PHYSICAL REVIEW D VOLUME 46, NUMBER 1 1 JULY 1992

SU(3)U(1) model for electroweak interactions

F. Pisano and V. Pleitez
Instituto de Fisica Tedrica, Unioersidade Estadual Paulista, Rua Pamplona, 1$$ CEP 01/05 Sa—o Paulo, Brazil

(Received 18 February 1992)

We consider a gauge model based on a SU(3)U(1) symmetry in which the lepton number is
violated explicitly by charged scalar and gauge bosons, including a vector field with double electric
charge.

PACS number(s): 12.15.Cc, 14.80.—j

I. INTRODUCTION

Some years ago, it was pointed out that processes such
as e e ~ W V in Fig. 1(a), if induced by right-
handed currents coupled to the vector V, imply vio-
lation of unitarity at high energies. Then, if the right-
handed currents are part of a gauge theory, it has been
argued that at least some neutrinos must have a nonzero
mass [1].

The argument to justify this follows exactly the same
way as in the usual electroweak theory for the process
vv ~ 8'+O' . The graph induced by an electron ex-
change has bad high-energy behavior; when the energy
goes to infinity, the respective amplitude violates unitar-
ity [2].

In Fig. 1(a) the lower vertex indicates a right-handed

w-

current which absorbs the right-handed antineutrino
coming from the upper vertex, which represents the left-
handed current of the electroweak standard model. The
part of the amplitude, corresponding to Fig. 1(a), in
which we are interested is

where U~ (U+) is the mixing matrix in the left- (right)-
handed current, and q is the four-momentum transfer [1].
The space-time structure of Eq. (1) is the same as the
charged-lepton exchange amplitude in the process vv ~
W+W [2]. Then, we must have the same bad high-
eaergy behavior of the last process. One way to avoid
this is to have a cancellation among the contributions
from the various v exchanges when we add them up;
at high energy and large q~, the latter dominates the
denominator in Eq. (1), and if we require that

&m

(2)

(a}

FIG. 1. (a) e e ~ W V process induced by right-
handed currents; L and 8 denote the handedness of the cur-
rent at the vertex, and q is the momentum transfer. (b) Dia-
gram for W H ~ e e vrith massive Majorana neutrinos;
both vertices are left-handed.

the amplitude in Eq. (1) vanishes even at low energies,
unless at least one of the masses M„ is nonzero. On the
other hand, the diagram in Fig. 1(a) or its time-reversed
one W W -+ e e appearing in Fig. 1(b), when both
vertices are left handed, proceeds via Majorana massive
neutrinos.

Here we are concerned with a gauge model based on a
SUI. (3)U~(1) symmetry. The original motivation lead-
ing to the st;udy of this model stemmed from the observa-
tion that a gauge theory must be consistent, that is, uni-
tary and renormalizable, independently of the values of
some parameters, such as mixing angles. Then, from this
point of view, instead of using the condition in Eq. (2) in
order to solve the problem of the graph in Fig. 1(a), we
prefer the introduction of a doubly charged gauge boson
w'hich, like the Z in the standard model, will restore the
good high-energy behavior.

Although there exist in the literature several models
based on a SU(3)U(1) gauge symmetry [3—7], our model
has a different representation content and a quite differ-
ent new physics at an, in principle, arbitrary mass scale.
The main new features of our model occur in processes
in which the initial electric charge is not zero. Even from

410 1992 The American Physical Society



46 SU(3) U(1) MODEL FOR ELECTROWEAK INTERACTIONS 411

the theoretical point of view, that sort of processes have
not been well studied; for instance, general results exist
only for zero initial charge [8].

The plan of this paper is as follows. Section II is
devoted to present the model. Some phenomenological
consequences are given in Sec. III. In this way we can
estimate the allowed value for the mass scale character-
izing the new physics. In Sec. IV we study briefly the
scalar potential and show that there is no mixing be-
tween the lepton-number-conserving and lepton-number-
violating scalar fields which could induce decays such as
the neutrinoless double P decay. The last section is de-
voted to our conclusions and some comments and in the
Appendix we give more details about the definition of the
charge-conjugation operation we have used in this work.

II. THE MODEL

As we said before, the gauge model that we shall con-
sider is one in which the gauge group is SUg(3)UN(1).
This is possibly the simplest way to enlarge the gauge
group SUg(2)Uy(1) in order to have doubly charged
gauge bosons, without losing the natural features of the
standard electroweak model. The price we must pay is
the introduction of exotic quarks, with electric charges
5/3 and -4/3.

In this model we have the processes appearing in
Figs. 2(a) and 2(b); the last diagram plays the same role
as the similar diagram with Zo in the standard model and
it restores the "safe" high-energy behavior of the model.
Both vector bosons V and U in Figs. 2(a) and 2(b)
are very massive, and their masses depend on the mass
scale of the breaking of the SUL, (3)U~(1) symmetry
into SUg(2)U~(1). Phenomenological bounds on this
mass scale will be given in the next section.

A. Yukawa interactions

We start by choosing the following triplet representa-
tions for the left-handed fields of the first family:

and

EL, = e (3, 0), Qgg = d (3, + ), (3)
&") g

"& (1~+s)~ "& ( ~ 3)~ » ( +3) (4)

for the respective right-handed fields. Notice that we

have not introduced right-handed neutrinos. The num-
bers 0, 2/3 in Eq. (3), and 2/3, —1/3, and 5/3 in Eq. (4)
are UN (1) charges. The electric charge operator has been
defined as

A3 — A8 + N, 5Q 1

2

where As and As are the usual Gell-Mann matrices; N is
proportional to the unit matrix. Then, the exotic quark
Jq has an electric charge +5/3.

The other two lepton generations also belong to triplet
representations:

f vp I (&r)
ML, = p, (3, 0), T'I, = ~ (» 0) (6)&").

The model is anomaly-free if we have equal number of
triplets and antitriplets, counting the color of SU(3)„
and furthermore requiring the sum of all fermion charges
to vanish. As in the model of Ref. [3], the anomaly can-
cellation occurs for the three generations together and
not generation by generation.

Then, we must introduce the antitriplets

(z, )
c (3', —s), Qsg ——

(s)
(&s'i

(3e 1)

YRi q

~R
(a}

(b}

(7)
also with the respective right-handed fields in singlets.
The quarks Jz and Js have both charge —4/3.

In order to generate fermion masses, we introduce the
following Higgs triplets, g, p, and y:

(~'l
g, (3,0), p (3, 1),

(~'i
(8)

x
(3, —1),

( x')
These Higgs triplets will produce the following hierarchi-
cal symmetry breaking:

SUI, (3)SU~(1) -. SUI.(2) g U~(1) '": U,~(1),

FIG. 2. Diagram for W V ~ e e due to the existence
of right-handed current (a} and doubly charged gauge boson

(b).
The Yukawa Lagrangian, without considering the mixed
terms between quarks, is



412 F. PISANO AND V. PI.EITEZ

—ZY ———) Gts' Qhgt, tlb+ Q1L(G„uRtl+ GsdRp+ Gzi JiRX)
t

+ (GcQ2LcR + GtQ3LtR) p + (GtQ2LsR + GbQ3L~R) 9 + (Gl2Q2L J2R + GJ3Q3L J3R) X + H C.

with t = e, p, r. Explicitly, we have, for the leptons,

(10)

2l IY =) Gl (IRIL IR—IL)tI —(vt'RIL IRv—&L)tl, + (vt'RIL IR—vtL)t12 + H.c.,
I

and, using the definition of charge conjugation g' = psCQ+ that we shall discuss in the Appendix, we can write
Eq. (11) as

Zt) = ) Gt( IRIL—g+ 'IRVLtli + vRILtI2 + H. c.).
l

In Eq. (12) there is lepton-number violation through the coupling with the t12+ Higgs scalar.
For the first and second quark generations we have the Yukawa interactions

~qY = Gtt(uLuRg + dLuRtli + J1LuR92 )
+ Gs(uLdRp+ + dLdRp + J1LdRp++)

+ Gc(J2LCRp + CLCRp + SLCRp )
+ Gt(J2LsRt12 + cLsRf/) + sLsRg )
+ Gjt (uL J1RX + dL J1RX + J1LJ1RX )
+ GJg(J2L J2RX + cL J2RX + sL J2RX ) + H C.

(12)

The Yukawa interactions for the third quark generation
are obtained from those of the second generation mak-
ing c -+ t, s ~ b, and J2 -+ Js. In Eq. (10), since the
neutrinos are massless there is no mixing between lep-
tons, so it is not necessary at all to consider terms such

the coupling constants h„~ = —h~„and H~'&l = d&btl .
The neutral component of the Higgs fields develops the

vacuum expectation value

I

Dirac neutrinos through their couplings with the g Higgs
triplet.

B. The gauge bosons

The gauge bosons of this theory consist of an octet W„'
associated with SUL(3) and a singlet B„associated with
Utv(1). The covariant derivatives are

(vv) 1 (0 )
(tl ) = 0 , (p ) = vp

~o) ko)
(14)

Dtttp; = Otttpt' + tg(Wtt A/2) p~ + tg N~ptBtt,

where N& denotes the N charge for the y Higgs multiplet,
tp = tv, p, X. Using Eqs. (14) in Eq. (16) we obtain the
symmetry-breaking pattern appearing in Eq. (9).

The gauge bosons ~2W+ = —(W —iW ), i/2V
—(W4 —iW5), and ~2U = —(Ws —iW ) have the
masses

So, the masses of the fermions are mt = Gt~~ for the

charged leptons and Miv: —g (v +v ), My ——g (v~+v ),

m„—G„, m, G, , mg-
V2 V& V2'

md
——Gg ~,

mg, = GJ, ~p, mg, = Gg, ~, mg, =Gg, ~
for the quarks. The exotic quarks obtain their masses
from the y triplet. Notice that, if we had intro-
duced right-handed neutrinos, we would have massive

Notice that even if vz
——vz v/~2, v being the usual

vacuum expectation value of the Higgs boson in the stan-
dard model, the e„must be large enough to keep the new

gauge bosons V+ and U++ suKciently heavy in order to
have consistency with low-energy phenomenology. On
the other hand, the neutral gauge bosons have the fol-
lowing mass matrix in the (W3, Ws, B) basis:



SU(3) U(1) MODEL FOR ELECIROWEAK INTERACTIONS 413

1' v„'+ v,'
M = —g ~(v„—vp)

].

4 3

~(v.' —v,')
(18)

and, since det M2 = 0, we must have a photon after
the symmetry breaking. If we had introduced a 6', the
matrix M~ in Eq. (18) would be such that det M2 g 0.
In fact, the eigenvalues of the matrix in Eq. (18) are

fying the electron charge as (see the Appendix)

gsin0 g'cos8

(1+3 sin 8) ~ (1+3 sin 8) ~
(23)

M~ ——0,
g2 g2 + 4gI2„(.„+,),g+ g

(19)

and the charged-current interactions are

) ~ vil, y" lI, W„+ + ll 7"vir. V„+

2 1 2Mz, -(g + 3g' )vz,
+II p"lr, U++ + H. c. ~. (24)

M' 1+4t'
Mi22, 1 + 3&s

' (20)

where t = g'/g = tan 8, and in order to obtain the usual
relation cos H~Mz2 ——M~, with cos 8~ 0.78, we must
have 8 54', i.e. , tan 8 11/6. Then, we can identify
Z as the neutral gauge boson of the standard model.

The neutral physical states are

(W„' —~SW„')t + B„,
(1+4~2):

where we have used vx » v~ „ for the case of Mz and
Mz~. Notice that the Z' boson has a mass proportional
to v& and, like the charged bosons V+, U++, must be
very massive. In the present model we have

Lg, w = —
I &n"der, ~„++Jiry"uL. ~„+

2&

+dsL, p" Jir. U + H c (25)

Notice that, as we have not assigned to the gauge bosons
a lepton number, we have explicit breakdown of this
quantum number induced by the V+, U++ gauge bosons.
A similar mechanism for lepton number violation was
proposed in Ref. [9] but in that reference the lepton-
number-violating currents are coupled to the standard
gauge bosons and they are proportional to a small pa-
rameter appearing in this model.

For the first generation of quarks we have the charged-
current interactions

Z„—,(1+3t ) ~ W„+,W„
(1+4g~) k " (1+3~2) k

Bp
t

(1+3t2)'

and, for the second generation of quarks we have

&q,g = —
l cl.V"dsI. W„—sel, v" J2yl. ~„

2 (
+cLy" J2pL, U + H. c. i. (26)

Concerning the vector bosons, we have the trilin-
ear interactions W+W N, V+V N, U++U N, and
S'+V+U, where N could be any of the neutral vector
bosons A, Z, or Z' .

C. Charged and neutral currents

The interactions among the gauge bosons and fermions
are read off from

L~ = Rip" (8„+ig'B„N)R

+Lip"
~
B„~ig'B„N+—A W„~ L, ('22)

Nc g Mz ~ 1 1
Vil T ViL, Z&— Zp2 Mw ( 3 gh(t) ") (27)

with h(t) = 1+4t2, for neutrinos and

The charge-changing interactions for the third genera-
tion of quarks are obtained from those of the second
generation, making c ~ t, s ~ 6, and J2 ~ Js.
We have mixing only in the Q= —si and Q= —s sec-
tors, then in Eqs. (25) and (26) ds, ss, and J2y mean
Cabibbo-Kobayashi-Maskawa states in the three- and
two-dimensional Qavor space d, 8, b and J2, J3 respec-
tively.

Similarly, we have the neutral currents coupled to both
Z and Z' massive vector bosons, according to the La-
grangian

where R represents any right-handed singlet and L any
left-handed triplet.

Let us consider first the leptons. For the charged lep-
tons, we have the electromagnetic interaction by identi-

Li ———— [lp" (vi + a&y )IZ„+ lp" (vi + a&p )IZ„'],

(28)



414 F. PISANO AND V. PLEITEZ 46

v~
——1/h(t), a~ = 1, (29a)

for the charged leptons, where we have used ll y"ll
—l~p" lR and defined

where i = u, c, t, d, s, b, J~, J2, J3, with

v = (3+4t )/3h(t), a = —1, (32a)

93/"(t) = " /3 (29b)
vD = —(3+8t')/3h(t), aD =1, (32b)

U 28
vsM ——1 ——sin 8~, asM ———1,U

3
(30)

The Lagrangian interaction among quarks and the Zo is

With t2 = 11/6, v~ and ai have the same values of the
standard model.

As it was said before, the quark representations in
Eqs. (3) and (7) are symmetry eigenstates; that is, they
are related to the mass eigenstates by Cabibbo-like an-
gles. As we have one triplet and two antitriplets, it
should be expected flavor-changing neutral currents ex-
ist. Notwithstanding, as we shall show below, when we
calculate the neutral currents explicitly, we find that all
of them, for the same charge sector, have equal factors
and the Glashow-Iliopoulos-Maiani (GIM) [2] cancella-
tion is automatic in neutral currents coupled to Z . Re-
member that, in the standard electroweak model, the
GIM mechanism is a consequence of having each charge
sector the same coupling with Z; for example, for the
charge +2/3 sector,

v
J' = —20t2/3h(t), a ' = 0, (32c)

v~~ = v~3 = 16t'/3h(t), a ' = a" = 0; (32d)

) [4;~"(v" + a"y')4, ]Z„',
4 Mgr

U and D mean the charge +2/3 and —1/3 respectively,
the same for Jq 2 q. Notice that, as was said above, there
is no flavor-changing neutral current coupled to the Z
field and the exotic quarks couple to Z only through
vector currents. It is easy to verify that for the Q = s, —

s
sectors the respective coefficients v and a also coincide
with those of the standard electroweak model if t2 =
11/6, as required to maintain the relation cos 8~Mz ——

Mgr .
The same cancellation does not happen with the corre-

sponding currents coupled to the Z' boson, each quark
having its respective coefficients. Explicitly, we have

where

v'" = —(1+8t')//3h(t), a'" = 1//3h(t), (34a)

v" = v" = (1 —2t )/+3h(t), a" = a" = —(1+6t )//3h(t), (34b)

v' = —(1+ 2t )//3h(t), a'" = —a", (34c)

v" = v' = gh(t)/3, a" = a' = —a'",

for the usual quarks, and

(34d)

1 —7t J 2 1+ 3$

~3 gh(t) ~3 gh(t)
' (35a)

IJ~ IJ3 1—
v = v a '=a '= —aIJ IJ IJ,

3 gh(t)

for the exotic quarks.

(35b)

III. THE SCALAR POTENTIAL

The most general gauge-invariant potential involving the three Higgs triplets is

V(n, u, X) = I ~ n'n+ S 21 't + I sX'X+ &i(n'n)'+»(u'u)'+ &s(&'&)'

+ (n'g) [&4n'u+ &5X'X] + &6(e' p)(X'X) + ).~*'"(fr t ~ Xk + H.c ).
sjk

(36)



46 SU(3) U(1) MODEL FOR ELECTROWEAK INTERACTIONS 415

The coupling f has dimension of mass. We can analyze
the scalar spectrum defining

R= I'(p ~ e v, v„)
r(&- ail)

(41)

g —vg + Hg + l1lg) p = v2+ H2+ ih2,

(37)

tests the nature of the lepton family number conserva-
tion, i.e. , additive vs multiplicative. Roughly we have

V3 + H3 + ih3,

where we have redefined v„/~2, vz/~2, and vz/K2 as
v~, vp, and v3 respectively, and for simplicity we are not
considering relative phases between the vacuum expecta-
tion values. Here we are only interested in the charged
scalars spectrum. Requiring that the shifted potential
has no linear terms in any of the H; and h; fields,
i = 1, 2, 3, we obtain in the tree approximation the con-
straint equations

pl + 2%i vl + &4vz + &sv3' + « fvl
'

V2v3 = 0,

p& + 2A&v& + A4vl + Asvs + Re fvl v& vs ——0,
p3 + 2A3v3 + Asvl + ~sv2 + Re fvlv2v3

Imf =0.
(38)

Gl = (-»~z + vs~ )/(vl + vs) '
1

G2 =(-»~l +»P )/("l+V') (39)

Then, it is possible to verify that there is a doubly
charged Goldstone boson and a doubly charged physi-
cal scalar. There are also two singly charged Goldstone
bosons,

A(3a) (Mw l
A(3b) ( Mv

where A(3a) and A(3b) are the amplitudes for the pro-
cesses in Fig. (3a ) and (3b ) respectively. Experimentally
R ( 5 x 10 3 [11];then we have that the occurrence of
the decay p ~ e v, v„ implies that My ) 2Mw.

In addition to decays, eKects such as el e& ~ v, l.v,~
will also occur in accelerators, but these events impose
constraints on the masses of the vector bosons which are
weaker than those coming from the decays. Notice that
the incoming negative charged lepton is right handed be-
cause the lepton-number-violating interactions with the
V+ vector boson in Eq. (24) is a right-handed current for
the electron.

The doubly charged vector boson U will produce
deviations from the pure QED Moiler scattering which
could be detected at high energies.

Stronger bounds on the masses of the exotic vector
bosons come from fiavor-changing neutral currents in-
duced by Z' . The contribution to the Kl-A& mass dif-
ference due to the exchange of a heavy neutral boson Z'
appears in Fig. 4. From Eq. (33) we have explicitly

and two singly charged physical scalars,

= (vsr)3 + vip )/(vl + v3)

p = (Vzr)l + Vlp )/(Vl + Vz)
(40)

cos 8~ sin Hc [ d7" (v'" + a' ps) s
4Mw

+dy" (v'* + a"7 )s]Z„, (42)

with masses ml — fv2(vl v3 + vl v3 ) and m&

fvs(vl V2 + V2 vl) respectively. We can see from
Eq. (40) that the mixing occurs between rlz and y
g& and p but not between g& and gz . This implies
that the neutrinoless double-P decay does not occur in
the minimal model. It is necessary to introduce two new

Higgs triplets, say cr and ~, with the quantum number
of g to have mixing between g& and g2 . In this case the
potential has terms with g ~ o, u in Eq. (36) and terms
which mix g, o, and ~. In particular the term e'& "g;0"~p

g, o, u with rb, o,~ [1o).

with v'"" and a'd" given in Eq. (34c,d) respectively, and
for simplicity we have assumed only two-family mixing.
Then, Eq. (42) produces at low energies the effective in-
teraction

IV. PHENOMENOLOGICAL CONSEQUENCES

In this model, the lepton number is violated in the
heavy charged vector bosons exchange but it is not in the
neutral exchange ones, because neutral interactions are
diagonal in the lepton sector. However, we have Qavor-
changing neutral currents in the quark sector coupled
to the heavy neutral vector boson Z' . All these heavy
bosons have a mass which depends on v& and this vacuum
expectation value is, in principle, arbitrary.

Processes such as p ~ e v, v„are the typical ones,
involving leptons, which are induced by lepton-number-
violating charged currents in the present model. It is well
known that the ratio

FIG. 3. (s) Lepton-number-conserving process.
Lepton-number- violating process.

(b)



416 F. PISANO AND V. PLEITEZ 46

FIG. 4. Z' exchange contribution to the effective La-

grangian for Kg-KL, mixing.

constraints on the masses of the exotic quarks J~ and
J2 3 with charge +3 and —

3 respectively, but they must
be too massive to be detected by present accelerators.
For the case of the heavy vector bosons, charged U, V,
and the neutral Z', rare decays constrain their masses
as we have shown before. It is interesting to note that no
extremely high-mass scale emerges in this model, making
possible its experimental test in future accelerators.

Vertices such as the following appear in the scalar-
vector sector:

g f Mz cos ec sin 8~ 5
' 2

&,tr —~, dy" (c„+c,p')s
16 (Mw gro

(43)

where we have defined

c„=v'" —v" = ——(1+3t2)/Qh(t),
c, —= a" —a" = —c„. (44)

The contribution of the c quark in the standard model
is [12]

sM GF n m,2

i/2 4& Mwz sin Hw

x sin Hg[dp" —,'(1 —p')s]',

cos Og

(45)

with gz/8Mwz —GF/i/2. We can obtain the constraint
upon the neutral Z' mass assuming, as usual, that any
additional contribution to the K&-KL mass difference
from the Z' boson cannot be much bigger than the con-
tribution of the charmed quark [13].Then, from Eqs. (43)
and (45) we get

(14m z Mw
Mzo& ~

——c, tan ew ~Mw,(2 n mz (46)

which implies the following lower bound on the mass of
the Z":

Mz ~ & &0 TeV

From this value and Eq. (19) we see that vx must satisfy

v„& ~ (40 TeV),3i/2 2

8GFMw 1+ 3t2

that is, v~ ) 12 TeV. As the vacuum expectation value

of the y Higgs boson is ( y &= vz/y 2 then we have
that (y » 8.4 TeV. This also implies, from Eq. (17),
that the masses of the charged vector bosons V, U
are larger than 4 TeV.

V. CONCLUSIONS

If we admit lepton-number violation, SU(3) could be a
good symmetry at high energies, at least for the lightest
leptons (v, e, e+). Assuming that this is a local gauge
symmetry, the rest of the model follows naturally, includ-
ing the exotic quarks J. To the best of our knowledge,
there are no laboratory or cosmological/astrophysical

2

[W„+(rl, D" rP —0"rt, g )
2

(47)

and also with g ~ o, ~, when these two new triplets
are added to the model. Then we have mixing in the
scalar sector which imply 1-loop contributions to the
(P|3)p involving the vector bosons V, U but these
are less than contributions at the tree level through
scalar exchange [10]. On the other hand, this model
cannot produce processes such as K ~ r+e p and
r ~ 1+x x with l = e p.

Notice that the definition of the charge-conjugation
transformation we have used in this work (see the Ap-
pendix) has physical consequences only in the Yukawa
interactions and in the currents coupled to the heavy
charged gauge bosons where an opposite sign appears
with respect to the usual definition of that transforma-
tion.

ACKNOWLEDGMENTS

We would like to thank the Conselho Nacional de De-
senvolvimento Cientifico e Tecnologico (CNPq) for full

(F.P.) and partial (V.P.) financial support, M.C. Tijero
for reading the manuscript, and finally C.O. Escobar, M.
Guzzo, and A.A. Natale for useful discussions.

APPENDIX

In this appendix we shall treat in more detail how it is
possible to get Yukawa interactions from Eq. (11).

In the present model we have put together in the same
multiplet the charged leptons and their respective charge-
conjugated field. That is, both of them are considered as
the two independent fermion degrees of freedom. If we

use the usual definition of the charge conjugation trans-
formation g' = Cg, g' = ETC ' the Yuk—awa cou-

plings in Eq. (11)vanish, including the mass terms. This
is a consequence of the degrees of freedom we have cho-
sen. Notwithstanding, it is possible to define the charge-
conjugation operation as

0'=0 G 'v'

This definition is consistent with quantum electrodynam-
ics since its only effect is to change the sign of the mass
term in the Dirac equation for the charge conjugated
spinor g' with respect to the mass term of the spinor

@, and it is well known that the sign of the mass term in



46 SU(3)U(1) MODEL FOR ELECTROWEAK INTERACTIONS 417

the Dirac equation has no physical meaning. With the
negative sign, the upper components of the spinor are
the "large" ones, and with the positive sign, the large
components are the lower ones [14].

Using this definition it is easy to verify that l&l&
—l~ll. instead of l&ll ——+l~ll. , which follows using the
usual definition of the charge conjugation transformation.

On the other hand, the definition of charge conjuga-
tion we have used in this work produces the same eR'ect as
the usual one in bilinear terms for the vector interaction.
Then, in the kinetic term and the vector interaction with
the photon, it is not possible to distinguish both defini-
tions. For example, the kinetic terms in the model are

with 1 = e, p, r, and this can be written as

) (lr, t pit, + IRt pIR) ~

1

where the right-handed electron has been interpreted as
the absence of a left-handed positron with ( E,——p).

For charged leptons we have the electromagnetic inter-
action

e(l—r, 7"It. —I&7"I&)A„;

and using I&7"II —— IRAQI"—IR we obtain the usual vec-
tor interaction eI—7"IA„; but, on the other hand, in the
charged currents we have v&'&ll. ———l~v~I. .

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