On the Higgs spectra of the 3-3-1 model Jo ao Paulo Pinheiroa C. A. de S. Piresb aDepartament de F sica Qu antica i Astrof sica and Institut de Ci encies del Cosmos

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On the Higgs spectra of the 3-3-1 model
Jo˜ao Paulo Pinheiroa, C. A. de S. Piresb
aDepartament de F´ısica Qu`antica i Astrof´ısica and Institut de Ci`encies del Cosmos,
Universitat de Barcelona, Diagonal 647,
E-08028 Barcelona, Spain,
bDepartamento de F´ısica,
Universidade Federal da Para´ıba,
Caixa Postal 5008, 58051-970,
Jo˜ao Pessoa, PB, Brazil,
(Dated: October 31, 2022)
Abstract
The minimal scalar sector of the 3-3-1 model is composed by the SU(3)Ltriplet scalars η,ρ,
χand its potential allows the trilinear term f
√2χηρ. Since fis an energy scale associated to the
explicit violation of Peccei-Quinn global symmetry, it is natural to consider in what energy scale
such symmetry is broken and its consequences in the spectrum of scalars of the model. Here, we
show that fdetermines the spectrum of scalars of the model. Hence, we develop the scalar sector
considering fbelonging to four energy regimes, namely f hηi0,hρi0;f=hηi0,hρi0;f=hχi0
and f hχi0and obtain the spectrum of scalars for each case. In the first and second cases the
spectrum of scalars presents a set of new scalars belonging to the electroweak scale, while in the
third case all new scalars belong to the 3-3-1 scale and the fourth case all the new scalars have
masses lying at fscale. All cases have a neutral CP-even scalar mimicking the standard Higgs.
1
arXiv:2210.05426v2 [hep-ph] 28 Oct 2022
I. INTRODUCTION
The detection of the standard Higgs boson by ATLAS[1] and CMS[2] and the search
for additional Higgs bosons by the LHC, motivates the study of the scalar sector of gauge
extensions of the standard model (SM) with the purpose of checking if the scalar spectra
of these models contain a CP even with 125 GeV of mass that mimics the Higgs of the SM
and verify if the spectra of new scalars is reachable by current and future colliders.
The SU (3)C×SU (3)L×U(1)Nmodel with right-handed neutrinos (331RHNs)[3–6] is
a well-motivated gauge extension of the standard model. It provides an understanding for
family replication[7–9], solve the strong CP-problem[10–12], gives an explanation for the
pattern of quantization of electric charges[13, 14] and has a rich phenomenology in what
concern flavor physics[7, 8][15–24], dark matter [25–30] and provide a robust material for
collider physics as, for example, new neutral and charged gauge bosons belonging to the
TeV scale[31].
Here we are interested in the Higgs sector of the 331RHNs. Besides this model has
received considerable attention in the last decades, its scalar sector is poorly developed
and its mass spectra is unknown[32–36] . In this work we develop the scalar sector of the
331RHNs and obtain its spectra of scalars varying the energy scale f.
This work is organized as follow: In Sec. II we make a brief review of the scalar sector
where we present the texture of all the four mass matrices involving the scalars of the model.
In Sec. III we obtain the spectrum of scalars for four scenario characterized by the regime
of energy that fis supposed to belong. In Sec. IV we present our conclusions.
II. GENERAL ASPECT OF THE SCALAR SECTOR
The scalar sector of the 331RHNs is composed by the following three triplets [6],
η=
η0
η−
η00
, ρ =
ρ+
ρ0
ρ0+
, χ =
χ0
χ−
χ00
,(1)
with ηand χtransforming as (1 ,3,−1/3) and ρas (1 ,3,2/3). After spontaneous breaking
of the symmetry, such content of scalar generates masses for all massive particles of the model
2
including fermions and gauge bosons1.
The most general potential that conserve lepton number involves the following terms[10]:
V(η, ρ, χ) = µ2
χχ2+µ2
ηη2+µ2
ρρ2+λ1χ4+λ2η4+λ3ρ4+
λ4(χ†χ)(η†η) + λ5(χ†χ)(ρ†ρ) + λ6(η†η)(ρ†ρ) +
λ7(χ†η)(η†χ) + λ8(χ†ρ)(ρ†χ) + λ9(η†ρ)(ρ†η)
−f
√2ijkηiρjχk+ H.c. .(2)
In order to avoid spontaneous breaking of the lepton number, we assume that only η0, ρ0,
and χ00develop VEV2. Shifting the fields in the usual way,
η0, ρ0, χ00=1
√2(vη ,ρ ,χ0+Rη ,ρ ,χ0+iIη ,ρ ,χ0),(3)
the potential above provides the following set of equations that guarantee the potential
develop a minimum
µ2
χ+λ1v2
χ0+λ4
2v2
η+λ5
2v2
ρ−f
2
vηvρ
vχ0
= 0,
µ2
η+λ2v2
η+λ4
2v2
χ0+λ6
2v2
ρ−f
2
vχ0vρ
vη
= 0,
µ2
ρ+λ3v2
ρ+λ5
2v2
χ0+λ6
2v2
η−f
2
vηvχ0
vρ
= 0.(4)
After solving the minimum conditions of the potential, we are able to obtain the mass
matrices of the scalars of the model. We remember that v2
η+v2
ρ= (246)2GeV2, while vχ0
characterize the energy scale of the breaking of the SU (3)L×U(1)Nsymmetry. Current
LHC bounds imposes vχ0≥4TeV[41, 42].
The potential above and the constraints in Eq. (4) provide the following mass matrix for
the CP-even neutral scalars according the basis (Rχ0, Rη, Rρ):
M2
R=
λ1v2
χ0+fvηvρ/4vχ0λ4vχ0vη/2−fvρ/4λ5vχ0vρ/2−fvη/4
λ4vχ0vη/2−fvρ/4λ2v2
η+fvχ0vρ/4vηλ6vηvρ/2−f vχ0/4
λ5vχ0vρ/2−fvη/4λ6vηvρ/2−f vχ0/4λ3v2
ρ+fvχ0vη/4vρ
,(5)
After diagonalization, the eigenvalues of this matrix cannot assume a simple analytic form
due to the parameter f. Hence, we can only find eigenvalues and eigenvectors associated to
1For the development of the other sectors of the model, see Refs.[23, 37–39]
2For the case where the other neutral scalars develop VEVs, see Ref. [40]
3
M2
Rafter assuming some approximations among fand the others energy scales vχ0,vηand
vρ. Here we consider four scenarios we think that are important.
Now, let us move to the CP-odd scalars. Considering the basis (Iχ0, Iη, Iρ), we obtain the
following mass matrix for these scalars:
M2
I=
fvηvρ/4vχ0f vρ/4fvη/4
fvρ/4f vχ0vρ/4vηfvχ0/4
fvη/4f vχ0/4fvχ0vη/4vρ
.(6)
This matrix allows an analytical solutions to their eigenvalues and eigenvectors. Assuming
vχ0vη, vρwe get two null eigenvalues, which correspond to the eigenstates Iχ0and
G=vη
√v2
η+v2
ρ
Iη−vρ
√v2
η+v2
ρ
Iρ, and a massive one with mass expression given by m2
A=fvχ0
4(vηvρ
v2
χ0+
vη
vρ+vρ
vη) corresponding to the eigenstate A=vρ
√v2
η+v2
ρ
Iη+vη
√v2
η+v2
ρ
Iρ. Observe that if f= 0
then mA= 0 which means that Ais a pseudo Goldstone associate to some U(1)Xglobal
symmetry. In fact it was showed in [10] that U(1)Xis the Peccei-Quinn symmetry and Ais
the Weinberg-Wilczek axion[43, 44]. Of course that this Axion must be avoided. The triliner
term fηρχ is the most economical way of breaking explicitly Peccei-Quinn symmetry and
still conserve lepton number3.
The other neutral scalars are χ0an η00. They carry two units of lepton number each[25].
As we assume lepton number is preserved, they do not mix with η0,ρneither χ00. Considering
the basis (χ0, η00) we obtain the following mass matrix
M2
χη0=
λ7v2
η/4 + fvηvρ/4vχ0−λ7vηvχ0/4−f vρ/4
−λ7vηvχ0/4−fvρ/ λ7v2
χ0/4 + fvρvχ0/4vη
,(7)
After diagonalizing it we obtain G3=vχ0
qv2
η+v2
χ0
χ0+vη
qv2
η+v2
χ0
η00that is the Goldstone
eaten by the non-hermitian gauge bosons U0and U0†. The other neutral scalar is G4=
−vη
qv2
η+v2
χ0
χ0+vχ0
qv2
η+v2
χ0
η00whose mass is given by the expression m2
G4=λ7
4(v2
χ0+v2
η)+ f vρ
4(vχ0
vη+
vη
vχ0). This mass depends of fand vχ0which means that this scalar is a heavy particle whatever
is the approximation assumed on f.
Now, let us consider the charged scalars. According to the basis (χ−, ρ0−, η−, ρ−), we
3For the implementation of Peccei-Quinn symmetry with a safe axion in the 331RHNs, see [11]
4
摘要：

OntheHiggsspectraofthe3-3-1modelJo~aoPauloPinheiroa,C.A.deS.PiresbaDepartamentdeFsicaQuanticaiAstrofsicaandInstitutdeCienciesdelCosmos,UniversitatdeBarcelona,Diagonal647,E-08028Barcelona,Spain,bDepartamentodeFsica,UniversidadeFederaldaParaba,CaixaPostal5008,58051-970,Jo~aoPessoa,PB,Brazil,...

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