Minimal scalar sector of 3-3-1 models without exotic electric charges
William A. Ponce and Yithsbey Giraldo
Instituto de Fı
´
sica, Universidad de Antioquia, A.A. 1226, Medellı
´
n, Colombia
Luis A. Sa
´
nchez
Instituto de Fı
´
sica, Universidad de Antioquia, A.A. 1226, Medellı
´
n, Colombia
and Escuela de Fı
´
sica, Universidad Nacional de Colombia, A.A. 3840, Medellı
´
n, Colombia
共Received 2 September 2002; published 4 April 2003兲
We study the minimal set of Higgs scalars, for models based on the local gauge group
SU(3)
c
丢 SU(3)
L
丢 U(1)
X
, which do not contain particles with exotic electric charges. We show that only two
Higgs SU(3)
L
triplets are needed in order to properly break the symmetry. The exact tree-level scalar mass
matrices resulting from symmetry breaking are calculated at the minimum of the most general scalar potential,
and the gauge bosons are obtained, together with their couplings to the physical scalar fields. We show how the
scalar sector introduced is enough to produce masses for fermions in a particular model. By using experimental
results we constrain the scale of new physics to be above 1.5 TeV.
DOI: 10.1103/PhysRevD.67.075001 PACS number共s兲: 12.60.Cn, 14.80.Cp
I. INTRODUCTION
The standard model 共SM兲 based on the local gauge group
SU(3)
c
丢 SU(2)
L
丢 U(1)
Y
关1兴 can be extended in several
different ways: first, by adding new fermion fields 共adding a
right-handed neutrino field constitutes its simplest extension
and has profound consequences, such as, for example, the
implementation of the seesaw mechanism and the enlarging
of the possible number of local gauge Abelian symmetries
that can be gauged simultaneously兲; second, by augmenting
the scalar sector to more than one Higgs representation; and
third, by enlarging the local gauge group. In this last direc-
tion SU(3)
L
丢 U(1)
X
as a flavor group has been studied pre-
viously in the literature 关2–10兴 by many authors who have
explored possible fermion and Higgs-boson representation
assignments. From now on, models based on the local gauge
group SU(3)
c
丢 SU(3)
L
丢 U(1)
X
are going to be called
3-3-1 models.
There are in the literature several 3-3-1 models; the most
popular one, the Pleitez-Frampton model 关2兴, is far from be-
ing the simplest construction. Not only is its scalar sector
quite complicated and messy 共three triplets and one sextet
关3兴兲, but its physical spectrum is plagued with particles with
exotic electric charges; namely, double charged gauge and
Higgs bosons which induce tree-level flavor changing neutral
currents 共FCNC兲 in the lepton sector 关4兴 and exotic quarks
with electric charges 5/3 and ⫺ 4/3. Other 3-3-1 models in
the literature are just introduced or merely sketched in a few
papers 关5–8兴, with a detailed phenomenological analysis of
them still lacking. In particular, there are no published papers
related to the study of the scalar sector for models 关5–8兴.
All possible 3-3-1 models without exotic electric charges
are presented in Ref. 关9兴 and summarized in the next section.
As shown, there are just a few anomaly-free models for one
or three families which all share in common the same gauge-
boson content and, as we are going to see in this paper, they
also may share a common scalar sector 共which does not con-
tain scalars with exotic electric charges either兲.
This paper is organized as follows. In Sec. II we review
all possible 3-3-1 models without exotic electric charges for
one and three families. In Sec. III we study the common
scalar sector for all those models, including the analysis of
its mass spectrum. In Sec. IV we analyze the gauge boson
structure common to all the models considered. In Sec. V we
present the couplings between the neutral scalar fields in the
models and the SM gauge bosons, and in Sec. VI we make
general remarks and present the conclusions. An Appendix at
the end shows how the Higgs scalars that are used to break
the symmetry can also be used to produce a consistent mass
spectrum for the fermion fields in the particular model which
is an E
6
subgroup 关7兴.
II. A REVIEW OF THE MODEL
Let us start with a summary of Refs. 关9兴 and 关10兴.
First we assume that the electroweak group is
SU(3)
L
丢 U(1)
X
傻 SU(2)
L
丢 U(1)
Y
and that the left-handed
quarks and left-handed leptons transform as the two funda-
mental representations of SU(3)
L
共the 3 and 3
*
). The right-
handed fields are singlets under SU(3)
L
, and SU(3)
c
is vec-
torlike as in the SM.
The most general electric charge operator in
SU(3)
L
丢 U(1)
X
is a linear combination of the three diago-
nal generators of the gauge group
Q⫽ aT
3L
⫹
2
冑
3
bT
8L
⫹ XI
3
, 共1兲
where T
iL
⫽
iL
/2,
iL
is the Gell-Mann matrices for
SU(3)
L
normalized as usual, I
3
⫽ Dg(1,1,1) is the diagonal
3⫻ 3 unit matrix, a⫽ 1 gives the usual isospin of the elec-
troweak interactions, and b is a free parameter. The X values
are fixed by anomaly cancellation 关9,10兴 and an eventual
coefficient for XI
3
can be absorbed in the hypercharge defi-
nition.
There are a total of 17 gauge bosons in the gauge group
under consideration. They are one gauge field B
associated
with U(1)
X
, the 8 gluon fields associated with SU(3)
c
PHYSICAL REVIEW D 67, 075001 共2003兲
0556-2821/2003/67共7兲/075001共10兲/$20.00 ©2003 The American Physical Society67 075001-1
which remain massless after breaking the symmetry, and an-
other 8 gauge fields associated with SU(3)
L
. We may write
the latter in the following way:
1
2
␣
L
A
␣
⫽
1
冑
2
冉
D
1
0
W
⫹
K
(1/2⫹ b)
W
⫺
D
2
0
K
⫺ (1/2⫺b)
K
⫺ (1/2⫹b)
K
¯
1/2⫺ b
D
3
0
冊
,
where D
1
0
⫽ A
3
/
冑
2⫹ A
8
/
冑
6, D
2
0
⫽⫺A
3
/
冑
2⫹ A
8
/
冑
6, and D
3
0
⫽⫺2A
8
/
冑
6. The upper indices of the gauge
bosons in the former expression stand for the electric charge
of the corresponding particle, some of them are functions of
the b parameter, as they should be 关10兴. Notice that the gauge
bosons have integer electric charges only for b⫽⫾n/2, n
⫽ 1,3,5,7,..., and as shown in Refs. 关9,10兴, a negative
value for b can be related to a positive value. So b⫽ 1/2
avoids exotic electric charges in the gauge sector of the
theory.
For the 3-3-1 models both, SU(3)
L
and U(1)
X
are
anomalous
关
SU(3)
c
is vectorlike兴. So, special combinations
of multiplets must be used in each particular model in order
to cancel the several possible anomalies, and end with physi-
cal acceptable models. The triangle anomalies we must take
care of are as follows:
关
SU(3)
L
兴
3
,
关
SU(3)
c
兴
2
U(1)
X
,
关
SU(3)
L
兴
2
U(1)
X
,
关
grav
兴
2
U(1)
X
共the gravitational
anomaly兲, and
关
U(1)
X
兴
3
.
In order to present specific examples let us see how the
charge operator in Eq. 共1兲 acts on the representations 3 and
3
*
of SU(3)
L
:
Q
关
3
兴
⫽ Dg
冉
1
2
⫹
b
3
⫹ X,⫺
1
2
⫹
b
3
⫹ X,⫺
2b
3
⫹ X
冊
,
Q
关
3
*
兴
⫽ Dg
冉
⫺
1
2
⫺
b
3
⫹ X,
1
2
⫺
b
3
⫹ X,
2b
3
⫹ X
冊
.
Notice from these expressions that if we accommodate the
known left-handed quark and lepton isodoublets in the two
upper components of 3 and 3
*
共or 3
*
and 3兲, and forbid the
presence of exotic electric charges in the possible models,
then b⫽⫾1/2 is mandatory.
To see this, let us take b⫽ 3/2, then Q
关
3
兴
⫽ Dg(1
⫹ X,X,X⫺ 1) and Q
关
3
*
兴
⫽ Dg(X⫺ 1,X,1⫹ X). Then the fol-
lowing multiplets are associated with the respective
关
SU(3)
c
,SU(3)
L
,U(1)
X
兴
quantum numbers: (e
⫺
,
e
,e
⫹
)
L
T
⬃(1,3
*
,0), (u,d,j)
L
T
⬃(3,3,⫺ 1/3), and (d,u,k)
L
T
⬃(3,3
*
,2/3), where j and k are isosinglet exotic quarks of
electric charges ⫺ 4/3 and 5/3, respectively. This multiplet
structure is the basis of the Pleitez-Frampton model 关2兴 for
which the anomaly-free arrangement for the three families is
given by
L
a
⫽
共
e
a
,
a
,e
ca
兲
L
T
⬃
共
1,3
*
,0
兲
,
q
L
i
⫽
共
u
i
,d
i
,j
i
兲
L
T
⬃
共
3,3,⫺ 1/3
兲
,
q
L
1
⫽
共
d
1
,u
1
,k
兲
L
T
⬃
共
3,3
*
,2/3
兲
,
u
L
ca
⬃
共
3
*
,1,⫺ 2/3
兲
, d
L
ca
⬃
共
3
*
,1,1/3
兲
,
k
L
a
⬃
共
3
*
,1,⫺ 5/3
兲
, j
L
ci
⬃
共
3
*
,1,⫺ 4/3
兲
,
where the upper c symbol stands for charge conjugation, a
⫽ 1,2,3 is a family index, and i⫽ 2,3 is related to two of the
three families.
Now, for a⫽ 1 and b⫽ 1/2 in Eq. 共1兲 let us introduce the
following closed sets of fermions 共closed in the sense that
each one includes the antiparticles of the charged particles兲:
S
1
⫽
关
(
␣
,
␣
⫺
,E
␣
⫺
);
␣
⫹
;E
␣
⫹
兴
with quantum numbers
关
(1,3,⫺ 2/3); (1,1,1); (1,1,1)].
S
2
⫽
关
(
␣
⫺
,
␣
,N
␣
0
);
␣
⫹
兴
with quantum numbers
关
(1,3
*
,
⫺ 1/3); (1,1,1)].
S
3
⫽
关
(d,u,U);d
c
;u
c
;U
c
兴
with quantum numbers
(3,3
*
,1/3); (3
*
,1,1/3); (3
*
,1,⫺ 2/3); and (3
*
,1,⫺ 2/3), re-
spectively.
S
4
⫽
关
(u,d,D);u
c
;d
c
;D
c
兴
with quantum numbers (3,3,0);
(3
*
,1,⫺ 2/3); (3
*
,1,1/3); and (3
*
,1,1/3), respectively.
S
5
⫽
关
(e
⫺
,
e
,N
1
0
);(E
⫺
,N
2
0
,N
3
0
);(N
4
0
,E
⫹
,e
⫹
)
兴
with quan-
tum numbers (1,3
*
,⫺ 1/3); (1,3
*
,⫺ 1/3); and (1,3
*
,2/3),
respectively.
S
6
⫽
关
(
e
,e
⫺
,E
1
⫺
);(E
2
⫹
,N
1
0
,N
2
0
);(N
3
0
,E
2
⫺
,E
3
⫺
);e
⫹
; E
1
⫹
;
E
3
⫹
] with quantum numbers
关
(1,3,⫺ 2/3); (1,3,1/3); (1,3,
⫺ 2/3); (1,1,1); (1,1,1); (1,1,1)].
The quantum numbers in parentheses refer to
关
SU(3)
c
,SU(3)
L
,U(1)
X
兴
representations. The several
anomalies for the former six sets are presented in Table I,
which in turn allows us to build anomaly-free models for one
or more families.
A. One family models
There are just two anomaly-free one family structures that
can be extracted from Table I. They are as follows:
Model A:(S
4
⫹ S
5
). This model is associated with an E
6
subgroup and has been partially analyzed in Ref. 关7兴共see
also the Appendix at the end of this paper兲.
Model B:(S
3
⫹ S
6
). This model is associated with an
SU(6)
L
丢 U(1)
X
subgroup and has been partially analyzed in
Ref. 关8兴.
B. Three family models
Model C:(3S
2
⫹ S
3
⫹ 2S
4
). This model deals with the
following multiplets associated with the given quantum num-
bers: (u,d,D)
L
T
⬃(3,3,0), (e
⫺
,
e
,N
0
)
L
T
⬃(1,3
*
,⫺ 1/3), and
(d,u,U)
L
T
⬃(3,3
*
,1/3), where D and U are exotic quarks
TABLE I. Anomalies for S
i
.
Anomalies S
1
S
2
S
3
S
4
S
5
S
6
关
SU(3)
c
兴
2
U(1)
X
000000
关
SU(3)
L
兴
2
U(1)
X
⫺ 2/3 ⫺ 1/3 1 0 0 ⫺ 1
关
gra
v 兴
2
U(1)
X
000000
关
U(1)
X
兴
3
10/9 8/9 ⫺ 12/9 ⫺ 6/9 6/9 12/9
关
SU(3)
L
兴
3
1 ⫺ 1 ⫺ 33⫺ 33
PONCE, GIRALDO, AND SA
´
NCHEZ PHYSICAL REVIEW D 67, 075001 共2003兲
075001-2
with electric charges ⫺ 1/3 and 2/3, respectively. With such a
gauge structure the three family anomaly-free model is given
by
⬘
L
a
⫽
共
e
⫺ a
,
a
,N
0a
兲
L
T
⬃
共
1,3
*
,⫺ 1/3
兲
,
e
L
⫹ a
⬃
共
1,1,1
兲
,
q
L
⬘
i
⫽
共
u
i
,d
i
,D
i
兲
L
T
⬃
共
3,3,0
兲
,
q
L
⬘
1
⫽
共
d
1
,u
1
,U
兲
L
T
⬃
共
3,3
*
,1/3
兲
,
u
L
ca
⬃
共
3
*
,1,⫺ 2/3
兲
, d
L
ca
⬃
共
3
*
,1,1/3
兲
,
U
L
c
⬃
共
3
*
,1,⫺ 2/3
兲
, D
L
ci
⬃
共
3
*
,1,1/3
兲
,
where a⫽ 1,2,3 and i⫽ 1,2 as above. This model has been
analyzed in the literature in Ref. 关5兴. If needed, the fermion
sector can be augmented with an undetermined number of
neutral Weyl states N
L
0j
⬃(1,1,0), j⫽ 1,2,... without vio-
lating the anomaly cancellation.
Model D:(3S
1
⫹ 2S
3
⫹ S
4
). It makes use of the same mul-
tiplets used in the previous model arranged in a different
way, plus a new lepton multiplet (
e
,e
⫺
,E
⫺
)
L
T
⬃(1,3,
⫺ 2/3). The family structure of this new anomaly-free model
is given by
L
⬙
a
⫽
共
a
,e
a
,E
a
兲
L
T
⬃
共
1,3,⫺ 2/3
兲
,
e
L
ca
⬃
共
1,1,1
兲
, E
L
ca
⬃
共
1,1,1
兲
,
q
L
⬙
1
⫽
共
u
1
,d
1
,D
兲
L
T
⬃
共
3,3,0
兲
,
q
L
⬙
i
⫽
共
d
i
,u
i
,U
i
兲
L
T
⬃
共
3,3
*
,1/3
兲
,
u
L
ca
⬃
共
3
*
,1,⫺ 2/3
兲
, d
L
ca
⬃
共
3
*
,1,1/3
兲
,
D
L
c
⬃
共
3
*
,1,1/3
兲
, U
L
ci
⬃
共
3
*
,1,2/3
兲
.
This model has been analyzed in Ref. 关6兴.
Model E:(S
1
⫹ S
2
⫹ S
3
⫹ 2S
4
⫹ S
5
).
Model F:(S
1
⫹ S
2
⫹ 2S
3
⫹ S
4
⫹ S
6
).
Besides the former four three family models, another four,
carbon copies of the two one family models A and B can also
be constructed. They are as follows:
Model G:3(S
4
⫹ S
5
).
Model H:3(S
3
⫹ S
6
).
Model I:2(S
4
⫹ S
5
)⫹ (S
3
⫹ S
6
).
Model J:(S
4
⫹ S
5
)⫹ 2(S
3
⫹ S
6
).
There are a total of eight different three-family models,
each one with a different fermion field content. Notice in
particular that in models E and F each one of the three fami-
lies is treated differently. As far as we know the last six
models have not yet been studied in the literature.
III. THE SCALAR SECTOR
If we pretend to use the simplest SU(3)
L
representations
in order to break the symmetry, at least two complex scalar
triplets, equivalent to 12 real scalar fields, are required. For
b⫽ 1/2 there are just two Higgs scalars 共together with their
complex conjugates兲 which may develop nonzero vacuum
expectation values 共VEV兲; they are
1
(1,3
*
,⫺ 1/3)
T
⫽ (
1
⫺
,
1
⬘
0
,
1
0
) with VEV
具
1
典
T
⫽ (0,
v
1
,V) and
2
(1,3
*
,2/3)
T
⫽ (
2
0
,
2
⫹
,
2
⬘
⫹
) with VEV
具
2
典
T
⫽ (
v
2
,0,0). As we will see ahead, to reach consistency with
phenomenology we must have the hierarchy V⬎
v
1
⬃
v
2
.
Our aim is to break the symmetry in one single step:
SU
共
3
兲
c
丢 SU
共
3
兲
L
丢 U
共
1
兲
X
→SU
共
3
兲
c
丢 U
共
1
兲
Q
,
which implies the existence of eight Goldstone bosons in-
cluded in the scalar sector of the theory 关11兴. For the sake of
simplicity we assume that the VEV are real. This means that
the CP violation through the scalar exchange is not consid-
ered in this work. Now, for convenience in reading we re-
write the expansion of the scalar fields which acquire VEV
as
1
0
⫽ V⫹
H
1
0
⫹ iA
1
0
冑
2
,
1
⬘
0
⫽
v
1
⫹
H
1
⬘
0
⫹ iA
1
⬘
0
冑
2
, 共2兲
2
0
⫽
v
2
⫹
H
2
0
⫹ iA
2
0
冑
2
.
In the literature, a real part H is called a CP-even scalar and
an imaginary part A a CP-odd scalar or pseudoscalar field.
Now, the most general potential which includes
1
and
2
can then be written in the following form:
V
共
1
,
2
兲
⫽
1
2
1
†
1
⫹
2
2
2
†
2
⫹
1
共
1
†
1
兲
2
⫹
2
共
2
†
2
兲
2
⫹
3
共
1
†
1
兲
共
2
†
2
兲
⫹
4
共
1
†
2
兲
共
2
†
1
兲
. 共3兲
Requiring that in the shifted potential V(
1
,
2
), the linear
terms in fields must be absent, we get in the tree-level ap-
proximation the following constraint equations:
1
2
⫹ 2
1
共v
1
2
⫹ V
2
兲
⫹
3
v
2
2
⫽ 0,
共4兲
2
2
⫹
3
共v
1
2
⫹ V
2
兲
⫹ 2
2
v
2
2
⫽ 0.
The analysis to the former equations shows that they are
related to a minimum of the scalar potential with the value
V
min
⫽⫺
v
2
4
2
⫺
共v
1
2
⫹ V
2
兲
关共v
1
2
⫹ V
2
兲
1
⫹
v
2
2
3
兴
⫽ V
共v
1
,
v
2
,V
兲
,
where V(
v
1
⫽ 0,
v
2
,V)⬎ V(
v
1
⫽” 0,
v
2
,V), implying that
v
1
⫽0 is preferred. Substituting Eqs. 共2兲 and 共4兲 in Eq. 共3兲 we
get the following mass matrices.
MINIMAL SCALAR SECTOR OF 3-3-1 MODELS... PHYSICAL REVIEW D 67, 075001 共2003兲
075001-3
A. Spectrum in the scalar neutral sector
In the (H
1
0
,H
2
0
,H
1
⬘
0
) basis, the square mass matrix can be calculated using M
ij
2
⫽ 2
关
2
V(
1
2
)/
H
i
0
H
j
0
兴
. After im-
posing the constraints in Eq. 共4兲 we get
M
H
2
⫽ 2
冉
2
1
V
2
3
v
2
V 2
1
v
1
V
3
v
2
V 2
2
v
2
2
3
v
1
v
2
2
1
v
1
V
3
v
1
v
2
2
1
v
1
2
冊
, 共5兲
which has zero determinant, providing us with a Goldstone boson G
1
and two physical massive neutral scalar fields H
1
and H
2
with masses
M
H
1
,H
2
2
⫽ 2
共v
1
2
⫹ V
2
兲
1
⫹ 2
v
2
2
2
⫾ 2
冑
关共v
1
2
⫹ V
2
兲
1
⫹
v
2
2
2
兴
2
⫹
v
2
2
共v
1
2
⫹ V
2
兲
共
3
2
⫺ 4
1
2
兲
,
where real lambdas produce positive masses for the scalars only if
1
⬎ 0 and 4
1
2
⬎
3
2
共which implies
2
⬎ 0).
We may see from the former equations that in the limit V⬎
v
1
⬃
v
2
, and for lambdas of order one, there is a neutral Higgs
scalar with a mass of order V and another one with a mass of the order of
v
1
⬃
v
2
, which may be identified with the SM scalar
as we will see ahead.
The physical fields are related to the scalars in the weak basis by the lineal transformation:
冉
H
1
0
H
2
0
H
1
⬘
0
冊
⫽
冉
v
2
V
S
1
v
2
V
S
2
⫺
v
1
冑
v
1
2
⫹ V
2
M
H
1
2
⫺ 4
共v
1
2
⫹ V
2
兲
1
2S
1
3
⫺
共
M
H
1
2
⫺ 4
v
2
2
2
兲
2S
2
3
0
v
1
v
2
S
1
v
1
v
2
S
2
V
冑
v
1
2
⫹ V
2
冊
冉
H
1
H
2
G
1
冊
, 共6兲
where we have defined
S
1
⫽
冑
v
2
2
共v
1
2
⫹ V
2
兲
⫹
关
M
H
1
2
⫺ 4
共v
1
2
⫹ V
2
兲
1
兴
2
/4
3
2
,
S
2
⫽
冑
v
2
2
共v
1
2
⫹ V
2
兲
⫹
共
M
H
1
2
⫺ 4
v
2
2
2
兲
2
/4
3
2
.
B. Spectrum in the pseudoscalar neutral sector
The analysis shows that V(
1
,
2
) in Eq. 共3兲, when ex-
panded around the most general vacuum given by Eqs. 共2兲
and using the constraints in Eq. 共4兲, does not contain pseu-
doscalar fields A
i
0
. This allows us to identify another three
Goldstone bosons G
2
⫽ A
1
0
, G
3
⫽ A
2
0
, and G
4
⫽ A
⬘
1
0
.
C. Spectrum in the charged scalar sector
In the basis (
1
⫹
,
2
⫹
,
2
⬘
⫹
) the square mass matrix is
given by
M
⫹
2
⫽ 2
4
冉
v
2
2
v
1
v
2
v
2
V
v
1
v
2
v
1
2
v
1
V
v
2
V
v
1
VV
2
冊
, 共7兲
which has two eigenvalues equal to zero equivalent to four
Goldstone bosons (G
5
⫾
,G
6
⫾
) and two physical charged Higgs
scalars with large masses given by
4
(
v
1
2
⫹
v
2
2
⫹ V
2
), with
the new constraint
4
⬎ 0.
Our analysis shows that after symmetry breaking, the
original 12 degrees of freedom in the scalar sector have be-
come eight Goldstone bosons 共four electrically neutral and
four charged兲, and four physical scalar fields, two neutrals
共one of them the SM Higgs scalar兲 and two charged ones.
The eight Goldstone bosons must be swallowed up by eight
gauge fields as we will see in the next section.
IV. THE GAUGE BOSON SECTOR
For b⫽ 1/2 the nine gauge bosons in SU(3)
L
丢 U(1)
X
,
when acting on left-handed triplets, can be arranged in the
following convenient way:
A
⫽
1
2
g
␣
L
A
␣
⫹ g
⬘
XB
I
3
⫽
g
冑
2
冉
Y
1
0
W
⫹
K
⫹
W
⫺
Y
2
0
K
0
K
⫺
K
¯
0
Y
3
0
冊
,
where Y
1
0
⫽ A
3
/
冑
2⫹ A
8
/
冑
6⫹
冑
2(g
⬘
/g)XB
, Y
2
0
⫽
⫺ A
3
/
冑
2⫹ A
8
/
冑
6⫹
冑
2(g
⬘
/g)XB
, and Y
3
0
⫽⫺2A
8
/
PONCE, GIRALDO, AND SA
´
NCHEZ PHYSICAL REVIEW D 67, 075001 共2003兲
075001-4
冑
6⫹
冑
2(g
⬘
/g)XB
. X is the hypercharge value of the given
left-handed triplet, and g and g
⬘
are the gauge coupling con-
stants for SU(3)
L
and U(1)
X
, respectively. After breaking
the symmetry with
具
i
典
, i⫽ 1,2, and using for the covariant
derivative for triplets D
⫽
⫺ iA
, we get the following
mass terms in the gauge boson sector.
A. Spectrum in the charged gauge boson sector
In the basis (K
⫾
,W
⫾
) the square mass matrix produced is
M
⫾
2
⫽
g
2
2
冉
共
V
2
⫹
v
2
2
兲
v
1
V
v
1
V
共v
1
2
⫹
v
2
2
兲
冊
. 共8兲
The former symmetric matrix give us the masses M
W
⬘
2
⫽ g
2
v
2
2
/2 and M
K
⬘
2
⫽ g
2
(
v
1
2
⫹
v
2
2
⫹ V
2
)/2, related to the physi-
cal fields W
⬘
⫽
(
v
1
K
⫺ VW
), and K
⬘
⫽
(VK
⫹
v
1
W
) associated with the known charged weak current
W
⬘
⫾
, and with a new one K
⬘
⫾
predicted in the context of
this model (
⫺ 2
⫽
v
1
2
⫹ V
2
is a normalization factor兲. From
the experimental value M
W
⬘
⫽ 80.419⫾ 0.056 GeV 关12兴 we
obtain
v
2
⯝174 GeV as in the SM.
B. Spectrum in the neutral gauge boson sector
For the five electrically neutral gauge bosons we get first
that the imaginary part of K
0
⫽ (K
R
0
⫹ iK
I
0
)/
冑
2 decouples
from the other four electrically neutral gauge bosons, acquir-
ing a mass M
K
I
0
2
⫽ g
2
(
v
1
2
⫹ V
2
)/2. Then, in the basis
(B
,A
3
,A
8
,K
R
0
), the following squared mass matrix is ob-
tained:
M
0
2
⫽
冢
g
⬘
2
9
共v
1
2
⫹ V
2
⫹ 4
v
2
2
兲
⫺
gg
⬘
6
共v
1
2
⫹ 2
v
2
2
兲
⫺
gg
⬘
3
冑
3
共
V
2
⫹
v
2
2
⫺
v
1
2
/2
兲
gg
⬘
v
1
V/3
⫺
gg
⬘
6
共v
1
2
⫹ 2
v
2
2
兲
g
2
共v
1
2
⫹
v
2
2
兲
/4
g
2
4
冑
3
共v
2
2
⫺
v
1
2
兲
⫺ g
2
v
1
V/4
⫺
gg
⬘
3
冑
3
共
V
2
⫹
v
2
2
⫺
v
1
2
/2
兲
g
2
4
冑
3
共v
2
2
⫺
v
1
2
兲
g
2
12
共v
1
2
⫹
v
2
2
⫹ 4V
2
兲
⫺ g
2
v
1
V/
共
4
冑
3
兲
gg
⬘
v
1
V/3 ⫺ g
2
v
1
V/4
⫺ g
2
v
1
V/
共
4
冑
3
兲
g
2
共v
1
2
⫹ V
2
兲
/4
冣
. 共9兲
This matrix has a determinant equal to zero, which implies
that there is a zero eigenvalue associated with the photon
field with the eigenvector
A
⫽ S
W
A
3
⫹ C
W
冋
T
W
冑
3
A
8
⫹
共
1⫺ T
W
2
/3
兲
1/2
B
册
, 共10兲
where S
W
⫽
冑
3g
⬘
/
冑
3g
2
⫹ 4g
⬘
2
and C
W
are the sine and co-
sine of the electroweak mixing angle (T
W
⫽ S
W
/C
W
). Or-
thogonal to the photon field A
we may define another two
fields
Z
⫽ C
W
A
3
⫺ S
W
冋
T
W
冑
3
A
8
⫹
共
1⫺ T
W
2
/3
兲
1/2
B
册
,
Z
⬘
⫽⫺
共
1⫺ T
W
2
/3
兲
1/2
A
8
⫹
T
W
冑
3
B
, 共11兲
where Z
corresponds to the neutral current of the SM and
Z
⬘
is a new weak neutral current predicted for these mod-
els.
We may also identify the gauge boson Y
associated with
the SM hypercharge in U(1)
Y
as
Y
⫽
冋
T
W
冑
3
A
8
⫹
共
1⫺ T
W
2
/3
兲
1/2
B
册
.
In the basis (Z
⬘
,Z
,K
R
0
) the mass matrix for the neutral
sector reduces to
g
2
4C
W
2
冉
␦
2
共v
1
2
C
2W
2
⫹
v
2
2
⫹ 4V
2
C
W
4
兲
␦
共v
1
2
C
2W
⫺
v
2
2
兲
␦
C
W
v
1
V
␦
共v
1
2
C
2W
⫺
v
2
2
兲
v
1
2
⫹
v
2
2
⫺ C
W
v
1
V
␦
C
W
v
1
V ⫺ C
W
v
1
VC
W
2
共v
1
2
⫹ V
2
兲
冊
, 共12兲
MINIMAL SCALAR SECTOR OF 3-3-1 MODELS... PHYSICAL REVIEW D 67, 075001 共2003兲
075001-5
