arXiv:hep-ph/0701088v1 11 Jan 2007
A supersymmetric 3-4-1 model
M. C. Rodriguez
Funda¸ c˜ ao Universidade Federal do Rio Grande-FURG
Departamento de F´ ısica
Av. It´ alia, km 8, Campus Carreiros
96201-900, Rio Grande, RS
We build the complete supersymmetric version of a 3-4-1 gauge
model using the superfield formalism. We point out that a discrete
symmetry, similar to the R-symmetry in the minimal supersymmetric
standard model, is possible to be defined in this model. Hence we
have both R-conserving and R-violating possibilities. We also discuss
some phenomenological results coming from this model.
PACS numbers: 12.60.-i, 12.60.Jv
The full symmetry of the so called Standard Model (SM) is the gauge group
SU(3)c⊗ SU(2)L⊗ U(1)Y. Nevertheless, the SM is not considered as the
ultimate theory since neither the fundamental parameters, masses and cou-
plings, nor the symmetry pattern are predicted. Even though many aspects
of the SM are experimentally supported to a very accuracy, the embedding
of the model into a more general framework is to be expected.
Some of these possibilities is that, at energies of a few TeVs, the gauge
symmetry may be SU(3)c⊗ SU(3)L⊗ U(1)N (3-3-1 for shortness) [1, 2,
3]. Recently, the supersymmetric version of these model have alreday benn
constructed in [4, 5]. These 3-3-1 models can be embedded in a model with
3-4-1, its mean SU(3)c⊗ SU(4)L⊗ U(1)Ngauge symmetry .
In SU(4)L⊗ U(1)N, the most general expression for the electric charge
generator is a linear combination of the four diagonal generators of the gauge
where λi, being the Gell-Mann matrices for SU(4)L, see [7, 8], normalized as
Tr(λiλj) = 2δij, I4×4= diag(1,1,1,1) is the diagonal 4 × 4 unit matrix, and
a, b and c are free parameters to be fixed next. Therefore, there is an infinite
number of models can, in principle, be constructed.
A model with the SU(4) ⊗ U(1) symmetry in the lepton sector, quarks
were not considered on this work, was suggested some years ago in Ref. ,
in wchich the magnetic moment of neutrinos arises as the result of charged
scalars that belong to an SU(4) sextet, and the mass of neutrino arises at
two-loop level as the result of electroweak radiative correction.
The 3-4-1 model in Ref.  contain exotic electric charges only in the
quark sector, while leptons have ordinary electric charges and gauge bosons
have integer electric charges. The best feature of this model is that it provides
us with an alternative to the problem of the number Nfof fermion families.
These sort of models are anomaly free only if there are equal number of
quadriplet and anti-quadriplet (considering the color degrees of freedom),
and furthermore requiring the sum of all fermion charges to vanish. Two of
the three quark generations transform identically and one generation, it does
not matter which one, transforms in a different representation of SU(4)L⊗
U(1)N. This means that in these models as in the SU(3)c⊗SU(3)L⊗U(1)N
ones , in order to cancel anomalies, the number of families (Nf) must be
divisible by the number of color degrees of freedom (n). This fact, together
with asymptotic freedom in QCD, the model predicts that the number of
generations must be three and only three.
On the other hand, at low energies these models are indistinguishable from
the SM. There is a very nice review about this kind of model see [10, 11].
This make 3-4-1 model interesting by their own. In this article we construct
the supersymmetric version of the model in Ref .
The outline of the paper is as follows. In Sec. 2 we present the represen-
tation content of the supersymmetric 3-4-1 model. We build the lagrangian
in Sec. 3. While in Sec. 4, we discuss the double charged charginos inthis
model, while in the last section we present our conclusion.
2 The model
In this section (Sec. 2.1) we review the non-supersymmetric 3-4-1 model of
Refs.  and add the superpartners (Sec. 2.2) of the usual particles of the non
supersimmetric model. The superfields, useful to construct the supersimmet-
ric lagrangian of the model, associated with the particles of this model are
introduced in section (Sec. 2.3).
2.1 The representation content
In the model of Ref. , the free parameters for the eletric charge generators
a = 1,b = −1,
and Eq.(1) can be rewritten as
c = −4,(2)
= diag(N,N − 1,N,N + 1).(3)
However, let us first consider the particle content of the model without su-
persymmetry. We have the leptons transforming in the lowest representation
of SU(4)Lthe quartet1in the following way
∼ (1,4,0), a = 1,2,3.(4)
In parenthesis it appears the transformations properties under the respective
In the quark sector, one quark family is also put in the quartet represen-
1In the same way as proposed by Voloshin  in order to understand the existence of
neutrinos with large magnetic moment and small mass.
and the respective singlets are given by
writing all the fields as left-handed; u′and J are new quarks with charge
+2/3 and +5/3 respectively.
The others two quark generations, as we have explained in the introduc-
tion, we put in the anti-quartet representation
and also with the respective singlets,
jβ and d′
tively, while α = 2,3 is the familly index for the quarks. We remind that in
Eqs. (4,5,6,7,8) all fields are still symmetry eigenstates.
On the other hand, the scalars, in quartet, which are necessary to generate
the quark masses are
β, β = 1,2 are new quarks with charge −4/3 and −1/3 respec-
From the phenomenological point of view there are several possibilities.
Since it is possible to define the R-parity symmetry, the phenomenology of
this model with R-parity conserved has similar features to that of the R-
conserving MSSM: the supersymmetric particles are pair-produced and the
lightest neutralino is the lightest supersymmetric particle (LSP).
While in the case that R-parity is not conserved we can induce masses to
neutrinos of the model in the same way as in the MSSM. We also studied the
proton decay problem in this model, and we show that it is in asgreement
with the experimental data.
This work was supported by Conselho Nacional de Ciˆ encia e Tecnologia
(CNPq) under the processes 309564/2006-9.
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