β = V higgs = µ χi χ i + µ ρi ρ i + µ ηi η i + f χ i ρ j η k ε ijk + h.c + λ χ i χ i + λ ρ i ρ i + λ η i η i + λ 4 χ i χ i ρ j ρ j + λ 5 χ i χ i η j η j + λ 6 ρ i ρ i η j η j + λ 7 χ i η i η j χ j + λ 8 χ i ρ i ρ j χ j + λ 9 η i ρ i ρ j η j + λ η i χ i η j ρ j + h.c For β arbitrary different from ±, ±/ V higgs = µ χi χ i + µ ρi ρ i + µ ηi η i + f χ i ρ j η k ε ijk + h.c. + λ χ i χ i + λ ρ i ρ i +λ η i η i + λ 4 χ i χ i ρ j ρ j + λ 5 χ i χ i η j η j + λ 6 ρ i ρ i η j η j + λ 7 χ i η i η j χ j +λ 8 χ i ρ i ρ j χ j + λ 9 η i ρ i ρ j η j Higgs sextet In some circumstances, the choice of three triplets may be not enough to provide all leptons with masses [4,,. Hence, an additional sextet is considered, which should be compatible with any of the solutions of the table 4. The choice of one of these solutions depend on the fermionic sector to which we want to give masses. Once again, when we evaluate all possible linear combinations among the four fields χ, ρ, η and the sextet S, additional terms are obtained which should be added to the Higgs potentials found for the case of three triplets, and according to the chosen value of β. They are shown in tables 6-.

15 The scalar sector of the model 5 X S = V S = µ 5 Sij S ij + f χ i S ij χ j + h.c + λ 5 χi S ij η k ρ l ε jkl + h.c +S ij S ij λ6 χ k χ k + λ 7 ρ k ρ k + λ 8 η k η k + λ9 S ij S ij ρ k χ k + h.c +λ χ i S ij S jk χ k + λ ρ i S ij S jk ρ k + λ η i S ij S jk η k + λ ρ i S ij S jk χ k + h.c +λ 4 S ij S ij + λ 5 S ij S jk S kl S li X S = 6 X S = V S = µ 5 Sij S ij + f η i S ij ρ j + h.c + λ 5 ρ i S ij ρ k χ l ε jkl +λ 6 η i S ij η k χ l ε jkl + h.c + S ij S ij λ7 χ k χ k + λ 8 ρ k ρ k + λ 9 η k η k +λ S ij S ij ρ k χ k + h.c + λ χ i S ij S jk χ k + λ ρ i S ij S jk ρ k +λ η i S ij S jk η k + λ 4 ρ i S ij S jk χ k + h.c + λ 5 S ij S ij + λ 6 S ij S jk S kl S li V S = µ 5 Sij S ij + f η i S ij η j + h.c + λ 5 ηi S ij ρ k χ l ε jkl + h.c +S ij S ij λ6 χ k χ k + λ 7 ρ k ρ k + λ 8 η k η k +λ 9 χ i S ij S jk χ k + λ ρ i S ij S jk ρ k + λ η i S ij S jk η k +λ S ij S ij + λ S ij S jk S kl S li Table 6: Additional terms in the Higgs potential of Eq. 4. for β =, when a Higgs sextet is included. X S = V S = µ 5 Sij S ij + f χ i S ij χ j + h.c + λ 5 χi S ij η k ρ l ε jkl + h.c +S ij S ij λ6 χ k χ k + λ 7 ρ k ρ k + λ 8 η k η k +λ 9 S ij S ij η k χ k + h.c + λ χ i S ij S jk χ k + λ ρ i S ij S jk ρ k +λ η i S ij S jk η k + λ η i S ij S jk χ k + h.c + λ 4 S ij S ij + λ 5 S ij S jk S kl S li X S = 6 X S = V S = µ 5 Sij S ij + f η i S ij ρ j + h.c + λ 5 ρ i S ij ρ k χ l ε jkl +λ 6 η i S ij η k χ l ε jkl + h.c + S ij S ij λ7 χ k χ k + λ 8 ρ k ρ k + λ 9 η k η k +λ S ij S ij η k χ k + h.c + λ χ i S ij S jk χ k + λ ρ i S ij S jk ρ k + λ η i S ij S jk η k +λ 4 η i S ij S jk χ k + h.c + λ 5 S ij S ij + λ 6 S ij S jk S kl S li V S = µ 5S ij S ij + f ρ i S ij ρ j + h.c + λ 5 ρi S ij η k χ l ε jkl + h.c +S ij S ij λ6 χ k χ k + λ 7 ρ k ρ k + λ 8 η k η k +λ 9 χ i S ij S jk χ k + λ ρ i S ij S jk ρ k + λ η i S ij S jk η k +λ S ij S ij + λ S ij S jk S kl S li Table 7: Additional terms in the Higgs potential of Eq. 4. for β =, when a Higgs sextet is included.

16 6 R. A. Diaz, R. Martinez, and F. Ochoa X S = X S = V S = µ 5S ij S ij + S ij S ij λ5 χ k χ k + λ 6 ρ k ρ k + λ 7 η k η k +λ 8 χ i S ij S jk χ k + λ 9 ρ i S ij S jk ρ k + λ η i S ij S jk η k +λ S ij S ij + λ S ij S jk S kl S li V S = µ 5 Sij S ij + f η i S ij η j + h.c + λ 5 ηi S ij ρ k χ l ε jkl + h.c +S ij S ij λ7 χ k χ k + λ 8 ρ k ρ k + λ 9 η k η k + λ χ i S ij S jk χ k +λ ρ i S ij S jk ρ k + λ η i S ij S jk η k + λ S ij S ij + λ 4 S ij S jk S kl S li X S = V S = µ 5 Sij S ij + f ρ i S ij χ j + h.c + λ 5 χ i S ij η k χ l ε jkl +λ 6 ρ i S ij η k ρ l ε jkl + h.c + S ij S ij λ7 χ k χ k + λ 8 ρ k ρ k + λ 9 η k η k +λ χ i S ij S jk χ k + λ ρ i S ij S jk ρ k + λ η i S ij S jk η k +λ S ij S ij + λ 4 S ij S jk S kl S li Table 8: Additional terms in the Higgs potential of Eq. 4. for β =, when a Higgs sextet is included. V S = µ 5 Sij S ij + S ij S ij λ5 χ k χ k + λ 6 ρ k ρ k + λ 7 η k η k X S = X S = +λ 8 χ i S ij S jk χ k + λ 9 ρ i S ij S jk ρ k + λ η i S ij S jk η k +λ S ij S ij + λ S ij S jk S kl S li V S = µ 5S ij S ij + f ρ i S ij ρ j + h.c + λ 5 ρi S ij η k χ l ε jkl + h.c +S ij S ij λ7 χ k χ k + λ 8 ρ k ρ k + λ 9 η k η k +λ χ i S ij S jk χ k + λ ρ i S ij S jk ρ k + λ η i S ij S jk η k +λ S ij S ij + λ 4 S ij S jk S kl S li X S = V S = µ 5S ij S ij + f η i S ij χ j + h.c + λ 5 χ i S ij ρ k χ l ε jkl +λ 6 η i S ij η k ρ l ε jkl + h.c + S ij S ij λ7 χ k χ k + λ 8 ρ k ρ k + λ 9 η k η k +λ χ i S ij S jk χ k + λ ρ i S ij S jk ρ k + λ η i S ij S jk η k +λ S ij S ij + λ 4 S ij S jk S kl S li Table 9: Additional terms in the Higgs potential of Eq. 4.4 for β =, when a Higgs sextet is included. V S = µ 5S ij S ij + S ij S ij λ5 χ k χ k + λ 6 ρ k ρ k + λ 7 η k η k β arbitrary +λ 8 χ i S ij S jk χ k + λ 9 ρ i S ij S jk ρ k + λ η i S ij S jk η k +λ S ij S ij + λ S ij S jk S kl S li Table : Additional terms in the Higgs potential of Eq. 4.5 for β ± and β ±, when a Higgs sextet is included.

17 The scalar sector of the model 7 5 model with β = / 5. Fermionic content Before describing in detail the model with β = /, let us check that it could provide a realistic scenario. As we saw in section, we found the Higgs structures and vacuum alignments that satisfy the gauge invariance, the symmetry breaking pattern and the adequate assignment of masses. The quantum numbers associated to U X for three Higgs triplets in the specific case of β = /, can be determined from Eq..7 for the triplet associated to the first transition and from table for the two triplets of the second transition. On the other hand, gauge invariance in the Yukawa sector provides some relations among the X Φ and X f numbers associated to U X, corresponding to scalars and fermions respectively. Further, cancelation of fermionic anomalies and the requirement that the first two components of the fermion triplets of SU transform as doublets of the SM, provide the complete requirements to write down the quantum number assignments for the fermionic sector of the model. From the fermionic content described in table with β = / we get the quantum numbers of table for a one family framework. This table shows us that the standard model fermions acquire the appropiate quantum numbers and charges, and that the exotic fermions do not have exotic charges either. 5. Scalar mass spectrum for β = / The masses of the scalar sector arise from the implementation of the SSB over the Higgs potentials. A detailed calculation of the scalar spectrum is done in Ref. [ in models with β = and β =, that coincide for these values of β, with the solutions of table 5 for the triplets, and of table 4 for the sextet. In this article, we examine in detail a model with β =, which has no exotic charges. With three Higgs triplets, the most general potential is given by Eq. 4., where the solution in table 5 is presented more explicitly in table. The conditions for the minimum of the potential are given by V higgs ν χ =, V higgs ν ρ =, V higgs ν ρ =, and evaluated when all fields are equal to zero, we get µ = λ ν χ λ 4 ν η λ 5 ν ρ + ν ρ λ ν χ ν ρ f V higgs =, 5. νη ν ρ ν χ ν ρ + ν ρ ν χ µ = λ ν ρ + ν ρ λ5 νχ λ 6 νη λ ν χ ν ρ f ν χ, ν ρ µ = λ νη λ 4νχ λ 6 ν ρ + ν ρ λ ν χ ν ρ f ν ρ ν χ, µ 4 = λ 8 ν χ ν ρ λ νχ λ ν ρ + ν ρ λ νη λ ν χ ν ρ + f ν ρ. 5. ν ρ,

18 8 R. A. Diaz, R. Martinez, and F. Ochoa representation Q ψ Y ψ X ψ u q = d : Xq = J 6 6 q R = u R : q R = d R : q l = R = J R : l = l = e ν e E E E + e + E E E 4 : : : X R u = X R d = X R J = X l = X l = X l = Table : Particle content for the fermionic sector of the model with β = / for a one family scenario. χ = ρ = η = χ ± ξ χ ± iζ χ ξ χ ± iζ χ ρ ± ξ ρ ± iζ ρ ξ ρ ± iζ ρ ξ η ± iζ η η χ Q Φ Y Φ X Φ Φ ± ± ± ± ± ± ν χ ν ρ ν ρ Table : Quantum numbers of three scalar triplets for β =

19 The scalar sector of the model 9 These parameters are again replaced in the Higgs potential. If the potential is derived twice and then evaluated for all scalar fields such that Φ i =, we find the square mass terms. For the neutral scalar fields we have that M Φ i Φ j = V higgs Φ i Φ j Φ i = 5. with Φ i = ζ χ, ζ χ, ζ ρ, ζ ρ, ζ η pseudoscalars and Φ i = ξ χ, ξ χ, ξ ρ, ξ ρ, ξ η real. For the charged components we have M Φ i Φ j = V higgs Φ i Φ j Φ i = 5.4 with Φ i = χ +, ρ +, η +, η +. From the equations above, we obtain the mass matrices M ζζ for the imaginary sector, M ξξ for the scalar real sector, and M φ ± for the charged scalar sector. They are written explicitly in appendix A Eqs. A., A. and A., respectively. 5.. Diagonalization of the charged sector To obtain the physical spectrum of scalar particles, we should diagonalize the mass matrices. For the matrix M φ in A. it is obtain that detm ± φ =, which guarantees null eigenvalues associated to the would be Goldstone bosons. The eigenvalue ± equation is detmφ P ii =. 5.5 ± Resolving this equation, a degeneration with P = P =, is observed; generating two would be Goldstone bosons associated to two charged gauge fields that acquire mass W + µ, K + µ. The eigenvectors V i associated to each eigenvalue are obtained from M φ ± IP i Vi =, 5.6 When solving this equation for the null eigenvalue modes see appendix A., and using the basis χ ±, ρ ±, η ±, η ±, it leads to the following combinations, except for a normalization factor φ ± ν ρ ρ ± + η ±, φ ± ν χ χ ± + ν ρ ρ ± η ±, 5.7 In the R ζ gauge, µ X µ αm X G X /α, the would be Goldstone boson G X of the gauge field X µ can be defined if we demand from the bilinear term of the gauge fixing to be canceled with the bilinear term coming from the kinetic scalar agrangian. In this case we observe that the equation above coincides directly with the first two relations of the Eq. B. in appendix B; where the would be Goldstone

20 R. A. Diaz, R. Martinez, and F. Ochoa bosons of the theory have been obtained from the bilinear term of scalar-gauge fields that appear in the kinetic term of the scalar fields. Therefore, it is deduced that φ ± endows the electroweak charged gauge bosons W ± µ with mass; while φ ± gives masses to the exotic gauge bosons K ± µ. The rest of the eigenvalues are complicate expressions no easy to visualize directly. To find a solution with a clear significance, we should bear in mind that the exotic particles non observed phenomenologically at low energies acquire their masses in the first transition of SSB, which happens at very high energies respect to the second transition. It means that [: χ ρ, η. 5.8 Under this approximation, it is valid to keep only terms involving ν χ in the matrix A.. If the aproximation of only quadratic terms ν χ is held, it is obtain the matrix shown in appendix A. Eq. A.. As it is observed, this approximation cancels all the components, except the term in the position M φ ± 4, 4, corresponding to the field η ±, from which information about one of the non-zero eigenvalues is lost. To obtain a better approximation we should allow additional terms but without demanding linear orders in ν χ. One way to have this, is by considering the coefficient f of the cubic term in the scalar potential eq. 4. under the following approximation [: f ν χ. 5.9 It means that terms of the form fν χ in the mass matrix Eq. A. are of the order O νχ, and should be preserved in the aproximation, from which we get the matrix of Eq. A.. As it was mentioned in Ref. [, the approximation 5.9 avoids the introduction of another mass scale apart from the ones defined by the two transitions in Eq... Effectively, when we study the first transition only, we obtain Higgs square masses of the order of νχ and fν χ, which can be considered of the same order of magnitude if f ν χ. In the matrix A., it is observed that the component η ± decouples, and the mass matrix is reduced to M ρη Eq. A. of appendix A.. Getting the eigenvalue P = λ 7 νχ fν ν ρ χ, i.e., and the last eigenvalue is h ± = η±, M h ± P 4 fν χ νη ν ρ whose eigenvector in the basis ρ ±, η ± is: λ 7 νχ fν ν ρ χ ν ρ, 5. h ± C θρ ± + S θ η ±, 5.

21 The scalar sector of the model with S θ = sin θ = ν ρ ν ρ + ν η, C θ = cosθ = ν ρ + ν η. 5. Using the hierarchy of the VEV s in Eq. 5.8 we finally find the particles spectrum of the scalar charged sector and their masses which we summarize in table. The fields φ ± and φ ± correspond to four charged and massless would-be Goldstone bosons, associated to four charged gauge bosons W ± µ and K± µ respectively; while h ± and h ± appears in the spectrum as four charged Higgs bosons with the masses indicated in table, of the order of the scale of the first SSB. The Higgs bosons h ± have real and positive masses if f <. Charged scalars Squared masses Features φ ± = S θρ ± + C θ η ±, M φ ± φ ± χ± M φ ± = = h ± η ± M λ h ± 7 ν ν χ fν ρ χ ν η h ± C θ ρ ± + S θ η ± M fν νη h ± χ + ν ρ ν ρ Goldstone associated to W ± µ Goldstone associated to K ± µ Higgs Higgs Table : Physical spectrum of charged scalar particles, for three Higgs triplets and β = /. 5.. Diagonalization of the imaginary sector Now we proceed to diagonalize the matrix Mζζ of Eq. A. appendix A, which has null determinant associated to the would be Goldstone bosons of the neutral massive gauge fields. Solving the eigenvalues equation for Mζζ we get the eigenvalues indicated in appendix A. Eq. A.4; three of them are degenerate with eigenvalue zero. Since the triplets of the scalar fields χ and ρ have the same quantum numbers they can be rotated to eliminate one VEV [4. With this basis it is simpler to find the mass eigenstates of the scalar sector ν χ ν χ, ν χ ν ρ ν ρ ν χ ν ρ. 5.4

22 R. A. Diaz, R. Martinez, and F. Ochoa The solution of the characteristic equation generates the eigenvectors written in their complete form in Eq. A.5. For the first three degenerate eigenvalues P = P = P =, we have the corresponding eigenvectors associated to the three would be Goldstone bosons. Using the basis of the matrix Mζζ Eq. A., we have φ = N ν ρ ζ ρ + ν χζ χ, φ = N ν χ ζ χ ζ η, φ = N ν ρ ζ ρ + ζ η 5.5 When we compare the first and the third of these combinations with the ones obtained in the appendix B Eq.B.4, they could be identified with φ and φ which are associated to the longitudinal modes of the gauge fields K µ and Z µ respectively. As can be seen in Eq. 5.5, the vector φ is not orthogonal to φ. Orthogonalizing, we get φ = N ν χ ζ χ ν ρ ζ ν ρ + ν ρ η ν ρ ν η ζ ν ρ + ν η. 5.6 η which correspond to the longitudinal component of Z µ. The other two eigenvalues different from zero in Eq. A.4, correspond to two massive neutral Higgs bosons, whose combination with respect to the basis of the matrix A. are the vectors φ 4 and φ 5 which we rename as h and h respectively. h = N 4 ν ρ ζ χ + ν χ ζ ρ + ν χ ν ρ ζ η, h = N 5 ν χ ζ ρ ν ρ ζ χ. 5.7 If additionally, we implement the approximations of Eqs. 5.8 and 5.9 in the results obtained, the particle spectrum can be written as indicated in table 4. The fields φ, φ and φ are the three neutral and massless would be Goldstone bosons, that dissapear from the particle spectrum to give masses to the three neutral gauge bosons K µ, Z µ and Z µ, respectively; while h and h appear in the spectrum as two neutral Higgs bosons with the masses indicated in table 4. It is taken f < to define the mass of h real and positive. 5.. Diagonalization of the real sector Finally, we take the matrix Mξξ in A. with null determinant too. The eigenvalues obtained are not simple. So we use the approximations of the Eqs. 5.8 and 5.9, obtaining the matrix given in appendix A. Eq. A.6, which decouples in a null eigenvalue would-be Goldstone boson and the two submatrices Mξ ρ ξ η and Mξ χξ ρ indicated in A.7. In this manner, it is defined φ 4 ξ χ, 5.8

23 The scalar sector of the model Imaginary scalars Square masses Features φ ζ χ, M φ = φ ζ χ M φ = φ = S θζ ρ + C θ ζ η M φ = h C θζ ρ + S θ ζ η M h h ζ ρ M h fνχ ν ρ + ν ρ fνηνχ ν ρ Goldstone asociado a K µ Goldstone asociado a Z µ Goldstone asociado a Z µ Higgs + λ 8 λ ν χ Higgs Table 4: Physical spectrum of the pseudoscalar fields, for three Higgs triplets and β = /. when comparing with φ 4 of Appendix B Eq. B.4 it is deduced a coupling to the gauge boson K µ. The submatrix M ξ ρ ξ η contains an eigenvalue zero. However, it does not correspond to a Goldstone boson. Such eigenvalue arose from the effect of the quadratic approximation in A.6. To extract information for this eigenvalue, we take all the terms of the sector Mξ ρ ξ η from Mξξ, obtaining an exact submatrix written in A.8. Based on this we get the eigenvalues P, P associated to the matrix Mξ ρ ξ η of A.8 and P 4, P 5 for the matrix Mξ χξ of A.7: ρ P = [ M + M + M M + 4M [M D + M C, P = [ M + M M M + 4M [M D M C, [ P 4 = Λ + + Λ + 6λ νχ, [ P 5 = Λ + Λ + 6λ νχ, 5.9 with Λ ± 4λ ± λ 8 + λ fν η. 5. ν χ ν ρ and M ij denote the elements of the matrix M ξ ρ ξ η. The non-normalized eigenvectors are written respectively as

24 4 R. A. Diaz, R. Martinez, and F. Ochoa h = S α ξ ρ + C α ξ η, h 4 = C α ξ ρ S α ξ η, h 5 Λ + Λ + 6λ ξ χ + 4λ ξ ρ, h 6 Λ Λ + 6λ ξ χ + 4λ ξ ρ, 5. where S α sin α, C α cosα. Being α a new mixing angle defined as: sin α = cos α = M M M + 4M M M. 5. M M + 4M the equations above define finally the physical spectrum of the scalar real fields, as indicated in table 5. Scalars Square masses Features φ 4 ξ χ M Goldstone = φ 4 of K µ h S [ αξ ρ + C α ξ η M = h MD + M C, Higgs h 4 C [ αξ ρ S α ξ η M = h MD M 4 C Higgs h 5 Λ ξ χ + 4λ ξ ρ M Λ h νχ 5 Higgs h 6 Λ ξ χ + 4λ ξ ρ M Λ h νχ 6 Higgs Table 5: Physical spectrum of masses for the real scalar fields, with three Higgs triplets and with β = /. We use the definitions of Eqs. 5.. φ 4 is a neutral would-be Goldstone boson associated to the gauge field K µ and h, h 4, h 5, h 6 are four neutral Higgs bosons with masses given in table 5; with the condition f < in order for h 4 to acquire a positive defined mass. Finally, we are able to summarize the particle spectrum of the scalar sector and the would be Goldstone bosons in table 6. Taking into account the following definitions

25 The scalar sector of the model 5 Λ ± Λ + 6λ Λ ± = 4λ ± λ 8 + λ fν η ν χ ν ρ ν χ ν ρ, ν ρ, ; f ν χ ; f <, Λ, S θ ν ρ ν ρ + ν η, C θ ν ρ + ν η sin α M MC, cos α M M MC,, M D M + M ; M C = M M + 4M M ij = M ξρξη ij 5. We see then that the scalar spectrum consists of eight would be Goldstone bosons four charged and four neutral ones plus ten physical Higgs bosons four charged and six neutral ones. 6 Trilinear Higgs-Gauge Bosons terms for β = / The term in Eq. B. contains cubic couplings between scalar and gauge bosons. We consider these couplings for the SM gauge bosons: W ± µ, Z µ and A µ, with masses given by [8: M W ± = g ν ρ + ν η M Z = g CW ν ρ + ν η. 6. In Eq. B. there are not cubic couplings of electroweak bosons with the scalar boson χ that acquire VEV in the rd component, as can be seen in Eq. B.5, where the SM gauge bosons are in the superior components, while in the third component rd row and column are the exotic gauge bosons which get their masses

26 6 R. A. Diaz, R. Martinez, and F. Ochoa Scalars Square masses Feature φ ± = S θ ρ ± + C θ η ± M = φ ± Goldstone φ ± χ± M = φ ± Goldstone φ ζ χ M = φ Goldstone φ ζ χ M φ = Goldstone φ = S θζ ρ + C θ ζ η M φ = Goldstone φ 4 ξ χ M φ 4 = Goldstone h ± η ± M h ± λ 7 νχ fν ν ρ χ Higgs h ± C θ ρ ± + S θ η ± M fν νη h ± χ ρ Higgs ν ρ h C θζ ρ + S θ ζ η M fνχ h ν + νρ Higgs ρ h ζ ρ M fνηνχ + λ h ν ρ 8 λ νχ Higgs h S αξ ρ + C α ξ η M ν h η ν ρ Higgs h 4 C αξ ρ S α ξ η M h 4 fν χ Higgs h 5 Λ ξ χ + 4λ ξ ρ M h 5 Λ ν χ Higgs h 6 Λ ξ χ + 4λ ξ ρ M h 6 Λ ν χ Higgs Table 6: Physical scalar spectrum after both transitions, for three Higgs triplets with β = /. Taking into account the definitions in Eqs. 5.. through v χ. Using Eqs. B.5 and B.7 for Φ = ρ and η in Eq. B., it is found SM trilinear = g ν ρ ξ ρ + ξ η W µ W µ + + a 4 ν ρ ξ ρ + ξ η Z µ Z µ + g a νρ ρ + η A µ W µ + + h.c + g a + a 4 ν ρ ρ + η Z µ W µ + ig µ ρ W + µ ξ ρ ig µ ξ ρ W µ ρ + ig µ η + W µ ξ η ig µ ξ η W µ + η + h.c, 6. where a, a and a 4 are defined in Eq. B.7. The terms in Eq. 6. are in the weak basis, which are related to the physical basis through table 6. In the physical basis, and after some algebraic manipulation using Eq. B.7 in Eq. 6., it is finally

27 The scalar sector of the model 7 found SM trilinear = gm W ± cos θ α W µ W µ + h + gm W ± sin θ αw µ W µ + h 4 g + M Z cos θ αz µ Z µ h C + g M Z sin θ αz µ Z µ h 4 W C W em W ±A µ W µ + φ + gm ZSW Zµ W µ + φ + h.c g p kµ cos θ αw + µ h h 4 + g p qµ sin θ αw + µ h h ig p rµ W µ + h h + h.c, 6. where the electric charge has been defined as e = gs W, and θ and α are given by Eqs. 5. and 5.. It worths to note that these vertices are the same as the ones obtained in the Standard Model with two Higgs doublets HDM. In the notation of Ref. [, the Higgs bosons of the HDM have the following correspondence with the Higgs notation shown here h h, h 4 H, h A, h ± H±, φ ± G± W, φ G Z The couplings of the charged Higgs bosons to the vertex A µ W µ vanish because of the conservation of electromagnetic charge and only the couplings to the would-be Goldstone bosons can exist since they are not physical fields. In fact, we can realize that the aproximation fν x >> ν ρ,ν η implies that the boson h 4 very massive Higgs decouple from the vertices, and that h couple in the same way as the SM Higgs boson, without any other constraint. In this limit, it is obtained that tanα ν ρ 6.4 νη ν ρ i.e. θ α, and the SM trilinear SM is given by trilinear = gm W ±W µ W µ + h + g M Z Z µ Z µ h C W em W ±A µ W µ + φ gm ZSW Zµ W µ + φ + h.c g p kµ W µ + h h 4 + ig p rµ W µ + h h + h.c Where Eq. 6.5 shows explicitly the fact that the couplings of h are SM-like when the limit fν x >> ν ρ,ν η ; is taken. 7 ow energy limit for β = / We shall discuss briefly some important properties of the low energy limit of this model. As we see in table 6, the Higgs scalar masses depend on some parameters of the Higgs potential, the vacuum expectation value ν χ at the higher scale, and

28 8 R. A. Diaz, R. Martinez, and F. Ochoa the vacuum expectation values at the electroweak scale, ν ρ as it must be. Notwithstanding, there is a crucial parameter that determines the scale in which the Higgs bosons lie, i.e. the trilinear coupling constant f defined in Eq. 4.. For arbitrary values of it or more precisely for arbitrary values of fν χ some of the Higgs bosons would lie on a new scale different from the breakdown ones. If we argue a naturalness criterium, we could assume that fν χ should lie either in the electroweak or SU U X breaking scale. Since these models break to a SU U Y two Higgs doublet model; the value assumed for fν χ determines the scales for the SU U Y Higgs bosons which can be identified as the h ±, h, h and h 4 fields of table 6, where φ± and φ are the corresponding would-be Goldstone bosons. We also see in table 6 that h is the only Higgs field that depends exclusively on the vacuum expectation values at the electroweak scale, so it corresponds to the SM Higgs. The scale of the other SU U Y Higgs bosons depend on the value assumed by fν χ. If fν χ O νχ the remaining SU U Y Higgs bosons belong to the higher breaking scale and for sufficiently large values of νχ, the h ±, h and h 4 fields become heavy modes, in addition by taking fν χ large enough we obtain automatically that the two mixing angles of the Higgs sector in the remnant two Higgs doublet model become equal, from which the decoupling limit at the electroweak scale to the minimal SM follows naturally[5. Otherwise, if we assume fν χ O νew the diagonalization process for the scalar sector would be much more complicated, but a naive analysis shows that we would get a spectrum of SU U Y Higgs bosons lying roughly on the EW scale, obtaining a nondecoupling two Higgs doublet model. 8 Conclusions For the models, the set of equations arising from anomalies predicts that the number of triplets and antitriplets must be equal in order to cancel anomalies. These equations lead to an infinite set of possible solutions. If we impose for the model not to have exotic electromagnetic charges, the number of solutions is reduced to only two. On the other hand, the quantum numbers may be restricted by demanding gauge invariance for the Yukawa couplings of the scalar fields to fermions classical constraints as well as from the cancelation of fermionic anomalies quantum constraints, obtaining models with ordinary and exotic charges. etting free the parameters X and β that define the electromagnetic charge, we get all possible vacuum alignments in both symmetry breakings. When we choice the VEV s according to the breakdown scheme the values for X and β are found, obtaining the models already studied in the literature plus other ones with exotic charges. Moreover, it is found that one scalar triplet is necessary for the first breaking and two scalar triplets for the second one, the two triplets for the second breaking contain two doublets of SU for the second breaking that give masses to the up and down quarks respectively. In some cases it is necessary to introduce an additional

29 The scalar sector of the model 9 scalar sextet in order to endow the neutrinos with masses. Furthermore, we present the most general Higgs potentials for the models with β = ±, β = ±/ and β arbitrary with three scalar triplets and one scalar sextet. In particular, the case of β = / is analized, getting the scalar spectrum in the R ζ gauge; such spectrum consists of ten massive Higgs bosons as well as eight would be Goldstone bosons necessary to break the eight generators of the SSB scheme demanded. Besides, in the framework of the β = / model, we calculate the trilinear couplings of the Higgs bosons to the SM gauge bosons, finding that such couplings are identical to the ones of the standard two Higgs doublet model HDM. It worths to say that A µ W µ ± are not coupled to the charged Higgs bosons in consistence with electromagnetic charge conservation. As for the low energy limit νx >> ν ρ, ν η, the Higgs potential of the model with β = / is also reduced to the Higgs potential of the HDM in this limit. Thus, this model ends up in a SU U Y two Higgs doublet model after the first symmetry breaking. We see that the behavior of the scalar sector of these models at low energies is strongly determined by the trilinear coupling f of the Higgs potential. For instance, by setting fν χ O νew all the SU U Y Higgs bosons belong to the electroweak scale, obtaining a non-decoupling two Higgs doublet model. By contrast, the assumption fν χ O νχ leaves one of these Higgs bosons at the electroweak scale the SM Higgs, and the other four ones would lie on the scale O ν χ of the first transition; additionally, if the scale O ν χ is sufficiently large, we obtain that the mixing angles of the Higgs potential are equal, from which the couplings of the light Higgs boson to fermions and gauge fields become identical to the SM couplings. Therefore, the natural assumption fν χ O νχ >> ν ρ, ν η leads automatically to the minimal SM at low energies. Acknowledgments We thank to Fundación Banco de la República for its financial support.

30 R. A. Diaz, R. Martinez, and F. Ochoa A Scalar masses i Imaginary sector The mass matrix M ζζ in the basis ζ χ, ζ ρ, ζ ρ, ζ η, ζ χ is given by M ζζ, = λ 8 4λ ν ρ fν ρ ν χ + ν ρ ν χ ν ρ M ζζ, = M ζζ, = f M ζζ, = M ζζ, = λ 8 + 4λ ν chi ν ρ + f ν ρ ν eta ν ρ M ζζ, 4 = M ζζ 4, = fν ρ M ζζ, 5 = M ζζ 5, = λ 8 4λ ν ρ ν ρ M ζζ, = f ν χ ν ρ Mζζ, = M ζζ, = M ζζ, 4 = M ζζ 4, = fν χ M ζζ, 5 = M ζζ 5, = f ν ρ ν ρ M ζζ, = λ 8 4λ ν χ f ν χ ν ρ M ζζ, 4 = M ζζ4, = M ζζ, 5 = M ζζ5, = λ 8 4λ ν χ f ν χ ν ρ M ζζ 4, 4 = f ν χν ρ Mζζ4, 5 = Mζζ5, 4 = fν ρ Mζζ5, 5 = λ 8 4λ ν ρ f ν ρ ν χ + ν ρ ν χ ν ρ A.

31 The scalar sector of the model ii. Real sector The mass matrix M ξξ in the basis ξ χ, ξ ρ, ξ ρ, ξ η, ξ χ is given by Mξξ, = 8λ νχ + 8λ ν χ ν ρ + λ 8 + 4λ ν ρ ν ρ f + ν ην ρ ν χ ν χ ν ρ M ξξ, = M ξξ, = 4λ 5ν χ ν ρ + 4λ ν ρ ν ρ + f M ξξ, = M ξξ, = 4λ 5 + λ 8 + 4λ ν χ ν ρ + 4λ ν χ + 4λ ν ρ + f ν ρ M ξξ, 4 = M ξξ4, = 4λ 4 ν χ + 4λ M ξξ, 5 = M ξξ 5, = λ 8 + 4λ ν ρ ν ρ + 4λ ν χ ν ρ M ξξ, = 8λ ν ρ f ν χ ν ρ M ξξ, = M ξξ, = 8λ ν ρ ν ρ + 4λ ν χ ν ρ M ξξ, 4 = M ξξ 4, = 4λ 6 ν ρ + fν χ M ξξ, 5 = M ξξ 5, = 4λ ν ρ + f ν ρ ν ρ M ξξ, = λ 8 + 4λ ν χ + 8λ ν χ ν ρ + 8λ ν ρ f ν χ ν ρ M ξξ, 4 = M ξξ4, = 4λ 6 ν ρ + 4λ ν χ M ξξ, 5 = M ξξ 5, = λ 8 + 4λ ν χ ν ρ + 4λ ν ρ ν ρ f M ξξ4, 4 = 8λ ν η f ν χν ρ M ξξ 4, 5 = M ξξ 5, 4 = 4λ ν ρ fν ρ ν ρ M ξξ 5, 5 = λ 8 + 4λ ν ρ fν ρ ν χ + ν ρ A. ν χ ν ρ iii. Charged sector The mass matrix M φ± in the basis χ±, ρ ±, η ±, η ± reads

32 R. A. Diaz, R. Martinez, and F. Ochoa M φ±, = λ 7ν η f ν ρ ν χ M φ±, = M φ±, = λ 4ν η + f ν ρ ν ρ M φ±, = M φ±, = λ 4 ν ρ + fν ρ + ν ρ ν χ ν ρ M φ±, 4 = M φ±4, = λ 7 ν χ + λ 4 ν ρ fν ρ M φ±, = λ 9 ν η f ν χ ν ρ M φ±, = M φ±, = λ 9 ν ρ fν χ M φ±, 4 = M φ±4, = λ 9 ν ρ + λ 4 ν χ M φ±, = λ 9 ν ρ f ν ρ ν χ M φ±, 4 = M φ± 4, = λ 9ν ρ ν ρ + λ 4 ν χ ν ρ M φ± 4, 4 = λ 7ν χ + λ 4ν χ ν ρ + λ 9 ν ρ f ν χν ρ A. The previous matrices are singular, it is associated to the would be Goldstone bosons of the theory at least one null eigenvalue. detm ξξ = detm ζζ = detm φ± =. A.4 A. Diagonalization of the charged sector The matrix Mφ± in Eq. A. presents null eigenvalues. For the eigenvalue P =, the equation 5.6 is block reduced to: ν χ x ν ρ ν ρ x x =, A.5 x 4 it produces two rows of zeros because of the fact that the zero eigenvalue is two-fold degenerate, such that only two of the four equations defined by 5.6 are linearly independent, as can be seen in Eq. A.5. Since there are four variables, we take a particular solution among the infinite ones, for instance and when we replace in A.5, it is found x = ; x 4 =, A.6 x = ; x = ν ρ. A.7

33 The scalar sector of the model obtaining the solution φ ± written in Eq To find out the solution of the other degenerate eigenvalue P =, we choose and from Eq. A.5, we find x = ; x 4 =, A.8 x = ν χ ; x = ν ρ. A.9 obtaining φ ± in Eq For the other two eigenvalues we utilize the approximation given by Eqs Keeping only quadratic order in ν χ, it is obtained from the matrix in Eq. A.: M φ± χ ± ρ ± η ± η ± A.. A. λ 7 ν χ with only one eigenvalue. To be able to find out the other eigenvalue, we demand the condition of the Eq. 5.9, and the matrix A. can be approximated at order ν χ, fν χ to the form M φ± which is block reduced in the following way χ ± ρ ± η ± η ± f νηνχ fν ν ρ χ fν χ f ν ρ ν χ λ 7 νχ f νχν ρ M χ ± = ρ ± η ±, A. [ f ν χ M ρη ν ρ fν χ M η ± λ 7ν χ f ν χν ρ fν χ f ν ρ ν χ, A. The eigenvalue P = λ 7 ν χ fν χ ν ρ, is associated to the decoupled component η ± as it is written in Eq. 5.. The square matrix in Eq. A. contains one of the null degenerate eigenvalues. The other one is the P 4 eigenvalue written in Eq. 5., whose eigenvector is given by Eq. 5..