Supersymmetric 3-3-1 model

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 toRsymmetry in the minimal supersymmetric standard model, is possible to be defined in this model. Hence we have bothR-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)_N ~3-3-1 for shortness! instead of that of the SM

@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 e2_{, n}

e, and e1. 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 theU(1)_N

and SU(3)_L coupling constants, g

8

andg, respectively, are related by

t2[g

8

2

g2 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, thenSU(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_{5B1L 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 (ne,e,nec,ec)L would beSU(4)L^U(1)N @15#. This

is possibly the last symmetry among leptons. There is no room forSU(5)_L^_U(1)

Nif 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 largerSU(4)_L symmetry@15#.

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 ofC Pviolation. 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 softC Pviolation@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 (124s_W2)1/2 but with some quarks there are enhancements

because of the factor (124s_W2)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 sin2uW,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_l5

S

nl

l

lc

D

_L

;~1,3,0!, l5e,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

Q1L5

S

u1

d1

J

D

L

;

S

3,3,2

3

D

, ~3!

and the respective singlets are given by

u₁c_L;

S

3*,1,22

3

D

, d1L 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

Q2L5

S

d2

u2

j1

D

_L

, Q3L5

S

d3

u3

j2

D

_L

;

S

3,3*,21

3

D

, ~5!

and also with the respective singlets,

u2L c _,

u3L

c _;

S

_3*,1,

22

3

D

, d2L c _,

d3L

c _;

S

_3*,1,1

3

D

,

j₁c_L,jc_2L;

S

3*,1,4

3

D

. ~6!

On the other hand, the scalars which are necessary to generate the fermion masses are

h5

S

h0

h₁2

h₂1

D

;~1,3,0!, r5

S

r1

r0 r11

D

;~1,3,11!,

x5

S

x2 x22

x0

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:

S5

S

s₁0 h2 1

A

2 h₁2

A

2

h₂1

A

2

H₁11 s2 0

A

2

h₁2

A

2 s₂0

A

2

H₂22

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 antisextetS.

Summarizing, we have in the 3-3-1s model the following superfields: Lˆ_e,m,t, Qˆ1,2,3, hˆ, rˆ, xˆ, Sˆ; hˆ

8

, rˆ

8

, xˆ

8

, Sˆ

8

; uˆ_1,2,3c , dˆ_1,2,3c , Jˆc, andjˆ_1,2c , i.e., 23 chiral superfields, and 17 vector superfields: Vˆa, Vˆa, 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:

L₃₃₁₅L

HereL

SUSY is the supersymmetric piece, whileLso 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_SUSY5L_{Le pton1}L_Quarks1L_Gauge1L_Scalar, ~11!

where each term is given by

L

Le pton5

E

d4u@L¯ˆ e2gV ˆ

Lˆ#, ~12!

L_Quarks5

E

_d4_u@Q¯ˆ_1exp$@2g~Vˆc1Vˆ!1~2g

8

/3!Vˆ

8

#%Qˆ1

1Q¯ˆaexp$@2g~Vˆc1V¯ˆ!2~g

8

/3!Vˆ

8

#%Qˆa

1¯ˆu_icexp$@2gV¯ˆc2~2g

8

/3!Vˆ

8

#%uˆic

1¯ˆd_icexp$@2gV¯ˆc1~g

8

/3!Vˆ

8

#%dˆic

1¯ˆJcexp$@2gV¯ˆc2~5g

8

/3!Vˆ

8

#%Jˆc

1¯ˆjbcexp$@2gV¯ˆc1~4g

8

/3!Vˆ

8

#%jˆbc# ~13!

and

L

Gauge5

1 4

F

E

d

2_u@W

c a_W

c a

1Wa_Wa

1W

₈

W

₈

#

1

E

d2¯u_@W¯ c a_W¯

c a

1W¯aW¯a1W¯

₈

W¯

₈

#

G

, ~14!

where Vˆc5TaVˆac, V¯ˆc5T¯aVˆac, Vˆ5TaVˆa, V¯ˆ5T¯aVˆa, and Ta

5la_{/2 (¯T}a

52l*a/2) are the generators of triplets~ antitrip-lets!ofSU(3), i.e.,a51, . . . ,8, andgandg

8

are the gauge couplings of SU(3) and U(1), respectively. W_ca, Wa, and W

₈

are the strength fields, and they are given by

Waac52

1 8gD¯ D¯ e

22gVˆc_D

ae2gV ˆ

c_,

W_aa52 1 8gD¯ D¯ e

22gVˆ_D ae2gV

ˆ

,

Wa

8

52

1

4D¯ D¯ DaVˆ

8

. ~15!

Finally

L_Scalar5

E

_d4_u@h¯ˆ e2gVˆ_hˆ₁¯ˆ e_r 2gVˆ1g₈Vˆ_{8rˆ1x¯ˆ e}2gVˆ2g₈Vˆ_8xˆ

1¯ˆ eS 2gVˆSˆ1h¯ˆ

8

e2gVChˆ

₈

₁¯ˆ_r

₈

_e2gVC2g₈Vˆ_8rˆ

8

1x¯ˆ

8

e2gV¯ˆ1g₈Vˆ_8xˆ

₈

1¯ˆS

₈

e2gV¯ˆ_Sˆ

₈

_#

1

E

d2uW

1

E

d2_{¯ Wu¯ , ~16!}

whereWis the superpotential, which we discuss in the next subsection.

B. Superpotential

The superpotential of our model is given by

W5W2 2 1

W3

3 , ~17!

with W2 having only two chiral superfields and the terms permitted by our symmetry are

W25m0Lˆhˆ

8

1mhhˆhˆ

8

1mrrˆrˆ

8

1mxxˆxˆ

8

1mSSˆSˆ

8

,

~18!

and in the case of three chiral superfields the terms are

W35l₁eLˆLˆLˆ1l₂eLˆLˆhˆ_1l

3LˆLˆSˆ1l4eLˆxˆrˆ1f1erˆxˆhˆ

1f2hˆhˆ Sˆ1f3xˆrˆ Sˆ1f1

8

erˆ

8

xˆ

8

hˆ

8

1f2

8

hˆ

8

hˆ

8

Sˆ

8

1f₃

8

xˆ

₈

_rˆ

₈

_Sˆ

₈

₁

₍

i

@k1iQˆ1hˆ

8

uˆic1k2iQˆ1rˆ

8

dˆic#

1k3Qˆ1xˆ

8

Jˆc1

(

ai

@k₄aiQˆahˆ dˆi c

1k5aiQˆarˆ uˆi c_#

1

(

ab k6abQ

ˆ_{axˆ jˆb}c

1

(

ai j

k7ai jQˆaLˆidˆjc

1

(

i,j,k j1i jkdˆi c

dˆc_juˆ_kc1

(

i jb j2i jbuˆi c

uˆ_jcjˆbc1

(

ib j3ibdˆi

c Jˆcjˆbc,

~19!

withi,j,k51,2,3, a52,3, andb51,2.

All the seven neutral scalar components h0_,r0_,x0_,s

2 0_,h

8

0,r

₈

0_{, andx}

8

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

^

h0

_&

5v_h/

A

2,

^

h0

8

&

₅v_h

8

/

A

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

Gu

5 1

A

2

S

k11v_h

8

k₁₂v_h

8

k₁₃v_h

8

k₅₂₁v_r k₅₂₂v_r k₅₂₃v_r k₅₃₁v_r k₅₃₂v_r k₅₃₃v_r

D

, ~20!

Gd

5 1

A

2

S

k21v

r

8

k22v

r

8

k23v

r

8

k421v_h k₄₂₂v_h k₄₂₃v_h k431v_h k₄₃₂v_h k₄₃₃₃v_h

D

, ~21!

for the d quarks, and for the exotic quarks, J and j1,2, we have MJ5k3v_x

8

and

Gj5 v_x

A

2

S

k621 k622

k₆₃₁ k₆₃₂

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}l5v_s

2l3i j/

A

2, where vs2 is the VEV of the

^

s2 0

&

component of the antisextetSin Eq.~8!. However,v_s

1,2

8

can both be zero since the sextetS

₈

does not couple to leptons at all.

The terms which are proportional to the following con-stants: m0 in Eq.~18! andl1, l4, f2, f2

8

, k7, andj1,2,3 in Eq. ~19! violate the conservation of the F_{5B1L quantum}

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

symmetry, it allows us to introduce the R-conserving sym-metry @37#, defined asR5(21)3F₁2S_{. The Fnumber}

attri-bution is

F_~U22_!

5F_~V2_!

52F_~J1!5F_~J2,3!5F_~r22_!

5F_~x22_!

5F_~x2_! 5F_~h

2 2_!

5F_~s 1 0_!

52,

~23!

withF_{50 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 Rparity 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 t52

1

2

F

mlc

(

_a lc

a_l c a

1ml

(

a

~ la_l

a!1m

8

ll1H.c.

G

2mL2L˜†˜L2mQ2₁Q˜1†Q˜11

(

i

@˜u_ic†m_u

i

2_˜u

i c

2˜d_ic†m_d

i

2_d˜

i c_#

2m_J2˜Jc†˜Jc2

(

b j

˜_bc†_m

j_b

2_˜j

b c

2

(

a mQa

2 _Q˜

a

†_Q˜

a2mh2h†h2mx2x†x2m2rr†r2ms2Tr~S†S!2mh₈

2 h

₈

†_h

8

2m_r 8

2 r

₈

†_r

8

2m_x 8

2 x

₈

†_x

8

2m_s 8

2

Tr~S

₈

†S

₈

!2

F

k1erxh1k2hhS†1k3xrS†1k

8

1er

8

x

8

h

8

1k2

8

h

8

h

8

S

8

†1k3

8

x

8

r

8

S

8

† 1M2L˜h†

1«₀

(

i jk ei jkL ˜

iL˜jL˜k1«1

(

i jk ei jkL ˜

iL˜jhk1«2

(

i j L ˜

iL˜jSi j1«3

(

i jk ei jkL ˜

ixjrk

1

(

i Q ˜

1~z1ih

8

˜uic1z2ir

8

˜dic1z3Jx

8

˜Jc!1

(

a Q

˜_a

F

₍

i

S

v1ai

h˜d_ic1v2airu˜ic1

(

j v3ai j

L ˜

i˜djc

D

1

(

b v4abxj

˜

b c

G

1

(

i

S

(

jk

§_{1i jk}˜d_ic˜d_jc˜u_kc1

(

a

F

§_2iad˜_ic˜Jc˜jac1

(

j

§_{3i ja}˜u_ic˜uc_j˜jac

GD

1H.c.

G

. ~24!

The terms proportional to k2, k2

8

, M2,«0, «3, v3, and

§_1,2,3 violates theR-parity symmetry. The SU(3) invariance tell us that

mL5

S

m˜n 0 0

0 m_l˜ 0 0 0 m_l˜c

D

, ~25!

and we need ml˜c5m_l˜5m_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

V3315VD1VF1Vsoft, ~26a!

VD52LD5

1 2~D

a_Da

1DD!g

8

2

2 ~r †_r

2r

8

†_r

₈

_2x†_x1x

₈

†_x

₈

_!2 1g

2

8

(

i,j ~

h_i†_l

i j a_h

j1ri

†_l

i j a_r

j1xi

†_l

i j a_x

j

1S_{i j}†la_jkS_kl2hi

8

†li j*ahj

8

2ri

8

†li j*arj

8

2xi

8

†li j*ax

8

j2Si j

8

†ljk*aSkl

8

!2, ~26b!

VF52LF5

(

m Fm

*Fm5

(

i,j,k

F

U

m_h 2 hi

8

1

f1

3 ei jkrjxk1 2f2

3 hiSi j

U

2

1

U

mr 2 ri

8

1

f1

3 ei jkxjhk1 f3 3 xiSi j

U

2

1

U

mx 2 xi

8

1

f1

3 ei jkrjhk1 f3

3 riSi j

U

2

1

U

ms 2 Si j

8

1

f2 3 hihj1

f3 3 xirj

U

2

1

U

mh 2 hi1

f1

8

3 ei jkr

8

jxk

8

1

2f2

8

3 hi

8

Si j

8

U

2

1

U

mr 2 ri

1 f₁

8

3 ei jkx

8

jhk

8

1 f₃

8

3 xi

8

Si j

8

U

2

1

U

mx 2 xi1

f₁

8

3 ei jkr

8

jhk

8

1 f₃

8

3 ri

8

Si j

8

U

2

1

U

ms 2 Si j1

f₂

8

3 hi

8

h

8

j1 f₃

8

3 xi

8

rj

8

U

2

G

, ~26c!

Vso f t52Lso f t

scalar

5m_h2h†_h

1mr2r†r1mx2x†x1ms

2_Tr~

S†S!1m_h 8

2 _h

8

†_h

₈

1m_r

8

2 r

8

†_r

₈

1m_x 8

2 x

8

†_x

₈

1m_s 8

2

Tr~S

8

†_S

₈

_!

1@k1e rxh1k2hS†h1k3xTS†r1k1

8

e r

8

x

8

h

8

1k2

8

h

8

S

8

†h

8

1k3

8

x

8

TS

8

†r

8

1H.c.#. ~26d!

Note that k1,2,3,k1,2,3

8

has dimension of mass and that the terms which are proportional to k2,k2

8

and f2,f2

8

violate the Rparity.

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

V_D5g

8

2

2 ~r †_r

2x†x2r

8

†r

₈

1x

8

†x

₈

!21g 2

8

F

4 3

(

i

~X_i†X_i!2

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~TrS

†_S!2

1pt

G

, ~27!

where ‘‘pt’’ in the expression above denotes the replacements X_i

8

↔_X

ibut not ing

8

which is always the coupling constant

of theU(1)_Nfactor. In the same way we rewrite Eq.~26c!as

V_F5

(

i

m_X

i

2

4 ~Xi†Xi1pt!1

um_S2_u 4 @Tr~S

†_S!

1pt#11₆

(

iÞjÞk

H

m_X

i

*

6 @f1ei jkXi

8

†XjXk1f1

8

ei jkXi†X

8

jXk

8

#

1f₃2

F

mh*h

8

†Sh1

m_S*

2 h †_S

8

h

G

1pt1f₆3@m_r*r

₈

†_Sx

1mx*x

8

†Sr1m*Sx†S

8

r#1pt1H.c.

J

1uf1u2

F

(

iÞj

~X_i†Xi!~Xj

†

Xj!2~Xj

† Xi!~Xi

†

Xj!

G

1pt1

4uf2u2 9 @~hS!

†_~hS!

1~h†_h!2_#

1pt1uf3u 2

9 @~xS! †_~xS!

1~rS!†~rS!1~x†_x!†_~r†_r!# 1pt1

2f₁*f2 9 e~rx!

†_hS 1pt

1

F

f1

*f3 9 e~xh!

†_xS1pt1f2*f3 9 ~h

†_x!~h†_r!

1pt1H.c.

G

, ~28!

where ‘‘pt’’ in the expression above denotes as before the replacementsX_i

8

↔_X

i, andf1,2,3

8

↔f1,2,3, andk1,2,3

8

↔k1,2,3. We have omitted SU(3) indices since we have denoted the unprimed triplets wherever it is possible as Xi5h,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_X1H_X1iF_X) ~similarly for the case of the primed fields!for all the neutral scalar fields of the multiplets X_i. Note thatH_X andF_X are not mass eigenstates yet. We will denote Hi,i51, . . . ,8 and Ai,i51, . . . ,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:

f15f351, f₁

8

5f

8

₃51026 _{~dimensionless!, ~29!}

and

2k15k1

8

510, k35k3

8

52100,

2mh5mr52ms5mx51000 ~in GeV!, ~30!

we also use the constraintV_h21V_r212V₂25(246 GeV)2 com-ing from MW, where, we have defined Vh25v2_h1v_h

8

2, V_r2 5v_r21v_r

8

2, and V₂25v_s

2 2

1v

s₂₈

2

. Assuming that v_h520, v_x 51000, v_s

2510, and vh

8

5vr

8

5vs

8

25

v_x

8

51 in GeV, the value ofv_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_i5121.01,

277.14,515.26,963.68,1218.8,1243.24,3797.86, and 4516.43, wherei51, . . . ,8 andMH_j.MH_i withj.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_i5276.4,515.3,963.65,1243.24,3797.85, and

4516.43.

The behavior of the lightest scalar (H1) and pseudoscalar (A1) as a function ofv_xis 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 MH₁,124.5

GeV and that for these values of the parameters MA₁

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

values for the upper limit of MH₁. 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 MA₁.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) andE_Lc;(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)

Yoccurred 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 s₂

8

0 and s₁0) 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; MH₁ is the mass of the lightest scalar andMAthe 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 withoutR-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 theU-vector boson with the doubly charged Higgsinos. This implies new interactions that are not present in the MSSM, for instance x˜22˜_x0_U11_, x

˜2_x˜2_U11_{, l˜}2_l2_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 theU-bilepton in theschannel. Due to this fact we expect that there will be an enhancement in the cross section of production of these particles in e2_e2 _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_l1_x˜0_{, n˜}

ll2˜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 scalarC P-even sector; the constraint equations given

in the Appendix C have been already taken into account:

M115

g2v_h2

3 1

1 18

A

2v_h

~f1f3v_r2v_s

2218k1vrvx13f1vr

8

mrvx

1f1f3v_s

2vx 2

23f₁

8

mhv_r

8

v_x

8

13f₁m_xv_rv_x

8

!,

M1252

g2v_hv_r

6 1

1

9

A

2

S

A

2f1 2_v

hv_r2f₁f₃v_rv_s

2

19k1v_x2 3 2mxvx

8

D

,

M₁₃52g

2_v

hv_r

6 1

1 9

A

2

S

9k1

v_r2 3 2f1mrvr

8

1

A

2f12v_hv_x2f₁f₃v_s

2vx

D

,

M₁₄5 g2v_hv_s

2

6 2

f1f3

18

A

2~

v_r21v_x2!, M₁₅52 g2v_hv_h

8

3 ,

M165

g2v_hv

r

8

6 2

1 6

A

2~mr

v_x2m_hv

x

8

!,

M175

g2v_hv_x

8

6 1

1 6

A

2~f1

8

mh

v_r

8

2f₁m_xv_r!,

M185

g2v_hv_s

2

6 ,

M225

S

g2 3 1g

8

2

D

_v

r

2 2

v_x 12

A

2v_r

~12k1v_h16

A

2k₃v_s

2

12f₁m_hv_h

8

1

A

2f₃m_sv_s

2

8

!1 v

x

8

12

A

2v_r

~2f₁

8

mrv_h

8

2

A

2f₃

8

mrv

s₂

8

12f1mxv_h2

A

2m_xv_s

2vx

8

!,

M2352

S

g2 6 1g

8

2

D

_v

rv_x1 1 12

A

2~12k1

v_h16

A

2k₃v_s

2

12f

₈

mhv_h

8

1

A

2f₃m_sv_s

2

8

!1 v_rv_x

9 ~f1 2

1f₃2!,

M245

g2 12vrvx1

1

18

A

2

S

22f1f3

v_hv_r1

A

2f 3 2_v

s₂v_r

19

A

2k3v_x1 3

A

2f3mx v

x

8

D

,

M255

g2 6 vh

8

vr1

1 6

A

2~f1mh

v_x2f 1

8

mrv

M2652

S

g2 3 1g

8

2

D

vrvr

8

3 ,

M275

S

g2 6 1g

8

2

D

_v

rv_x

8

2 mr

12

A

2~2f1

8

v_h

8

2f₃

8

v_s

2

8

!

2 mx 12

A

2~2f1

v_h2f₃v_s

2!,

M2852

g2v_rv_s

2

8

12 1

1

12~f3msvx1f3

8

mrv_x

8

!,

M₃₃5

S

g 2

3 1g

8

2

D

_v

x

2

2 1

12

A

2v_x

S

12k1v_hv_r1 12

A

2k3 v_rv_s

2

12f1mhv

8

_hv_r22f₁m_rv_hv_r

8

1

A

2f₃m_rv_r

8

v_s

2

1

A

2f3msv_rv_s

2

8

22f1

8

mxv_h

8

v

8

_r1

A

2f₃m_xv_r

8

v_s

2

8

D

,

M345

g2

12vs2vx1 1 18

A

2

S

9

A

2k3 v_r1

3

A

2f3mr v_r

8

22f1f3v_hv_x1

A

2f₃2v_s

2vx

D

,

M₃₅5g 2

6 vh

8

vx1 1 6

A

2~f1mr

v_r2f 1

8

mxv_r

8

!,

M365

S

g2 6 1g

8

2

D

_v

r

8

v_x1

1

12

A

2~22f1mr

v_h1

A

2f₃m_rv_s

2

22f₁

8

mxv_h

8

1

A

2f 3

8

mxv

s₂

8

!,

M₃₇52

S

g 2

3 1g

8

2

D

_v

xv_x

8

,

M3852

g2

12vs

8

2vx1 1

12~f3msvr1f3

8

mxv_r

8

!,

M₄₄5g 2

12vs2 2

1 1

18

A

2v_s

2

S

f1f3v_hv_r22 18

A

2k3 v_rv_x

2 3

A

2f3mr

v_r

8

v_x1f₁f₃v_hv_r21 3

A

2f3

8

ms v_r

8

v_x

8

2 3

A

2f3mx v_rv_x

8

D

,

M455

g2 6 vh

8

vs2,

M4652

g2

12vr

8

vs21 1

12~f3mrvx1f3

8

msv_x

8

!,

M4752

g2

12vs2vx

8

1 1

12~f3

8

msv

8

r1f3mxvr!,

M₄₈52g

2

12vs2vs

8

2,

M555

g2 3 vh

8

2

1 1

18

A

2v_h

8

~f₁

8

f₃

8

v_r

8

2v_s

2

8

23f1mrv_rv_h

13f1

8

mxv_r

8

v_x218k₁

8

v_r

8

v_x

8

13f₁

8

m_rv_rv_x

8

1f₁

8

f₃v_s

2

8

v_x

8

2!,

M5652

g2 6 vh

8

vr

8

1

1

9

A

2

S

A

2f1

8

2

v_h

8

v_r

8

2f 1

8

f₃

8

v_r

8

v_s

2

8

23

2f1

8

mxv_x19k 1

8

v_x

8

D

,

M5752

g2 6 vh

8

vx

8

1

1 9

A

2

S

9k1

8

v

r

8

23

2f1

8

mrvr1

A

2f1

8

2_v

h

8

v

x

8

2f₁

8

f₃

8

v_s

2vx

8

D

,

M₅₈52g

2

6 vh

8

vs

8

22

f₁

8

f₃

8

18

A

2~

v_r

8

21v_x

8

2!,

M₆₆5

S

g 2

3 1g

8

2

D

_v

r

8

21 1

12

A

2v_r

8

S

2f1mrv_hv_x

2

A

2f₃mrv_s

2vx12f1

8

mxvh

8

vx2

A

2f3

8

mxvs

8

2vx

212k₁

8

v_h

8

v_x

8

2 12

A

2k3

8

v_s

2

8

v_x

8

22f 1

8

mhv_hv_x

8

2

A

2f₃

8

msv_s

2vx

8

D

,

M6752

S

g2 6 1g

8

2

D

_v

r

8

v_x

8

1

1

12

A

2

S

12k1

8

v_h

8

1

12

A

2k3

8

v_s

2

8

12f1

8

mhv_h1

A

2f₃

8

m_sv_s

21 3

A

2f1

8

2

v_r

8

v_x

8

1 3

A

2f3

8

2

v_r

8

v_x

8

D

,

M685

g2

12vr

8

vs

8

₂1

f₃

8

v

r

8

18

A

2~

A

2f3

8

v

s₂

M775

S

g2 3 1g

8

2

D

_v

x

8

22 1

12

A

2v_x

8

~12k₁

8

v_h

8

v

r

8

112k₃

8

v

r

8

v

s₂

8

12f1

8

mhv_hv_r

8

22f₁

8

m_rv_h

8

v_r1

A

2f₃

8

m_rv_rv_s

2

1

A

2f₃

8

m_sv

r

8

v_s

222f1mxvhvr1

A

2f3mxvrvs2!,

M785

g2

12vs

8

2vx

8

1 1

12~6k3

8

vr

8

1f3

8

mrvr!

1 f3

8

v_x

8

18

A

2~f3

8

v_s

2

8

2f₁

8

v_h

8

!,

M885

g2 12vs

8

2₂2

1 18

A

2v_s

2

8

S

f1

8

f3

8

v_h

8

v

r

8

2

1 3

A

2f3ms v_rv_x

1 3

A

2f3

8

mx v_r

8

v_x1

18

A

2k3

8

v_rv_x

8

1

3

A

2f3

8

mx v_r

8

v_x

1 18

A

2k3

8

v_r

8

v_x

8

1

3

A

2f3

8

mr

v_rv_x

8

2f₁

8

f₃

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 scalarH1with thev_x, the largest VEV in the model.

APPENDIX B: MASS MATRIX OF THE PSEUDOSCALAR NEUTRAL FIELDS

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

M₁₁₅ 1

18

A

2v_h

~f1f3v

r

2 v_s

2218k1vrvx13f1mrvr

8

vx

8

!,

M₁₂₅ 1

6

A

2~f1mx26f1 v_x!,

M₁₃₅ 1

6

A

2~f1mr

v_r

8

26k₁v_r!,

M₁₄₅ f1f3

18

A

2~

v_r21v_x2!,

M_1550,

M₁₆₅ 1

6

A

2~f1mh v

x

8

2f₁

8

mrv_x!,

M₁₇₅ 1

6

A

2~f1

8

mh

v_r

8

2f₁m_xv_r!, M₁₈50,

M₂₂₅ 1

6

A

2v_r

S

26k1v_hv_x2 6

A

2k3 v_s

2vx2f1mhvh

8

vx

2 1

A

2f3ms v

s₂

8

v_x1f

1

8

mrv_h

8

v

x

8

2 1

A

2f3

8

mr v

s₂

8

v

x

8

1f1mxv_hv_x

8

2 1

A

2f3mx v_s

2vx

8

D

,

M₂₃₅ 1

6

A

2

S

26k1 v_h2

6

A

2k3 v_s

2

2f₁m_hv_h

8

2 1

A

2ms v_s

2

8

D

,

M₂₄₅1

12~6k3vx1f3mxvx

8

!,

M₂₅₅ 1

6

A

2~f1mh v_x2f

1

8

mrv_x

8

!, M₂₆50,

M₂₇₅ 1

12

A

2~22f1

8

mr

v_h

8

₁

A

2f 3

8

mrv

s₂

8

22f1mxv_h1

A

2f₃m_xv_s

2!,

M₂₈₅ 1

12~f3msvx1f3

8

mrv_x

8

!,

M₃₃₅ 1

6

A

2v_x

S

26f1k1v_hv_r2 6

A

2k3 v_rv_s

22f1mhvh

8

vr

1f1mrv_hv

r

8

2 1

A

2f3mr v

r

8

v_s

22 1

A

2f3ms v_rv

s₂

8

1f₁

8

mxv_h

8

v

r

8

2 1

A

2f3

8

mx v

r

8

v

s₂

8

D

,

M₃₄₅1

12~6k3vr1f3mrvr

8

!,

M₃₅₅ 1

A

2~f1mh v_r2f

1

8

mxv

x

8

!,

M₃₆₅ 1

12

A

2~22f1mr

v_h1

A

2f 3mrv_s

2

22f₁

8

mxv_h

8

₁

A

2f 3

8

mxv

s₂

8

!,

M₃₇₅0, M₃₈₅1

M₄₄₅ 1

18

A

2v_s

2

S

f1f3v_hv

r

2 2 18

A

2k3 v_rv_x2

3

A

2mr v

r

8

v_x

118

A

2k3 v_rv_x2

3

A

2f3mr

v_r

8

v_x1f₁f₃v_hv_x2

2 3

A

2f3

8

ms v_r

8

v_x

8

2

3

A

2f3mx v_rv_x

8

D

,

M₄₅₅0, M₄₆₅₂ 1

12~f3mrvx1f3

8

msvx

8

!,

M₄₇₅₂ 1

12~f3

8

mrv_r

8

1f₃m_xv_r

8

!, M₄₈50,

M₅₅₅ 1

18

A

2v_h

8

~f₁

8

f₃

8

v

r

8

2_v

s₂

8

23f1mhv_rv_x13f 1

8

mxv

r

8

v_x

218k₁

8

v_r

8

v_x

8

13f 1

8

mrv_rv_x

8

1f 1

8

f₃

8

v_s

2

8

v_x

8

2

13f1

8

mrv_rv_x

8

1f₁

8

f₃

8

v_s

2

8

v_x

8

2!,

M₅₆₅ f1f3

18

A

2~mx v_x2k

1

8

v

x

8

!,

M₅₇₅ 1

6

A

2~26k1

8

v

r

8

1f₁

8

mrv_r!,

M₅₈₅ f1

8

f3

8

18

A

2~

v_r

8

22v_x

8

2!,

M₆₆₅ 1

12

A

2v_r

8

S

2f1mrv_hv_x2

A

2f₃m_rv_s

2vx

12f₁

8

mxv_h

8

v_x2

A

2f 3

8

mxv

s₂

8

v_x212k 1

8

v_h

8

v

x

8

2 12

A

2k3

8

v_s

2

8

v_x

8

22f 1

8

mhv_hv_x

8

2

A

2m_sv_s

2vx

8

D

,

M₆₇₅ 1

12

A

2

S

212k1

8

v_h

8

2

12

A

2k3

8

v_s

2

8

22f₁

8

mhv_h

2

A

2f3

8

msv_s

2

D

,

M₆₈₅1

12~2f3

8

mxvx16k3

8

vx

8

!,

M₇₇₅ 1

12

A

2v_x

8

S

212k₃

8

v_h

8

v

r

8

2 12

A

2k3

8

v

r

8

v

s₂

8

22f₁

8

mhv_hv

r

8

12f₁

8

mrv_h

8

v_r2

A

2f 3

8

mrv_rv

s₂

8

D

,

M₇₈₅ 1

12~6k3

8

vr

8

2f3

8

mrvr!,

M₈₈₅ 1

18

A

2v_s

2

8

S

f1

8

f3

8

v_h

8

v

r

8

2

2 3

A

2ms v_rv_x2

3

A

2mx v

r

8

v_x

2 18

A

2k3

8

v_r

8

v_x

8

2

3

A

2f3

8

mr v_rv_x

8

1f

1

8

f₃

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 scalarA1as a function ofv_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

th v_h5

g2 12~2vh

2

22v_h

8

22v2_r1v_r

8

22v_s

2 2

1v_s

2

8

2

2v_x21v_x

8

2!1m_h2

11 4mh

2 1 f1

2

18~vr 2

1v_x2!1 f1f3

18

A

2 v_s

2

v_h ~ v_r21v_x2!

1k1

A

2 v_rv_x

v_h 1

1 6

A

2v_h

~f₁mrv_r

8

v_x1f₁m_xv_rv_x

8

1f₁

8

mhv_r

8

v_x

8

!,

t_r

vr5

g2 12

S

2vh

2

1v_h

8

2₁2v

r

2 22v

r

8

2

11₂v

s₂

2 21₂v

s₂

8

2

2v

x

2

1v_x

8

2

D

1 g

₈

2

2 ~vr 2

2v_r

8

22v_x21v_x

8

2!1 f₁2 18~vh

2 1v_x2!

1 f3 2

36~vs2 2

12v_x2!2 f1f3

9

A

2 v_hv_s

21mr 2

11 4mr

2

1 k1

A

2 v_hv_x

v_r 1

k3 2

v_s

2vx

v_r 1

f1

6

A

2v_r

~m_hv_h

8

v_x1m_xv_hv_x

8

!

1 f3 12v_r~ms

v_s

2

8

v_x1m_xv_s

2vx

8

!1

f₃

8

12v_rmr v_s

2