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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

Brazil

Abstract

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

1Introduction

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 [6].

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

group

Q =

1

2

?

aλ3+

b

√3λ8+

c

√6λ15

?

+ NI4×4

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= diag

?1

2

?

a +b

3+c

6

?

+ N,1

2

?

−a +b

3+c

6

?

+ N,1

2

?−2b

3

+c

6

?

+ N,−c

4

+ N

?

(1)

,

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. [9],

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. [6] 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 [1], 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 [6].

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.

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2 The model

In this section (Sec. 2.1) we review the non-supersymmetric 3-4-1 model of

Refs. [6] 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. [6], the free parameters for the eletric charge generators

are

a = 1,b = −1,

and Eq.(1) can be rewritten as

c = −4,(2)

Q =

1

2

?

λ3−

1

√3λ8−

4

√6λ15

?

+ NI4×4,

= 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

LaL=

νa

la

νc

lc

a

a

L

∼ (1,4,0), a = 1,2,3.(4)

In parenthesis it appears the transformations properties under the respective

factors (SU(3)C,SU(4)L,U(1)N).

In the quark sector, one quark family is also put in the quartet represen-

tation

Q1L=

u1

d1

u′

J

L

∼

?

3,4,2

3

?

,(5)

1In the same way as proposed by Voloshin [9] in order to understand the existence of

neutrinos with large magnetic moment and small mass.

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and the respective singlets are given by

uc

1L

∼

?

?

3∗,1,−2

3∗,1,−2

3

?

?

,dc

1L∼

?

3∗,1,−5

3∗,1,1

3

?

,

u′c

L

∼

3

,Jc

L∼

?

3

?

,(6)

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

Q2L=

d2

u2

d′

j1

1

L

∼

?

3,4∗,−1

3

?

,Q3L=

d3

u3

d′

j2

2

L

∼

?

3,4∗,−1

3

?

,

(7)

and also with the respective singlets,

uc

αL

∼

?

?

3∗,1,−2

3∗,1,1

3

?

?

,

,dc

αL∼

?

3∗,1,4

3∗,1,1

3

?

,

d′c

βL ∼

3

jc

βL∼

?

3

?

, (8)

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-

η =

η0

η−

1

η0

η+

2

ρ+

ρ0

ρ+

ρ++

1

2

∼ (1,4,0)

ρ =

1

2

∼ (1,4,1)

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χ =

χ−

χ−−

χ−

χ0

1

2

∼ (1,4,−1). (9)

In order to avoid mixing among primed and unprimed quarks, we have to

introduce an extra scalar transforming like η but with different vacuum ex-

pectation value (VEV)

φ =

φ0

φ−

φ0

φ+

1

1

2

2

∼ (1,4,0) . (10)

In order to obtain massive charged leptons it is necessary to introduce the

following symmetric anti-decuplet

H =

H0

H+

H0

H−

1

H+

H++

1

H+

H0

1

H0

H+

H0

H−

2

H−

H0

H−

H−−

2

2

1

3

3

2

3

4

4

2

3

4

∼ (1,10∗,0).(11)

then the charged leptons get a mass but neutrinos remain massless, at least

at tree level.

2.2 Supersymmetric partners

Now, we introduce the minimal set of particles in order to implement the

supersymmetry [12]. We have the sleptons corresponding to the leptons

in Eq. (4); squarks related to the quarks in Eqs.(6)-(8); and the Higgsinos

related to the scalars given in Eqs. (9) and (11). Then, we have to introduce

the following additional particles

˜Q1L =

˜ u1

˜d1

˜ u′

˜J

L

∼

?

3,4,2

3

?

,

˜QαL=

˜dα

˜ uα

˜d′

˜jβ

β

L

∼

?

3,4∗,−1

3

?

,