**Supersymmetric 3-3-1 model**

J. C. Montero, V. Pleitez, and M. C. Rodriguez

*Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona, 145, 01405-900 Sa˜o Paulo, SP, Brazil*

~Received 15 December 2000; published 10 January 2002!

We build a complete supersymmetric version of a 3-3-1 gauge model using the superfield formalism. We
point out that a discrete symmetry, similar to*R*symmetry in the minimal supersymmetric standard model, is
possible to be defined in this model. Hence we have both*R-conserving andR-violating possibilities. Analysis*
of the mass spectrum of the neutral real scalar fields show that in this model the lightest scalar Higgs boson has
a mass upper limit, and at the tree level it is 124.5 GeV for a given illustrative set of parameters.

DOI: 10.1103/PhysRevD.65.035006 PACS number~s!: 11.30.Pb, 12.60.Jv

**I. INTRODUCTION**

Although the standard model ~SM!, based on the gauge
symmetry *SU*(3)* _{c}*^

_{SU}_{(2)}

*L*^*U*(1)*Y*, describes the

ob-served properties of charged leptons and quarks, it is not the ultimate theory. However, the necessity to go beyond it, from the experimental point of view, comes at the moment only from neutrino data @1#. If neutrinos are massive then new physics beyond the SM is needed. From the theoretical point of view, the SM cannot be a fundamental theory since it has so many parameters and some important questions such as that of the number of families do not have an answer in its context. On the other hand, it is not clear what the physics beyond the SM should be. An interesting possibility is that at the TeV scale physics would be described by models which share some of the flaws of the SM but give some insight concerning some questions which remain open in the SM context.

One of these possibilities is that, at energies of a few
TeVs, the gauge symmetry may be *SU*(3)* _{c}*^

_{SU}_{(3)}

*L*
^* _{U}*(1)

*~3-3-1 for shortness! instead of that of the SM*

_{N}@2,3#. In fact, this may be the last symmetry involving the
lightest elementary particles: leptons. The lepton sector is
exactly the same as in the SM but now there is a symmetry,
at large energies among, say *e*2_{,} _{n}

*e*, and *e*1. Once this

symmetry is imposed on the lightest generation and extended to the other leptonic generations it follows that the quark sector must be enlarged by considering exotic charged quarks. It means that some gauge bosons carry lepton and baryon quantum number. Although this model coincides at low energies with the SM it explains some fundamental questions that are accommodated, but not explained, in the SM. These questions are the following.

~i!The family number must be a multiple of three in order
to cancel anomalies @2,3#. This result comes from the fact
that the model is anomaly free only if we have equal number
of triplets and antitriplets, counting the *SU*(3)* _{c}* colors, and
furthermore requiring the sum of all fermion charges to
van-ish. However, each generation is anomalous, the anomaly
cancellation occurs for the three, or multiple of three,
to-gether and not generation by generation like in the SM. This
may provide a first step towards answering the flavor
ques-tion.

~ii! Why sin2u* _{W}*,14 is observed. This point comes from
the fact that in the model of Ref.@2#we have that the

*U*(1)

_{N}and *SU*(3)* _{L}* coupling constants,

*g*

### 8

and*g*, respectively, are related by

*t*2[*g*

### 8

2

*g*2 5

sin2_{u}

*W*

124 sin2_{u}

*W*

. ~1!

Hence, this 3-3-1 model predicts that there exists an energy
scale, saym, at which the model loses its perturbative
char-acter. The value of m can be found through the condition
sin2_{u}

*W*(m)51/4. However, it is not clear at all what is the

value ofm; in fact, it has been argued that the upper limit on the vector bilepton masses is 3.5 TeV@4#instead of the 600 GeV given in Ref.@5#. Hence, if we want to implement su-persymmetry in this model, and to remain in a perturbative regime, it is natural that supersymmetry is broken at the TeV scale. This is very important because one of the motivations for supersymmetry is that it can help to understand the hier-archy problem: if it is broken at the TeV scale. Notwithstand-ing, in the context of the SM it is necessary to assume that the breakdown of supersymmetry happens at the TeV scale. However, other 3-3-1 models, i.e., with different representa-tion content, have a different upper limit for the maximal energy scale @6#.

~iii! The quantization of the electric charge is possible even in the SM context. This is because of the classical~ hy-percharge invariance of the Yukawa interactions! and quan-tum constraints ~anomalies! @7#. However, this occurs only family by family and if there are no right-handed neutrinos

~neutrinos if massive must be Majorana fields!; or, when the three families are considered together the quantization of the electric charge is possible only if right-handed neutrinos with Majorana mass term are introduced @8# or another Higgs doublet@9#or some neutral fermions@10#are introduced. On the other hand, in the 3-3-1 model @2,3,6#the charge quanti-zation in the three families case does not depend on if neu-trinos are massless or massive particles @11#.

ought to be also Majorana particles in order to retain both features of the electromagnetic interactions @8,12#. On the other hand, in all sorts of 3-3-1 models the quantization of the charge and the vectorial nature of the electromagnetic interactions are related one to another and are also indepen-dent of the nature of the neutrinos @13#.

Last but not least,~v!if we accept the criterion that
par-ticle symmetries are determined by the known leptonic
sec-tor, and if each generation is treated separately, then*SU*(3)*L*

is the largest chiral symmetry group to be considered among
(n,*e*,*ec*)* _{L}*. The lepton family quantum number is gauged;
only the total lepton number,

*L*, remains a global quantum number~or equivalently we can defineF

_{5}

_{B}_{1}

_{L}_{as the }glo-bal conserved quantum number where

*B*is the baryonic number @14#!. On the other hand, if right-handed neutrinos do exist, as it appears to be the case @1#, the symmetry among (n

*e*,

*e*,n

*ec*,

*ec*)

*L*would be

*SU*(4)

*L*^

*U*(1)

*N*@15#. This

is possibly the last symmetry among leptons. There is no
room for*SU*(5)* _{L}*^

_{U}_{(1)}

*N*if we restrict ourselves to the case

of leptons with electric charges 61,0 @16#. Hence, in this
case all versions of the 3-3-1 model, for instance the one in
Refs. @2,3,17#, and the one in Refs. @6,18,19#, are different
*SU*(3)* _{L}*-projections of the larger

*SU*(4)

*symmetry@15#.*

_{L}Besides the characteristic features given above, which we
can consider predictions of the model, the model has some
interesting phenomenological consequences.~a!An extended
version of some 3-3-1 models may solve the strong *C P*
problem. It was shown by Pal @20# that in 3-3-1 models

@2,3,6,19# the more general Yukawa couplings admit a
Peccei-Quinn symmetry @21#and that symmetry can be
ex-tended to the Higgs potential and, therefore, making it a
symmetry of the entire Lagrangian. This is obtained by
in-troducing extra Higgs scalar multiplets transforming under
the 3-3-1 symmetry as D;(**1**,**10**,23), for the model of
Refs.@2,3#, orD;(**1**,**10**,21), for the model of Refs.@6,19#.
In this case the resulting axion can be made invisible. The
interesting thing is that in those sort of models the
Peccei-Quinn symmetry is an automatic symmetry, in the sense that
it does not have to be imposed separately on the Lagrangian
but it is a consequence of the gauge symmetry and a discrete
symmetry.~b!There exist new contributions to the
neutrino-less double beta decay in models with three scalar triplets

@14# or in the model with the sextet @22#. If the model is
extended with a neutral scalar singlet it is possible to have a
safe Majoron-like Goldstone boson and there are also
contri-butions to that decay with Majoron emission @22#. ~c!It is
the simplest model that includes bileptons of both types:
sca-lar and vector ones. In fact, although there are several models
which include doubly charged scalar fields, not many of
them incorporate doubly charged vector bosons: this is a
par-ticularity of the 3-3-1 model of Refs. @2,3#. ~d! The model
has several sources of*C P*violation. In the 3-3-1 model@2,3#

we can implement the violation of the *C P* symmetry,
spon-taneously @23,24# or explicitly @25#. In models with exotic
leptons it is possible to implement soft*C P*violation@26#.~e!

The extra neutral vector boson *Z*

_{8}

conserves flavor in the
leptonic but not in the quark sector. The couplings to the
leptons are leptophobic because of the suppression factor
(124*s*2)1/2 but with some quarks there are enhancements

_{W}because of the factor (124*s _{W}*2)21/2

_{@}

_{27}

_{#}

_{.}

_{~}

_{f}

_{!}

_{Although the}minimal scalar sector of the model is rather complicated, with at least three triplets, we would like to stress that it contains all extensions of the electroweak standard model with extra scalar fields: two or more doublets @28#, neutral gauge singlet @29#, or doubly charged scalar fields@30#, or a combination of all that. However, some couplings which are allowed in the multi-Higgs extensions are not in the present model when we consider an

*SU*(2)^

_{U}_{(1) subgroup. }In-versely, there are some interactions that are allowed in the present models that are not in the multi-Higgs extensions of the SM, for instance, trilinear couplings among the doublets which have no analog in the SM, or even in the minimal supersymmetric standard model ~MSSM!. It means that the model preserves the memory of the 3-3-1 original symmetry. Hence, in our opinion, the large Higgs sector is not an intrin-sic trouble of this model.~g!Even if we restrict ourselves to leptons of charge 0,61 we can have exotic neutral @18# or charged heavy leptons@17#.~h!Neutrinos can gain Majorana masses if we allow one of the neutral components of the scalar sextet to gain a nonzero vacuum expectation value

@31#, or if we introduce right-handed neutrinos@32#, or if we add either terms in the scalar potential that break the total lepton number or an extra charged lepton transforming as a singlet under the 3-3-1 symmetry@33#.

Of course, some of the benefits of this type of model, such as items~i!,~ii!, and~iv!above, can be considered only as a hint to the final resolution of those problems: they depend on the representation content and we can always ask ourselves what is the main principle behind the representation content. Anyway, we think that the 3-3-1 models have interesting features by themselves and that it is well motivated to gen-eralize them by introducing supersymmetry. In the present paper we built exhaustively the supersymmetric version of the 3-3-1 model of Refs.@2,3#. A first supersymmetric 3-3-1 model, without the introduction of the scalar sextet, was con-sidered some years ago by Duong and Ma @34#.

The outline of the paper is as follows. In Sec. II we present the representation content of the supersymmetric 3-3-1 model. We build the Lagrangian in Sec. III. In Sec. IV we analyze the scalar potential; in particular, we found the mass spectrum of the neutral scalar and have shown that the lightest scalar field has an upper limit of 124.5 GeV. Our conclusion and a comparison with the supersymmetric model of Ref. @34# are in the last section. In the Appendixes we show the scalar mass matrices and the constraint equations for completeness.

**II. THE SUPERSYMMETRIC MODEL**

The fact that in the 3-3-1 model of Refs.@2,3#we have the
constraint at tree level sin2u*W*,1/4 means, as we said before,

Refs. @34,35#and we will comment on these later.

However, let us first consider the particle content of the model without supersymmetry @2,3#. We have the leptons transforming as

*L _{l}*5

### S

n*l*

*l*

*lc*

### D

_{L};~**1**,**3**,0!, *l*5*e*,m,t. ~2!

In parentheses it appears the transformation properties under
the respective factors @*SU*(3)*C*,*SU*(3)*L*,*U*(1)*N*#. We have

not introduced right-handed neutrinos and for the moment we assume here that the neutrinos are massless; however, see

@31–33#.

In the quark sector, one quark family is also put in the triplet representation

*Q*1*L*5

### S

*u*1

*d*1

*J*

### D

*L*

;

### S

**3**,

**3**,2

3

### D

, ~3!and the respective singlets are given by

*u*_{1}*c _{L}*;

### S

**3***,

**1**,22

3

### D

,*d*1

*L*

*c*

_{;}

### S

_{3}_{*}

,**1**,1
3

### D

,*JL*

*c*_{;}

### S

_{3}_{*}

,**1**,25
3

### D

,~4!

writing all the fields as left handed.

The other two quark generations we put in the antitriplet representation

*Q*2*L*5

### S

*d*2

*u*2

*j*1

### D

_{L}, *Q*3*L*5

### S

*d*3

*u*3

*j*2

### D

_{L};

### S

**3**,

**3***,21

3

### D

, ~5!and also with the respective singlets,

*u*2*L*
*c* _{,}

*u*3*L*

*c* _{;}

### S

_{3}_{*}

_{,}

_{1}_{,}

22

3

### D

,*d*2

*L*

*c*

_{,}

*d*3*L*

*c* _{;}

### S

_{3}_{*}

_{,}

_{1}_{,}1

3

### D

,*j*_{1}*c _{L}*,

*jc*

_{2}

*;*

_{L}### S

**3***,

**1**,4

3

### D

. ~6!On the other hand, the scalars which are necessary to generate the fermion masses are

h5

### S

h0h_{1}2

h_{2}1

### D

;~**1**,**3**,0!, r5

### S

r1r0 r11

### D

;~**1**,**3**,11!,

x5

### S

x2 x22x0

### D

;~**1**,**3**,21!, ~7!

and one way to obtain an arbitrary mass matrix for the lep-tons is to introduce the following symmetric antisextet:

*S*5

## S

s_{1}0 *h*2
1

### A

2*h*

_{1}2

### A

2*h*_{2}1

### A

2*H*_{1}11 s2
0

### A

2*h*_{1}2

### A

2 s_{2}0

### A

2*H*_{2}22

## D

;~**1**,**6***,0!. ~8!

Now, we introduce the minimal set of particles in order to implement the supersymmetry @36#. We have the sleptons corresponding to the leptons in Eq.~2!, squarks related to the quarks in Eqs.~4!–~6!, and the Higgsinos related to the sca-lars given in Eqs. ~7! and ~8!. Besides, in order to cancel chiral anomalies generated by the superpartners of the sca-lars, we have to add the following Higgsinos in the respec-tive antimultiplets:

h
*˜*

_{8}

_{5}

### S

h
*˜*

_{8}

0
h
*˜*
1

### 8

1 h*˜*2

### 8

2### D

*L*

;~**1**,**3***,0!, *˜*r

### 8

5### S

r*˜*

_{8}

2
r
*˜*

_{8}

0
r
*˜*

_{8}

22### D

*L*

;~**1**,**3***,

21!,

x
*˜*

_{8}

_{5}

### S

x
*˜*

_{8}

1
x
*˜*

_{8}

11
x
*˜*

_{8}

0 ### D

*L*

;~**1**,**3***,11!, ~9a!

*S*
*˜*

_{8}

_{5}

## S

s
*˜*
1

### 8

0*h*

*˜*
2

### 8

2### A

2*h*

*˜*1

### 8

1### A

2*h*
*˜*
2

### 8

2### A

2*H*

*˜*

1

### 8

22 s*˜*2

### 8

0### A

2*h*
*˜*
1

### 8

1### A

2 s*˜*2

### 8

0### A

2*H*

*˜*

2

### 8

11## D

*L*

;~**1**,**6**,0!. ~9b!

There are also the scalar partners of the Higgsinos defined in Eq.~9!and we will denote them byh

_{8}

,r### 8

,x### 8

,*S*

### 8

. This is the particle content which we will consider as the minimal 3-3-1s model if the charged lepton masses are generated by the antisextet*S*.

Summarizing, we have in the 3-3-1s model the following
superfields: *Lˆ _{e}*

_{,}m,t,

*Qˆ*1,2,3, h

*ˆ*, r

*ˆ*, x

*ˆ*,

*Sˆ*; h

*ˆ*

### 8

, r*ˆ*

### 8

, x*ˆ*

### 8

,*Sˆ*

### 8

;*uˆ*

_{1,2,3}

*c*,

*dˆ*

_{1,2,3}

*c*,

*Jˆc*, and

*jˆ*

_{1,2}

*c*, i.e., 23 chiral superfields, and 17 vector superfields:

*Vˆa*,

*Vˆ*a, and

*Vˆ*

### 8

. In the MSSM there are 14 chiral superfields and 12 vector superfields.**III. THE LAGRANGIAN**

With the superfields introduced in the last section we can build a supersymmetric invariant Lagrangian. It has the fol-lowing form:

L_{3315}L

HereL

*SUSY* is the supersymmetric piece, whileL*so f t*

explic-itly breaks SUSY. Below we will write each of these Lagrangians in terms of the respective superfields.

**A. The supersymmetric term**

The supersymmetric term can be divided as follows:

L_{SUSY}_{5}L_{Le pton}_{1}L_{Quarks}_{1}L_{Gauge}_{1}L* _{Scalar}*, ~11!

where each term is given by

L

*Le pton*5

### E

*d*4u@

*L¯ˆ e*2

*gV*

*ˆ*

*Lˆ*#, ~12!

L_{Quarks}_{5}

### E

*4*

_{d}_{u}

_{@}

_{Q}¯ˆ_{1}

_{exp$}

_{@}2

*g*~

*Vˆc*1

*Vˆ*!1~2

*g*

### 8

/3!*Vˆ*

### 8

#%*Qˆ*1

1*Q¯ˆ*aexp$@2*g*~*Vˆc*1*V¯ˆ*!2~*g*

### 8

/3!*Vˆ*

### 8

#%*Qˆ*a

1*¯ˆu _{i}c*exp$@2

*gV¯ˆc*2~2

*g*

### 8

/3!*Vˆ*

### 8

#%*uˆic*

1*¯ˆd _{i}c*exp$@2

*gV¯ˆc*1~

*g*

### 8

/3!*Vˆ*

### 8

#%*dˆic*

1*¯ˆJc*exp$@2*gV¯ˆc*2~5*g*

### 8

/3!*Vˆ*

### 8

#%*Jˆc*

1*¯ˆj*b*c*exp$@2*gV¯ˆc*1~4*g*

### 8

/3!*Vˆ*

### 8

#%*jˆ*b

*c*# ~13!

and

L

*Gauge*5

1 4

### F

### E

*d*

2_{u}_{@}_{W}

*c*
*a _{W}*

*c*
*a*

1*Wa _{W}a*

1*W*

_{8}

*W*

_{8}

#
1

### E

*d*2

*¯*u

_{@}

_{W}¯*c*

*a*

_{W}_{¯}*c*
*a*

1*W¯aW¯a*1*W¯*

_{8}

*W¯*

_{8}

#### G

, ~14!where *Vˆc*5*TaVˆac*, *V¯ˆc*5*T¯aVˆac*, *Vˆ*5*TaVˆa*, *V¯ˆ*5*T¯aVˆa*, and *Ta*

5l*a*_{/2 (}_{¯}_{T}a

52l**a*/2) are the generators of triplets~
antitrip-lets!of*SU*(3), i.e.,*a*51, . . . ,8, and*g*and*g*

### 8

are the gauge couplings of*SU*(3) and

*U*(1), respectively.

*W*,

_{c}a*Wa*, and

*W*

_{8}

are the strength fields, and they are given by
*W*a*ac*52

1
8*gD¯ D¯ e*

22*gVˆc _{D}*

a*e*2*gV*
*ˆ*

*c*_{,}

*W*_{a}*a*52 1
8*gD¯ D¯ e*

22*gVˆ _{D}*
a

*e*2

*gV*

*ˆ*

,

*W*a

### 8

521

4*D¯ D¯ D*a*Vˆ*

### 8

. ~15!Finally

L_{Scalar}_{5}

### E

*4*

_{d}_{u}

_{@}

_{h}

*¯ˆ e*2

*gVˆ*

_{h}

*ˆ*

_{1}

*¯ˆ e*

_{r}2

*gVˆ*1

*g*

_{8}

*Vˆ*

_{8}

_{r}

_{ˆ}_{1x}

*2*

_{¯ˆ e}*gVˆ*2

*g*

_{8}

*Vˆ*

_{8}

_{x}

_{ˆ}1*¯ˆ eS* 2*gVˆSˆ*1h*¯ˆ*

### 8

*e*2

*gV*Ch

*ˆ*

_{8}

_{1}

*¯ˆ*

_{r}

_{8}

*2*

_{e}*gV*C2

*g*

_{8}

*Vˆ*

_{8}

_{r}

_{ˆ}### 8

1x*¯ˆ*

### 8

*e*2

*gV¯ˆ*1

*g*

_{8}

*Vˆ*

_{8}

_{x}

_{ˆ}_{8}

1*¯ˆS*

_{8}

*e*2

*gV¯ˆ*

_{Sˆ}_{8}

_{#}

1

### E

*d*2u

*W*

1

### E

*d*2

_{¯ W}_{u}

_{¯}_{,}

_{~}

_{16}

_{!}

where*W*is the superpotential, which we discuss in the next
subsection.

**B. Superpotential**

The superpotential of our model is given by

*W*5*W*2
2 1

*W*3

3 , ~17!

with *W*2 having only two chiral superfields and the terms
permitted by our symmetry are

*W*25m0*Lˆ*h*ˆ*

### 8

1mhh*ˆ*h

*ˆ*

### 8

1mrr*ˆ*r

*ˆ*

### 8

1mxx*ˆ*x

*ˆ*

### 8

1m*SSˆSˆ*

### 8

,~18!

and in the case of three chiral superfields the terms are

*W*35l_{1}e*LˆLˆLˆ*1l_{2}e*LˆLˆ*h*ˆ*_{1}_{l}

3*LˆLˆSˆ*1l4e*Lˆ*x*ˆ*r*ˆ*1*f*1er*ˆ*x*ˆ*h*ˆ*

1*f*2h*ˆ*h*ˆ Sˆ*1*f*3x*ˆ*r*ˆ Sˆ*1*f*1

### 8

er*ˆ*

### 8

x*ˆ*

### 8

h*ˆ*

### 8

1*f*2

### 8

h*ˆ*

### 8

h*ˆ*

### 8

*Sˆ*

### 8

1*f*_{3}

### 8

x*ˆ*

_{8}

_{r}

*ˆ*

_{8}

_{Sˆ}_{8}

_{1}

_{(}

*i*

@k1*iQˆ*1h*ˆ*

### 8

*uˆic*1k2

*iQˆ*1r

*ˆ*

### 8

*dˆic*#

1k3*Qˆ*1x*ˆ*

### 8

*Jˆc*1

### (

a*i*

@k_{4}a*iQˆ*ah*ˆ dˆi*
*c*

1k5a*iQˆ*ar*ˆ uˆi*
*c*_{#}

1

### (

ab k6ab*Q*

*ˆ*_{a}_{x}_{ˆ jˆ}_{b}*c*

1

### (

a*i j*

k7a*i jQˆ*a*Lˆidˆjc*

1

### (

*i*,*j*,*k* j1*i jkdˆi*
*c*

*dˆc _{j}uˆ_{k}c*1

### (

*i j*b j2*i j*b*uˆi*
*c*

*uˆ _{j}cjˆ*b

*c*1

### (

*i*b j3

*i*b

*dˆi*

*c*
*Jˆcjˆ*b*c*,

~19!

with*i*,*j*,*k*51,2,3, a52,3, andb51,2.

All the seven neutral scalar components
h0_{,r}0_{,x}0_{,s}

2
0_{,h}

### 8

0,r_{8}

0_{, and}

_{x}

### 8

0 gain nonzero vacuum expec-tation values. This arises from the mass matrices for quarks. In fact, defining### ^

h0_{&}

5*v*_{h}/

### A

2,### ^

h0### 8

### &

_{5}

*v*

_{h}

### 8

/### A

2, etc, from the superpotential in Eq.~19!the following mass matrices arise:G*u*

5 1

### A

2### S

k11*v*_{h}

### 8

k_{12}

*v*

_{h}

### 8

k_{13}

*v*

_{h}

### 8

k_{521}

*v*

_{r}k

_{522}

*v*

_{r}k

_{523}

*v*

_{r}k

_{531}

*v*

_{r}k

_{532}

*v*

_{r}k

_{533}

*v*

_{r}

### D

, ~20!

G*d*

5 1

### A

2### S

k21*v*

r

### 8

k22*v*

r

### 8

k23*v*

r

### 8

k421*v*_{h} k_{422}*v*_{h} k_{423}*v*_{h}
k431*v*_{h} k_{432}*v*_{h} k_{4333}*v*_{h}

### D

, ~21!

for the *d* quarks, and for the exotic quarks, *J* and *j*1,2, we
have *MJ*5k3*v*_{x}

### 8

andG*j*5
*v*_{x}

### A

2### S

k621 k622

k_{631} k_{632}

### D

, ~22! respectively.From Eqs.~20!,~21!, and~22!we see that all the vacuum
expectation values~VEVs!have to be different from zero in
order to give mass to all quarks. Notice also that the *u*-like
and *d*-like mass matrices have no common VEVs. On the
other hand, the charged lepton mass matrix is already given
by *M _{i j}l*5

*v*

_{s}

2l3*i j*/

### A

2, where*v*s2 is the VEV of the

### ^

s2 0### &

component of the antisextet*S*in Eq.~8!. However,

*v*

_{s}

1,2

### 8

can both be zero since the sextet*S*

_{8}

does not couple to leptons at
all.
The terms which are proportional to the following
con-stants: m0 in Eq.~18! andl1, l4, *f*2, *f*2

### 8

, k7, andj1,2,3 in Eq. ~19! violate the conservation of the F_{5}

_{B}_{1}

_{L}_{quantum}

number. For instance, if we allow the j1 term it implies in
proton decay@35#. However, if we assume the global*U*(1)F

symmetry, it allows us to introduce the *R*-conserving
sym-metry @37#, defined as*R*5(21)3F_{1}2*S*_{. The} _{F}_{number }

attri-bution is

F_{~}* _{U}*22

_{!}

5F_{~}* _{V}*2

_{!}

52F_{~}_{J}_{1}_{!}_{5}F_{~}_{J}_{2,3}_{!}_{5}F_{~}_{r}22_{!}

5F_{~}_{x}22_{!}

5F_{~}_{x}2_{!}
5F_{~}_{h}

2
2_{!}

5F_{~}_{s}
1
0_{!}

52,

~23!

withF_{5}_{0 for the other Higgs scalar, while for leptons and}
the known quarks F _{coincides with the total lepton and}
baryon numbers, respectively. As in the MSSM this
defini-tion implies that all known standard model’s particles have
even *R*parity while their supersymmetric partners have odd
*R* parity. The termsj2,l4 were not considered in Ref.@35#.
However, the term withj2 involves an exotic quark~heavier
than the proton!so the analysis in that reference is still valid.
As usual, the supersymmetry breaking is accomplished by
including the most general renormalizable
soft-supersymmetry breaking terms but now, they must be also
consistent with the 3-3-1 gauge symmetry. We will also
in-clude terms which explicitly violate the *R*-like symmetry.
These soft terms are given by

L

*so f t*52

1

2

### F

*m*l

*c*

### (

*l*

_{a}*c*

*a*_{l}
*c*
*a*

1*m*l

### (

*a*

~ l*a*_{l}

*a*!1*m*

### 8

ll1H.c.### G

2*mL*2

*L˜*†

*˜L*2

*mQ*2

_{1}

*Q˜*1†

*Q˜*11

### (

*i*

@*˜u _{i}c*†

*m*

_{u}*i*

2_{˜}_{u}

*i*
*c*

2*˜d _{i}c*†

*m*

_{d}*i*

2_{d}_{˜}

*i*
*c*_{#}

2*m _{J}*2

*˜Jc*†

*˜Jc*2

### (

b *j*

*˜*_{b}*c*†_{m}

*j*_{b}

2_{˜}_{j}

b
*c*

2

### (

a *mQ*a

2 _{Q}_{˜}

a

†_{Q}_{˜}

a2*m*h2h†h2*m*x2x†x2*m*2rr†r2*ms*2Tr~*S*†*S*!2*m*h_{8}

2 h

_{8}

†_{h}

### 8

2*m*_{r}
8

2 r

_{8}

†_{r}

### 8

2*m*

_{x}8

2 x

_{8}

†_{x}

### 8

2*m*8

_{s}2

Tr~*S*

_{8}

†*S*

_{8}

!2### F

*k*1erxh1

*k*2hh

*S*†1

*k*3xr

*S*†1

*k*

### 8

1er### 8

x### 8

h### 8

1*k*2

### 8

h### 8

h### 8

*S*

### 8

†1*k*3

### 8

x### 8

r### 8

*S*

### 8

† 1*M*2

*L˜*h†

1«_{0}

### (

*i jk* e*i jkL*
*˜*

*iL˜jL˜k*1«1

### (

*i jk* e*i jkL*
*˜*

*iL˜j*h*k*1«2

### (

*i j* *L*
*˜*

*iL˜jSi j*1«3

### (

*i jk* e*i jkL*
*˜*

*i*x*j*r*k*

1

### (

*i*
*Q*
*˜*

1~z1*i*h

### 8

*˜uic*1z2

*i*r

### 8

*˜dic*1z3

*J*x

### 8

*˜Jc*!1

### (

a*Q*

*˜*_{a}

### F

_{(}

*i*

### S

v1a*i*

h*˜d _{i}c*1v2a

*i*r

*u˜ic*1

### (

*j*v3a

*i j*

*L*
*˜*

*i˜djc*

### D

1### (

b v4abx*j*

*˜*

b
*c*

### G

1

### (

*i*

### S

### (

*jk*

§_{1}* _{i jk}˜d_{i}c˜d_{j}c˜u_{k}c*1

### (

a

### F

§_{2}_{i}_{a}*d˜ _{i}c˜Jc˜j*a

*c*1

### (

*j*

§_{3}_{i j}_{a}*˜u _{i}c˜uc_{j}˜j*a

*c*

### GD

1H.c.### G

. ~24!The terms proportional to *k*2, *k*2

### 8

,*M*2,«0, «3, v3, and

§_{1,2,3} violates the*R*-parity symmetry.
The *SU*(3) invariance tell us that

*mL*5

### S

*m˜*n 0 0

0 *m _{l˜}* 0
0 0

*m*

_{l˜}c### D

, ~25!

and we need *ml˜c*5*m _{l˜}*5

*m*

_{n}

*˜*and the same for the other mass

parameters. More details of the Lagrangian are given else-where@38#.

**IV. THE SCALAR POTENTIAL**

The scalar potential is written as

*V*3315*VD*1*VF*1*V*soft, ~26a!

*VD*52L*D*5

1
2~*D*

*a _{D}a*

1*DD*!*g*

### 8

2

2 ~r
†_{r}

2r

### 8

†_{r}

_{8}

_{2x}†

_{x}

_{1x}

_{8}

†_{x}

_{8}

_{!}2 1

*g*

2

8

### (

*i*,

*j*~

h* _{i}*†

_{l}

*i j*
*a*_{h}

*j*1r*i*

†_{l}

*i j*
*a*_{r}

*j*1x*i*

†_{l}

*i j*
*a*_{x}

*j*

1*S _{i j}*†l

*a*2h

_{jk}S_{kl}*i*

### 8

†l*i j**

*a*h

*j*

### 8

2r*i*

### 8

†l*i j**

*a*r

*j*

### 8

2x*i*

### 8

†l*i j**

*a*x

### 8

*j*2

*Si j*

### 8

†l*jk**

*aSkl*

### 8

!2, ~26b!*VF*52L*F*5

### (

*m*

*Fm*

**Fm*5

### (

*i*,

*j*,

*k*

### F

### U

m_{h}
2 h*i*

### 8

1*f*1

3 e*i jk*r*j*x*k*1
2*f*2

3 h*iSi j*

### U

21

### U

mr 2 r*i*

### 8

1*f*1

3 e*i jk*x*j*h*k*1
*f*3
3 x*iSi j*

### U

2

1

### U

mx 2 x*i*

### 8

1*f*1

3 e*i jk*r*j*h*k*1
*f*3

3 r*iSi j*

### U

21

### U

m*s*2

*Si j*

### 8

1*f*2
3 h*i*h*j*1

*f*3
3 x*i*r*j*

### U

2

1

### U

mh 2 h*i*1

*f*1

### 8

3 e*i jk*r

### 8

*j*x

*k*

### 8

12*f*2

### 8

3 h*i*

### 8

*Si j*

### 8

### U

2

1

### U

mr 2 r*i*

1
*f*_{1}

### 8

3 e*i jk*x

### 8

*j*h

*k*

### 8

1*f*

_{3}

### 8

3 x*i*

### 8

*Si j*

### 8

### U

21

### U

mx 2 x*i*1

*f*_{1}

### 8

3 e*i jk*r

### 8

*j*h

*k*

### 8

1*f*

_{3}

### 8

3 r*i*

### 8

*Si j*

### 8

### U

21

### U

m*s*2

*Si j*1

*f*_{2}

### 8

3 h*i*

### 8

h### 8

*j*1

*f*

_{3}

### 8

3 x*i*

### 8

r*j*

### 8

### U

2### G

, ~26c!*Vso f t*52L*so f t*

scalar

5*m*_{h}2h†_{h}

1*m*r2r†r1*m*x2x†x1*ms*

2_{Tr}_{~}

*S*†*S*!1*m*_{h}
8

2 _{h}

### 8

†_{h}

_{8}

1*m*

_{r}

8

2 r

### 8

†_{r}

_{8}

1*m*_{x}
8

2 x

### 8

†_{x}

_{8}

1*m _{s}*
8

2

Tr~*S*

### 8

†

_{S}_{8}

_{!}

1@*k*1e rxh1*k*2h*S*†h1*k*3x*TS*†r1*k*1

### 8

e r### 8

x### 8

h### 8

1*k*2

### 8

h### 8

*S*

### 8

†h### 8

1*k*3

### 8

x### 8

*TS*

### 8

†r### 8

1H.c.#. ~26d!Note that *k*1,2,3,*k*1,2,3

### 8

has dimension of mass and that the terms which are proportional to*k*2,

*k*2

### 8

and*f*2,

*f*2

### 8

violate the*R*parity.

It is instructive to rewrite Eqs. ~26b!as follows:

*V _{D}*5

*g*

### 8

22 ~r
†_{r}

2x†x2r

### 8

†r_{8}

1x### 8

†x_{8}

!21*g*2

8

### F

4 3### (

*i*

~*X _{i}*†

*X*!2

_{i}12

### (

*i*,*j* ~*Xi*

†
*Xj*!~*Xj*

†

*Xi*!12

### (

*i*~

*Xi*

†

*S*!~*S*†*Xi*!

24 3

### (

*i*,

*j*

~*X _{i}*†

*Xi*!~

*Xj*

†
*Xj*!2

2
3Tr~*S*

†_{S}_{!}

_{(}

*i*

~*X _{i}*†

*Xi*!

12 Tr@~*S*†*S*!2#22
3~Tr*S*

†_{S}_{!}2

1pt

### G

, ~27!where ‘‘pt’’ in the expression above denotes the replacements
*X _{i}*

### 8

↔

_{X}*i*but not in*g*

### 8

which is always the coupling constantof the*U*(1)* _{N}*factor. In the same way we rewrite Eq.~26c!as

*V _{F}*5

### (

*i*

m_{X}

*i*

2

4 ~*Xi*†*Xi*1pt!1

um* _{S}*2

_{u}4 @Tr~

*S*

†_{S}_{!}

1pt#11_{6}

### (

*i*Þ*j*Þ*k*

### H

m_{X}

*i*

*

6 @*f*1e*i jkXi*

### 8

†*XjXk*1

*f*1

### 8

e*i jkXi*†

*X*

### 8

*jXk*

### 8

#1*f*_{3}2

### F

mh*h### 8

†*S*h1

m* _{S}**

2 h
†_{S}

### 8

h### G

1pt1*f*

_{6}3@m

_{r}*r

_{8}

†

_{S}_{x}

1mx*x

### 8

†*S*r1m*

*S*x†

*S*

### 8

r#1pt1H.c.### J

1u*f*1u2

### F

### (

*i*Þ*j*

~*X _{i}*†

*Xi*!~

*Xj*

†

*Xj*!2~*Xj*

†
*Xi*!~*Xi*

†

*Xj*!

### G

1pt14u*f*2u2
9 @~h*S*!

†_{~}_{h}_{S}_{!}

1~h†_{h}_{!}2_{#}

1pt1u*f*3u
2

9 @~x*S*!
†_{~}_{x}_{S}_{!}

1~r*S*!†~r*S*!1~x†_{x}_{!}†_{~}_{r}†_{r}_{!#}
1pt1

2*f*_{1}**f*2
9 e~rx!

†_{h}* _{S}*
1pt

1

### F

*f*1

**f*3
9 e~xh!

†_{x}_{S}_{1pt1}*f*2**f*3
9 ~h

†_{x}_{!~}_{h}†_{r}_{!}

1pt1H.c.

### G

, ~28!where ‘‘pt’’ in the expression above denotes as before the replacements*X _{i}*

### 8

↔

_{X}*i*, and*f*1,2,3

### 8

↔*f*1,2,3, and

*k*1,2,3

### 8

↔*k*1,2,3. We have omitted

*SU*(3) indices since we have denoted the unprimed triplets wherever it is possible as

*Xi*5h,r,x and

*Xi*

### 8

We can now work out the mass spectra of the scalar and
pseudoscalar fields by making a shift of the form *X*
→_{(1/}

### A

_{2)(}

*v*1

_{X}*H*1

_{X}*iF*) ~similarly for the case of the primed fields!for all the neutral scalar fields of the multiplets

_{X}*X*. Note that

_{i}*H*and

_{X}*F*are not mass eigenstates yet. We will denote

_{X}*Hi*,

*i*51, . . . ,8 and

*Ai*,

*i*51, . . . ,6 the

respec-tive massive fields; *G*_{1,2} will denote the two neutral
Gold-stone bosons. The mass matrices appear in the Appendix A,
for the real scalars, and in the Appendix B for the
pseudo-scalar case. The constraint equations are given in the
Appen-dix C. We will use the following set of parameters in the
scalar potential below:

*f*15*f*351, *f*_{1}

### 8

5*f*

### 8

_{3}51026

_{~}

_{dimensionless}

_{!}

_{,}

_{~}

_{29}

_{!}

and

2*k*15*k*1

### 8

510,*k*35

*k*3

### 8

52100,2mh5mr52m*s*5mx51000 ~in GeV!, ~30!

we also use the constraint*V*_{h}21*V*_{r}212*V*_{2}25(246 GeV)2
com-ing from *MW*, where, we have defined *V*h25*v*2_{h}1*v*_{h}

### 8

2,*V*

_{r}2 5

*v*

_{r}21

*v*

_{r}

### 8

2, and*V*

_{2}25

*v*

_{s}

2 2

1*v*

s_{2}_{8}

2

. Assuming that *v*_{h}520, *v*_{x}
51000, *v*_{s}

2510, and *v*h

### 8

5*v*r

### 8

5*v*s

### 8

25*v*_{x}

### 8

51 in GeV, the value of*v*

_{r}is fixed by the constraint above.

With this set of values for the parameters the real mass
eigenstates *H _{i}* are obtained by the diagonalization of the
mass matrix given in the Appendix A. Besides the constraint
equations ~Appendix C! and imposing the positivity of the
eigenvalues~mass square!, and the values for the parameters
given above, we obtain the following values for the masses
of the scalar sector ~in GeV!:

*MH*5121.01,

_{i}277.14,515.26,963.68,1218.8,1243.24,3797.86, and 4516.43,
where*i*51, . . . ,8 and*MH _{j}*.

*MH*with

_{i}*j*.

*i*. In the

pseudo-scalar sector we have verified analytically that the mass
ma-trix in the Appendix B, has two Goldstone bosons as it
should be. The other six physical pseudoscalars have the
following masses, with the same parameters as before,
in GeV, *MA _{i}*5276.4,515.3,963.65,1243.24,3797.85, and

4516.43.

The behavior of the lightest scalar (*H*1) and pseudoscalar
(*A*1) as a function of*v*_{x}is shown in Fig. 1 for a given choice
of the parameters, we see that, at the tree level, there is an
upper limit for the mass of the lightest scalar *MH*_{1},124.5

GeV and that for these values of the parameters *MA*_{1}

.*MZ*. Other values of the parameters give higher or lower

values for the upper limit of *MH*_{1}. Of course, radiative

cor-rections have to be taken into account, however, this has to be done in the context of the supersymmetric 3-3-1 model which is not in the scope of the present work. Hence the mass square of the lightest real scalar boson has an upper bound ~see Fig. 1!

*M _{H}*

1
2 _{<~}

124.51e!2 GeV2, ~31!

where 124.5 GeV is the tree value (e50). We recall that in
the MSSM if *MA*_{1}.*MZ* the upper limit on the mass of the

lightest neutral scalar is *MZ* at the tree level but radiative

corrections raise it to 130 GeV @39#.

**V. CONCLUSIONS**

We have built the complete supersymmetric version of the
3-3-1 model of Refs.@2,3#. Another possibility in this 3-3-1
model which avoids the introduction of the scalar sextet,*S*,
was considered some years ago by Duong and Ma, Ref.@34#,
who built the supersymmetric version of that model. The
sextet was substituted by a single charged lepton singlet *E _{L}*

;(**1**,1) and*E _{L}c*;(

**1**,21). Here we would like to point out the differences between our version of the supersymmetric 3-3-1 model and that of Ref. @34#. ~a! Duong and Ma as-sumed that the breaking of

*SU*(3)

*L*^

*U*(1)

*N*→

*SU*(2)

*L*^

_{U}_{(1)}

*Y*occurred before the breaking of supersymmetry and

the resulting model is a supersymmetric *SU*(2)*L*^*U*(1)*Y*

model. Even in this case the scalar potential involving dou-blets of the residual gauge symmetry does not coincide with the potential of the MSSM. In the present work, we have considered that the supersymmetry is broken at the same time as the 3-3-1 gauge symmetry. Hence, we have to con-sider the complete 3-3-1 scalar potential. It means that in the Duong and Ma supersymmetric model there are no doubly charged charginos and exotic charged squarks. ~b! In Ref.

@34#it was assumed that some of the VEVs have zero value,
unlikely we have considered all ~but s_{2}

### 8

0 and s_{1}0) of them different from zero. Hence we are able to obtain realistic quark and charged lepton masses, as can be seen from Eqs.

~20!,~21!, and~22!. In Ref.@34#some of these masses have to be generated by radiative corrections @40#. From ~a! and

~b!we see that the supersymmetric 3-3-1 model considered in this work has different phenomenological features from the supersymmetric 3-3-1 model of Duong and Ma.

On the other hand, we would like to recall that in the
MSSM with soft-breaking terms, it has been shown that at
the tree level the lightest scalar has a mass which is less than
*MZ* if the mass of the top quark has an upper bound of 200

FIG. 1. The eigenvalues of the mass matrix given in Appendix
A; *MH*_{1} is the mass of the lightest scalar and*MA*the mass of the

GeV@41# and this upper limit does not depend on the scale of supersymmetry masses and on the scale of supersymmetry breaking and holds independently of the short distance or large mass behavior of the theory@42#and it does not depend on the number of the Higgs doublets@43#. Radiative correc-tions@42#and the addition of extra Higgs multiplets@42,44#

shift the limit above depending on the top mass or on the type of extra scalars. However, in the 3-3-1s that we have built here the limit of Ref.@41#is avoided.

From the phenomenological point of view there are
sev-eral 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.
The mass spectra of all particles in this model are considered
in Ref. @38#. However, there are differences between this
model and the MSSM with or without*R*-parity breaking: due
to the fact that there are doubly charged scalar and vector
fields. Hence, we have doubly charged charginos which are
mixtures of the superpartners of the*U*-vector boson with the
doubly charged Higgsinos. This implies new interactions that
are not present in the MSSM, for instance x*˜*22*˜*_{x}0* _{U}*11

_{,}x

*˜*2_{x}* _{˜}*2

*11*

_{U}_{,}

*2*

_{l˜}*2*

_{l}_{x}

*11*

_{˜}_{, where}

_{x}

*11*

_{˜}_{denotes any doubly}charged chargino. Moreover, in the chargino production, be-sides the usual mechanism, we have additional contributions coming from the

*U*-bilepton in the

*s*channel. Due to this fact we expect that there will be an enhancement in the cross section of production of these particles in

*e*2

*2*

_{e}_{colliders,}such as the Next Linear Collider ~NLC! @38#. We will also have the singly charged charginos and neutralinos, as in the MSSM, where there are processes like

*l˜*2

*1*

_{l}_{x}

*˜*0

_{,}

_{n}

_{˜}*ll*2*˜*x1,

with *l˜* denoting any slepton; *˜*x2 _{denotes singly charged}
chargino and*˜*n_{L}_{denotes any sneutrino. The only difference}
is that in the MSSM there are five neutralinos and in the
3-3-1s model there are eight neutralinos.

Finally, we would like to call attention that, whatever the
energy scalem at which sin2_{u}

*W*(m)51/4 is in the

nonsuper-symmetric 3-3-1 model, when supersymmetry is added it will result in a rather different value form. In conclusion, we can say that the present model has a rich phenomenology that deserves to be studied in more detail.

**ACKNOWLEDGMENTS**

This work was supported by Fundac¸a˜o de Amparo a` Pes-quisa do Estado de Sa˜o Paulo~FAPESP!, Conselho Nacional de Cieˆncia e Tecnologia~CNPq!and by Programa de Apoio a Nu´cleos de Exceleˆncia ~PRONEX!. One of us ~M.C.R.!

would like to thank the Laboratoire de Physique Mathe´ma-tique et The´orique, Universite´ Montpellier II, for its kind hospitality and also M. Capdequi-Peyrane`re and G. Moultaka for useful discussions.

**APPENDIX A: MASS MATRIX OF THE SCALAR**
**NEUTRAL FIELDS**

Here we write down the complete symmetric mass matrix
in the scalar*C P*-even sector; the constraint equations given

in the Appendix C have been already taken into account:

**M11**5

*g*2*v*_{h}2

3 1

1 18

### A

2*v*

_{h}

~*f*1*f*3*v*_{r}2*v*_{s}

2218*k*1*v*r*v*x13*f*1*v*r

### 8

mr*v*x

1*f*1*f*3*v*_{s}

2*v*x
2

23*f*_{1}

### 8

mh*v*

_{r}

### 8

*v*

_{x}

### 8

13*f*

_{1}m

_{x}

*v*

_{r}

*v*

_{x}

### 8

!,**M12**52

*g*2*v*_{h}*v*_{r}

6 1

1

9

### A

2### S

### A

2*f*1 2

_{v}h*v*_{r}_{2}*f*_{1}*f*_{3}*v*_{r}*v*_{s}

2

19*k*1*v*_{x}2
3
2mx*v*x

### 8

### D

,**M _{13}**52

*g*

2_{v}

h*v*_{r}

6 1

1 9

### A

2### S

9*k*1

*v*_{r}2
3
2*f*1mr*v*r

### 8

1

### A

2*f*12

*v*

_{h}

*v*

_{x}2

*f*

_{1}

*f*

_{3}

*v*

_{s}

2*v*x

### D

,**M _{14}**5

*g*2

*v*

_{h}

*v*

_{s}

2

6 2

*f*1*f*3

18

### A

2~*v*_{r}21*v*_{x}2!, **M _{15}**52

*g*2

*v*

_{h}

*v*

_{h}

### 8

3 ,

**M16**5

*g*2*v*_{h}*v*

r

### 8

6 2

1 6

### A

2~mr*v*_{x}_{2}m_{h}*v*

x

### 8

!,**M17**5

*g*2*v*_{h}*v*_{x}

### 8

6 1

1 6

### A

2~*f*1

### 8

mh*v*_{r}

### 8

2*f*

_{1}m

_{x}

*v*

_{r}!,

**M18**5

*g*2*v*_{h}*v*_{s}

2

6 ,

**M22**5

### S

*g*2
3 1*g*

### 8

2

### D

_{v}r

2 2

*v*_{x}
12

### A

2*v*

_{r}

~12*k*1*v*_{h}16

### A

2*k*

_{3}

*v*

_{s}

2

12*f*_{1}m_{h}*v*_{h}

### 8

1### A

2*f*

_{3}m

_{s}v_{s}

2

### 8

!1*v*

x

### 8

12

### A

2*v*

_{r}

~2*f*_{1}

### 8

mr*v*

_{h}

### 8

2

### A

2*f*

_{3}

### 8

mr*v*

s_{2}

### 8

12*f*1mx

*v*

_{h}

_{2}

### A

2m_{x}

*v*

_{s}

2*v*x

### 8

!,**M23**52

### S

*g*2
6 1*g*

### 8

2

### D

_{v}r*v*_{x}_{1}
1
12

### A

2~12*k*1

*v*_{h}_{1}6

### A

2*k*

_{3}

*v*

_{s}

2

12*f*

_{8}

mh*v*

_{h}

### 8

1### A

2*f*

_{3}m

_{s}v_{s}

2

### 8

!1*v*

_{r}

*v*

_{x}

9 ~*f*1
2

1*f*_{3}2!,

**M24**5

*g*2
12*v*r*v*x1

1

18

### A

2### S

22*f*1

*f*3

*v*_{h}*v*_{r}_{1}

### A

2*f*3 2

_{v}s_{2}*v*_{r}

19

### A

2*k*3

*v*

_{x}1 3

### A

2*f*3mx

*v*

x

### 8

### D

,**M25**5

*g*2
6 *v*h

### 8

*v*r1

1 6

### A

2~*f*1mh

*v*_{x}_{2}*f*
1

### 8

mr*v*

**M26**52

### S

*g*2
3 1*g*

### 8

2

### D

*v*r

*v*r

### 8

3 ,**M27**5

### S

*g*2
6 1*g*

### 8

2

### D

_{v}r*v*_{x}

### 8

2 mr12

### A

2~2*f*1

### 8

*v*_{h}

### 8

2*f*

_{3}

### 8

*v*

_{s}

2

### 8

!2 mx 12

### A

2~2*f*1

*v*_{h}2*f*_{3}*v*_{s}

2!,

**M28**52

*g*2*v*_{r}*v*_{s}

2

### 8

12 1

1

12~*f*3m*sv*x1*f*3

### 8

mr*v*

_{x}

### 8

!,**M _{33}**5

### S

*g*2

3 1*g*

### 8

2### D

_{v}x

2

2 1

12

### A

2*v*

_{x}

### S

12*k*1*v*_{h}*v*_{r}1
12

### A

2*k*3

*v*

_{r}

*v*

_{s}

2

12*f*1mh*v*

### 8

_{h}

*v*

_{r}22

*f*

_{1}m

_{r}

*v*

_{h}

*v*

_{r}

### 8

1### A

2*f*

_{3}m

_{r}

*v*

_{r}

### 8

*v*

_{s}

2

1

### A

2*f*3m

*sv*

_{r}

*v*

_{s}

2

### 8

22*f*1

### 8

mx*v*

_{h}

### 8

*v*

### 8

_{r}1

### A

2*f*

_{3}m

_{x}

*v*

_{r}

### 8

*v*

_{s}

2

### 8

### D

,**M34**5

*g*2

12*v*s2*v*x1
1
18

### A

2### S

9

### A

2*k*3

*v*

_{r}1

3

### A

2*f*3mr

*v*

_{r}

### 8

22*f*1*f*3*v*_{h}*v*_{x}1

### A

2*f*

_{3}2

*v*

_{s}

2*v*x

### D

,**M _{35}**5

*g*2

6 *v*h

### 8

*v*x1 1 6

### A

2~*f*1mr

*v*_{r}2*f*
1

### 8

mx*v*

_{r}

### 8

!,**M36**5

### S

*g*2
6 1*g*

### 8

2

### D

_{v}r

### 8

*v*

_{x}

_{1}

1

12

### A

2~22*f*1mr

*v*_{h}_{1}

### A

2*f*

_{3}m

_{r}

*v*

_{s}

2

22*f*_{1}

### 8

mx*v*

_{h}

### 8

1### A

2*f*3

### 8

mx*v*

s_{2}

### 8

!,**M _{37}**52

### S

*g*2

3 1*g*

### 8

2### D

_{v}x*v*_{x}

### 8

,**M38**52

*g*2

12*v*s

### 8

2*v*x1 1

12~*f*3m*sv*r1*f*3

### 8

mx*v*

_{r}

### 8

!,**M _{44}**5

*g*2

12*v*s2
2

1 1

18

### A

2*v*

_{s}

2

### S

*f*1*f*3*v*_{h}*v*_{r}22
18

### A

2*k*3

*v*

_{r}

*v*

_{x}

2 3

### A

2*f*3mr

*v*_{r}

### 8

*v*

_{x}1

*f*

_{1}

*f*

_{3}

*v*

_{h}

*v*

_{r}21 3

### A

2*f*3

### 8

m*s*

*v*

_{r}

### 8

*v*

_{x}

### 8

2 3

### A

2*f*3mx

*v*

_{r}

*v*

_{x}

### 8

### D

,**M45**5

*g*2
6 *v*h

### 8

*v*s2,

**M46**52

*g*2

12*v*r

### 8

*v*s21 1

12~*f*3mr*v*x1*f*3

### 8

m*sv*

_{x}

### 8

!,**M47**52

*g*2

12*v*s2*v*x

### 8

1 112~*f*3

### 8

m*sv*

### 8

r1*f*3mx

*v*r!,

**M _{48}**52

*g*

2

12*v*s2*v*s

### 8

2,**M55**5

*g*2
3 *v*h

### 8

2

1 1

18

### A

2*v*

_{h}

### 8

~*f*_{1}

### 8

*f*

_{3}

### 8

*v*

_{r}

### 8

2*v*

_{s}

2

### 8

23*f*1mr

*v*

_{r}

*v*

_{h}

13*f*1

### 8

mx*v*

_{r}

### 8

*v*

_{x}218

*k*

_{1}

### 8

*v*

_{r}

### 8

*v*

_{x}

### 8

13*f*

_{1}

### 8

m_{r}

*v*

_{r}

*v*

_{x}

### 8

1*f*_{1}

### 8

*f*

_{3}

*v*

_{s}

2

### 8

*v*

_{x}

### 8

2!,**M56**52

*g*2
6 *v*h

### 8

*v*r

### 8

11

9

### A

2### S

### A

2*f*1

### 8

2*v*_{h}

### 8

*v*

_{r}

### 8

2*f*1

### 8

*f*

_{3}

### 8

*v*

_{r}

### 8

*v*

_{s}

2

### 8

23

2*f*1

### 8

mx*v*

_{x}19

*k*1

### 8

*v*

_{x}

### 8

### D

,**M57**52

*g*2
6 *v*h

### 8

*v*x

### 8

11 9

### A

2### S

9*k*1

### 8

*v*

r

### 8

232*f*1

### 8

mr*v*r1

### A

2*f*1

### 8

2

_{v}h

### 8

*v*

x

### 8

2*f*_{1}

### 8

*f*

_{3}

### 8

*v*

_{s}

2*v*x

### 8

### D

,**M _{58}**52

*g*

2

6 *v*h

### 8

*v*s

### 8

22*f*_{1}

### 8

*f*

_{3}

### 8

18

### A

2~*v*_{r}

### 8

21*v*

_{x}

### 8

2!,**M _{66}**5

### S

*g*2

3 1*g*

### 8

2### D

_{v}r

### 8

21 112

### A

2*v*

_{r}

### 8

### S

2*f*1mr*v*_{h}*v*_{x}

2

### A

2*f*

_{3}mr

*v*

_{s}

2*v*x12*f*1

### 8

mx*v*h

### 8

*v*x2

### A

2*f*3

### 8

mx*v*s

### 8

2*v*x

212*k*_{1}

### 8

*v*

_{h}

### 8

*v*

_{x}

### 8

2 12### A

2*k*3

### 8

*v*

_{s}

2

### 8

*v*

_{x}

### 8

22*f*1

### 8

mh*v*

_{h}

*v*

_{x}

### 8

2

### A

2*f*

_{3}

### 8

m*sv*

_{s}

2*v*x

### 8

### D

,**M67**52

### S

*g*2
6 1*g*

### 8

2

### D

_{v}r

### 8

*v*

_{x}

### 8

11

12

### A

2### S

12*k*1

### 8

*v*

_{h}

### 8

112

### A

2*k*3

### 8

*v*

_{s}

2

### 8

12*f*1

### 8

mh*v*

_{h}1

### A

2*f*

_{3}

### 8

m

_{s}v_{s}

21 3

### A

2*f*1

### 8

2*v*_{r}

### 8

*v*

_{x}

### 8

1 3

### A

2*f*3

### 8

2*v*_{r}

### 8

*v*

_{x}

### 8

### D

,**M68**5

*g*2

12*v*r

### 8

*v*s

### 8

_{2}1

*f*_{3}

### 8

*v*

r

### 8

18

### A

2~### A

2*f*3

### 8

*v*

s_{2}

**M77**5

### S

*g*2
3 1*g*

### 8

2

### D

_{v}x

### 8

22 112

### A

2*v*

_{x}

### 8

~12*k*_{1}

### 8

*v*

_{h}

### 8

*v*

r

### 8

112*k*

_{3}

### 8

*v*

r

### 8

*v*

s_{2}

### 8

12*f*1

### 8

mh*v*

_{h}

*v*

_{r}

### 8

22*f*

_{1}

### 8

m_{r}

*v*

_{h}

### 8

*v*

_{r}1

### A

2*f*

_{3}

### 8

m_{r}

*v*

_{r}

*v*

_{s}

2

1

### A

2*f*

_{3}

### 8

m

_{s}vr

### 8

*v*

_{s}

222*f*1mx*v*h*v*r1

### A

2*f*3mx

*v*r

*v*s2!,

**M78**5

*g*2

12*v*s

### 8

2*v*x

### 8

1 112~6*k*3

### 8

*v*r

### 8

1*f*3

### 8

mr*v*r!

1 *f*3

### 8

*v*

_{x}

### 8

18### A

2~*f*3

### 8

*v*_{s}

2

### 8

2*f*

_{1}

### 8

*v*

_{h}

### 8

!,**M88**5

*g*2
12*v*s

### 8

2_{2}2

1 18

### A

2*v*

_{s}

2

### 8

### S

*f*1

### 8

*f*3

### 8

*v*_{h}

### 8

*v*

r

### 8

21 3

### A

2*f*3m

*s*

*v*

_{r}

*v*

_{x}

1 3

### A

2*f*3

### 8

mx*v*

_{r}

### 8

*v*

_{x}1

18

### A

2*k*3

### 8

*v*

_{r}

*v*

_{x}

### 8

13

### A

2*f*3

### 8

mx*v*

_{r}

### 8

*v*

_{x}

1 18

### A

2*k*3

### 8

*v*

_{r}

### 8

*v*

_{x}

### 8

13

### A

2*f*3

### 8

mr*v*_{r}*v*_{x}

### 8

2*f*

_{1}

### 8

*f*

_{3}

### 8

*v*

_{h}

### 8

*v*

_{x}

### 8

2### D

.This mass matrix has no Goldstone bosons and eight mass
eigenstates. Some typical values of the masses of these
sca-lars, for a set of values of the parameters, are given in the
text. In Fig. 1 we show the behavior of the mass of the
lightest scalar*H*1with the*v*_{x}, the largest VEV in the model.

**APPENDIX B: MASS MATRIX OF THE PSEUDOSCALAR**
**NEUTRAL FIELDS**

The complete symmetric mass matrix in the*C P*-odd
sca-lar sector, with the constraint equations of Appendix C taken
into account, is given by

M_{11}_{5} 1

18

### A

2*v*

_{h}

~*f*1*f*3*v*

r

2
*v*_{s}

2218*k*1*v*r*v*x13*f*1mr*v*r

### 8

*v*x

### 8

!,M_{12}_{5} 1

6

### A

2~*f*1mx26

*f*1

*v*

_{x}!,

M_{13}_{5} 1

6

### A

2~*f*1mr

*v*_{r}

### 8

26*k*

_{1}

*v*

_{r}!,

M_{14}_{5} *f*1*f*3

18

### A

2~*v*_{r}21*v*_{x}2!,

M_{15}_{5}_{0,}

M_{16}_{5} 1

6

### A

2~*f*1mh

*v*

x

### 8

2*f*

_{1}

### 8

mr*v*

_{x}!,

M_{17}_{5} 1

6

### A

2~*f*1

### 8

mh*v*_{r}

### 8

2*f*

_{1}m

_{x}

*v*

_{r}!, M

**50,**

_{18}M_{22}_{5} 1

6

### A

2*v*

_{r}

### S

26*k*1*v*_{h}*v*_{x}_{2}
6

### A

2*k*3

*v*

_{s}

2*v*x2*f*1mh*v*h

### 8

*v*x

2 1

### A

2*f*3m

*s*

*v*

s_{2}

### 8

*v*

_{x}1

*f*

1

### 8

mr*v*

_{h}

### 8

*v*

x

### 8

2 1### A

2*f*3

### 8

mr*v*

s_{2}

### 8

*v*

x

### 8

1*f*1mx*v*_{h}*v*_{x}

### 8

2 1### A

2*f*3mx

*v*

_{s}

2*v*x

### 8

### D

,M_{23}_{5} 1

6

### A

2### S

26*k*1

*v*

_{h}2

6

### A

2*k*3

*v*

_{s}

2

2*f*_{1}m_{h}*v*_{h}

### 8

2 1### A

2m*s*

*v*

_{s}

2

### 8

### D

,M_{24}_{5}1

12~6*k*3*v*x1*f*3mx*v*x

### 8

!,M_{25}_{5} 1

6

### A

2~*f*1mh

*v*

_{x}2

*f*

1

### 8

mr*v*

_{x}

### 8

!, M**50,**

_{26}M_{27}_{5} 1

12

### A

2~22*f*1

### 8

mr*v*_{h}

### 8

_{1}

### A

2*f*3

### 8

mr*v*

s_{2}

### 8

22*f*1mx*v*_{h}1

### A

2*f*

_{3}m

_{x}

*v*

_{s}

2!,

M_{28}_{5} 1

12~*f*3m*sv*x1*f*3

### 8

mr*v*

_{x}

### 8

!,M_{33}_{5} 1

6

### A

2*v*

_{x}

### S

26*f*1*k*1*v*_{h}*v*_{r}_{2}
6

### A

2*k*3

*v*

_{r}

*v*

_{s}

22*f*1mh*v*h

### 8

*v*r

1*f*1mr*v*_{h}*v*

r

### 8

2 1### A

2*f*3mr

*v*

r

### 8

*v*

_{s}

22 1

### A

2*f*3m

*s*

*v*

_{r}

*v*

s_{2}

### 8

1*f*_{1}

### 8

mx*v*

_{h}

### 8

*v*

r

### 8

2 1### A

2*f*3

### 8

mx*v*

r

### 8

*v*

s_{2}

### 8

### D

,M_{34}_{5}1

12~6*k*3*v*r1*f*3mr*v*r

### 8

!,M_{35}_{5} 1

### A

2~*f*1mh

*v*

_{r}

_{2}

*f*

1

### 8

mx*v*

x

### 8

!,M_{36}_{5} 1

12

### A

2~22*f*1mr

*v*_{h}_{1}

### A

2*f*3mr

*v*

_{s}

2

22*f*_{1}

### 8

mx*v*

_{h}

### 8

_{1}

### A

2*f*3

### 8

mx*v*

s_{2}

### 8

!,M_{37}_{5}0, M_{38}_{5}1

M_{44}_{5} 1

18

### A

2*v*

_{s}

2

### S

*f*1*f*3*v*_{h}*v*

r

2 2 18

### A

2*k*3

*v*

_{r}

*v*

_{x}

_{2}

3

### A

2mr*v*

r

### 8

*v*

_{x}

118

### A

2*k*3

*v*

_{r}

*v*

_{x}2

3

### A

2*f*3mr

*v*_{r}

### 8

*v*

_{x}1

*f*

_{1}

*f*

_{3}

*v*

_{h}

*v*

_{x}2

2 3

### A

2*f*3

### 8

m*s*

*v*

_{r}

### 8

*v*

_{x}

### 8

23

### A

2*f*3mx

*v*

_{r}

*v*

_{x}

### 8

### D

,M_{45}_{5}0, M_{46}_{52} 1

12~*f*3mr*v*x1*f*3

### 8

m*sv*x

### 8

!,M_{47}_{52} 1

12~*f*3

### 8

mr*v*

_{r}

### 8

1*f*

_{3}m

_{x}

*v*

_{r}

### 8

!, M**50,**

_{48}M_{55}_{5} 1

18

### A

2*v*

_{h}

### 8

~*f*_{1}

### 8

*f*

_{3}

### 8

*v*

r

### 8

2

_{v}s_{2}

### 8

23*f*1mh

*v*

_{r}

*v*

_{x}

_{1}3

*f*1

### 8

mx*v*

r

### 8

*v*

_{x}

218*k*_{1}

### 8

*v*

_{r}

### 8

*v*

_{x}

### 8

13*f*1

### 8

mr*v*

_{r}

*v*

_{x}

### 8

1*f*1

### 8

*f*

_{3}

### 8

*v*

_{s}

2

### 8

*v*

_{x}

### 8

213*f*1

### 8

mr*v*

_{r}

*v*

_{x}

### 8

1*f*

_{1}

### 8

*f*

_{3}

### 8

*v*

_{s}

2

### 8

*v*

_{x}

### 8

2!,M_{56}_{5} *f*1*f*3

18

### A

2~mx*v*

_{x}

_{2}

*k*

1

### 8

*v*

x

### 8

!,M_{57}_{5} 1

6

### A

2~26*k*1

### 8

*v*

r

### 8

1*f*

_{1}

### 8

mr*v*

_{r}!,

M_{58}_{5} *f*1

### 8

*f*3

### 8

18

### A

2~*v*_{r}

### 8

22*v*

_{x}

### 8

2!,M_{66}_{5} 1

12

### A

2*v*

_{r}

### 8

### S

2*f*1mr*v*_{h}*v*_{x}2

### A

2*f*

_{3}m

_{r}

*v*

_{s}

2*v*x

12*f*_{1}

### 8

mx*v*

_{h}

### 8

*v*

_{x}

_{2}

### A

2*f*3

### 8

mx*v*

s_{2}

### 8

*v*

_{x}

_{2}12

*k*1

### 8

*v*

_{h}

### 8

*v*

x

### 8

2 12

### A

2*k*3

### 8

*v*

_{s}

2

### 8

*v*

_{x}

### 8

22*f*1

### 8

mh*v*

_{h}

*v*

_{x}

### 8

2### A

2m

_{s}v_{s}

2*v*x

### 8

### D

,M_{67}_{5} 1

12

### A

2### S

212*k*1

### 8

*v*

_{h}

### 8

212

### A

2*k*3

### 8

*v*

_{s}

2

### 8

22*f*

_{1}

### 8

mh*v*

_{h}

2

### A

2*f*3

### 8

m*sv*

_{s}

2

### D

,M_{68}_{5}1

12~2*f*3

### 8

mx*v*x16

*k*3

### 8

*v*x

### 8

!,M_{77}_{5} 1

12

### A

2*v*

_{x}

### 8

### S

212*k*_{3}

### 8

*v*

_{h}

### 8

*v*

r

### 8

2 12### A

2*k*3

### 8

*v*

r

### 8

*v*

s_{2}

### 8

22*f*_{1}

### 8

mh*v*

_{h}

*v*

r

### 8

12*f*

_{1}

### 8

mr*v*

_{h}

### 8

*v*

_{r}

_{2}

### A

2*f*3

### 8

mr*v*

_{r}

*v*

s_{2}

### 8

### D

,M_{78}_{5} 1

12~6*k*3

### 8

*v*r

### 8

2*f*3

### 8

mr*v*r!,

M_{88}_{5} 1

18

### A

2*v*

_{s}

2

### 8

### S

*f*1

### 8

*f*3

### 8

*v*_{h}

### 8

*v*

r

### 8

22 3

### A

2m*s*

*v*

_{r}

*v*

_{x}

_{2}

3

### A

2mx*v*

r

### 8

*v*

_{x}

2 18

### A

2*k*3

### 8

*v*

_{r}

### 8

*v*

_{x}

### 8

23

### A

2*f*3

### 8

mr*v*

_{r}

*v*

_{x}

### 8

1*f*

1

### 8

*f*

_{3}

### 8

*v*

_{h}

### 8

*v*

_{x}

### 8

2### D

.This mass matrix has two Goldstone bosons and six mass
eigenstates. In Fig. 1 we show the behavior of the mass of
the lightest pseudoscalar scalar*A*1as a function of*v*_{x}.
Typi-cal values for the masses in this sector for a set of values of
the parameters are given in the text.

**APPENDIX C: CONSTRAINT EQUATIONS**

The constraint equations are

**t**h
**v**_{h}5

*g*2
12~2*v*h

2

22*v*_{h}

### 8

22*v*2

_{r}1

*v*

_{r}

### 8

22*v*

_{s}

2 2

1*v*_{s}

2

### 8

22*v*_{x}21*v*_{x}

### 8

2!1*m*

_{h}2

11 4mh

2
1 *f*1

2

18~*v*r
2

1*v*_{x}2!1
*f*1*f*3

18

### A

2*v*

_{s}

2

*v*_{h} ~
*v*_{r}21*v*_{x}2!

1*k*1

### A

2*v*

_{r}

*v*

_{x}

*v*_{h} 1

1 6

### A

2*v*

_{h}

~*f*_{1}mr*v*_{r}

### 8

*v*

_{x}1

*f*

_{1}m

_{x}

*v*

_{r}

*v*

_{x}

### 8

1*f*_{1}

### 8

mh*v*

_{r}

### 8

*v*

_{x}

### 8

!,**t**_{r}

**v**r5

*g*2
12

### S

2*v*h

2

1*v*_{h}

### 8

2_{1}2

*v*

r

2
22*v*

r

### 8

211_{2}*v*

s_{2}

2
21_{2}*v*

s_{2}

### 8

22*v*

x

2

1*v*_{x}

### 8

2### D

1*g*

_{8}

2
2 ~*v*r
2

2*v*_{r}

### 8

22*v*

_{x}21

*v*

_{x}

### 8

2!1*f*

_{1}2 18~

*v*h

2
1*v*_{x}2!

1 *f*3
2

36~*v*s2
2

12*v*_{x}2!2
*f*1*f*3

9

### A

2*v*

_{h}

*v*

_{s}

21*m*r
2

11 4mr

2

1 *k*1

### A

2*v*

_{h}

*v*

_{x}

*v*_{r} 1

*k*3
2

*v*_{s}

2*v*x

*v*_{r} 1

*f*1

6

### A

2*v*

_{r}

~m_{h}*v*_{h}

### 8

*v*

_{x}1m

_{x}

*v*

_{h}

*v*

_{x}

### 8

!1 *f*3
12*v*_{r}~m*s*

*v*_{s}

2

### 8

*v*

_{x}1m

_{x}

*v*

_{s}

2*v*x

### 8

!1*f*_{3}

### 8

12*v*_{r}mr
*v*_{s}

2