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\title{Realistic extremely flat scalar potential in 3-3-1 models}
\author{Alex G. Dias}
\address{Instituto de F\'\i sica, Universidade de S\~ao Paulo, \\
C. P. 66.318, 05315-970\\ S\~ao Paulo, SP\\ Brazil} 
\author{V. Pleitez}
\address{Instituto de F\'\i sica Te\'orica, Universidade Estadual Paulista,\\
Rua Pamplona 145, \\
01405-900 S\~ao Paulo, SP \\Brazil} 
\author{M. D. Tonasse}  
\address{Instituto Tecnol\'ogico de Aeron\'autica, Centro T\'ecnico  
Aeroespacial\\ Pra\c ca Marechal do Ar Eduardo Gomes 50, 12228-901 \\
S\~ao Jos\'e 
dos Campos, SP\\ Brazil}  
\date{\today}  
\maketitle    
\begin{abstract}  
We show that in 3-3-1 models an extremely flat and
realistic scalar potential i.e., accommodate a tiny, or even zero, contribution to the
cosmological constant and still having realistic values of the Higgs scalar mass
spectra can be implemented. When loop corrections are considered, they impose 
constraints on the masses of other heavy particles present in the model. 
We also consider, at the tree level, the multi-Higgs extensions of the standard
model, i.e., two-Higgs doublets model with and without supersymmetry. A crucial
ingredient in 3-3-1 models is the existence of trilinear terms in the scalar
potential.  
\end{abstract}
%\vskip 1.1 pc]
%\bigskip\bigskip\bigskip   
\pacs{PACS numbers: 
12.60.-i Models beyond the standard model
\\
%12.60.Fr Extension of electroweak Higgs sector\\
12.80.Cp Non standard-model Higgs boson\\
%98.80.-k Cosmology\\
98.80.Es Observational cosmology (including Hubble constant, distance 
scale, cosmological constant, early Universe, etc)
}  

\section{Introduction}
\label{sec:intro}

It is well known that spontaneously broken gauge theories contribute 
to the vacuum energy i.e., it means a new large contribution to the cosmological
term in the Einstein's Equations~\cite{jdmv}. 
For a long time, even before the spontaneously broken gauge theories, particle
physicists and cosmologists have being concerned with the so-called
``cosmological constant problem'' (for a historical review see
Ref.~\cite{WE89}). This consists in the fact that the predicted value for the
vacuum energy density by quantum field theories is  
several orders of magnitude larger than the values suggested by the astronomical
observations. In the context of classical general relativity there is no such a
problem since we can always omit the $\Lambda$-term in the Lagrangian. On the
other hand, in quantum field theory only energy differences are measured, so we
can always redefine the vacuum energy. However, since gravitation is sensible to
the absolute value of the energy, through distortions in the space-time, all
matter and energy forms couple to gravitation, then the metric $g_{\mu\nu}$ can
be coupled as an external field to the bare action of a quantum field theory
playing the role of a source for the energy-momentum operator~\cite{ue}.
Therefore we cannot avoid the problem when both gravitational phenomena and
quantum field theory are taken into account: in this situation it is not
allowed to redefine the vacuum energy~\cite{ZE68,CS01}. 

Until the end of the nineties, astronomical data were able to give for the value
of $\Lambda$ only upper bounds. Because of the smallness of the bounds
suggested, compared with the theoretically predicted values, many 
attempts were made in order to find models in which the total cosmological
constant is exactly zero~\cite{DO97}. None of these attempts, however, are based
on some fundamental theory~\cite{WE89}.
At present only considerations based on the anthropic principle suggest a route
towards the response to the cosmological constant question. However, although
these considerations receive support from  the inflationary cosmological models,
they do not have yet experimental basis~\cite{WE89,WE00,CA00}. 

The effort spent to solve this (still open) problem is justified since the
geometry and the evolution of the Universe are closely related to
it~\cite{CS01,CA00} and, in fact, the cosmological constant problem is probably the
most pressing obstacle to significantly improve the models of elementary
particle physics derived from string theory~\cite{WI00}. Therefore, advances in
this problem may lead us to a better understanding of some of the crucial
problems in both cosmology and particle physics. 
In the latter, this problem is related to another not well understood 
question: the mechanism of mass generation via the spontaneous symmetry breaking
(SSB) through scalar Higgs fields with non-zero vacuum expectation values (VEV).
In this work we will be concerned only with the contributions of the SSB
mechanism to the vacuum energy. 

Independently of these theoretical issues, recent astronomical
observations~\cite{Pea99,FK98,Bea00,exp} strongly suggest a nonzero $\Lambda$. 
For instance observations of Type Ia supernova~\cite{Pea99},
gravitational lensing frequencies for quasars~\cite{FK98} and harmonics of
cosmic background radiation (CBR) anisotropies~\cite{Bea00} strongly suggest 
a tiny and positive $\Lambda$.    
A careful analysis of these astronomical data gives 
$0.6 < \Omega_\Lambda < 0.8$, where $\Omega_\Lambda = \Lambda/3H^2_0$, 
with $H_0 = 100h_0$ km s$^{-1}$Mpc$^{-1}$, $h_0 = 0.71$, 1 s = $1.52 \times 
10^{24}$ GeV$^{-1}$ and 1 pc = $3.086 \times 10^{16}$ m~\cite{Gea00}. Since
this implies $\Omega_\Lambda = 0.7$ which gives
$\Lambda =  4.8 \times 10^{-84}$ GeV$^2$, in our numerical calculations below we
will use simply $\Lambda=0$.  

The cosmological constant $\Lambda$ was initially postulated by Einstein. 
He included by hand the term
$\Lambda g_{\mu\nu}$ in his gravitational theory in order to obtain a static 
Universe solution. In subsequent works the meaning of this term was clarified
and it was shown that it is linked with the vacuum energy density. The
Einstein's  field equations with the $\Lambda$-term are
\begin{equation}
R_{\mu\nu} - \frac{R}{2}g_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi GT_{\mu\nu},
\end{equation} 
where $R_{\mu\nu}$ is the Ricci tensor, $g_{\mu\nu}$ is the metric tensor and
$R$ represents the scalar curvature of the space-time and $G = 6.71 \times 
10^{-39}$ GeV$^{-2}$ is the Newtonian gravitational constant $(\hbar = 
c = 1)$. As mentioned before here we will only consider gauge models with SSB
taken into account only the contributions of the scalar Higgs sector to the
vacuum energy density, avoiding in this way the difficult questions of how the
cosmological constant arises and why it has such a small positive
value~\cite{WE96}. 
In models with scalar fields, denoted collectively by $\Phi$, with the
respective scalar potential $V\left(\Phi\right)$, the energy momentum tensor is
\begin{equation}
T_{\mu\nu} = \frac{1}{2}\left(\partial_\mu\Phi\partial_\nu\Phi + 
g_{\mu\nu}g^{\alpha\beta}\partial_\alpha\Phi\partial_\beta\Phi\right) - 
g_{\mu\nu}V\left(\Phi\right).
\end{equation}
In order to obtain a vacuum expression for $T_{\mu\nu}$ we first take 
$\partial_\mu\Phi = 0$ and so we have $T^{\rm vac}_{\mu\nu} = -g_{\mu\nu}V
\left(\langle\phi\rangle\right)$, where $\langle\phi\rangle$ is the VEV of the
Higgs scalar field, therefore, the vacuum energy momentum tensor is  
\begin{equation}
T^{\rm vac}_{\mu\nu} = -g_{\mu\nu}V\left(\langle\phi\rangle\right) =
-g_{\mu\nu}\rho_\Lambda \equiv -g_{\mu\nu}\frac{\Lambda}{8\pi G},
\label{def}
\end{equation}
where $\rho_\Lambda$ is the vacuum energy density. However, in a given model of
elementary particles there are several contributions to the vacuum energy: 
\begin{enumerate}
\item $V_0\equiv V(\Phi=0)$, an arbitrary constant in the scalar potential.
Since it could be a model independent parameter, we will assume that $V_0=0$ in
all the models considered below. This condition will also be valid when
radiative corrections are taken into account in any renormalization scheme. 
Here we will consider only in one of the model the corrections to the scalar
potential at the 1 loop level.  
\item Quark condensates $\langle q\bar{q}\rangle$, $\Lambda_{\rm QCD}$, and
other quantum fluctuations typical of quantum field theories. Since their values
are smaller than the electroweak energy scale we will neglect them here.
\item Higher energy scales beyond the standard model. In particular the Planck
scale $\Lambda_{\rm Planck}$. Although they are larger than all the other
contributions, all of them also do not depend on the electroweak
model at energies below 1 TeV. So we can consider that all these
contributions, although they can be the dominant ones, are the same in all the
electroweak models that we will consider in this work.
\end{enumerate}
We are also not assuming any form of exotic dark matter inducing a highly
negative pressure~\cite{lips}.  

Hence, we will neglect all the above contributions to the
vacuum energy density and consider only the contributions of the Higgs scalars.
It means that we will compare, under the same assumptions, only the electroweak
contributions to the vacuum energy density in several electroweak models. 

In order to clarify the goal of this work let us consider this issue in the
context of the standard electroweak model. In this model the scalar potential is
\begin{equation}
V_{\rm SM} = \frac{1}{2}\left[\mu^2\Phi^\dagger\Phi + 
\frac{\lambda_{\rm SM}}{2}\left(\Phi^\dagger\Phi\right)^2\right],
\label{pot}\end{equation}\noindent
where $\mu^2 < 0$ and $\lambda_{\rm SM} > 0$ are parameters of the potential and
$\Phi = \left(\begin{array}{cc} \phi^+ & \phi^0 \end{array}\right)^T$ is the
only Higgs scalar doublet present in the model. The 
neutral component $\phi^0$ gets a non-zero VEV $v_W\approx 246$ GeV. 
When we take the potential at the minimum point we find 
\begin{equation}
V_{\rm SM}(v_W) =
 -\lambda_{\rm SM} v_W^4/4.
\label{sm}
\end{equation} 
Even if we ignore the wrong sign, we have
$\Lambda \sim -4 \times 10^{-17}$ GeV$^2$ unless we impose that the constant of
the scalar potential has an appropriate value: $\lambda_{\rm SM}
=3\times10^{-44}$.
Since $\lambda_{\rm SM}$ also determines the mass of the Higgs scalar
and it is given by $m_H = \sqrt{2\lambda_{\rm SM}}\,v_W$, we have a
rather light scalar $m_H\sim 3\times10^{-27}m_e$, where $m_e$ is the electron
mass~\cite{jdmv}. A Higgs boson with
mass $\lesssim 5$ GeV has been ruled out by a variety of arguments
derived from rare decays, static properties like the magnetic moment of the
muon, and nuclear physics~\cite{GH90}. So, in the SM we cannot use a fine-tuning
for obtaining a cosmological constant compatible with the observed value and at
the same time to obtain a scalar Higgs field with a mass of the order of 114
GeV~\cite{higgswg}. 

From the phenomenological point of view this is
not a  fault for any electroweak model. However, we wonder if it is possible to
built a model in which we have an arbitrary small (or even zero) contribution to
the cosmological constant coming from SSB sector and at the same time with a
realistic mass spectra.  

In principle, all  models with a
scalar potential of the form given in Eq.~(\ref{pot}) suffer from this problem:
they predict either a cosmological constant larger than the observed values or
$\Lambda < 0$ for realistic values of the VEVs.  However, in some models the
scalar potential is more complicated than the one in Eq.~(\ref{pot}).
In fact, we will show 3-3-1 models can accommodate an arbitrary small value (or
even zero) for the cosmological constant with reasonable values of the
parameters and at the same time the masses of the scalar particles having
typical values compatible with accelerator physics~\cite{higgswg}. 

The extended models which we study in this work are: (i) the non 
supersymmetric two Higgs doublet model~\cite{GH90}, (ii) the minimal 
supersymmetric extension (MSSM)~\cite{MA97,haber2} and (iii)  the so-called
3-3-1 model with only three Higgs triplets~\cite{PT93}. 
In the last one we consider also radiative correction to the scalar potential at
the 1 loop level. We also briefly comment the case of the 3-3-1 model with a
scalar sextet~\cite{331}. 

In the next section we discuss the effect of the cosmological constant in those
models mentioned in the last paragraph by calculating the value of the potential
at the point of minimum, under the general hypothesis given above, looking for
the sign of the cosmological constant and if it is possible to fine tune the
parameters of the scalar potential in such a way to obtain from Eq.~(\ref{def}),
$V(\langle \phi\rangle) \approx \Lambda/8\pi G$. Our conclusion appears in the
third section.

\section{Multi-Higgs models}
\label{sec:multi}

In this section we will consider three type of multi-Higgs scalars models.
One of them is the standard electroweak model with  two Higgs scalar doublets,
the second one is the minimal supersymmetric standard model and finally a 3-3-1
model with three Higgs scalar triplets. 

\subsection{Two Higgs scalar non-supersymmetric model}
\label{subsec:2h}

The Higgs potential for the non-supersymmetric two Higgs doublet model is 
given by
\begin{eqnarray}
V_{\rm TD} & = & \mu_1^2\,\Phi_1^\dagger\Phi_1 + \mu_2^2\,\Phi_2^\dagger\Phi_2 +
\beta_1\left(\Phi_1^\dagger\Phi_1\right)^2 + 
\beta_2\left(\Phi_2^\dagger\Phi_2\right)^2 + \beta_3\left(\Phi_1^\dagger
\Phi_1\right)\left(\Phi_2^\dagger\Phi_2\right)  \cr
&+& \beta_4\vert\Phi_1^\dagger\Phi_2\vert^2 + \frac{\beta_5}{2}\left[
\left(\Phi_1^\dagger\Phi_2\right)^2 + \left(\Phi_2^\dagger\Phi_1\right)^2
\right],
\label{pdd}\end{eqnarray}
with $\beta_1, \beta_2 > 0$ and $\beta_5 < 0$ from the positivity of the 
scalar masses. We have assumed invariance under $\Phi_i\to-\Phi_i,\,i=1,2$. 
By convenience, we can choose also $\beta_4 < 0$. The 
positivity of the scalar masses tells us also that the constraint 
$2\sqrt{\beta_1\beta_2} > \beta_3 + \beta_4 + \beta_5$ must be also 
satisfied \cite{SH89}. The two doublets are $\Phi_1 = 
\left(\begin{array}{cc} \phi_1^+ & \phi^0_1 \end{array}\right)^T$ and 
$\Phi_2 = \left(\begin{array}{cc} \phi_2^+ & \phi_2^0 \end{array}\right)^T$ 
with the VEVs $\langle\phi_1^0\rangle = u_1$ and $\langle\phi_1^0\rangle =
 u_2$. In the symmetry breaking process we expand the potential 
in Eq.~(\ref{pdd}) around its minimum. Thus, 
the vanishing of the linear terms in the neutral fields gives the constraints 
\begin{mathletters}
\begin{eqnarray}
\mu_1^2 & = & 2\beta_1u_1^2 + \left(\beta_3 + \beta_4 + \beta_5\right)u_2^2,
 \\
\mu_2^2 & = & 2\beta_2u_2^2 + \left(\beta_3 + \beta_4 + \beta_5\right)u_1^2. 
\end{eqnarray}\label{constrd}\end{mathletters}\noindent
Even by using the constraints in Eqs. (\ref{constrd}), the minimum of the  
expression in Eq. (\ref{pdd}) has still many parameters to allows us to extract 
useful information from it. However, if the two Higgs doublet 
model is a reasonable model we do not expect large differences among the 
$\beta$ constants in Eqs. (\ref{pdd}) and (\ref{constrd}) since they have 
a common origin. Therefore, in order of magnitude, we can assume that 
$\beta_1 = \beta_2 = \beta_3 = -\beta_4 = -\beta_5 \equiv \beta > 0$. 
Hence, using Eqs.~(\ref{constrd}) the minimum of the potential is 
\begin{equation}
V_{\rm TD}(u_1,v_W) = \beta\left[u_1^4 - \left(2u_1^2 - v_W^2\right)\left(v_W^2
-  u_1^2\right)\right],
\label{mh}
\end{equation}
where we have replaced $u_2^2 = v_W^2 - u_1^2$. 

From Eq.~(\ref{def}) we obtain 
\begin{equation}
\beta\left[u_1^4 - \left(2u_1^2 - v_W^2\right)\left(v_W^2 - 
u_1^2\right)\right]\approx\frac{\Lambda}{8\pi G}.
\label{f1}
\end{equation}

Assuming $u_1 = 100$ GeV, we find 
$\beta \sim 10^{-44}$. The typical square masses of the charged and the
pseudoscalar Higgs in this model are $m^2 \sim \beta v_W^2$ \cite{SH89}.
Therefore, as in the SM, it is not possible to obtain realistic values for the
Higgs mass spectrum in the non-supersymmetric two Higgs doublet model. 
A more general values for the $\beta$'s parameters will not change
this result since all the scalar masses depend mainly on $u_{1,2}$ and
this VEVs have an upper limit of 246 GeV. This is true even with a
$\mu_{12}\Phi^\dagger_1\Phi_2$ term in the scalar potential~\cite{santos}.


\subsection{Minimal supersymmetric standard model}
\label{subsec:mssm}

The minimal supersymmetric model has, in the Higgs sector, the two scalar 
doublets $H_1 = \left(\begin{array}{cc} H^{0*}_1 & -H^-_1 \end{array}
\right)^T$ and $H_2 = \left(\begin{array}{cc} -H_2^- & H_2^{0*} \end{array}
\right)^T$. The associated scalar potential is in this case
\begin{eqnarray}
V_{\rm S} & = & m_1^2\vert H_1\vert^2 + m_2^2\vert H_2\vert^2 - m_3^2
\left(\epsilon_{ij}H^i_1H^j_2 + {\mbox{H. c.}}\right) 
+ \frac{1}{8}\left[g^2\sum_i\vert H_1^\dagger\tau_i H_1 + H_2^\dagger
\tau_i H_2\vert^2 \right. \cr
&& \left. + g^{\prime2}\left(\vert H_1\vert^2 - \vert H_2\vert^2\right)^2\right],
\end{eqnarray}
where $m_1$, $m_2$ and $m_3$ are parameters, $\tau_i$ $\left(i = 1, 2, 3
\right)$ are the Pauli matrices, $g^\prime = g\tan{\theta_W}$, $g = 
2\sqrt{\sqrt{2}G_F}M_W$~\cite{SH89}. For the Weinberg mixing angle we take 
$\sin^2{\theta_W} = 0.2315$. $M_W = 80.33$ GeV is the mass of the charged 
standard gauge boson and $G_F = 1.166 \times 10^5$ GeV$^{-2}$ is the Fermi 
constant \cite{Gea00}. The vacuum structure is $\langle H^0_1\rangle = v_1$, 
$\langle H^0_2\rangle = v_2$, with $v_1^2 + v_2^2 = v_W^2$. By expanding the 
neutral fields around the minimum of the potential and eliminating linear 
terms in $V_{\rm S}$ (since it must be bounded from below), we have
\begin{mathletters}
\begin{eqnarray}
m_1^2 & = & \frac{1}{4v_1}\left[ v_1\left(g^2 + g^{\prime2}\right)\left(v_2^2 -
v_1^2\right) 
 + 4m_3^2v_2\right] , \\
m_2^2 & = & \frac{1}{4v_2}\left[v_2\left(g^2 + g^{\prime2}\right)\left(v_1^2 -
v_2^2\right)  
+ 4m_3^2v_1\right].
\end{eqnarray}\end{mathletters}
Therefore, taking the potential in the minimum we find
\begin{equation}
V_{\rm S}(v_1,v_W) = -\frac{g^2}{8\cos^2\theta_W}\left(2v_1^2 - v_W^2\right)^2.
\label{vs0}
\end{equation}

From Eqs. (\ref{def}) 
\begin{equation}
-\frac{g^2}{8\cos^2\theta_W}\left(2v_1^2 - v_W^2
\right)^2\approx \frac{\Lambda}{8\pi G}.
\label{f2}
\end{equation}

Although the fine-tuning $2v^2_1\approx v^2_W$ is possible, we see from 
Eq.~(\ref{vs0}) a negative value for $\Lambda$, in contradiction with the
observations~\cite{Gea00}. (Phases in the VEVs can be rotated away in this
model~\cite{haber2}.) On the other hand, the scalar mass spectra are
compatible with phenomenology since the masses are proportional to $v^2_1,v^2_2$
or $v_1v_2$. As in the previous case all parameters with dimension of mass are
constrained by phenomenology, for instance $m_3$ must be of the order of
100-200 GeV. It means that, as in the previous case, the masses of the different
scalars depend mainly on $v_1,v_2$ and $m_3$.

 
\subsection{3-3-1 model with only three  scalar triplets}
\label{subsec:331a}

Another model which we consider here is the three scalar triplet version of 
the 3-3-1 model \cite{PT93}. In this kind of models the gauge symmetry 
SU(3)$_C$\-$\otimes$\-SU(2)$_L$\-$\otimes$\-U(1)$_Y$ of the standard model 
is extended to SU(3)$_C$\-$\otimes$\-SU(3)$_L$\-$\otimes$\-U(1)$_N$. The 
pattern of symmetry breaking is SU(3)$_L$$\otimes$U(1)$_N$
$\stackrel{\langle\chi\rangle}{\longmapsto}$SU(2)$_L$$\otimes$U(1)$_Y$
$\stackrel{\langle\eta, \rho\rangle}{\longmapsto}$U(1)$_{\rm em}$ where
\begin{equation} 
\eta = 
\left(\begin{array}{c} \eta^0 \\ 
\eta_1^- \\ 
\eta_2^+ \end{array}\right), \quad  \rho = 
\left(\begin{array}{c} \rho^+ \\ 
\rho^0 \\ 
\rho^{++} \end{array}\right), \quad 
\chi = \left(\begin{array}{c} \chi^- \\ 
\chi^{--} \\
\chi^0 \end{array}\right) 
\label{trip}\end{equation}\noindent 
transform under the SU(3) group as $\left({\bf 3}, 0\right)$, $\left({\bf 3}, 
1\right)$ and $\left({\bf 3}^*, -1\right)$, respectively. The neutral scalar
fields develop the VEVs $\langle\eta^0\rangle = v_\eta/\sqrt2$, 
$\langle\rho^0\rangle =v_\rho/\sqrt2$ and $\langle\chi^0\rangle = v_\chi/\sqrt2$,
with $v_\eta^2 + v_\rho^2 = v_W^2$. According to the symmetry breaking pattern
we must have $v_\chi \gg  v_\eta, v_\rho$.  The representation content of the
3-3-1 model is given in Appendix~\ref{a1}. 

\subsubsection{The scalar potential at the tree level}
\label{subsubsec:331tl}

We take the more economical scalar potential which is 
renormalizable, i. e., 

\begin{eqnarray}
V_{\rm 331} & = & \mu_1^2\eta^\dagger\eta + \mu_2^2\rho^\dagger\rho + \mu_3^2
\chi^\dagger\chi + \lambda_1\left(\eta^\dagger\eta\right)^2 + 
\lambda_2\left(\rho^\dagger\rho\right)^2 + \lambda_3\left(\chi^\dagger\chi
\right)^2 + \left(\eta^\dagger\eta\right)\left[\lambda_4\left(\rho^\dagger
\rho\right) + \right. \cr 
&& \left. \lambda_5\left(\chi^\dagger\chi\right)\right] + \lambda_6
\left(\rho^\dagger\rho\right)\left(\chi^\dagger\chi\right) +  
 \lambda_7\left(\rho^\dagger\eta\right)\left(\eta^\dagger\rho\right) + 
\lambda_8\left(\chi^\dagger\eta\right)\left(\eta^\dagger\chi\right) +  
 \lambda_9\left(\rho^\dagger\chi\right)\left(\chi^\dagger\rho\right) \cr &+& 
\frac{1}{2}\left(f\epsilon^{ijk}\eta_i\rho_j\chi_k + \mbox{H. c.}\right),
\label{vt}
\end{eqnarray}
where the $\mu'$s, $\lambda'$s and $f$ are constants \cite{TO96}. As before,
conditions for the extremum of $V_T$, besides the trivial solutions
$v_\eta=v_\rho=v_\chi=0$, imposes

\begin{mathletters}\begin{eqnarray}
\mu^2_1+\lambda_1
v^2_\eta+\frac{\lambda_4}{2}v^2_\rho+
\frac{\lambda_5}{2}v^2_\chi+
\frac{A}{2\sqrt2  v^2_\eta} =0,
\label{mus1}\\
\mu^2_2+\lambda_2 v^2_\rho+\frac{\lambda_4}{2} v^2_\eta
+\frac{\lambda_6}{2}v^2_\chi+
\frac{A}{2\sqrt2 v^2_\eta} = 0, 
\label{mus2}\\  
\mu^2_3+\lambda_3v^2_\chi+\frac{\lambda_5}{2} v^2_\eta
+\frac{\lambda_6}{2} v^2_\rho+
\frac{A}{2\sqrt2 v^2_\chi}=0, 
\label{mus3}
\end{eqnarray}
\label{mus}
\end{mathletters}
We have defined $A\equiv fv_\eta v_\rho v_\chi$ and we have assume all VEVs and
$f<0$ to be real. (Otherwise a physical phase remains in the model, which we can
choose as the phase of $v_\chi$, and we have CP violation~\cite{cp3}.)   
Unlike the previous models the present one has two
parameters with dimension of mass which can be larger than 246 GeV: $v_\chi$ and
$f$. Hence, it is worth to consider in detail the scalar mass spectra. 

The scalar mass spectra have been
considered in Ref.~\cite{TO96} and the masses 
are given by
\begin{mathletters}
\label{mesca}
\begin{equation}
M^2_{++}=\frac{\lambda_9}{2}( v^2_\rho+v^2_\chi)-
\frac{A}{\sqrt2}\left(\frac{1}{v^2_\rho}+
\frac{1}{v^2_\chi}\right),
\label{m++}
\end{equation}
for the doubly charged scalar,
\begin{equation}
M^2_{+1}=\frac{\lambda_7}{2}(v^2_\eta+v^2_\rho)-
\frac{A}{\sqrt2}\left(\frac{1}{v^2_\rho}+
\frac{1}{v^2_\chi}\right),\;
M^2_{+2}=\frac{\lambda_8}{2}(v^2_\eta+v^2_\chi)-
\frac{A}{\sqrt2}\left(\frac{1}{v^2_\rho}+
\frac{1}{v^2_\chi}\right),\;
\label{m+12}
\end{equation}
for the two singly charged scalars, and
\begin{equation}
M^2_A=-\frac{A}{\sqrt2}\left(\frac{1}{v^2_\eta}+
\frac{1}{v^2_\rho}+\frac{1}{v^2_\chi} \right),
\label{ma1}
\end{equation}
for the pseudoscalar neutral boson. 

In the scalar neutral sector we have the mass matrix
\begin{equation}
M^2_0=\left(
\begin{array}{ccc}
2\lambda_1v^2_\eta-\frac{A}{\sqrt2 v^2_\eta} &
\lambda_4v_\eta v_\rho+\frac{A}{\sqrt2 v_\eta  v_\rho} &
\lambda_5v_\eta v_\chi+\frac{A}{\sqrt2  v_\eta v_\chi}\\
& 2\lambda_2v^2_\rho-\frac{A}{\sqrt2  v^2_\rho} &
\lambda_6v_\rho v_\chi+\frac{A}{\sqrt2 v_\rho v_\chi}\\
& & 2\lambda_3 v^2_\chi-\frac{A}{\sqrt2  v^2_\chi} 
\end{array}\right).
\label{m20}
\end{equation}
\end{mathletters}

and we have assumed real VEVs. All scalar masses in Eqs.(\ref{mesca}) are
analytic exact values.
   
On the other hand, the minimum of the potential i.e., $ V_{\rm 331 (min)}\equiv
V_{\rm 331}(v_\eta,v_\rho,v_\chi)$ is given by 
\begin{eqnarray}
V_{\rm 331 (min)}=  
 &-&\frac{1}{4}\left(\lambda_1v_\eta^2 +
 \frac{\lambda_4}{2}v_\rho^2+\frac{\lambda_5}{2}v_\chi^2
\right)v_\eta^2 - \frac{1}{4}\left(\lambda_2v_\rho^2
+\frac{\lambda_4}{2}v_\eta^2+
\frac{\lambda_6}{2}v_\chi^2\right)v_\rho^2\nonumber\\ \nonumber\\ 
&-&  \frac{1}{4}\left(\lambda_3v_\chi^2 + \frac{\lambda_5}{2}v_\eta^2+
\frac{\lambda_6}{2}v_\rho^2\right)v_\chi^2 - 
\frac{f}{2}v_\eta v_\rho v_\chi.
\label{3310}
\end{eqnarray}

We can now impose the extra constraint: 
\begin{equation}
V_{\rm 331 (min)}(v_\eta,v_\rho,v_\chi)=0,
\label{newc}
\end{equation}
which is, for practical purposes, equivalent to the condition
$V_{\rm 331 (min) }=\Lambda/8\pi G$. 
We can see if there some possibility of satisfying Eq.~(\ref{newc}) and still
obtain realistic scalar masses from Eqs.~(\ref{mesca}).

Notice that $f<0$, this minus sign is required by the positivity of the masses
of the scalar fields~\cite{TO96}. The condition in Eq.~(\ref{newc}) implies
\begin{eqnarray}
f&=&-\frac{1}{2v_\eta v_\rho v_\chi}\left[
\left(\lambda_1v_\eta^2 +\frac{\lambda_4}{2}v_\rho^2+
\frac{\lambda_5}{2}v_\chi^2 \right)v_\eta^2 
+ \left(\lambda_2v_\rho^2 + \frac{\lambda_4}{2}v_\eta^2+
\frac{\lambda_6}{2}v_\chi^2 \right)v_\rho^2 \right.
\nonumber \\ &+&\left. \left(\lambda_3v_\chi^2 + \frac{\lambda_5}{2}v_\eta^2+
\frac{\lambda_6}{2}v_\rho^2\right)v_\chi^2\right].
\label{app1}
\end{eqnarray}

As an illustration that it is possible to obtain a realistic mass spectra for
the physical scalar Higgs, see Eqs.~(\ref{mesca}), and at the same time 
the condition in Eq.~(\ref{newc}) is satisfied and if we assume that $\vert
f\vert\ll v_\chi$ and the condition that the mass matrix in Eq.~(\ref{m20}) is
already diagonal: 
\begin{equation}
\lambda_{4}=-\frac{f v_\chi}{\sqrt2 v_\eta v_\rho},\quad\lambda_5=-\frac{f
v_\rho} {\sqrt2v_\eta v_\chi},\quad\lambda_6=-\frac{f v_\eta}{\sqrt2 v_\rho
v_\chi}, 
\label{lp}
\end{equation}
which implies that
\begin{equation}
f=-\frac{\lambda_1v^4_\eta+\lambda_2v^4_\rho+\lambda_3v^4_\chi}
{(1+\frac{3}{2\sqrt2})2v_\eta v_\rho v_\chi},
\label{newf}
\end{equation}
and the following eigenvalues for the matrix in Eq.~(\ref{m20})
\begin{equation}
M^2_1=2\lambda_1 v^2_\eta-\frac{fv_\rho v_\chi}{\sqrt2v_\eta},
\;M^2_2=2\lambda_2v^2_\rho-\frac{f v_\eta v_\chi}{\sqrt2v_\rho},\;
M^2_3=2\lambda_3v^2_\chi-\frac{f v_\eta v_\rho}{\sqrt2v_\chi}.
\label{eigen}
\end{equation}

Using $\lambda_1=10^{-3},\lambda_2=10^{-2},\lambda_3=5\times10^{-3}$,
$v_\eta=100$ GeV, $v_\chi=1$ TeV and $v^2_\rho=[(246)^2-v^2_\eta]\,({\rm
GeV})^2$ from Eqs.~(\ref{lp}) we obtain  $\lambda_4=1.71,\lambda_5=0.09,
\lambda_6=0.02$ and from Eq.~(\ref{newf}) it follows $f=-54.25$ GeV. Next, from
Eqs.~(\ref{eigen}) we obtain (all masses below are in GeV)
$M_1=293.68$ $M_2=134.46$ and
$M_3=104.22$ for the neutral scalar Higgs bosons. Using
$\lambda_7=\lambda_8=\lambda_9=0.01$ from Eqs.~(\ref{m++}),
and (\ref{m+12}) we obtain $M_{++}=152.26,M_{+1}=135.03$ and $M_{+2}=151.60$.
Finally from Eq.~(\ref{ma1}) we get $M_A=322.73$. 
Moreover, we recall that all the masses above are only typical ones, it is
possible to obtain another set of values by choosing another 
values for the dimensionless constants $\lambda$'s (recall also that $v_\chi
\stackrel{<}{_{\sim}} 3.5$ TeV~\cite{JJ97}) and also by considering a general
form of the mass matrix in Eq.~(\ref{m20}). 
%daqui

\subsubsection{Radiative corrections to the scalar potential}
\label{subsubsec:331effpot}

Having showed that it is possible to get a flat potential at the tree level in a
3-3-1 model with the fine tuning did before, we now will be concerned with 
the effects of one loop radiative corrections to the nontrivial minimum in the
scalar potential of Eq.~(\ref{3310}). Since the value of the cosmological
constant is extraordinarily small, we must verify if it is possible to obtain a
zero contribution for the first order in perturbation theory to the nontrivial
minimum and still having a reasonable mass spectrum to the model. To do that we
appeal to the well known methods to calculate the effective potential at the 1
loop approximation~\cite{colemanweinberg,jackiw}.

At this level, all quantum corrections can be extracted
from  the quadratic part of the Lagrangian after shifting the neutral
component of the scalars fields i. e., we shift the real part of
these fields, $\xi_\varphi\rightarrow\xi_\varphi+u_\varphi$, to get the the
first leading term, of  $\hbar$ order, in a perturbative expansion,
$V=V^{(0)}+\hbar V^{(1)}+...$, for the potential~\cite{jackiw}. According to
this method, choosing a $R_\xi$ gauge with the Landau prescription~\cite{SH89},
a generic field $\varphi_i$  give us the following 1 loop level contributions
to the scalar potential
\begin{eqnarray}
V_i^{(1)}(u_{\varphi})&=&-\frac{i}{2}\int\frac{d^4k}{(2\pi)^4}\ln\hspace{0.1
cm}\det\left[i{{\cal{D}}^{-1}_{ab}(m_i(u_\varphi),k)}
\right]\nonumber\\\nonumber\\
&=&-\frac{n_i}{64\pi^2}\left[
\left(\frac{1}{\varepsilon}+\frac{3}{2}+\ln4\pi-
\gamma\right)m^4_i(u_\varphi)-\frac{m^4_i(u_\varphi)}{2}
\ln\frac{m^4_i(u_\varphi)}{\sigma^4}\right],
\label{Vi1}
\end{eqnarray}
where $n_i$ stands for the respective degrees of freedom times one for bosons
and minus one for fermions, $i{{\cal{D}}^{-1}_{ab}(m_i(u_\varphi) ,k)}$ 
is the inverse of the propagator and $\sigma$ is an appropriate energy scale. 
The second line in Eq.~(\ref{Vi1}) was
evaluated using dimensional regularization (see Appendix~\ref{a2}).
To remove the infinities in Eq.~(\ref{Vi1}) we adopt the
minimal subtraction scheme $\overline{MS}$. It means that all terms
proportional to $1/\varepsilon+\frac{3}{2}+\ln4\pi-\gamma$ will be
absorbed by renormalization counter terms. It must be pointed out that the mass
dimension functions $m_i(u_\varphi)$ above  are dependent on the $u_{\varphi}$,
which will be identified later with the real components of the neutral fields,
and only at the nontrivial minimum point it will assume the value of the
physical particle mass. Notice that there is not an infinite constant term,  
i.e., the counter term to the cosmological constant, which we
call $C_0$, and it is determined by a renormalization condition, is finite 
according to the regularized integral above. This is a characteristic of this
renormalization scheme. Thus, the total contribution at the 1 loop order to the
scalar potential, for a model with a number $n$ of fields, is given by  
\begin{eqnarray}
V^{(1)}=\frac{1}{64\pi^2}\sum_{i=1}^n\frac{n_i}{2}\,m^4_i(u_\varphi)
\hspace{0.05
cm}\ln\frac{m^4_i(u_\varphi)}{\sigma^4}+C_0.
\label{V1}
\end{eqnarray}
This equation includes also a part due the non physical fields which give rise
to the Goldstone bosons. In fact, the functions $m^2_i(u_\varphi)$ which came 
from the scalars fields, are the eigenvalues of the several mixing matrices
arising when the shift is realized in order to obtain the loop correction. It is
not difficult to see that when the constraints are imposed, there is no
contribution coming from these nonphysical bosons for the effective potential at
the non trivial minimum. The reason for such a thing is that Goldstone bosons
are massless.    
  
The energy scale parameter $\sigma$ will be chosen at the scale we discuss the
new physics~\cite{bando}. Changing $\sigma$ does not affect anything, since it
is equivalent to a reparametrization of the coupling constants. In our case, we
will take $\sigma=1\hspace{0.05 cm} TeV$. This is due to the fact
that the main contribution to the effective scalar potential is given by the
heaviest particles present in the model and these 
particles are expected to have masses, in the 3-3-1 model, of the order of 1
TeV. For the minimal standard model, it is easy to see  that a natural choice is
the vacuum expectation of the Higgs field i.e., 246 GeV~\cite{SH89}. Here the
situation is a little bit more complicated since we have four physical neutral
scalar fields. We have verified that there is not  significative change in the
final results for another choose of the energy scale, between the standard model
value and the one used here.   

Now that we have the effective potential up to order $\hbar$ adding to the
tree level part given in Eq.~(\ref{vt}), which we call now $V^{(0)}_{\rm 331}$, 
the quantity given in Eq.~(\ref{V1}) will be denoted by  $V^{(1)}_{\rm 331}$ and
the total scalar potential is given by $V_1=V^{(0)}_{\rm 331}+V^{(1)}_{\rm
331}$. We can fix the value of $C_0$ by the condition $V(0)=0$ order by order
i.e., in this case $V^{(0)}_{\rm 331 (min)}=V^{(1)}_{\rm 331(min)}=0$ and we
have 

\begin{eqnarray}
V_1= V^0_{331}(\xi_\varphi=u_\varphi) +\frac{1}{64\pi^2}\sum_{i=1}^nn_i\left[\frac{m^4_i(u_\varphi)}{2}\hspace{0.05
cm}\ln\frac{m^4_i(u_\varphi)}{\sigma^4}-\frac{m_j^4(0)}{2}\hspace{0.05
cm}\ln\frac{m^4_i(0)}{\sigma^4}\right],
\label{V-1}
\end{eqnarray}
with $m^4_i(0)$ meaning $m^4_i$ computed at the trivial minimum  i. e., at the
origin.  

New constraints arise, and they differ from the previous ones in Eq.~(\ref{mus})
only by corrections which came from Eq.~(\ref{V1}), i e. we add functions
$g_{v_\eta}$, $g_{v_\rho}$ and $g_{v_\chi}$ to Eqs.~(\ref{mus1}), (\ref{mus2})
and (\ref{mus3}), respectively, where $g_{v_\theta}$ stands for
\begin{eqnarray}
g_{v_\theta}=\frac{1}{32\pi^2}\sum_{i=1}^n
n_im_i^2(u_\varphi)
\left(\ln\frac{m^4_i(u_\varphi)}{\sigma^4}+1\right)\frac{d\hspace{0.05
cm}m^2_i(u_\varphi)}{d\hspace{0.05
cm}u_\theta^2}\Big{\vert}_{u_\theta=v_\theta}. 
\end{eqnarray}

We will denote the potential $V_1$ at the minimum by $V_{1\rm (min)}$, and it is
given by (using the Eq.~(\ref{newc}) )  
\begin{eqnarray}
V_{1\rm (min)}&=&-\frac{1}{64\pi^2}\left( \sum_{\varphi(phys)} n_\varphi
\left[(\ln\frac{m_\varphi^4}{\sigma^4}+1)m_\varphi^2\hspace{0.05
cm}v_\theta^2\frac{d\hspace{0.05 cm}m_\varphi^2}{d\hspace{0.05
cm}v_\theta^2}-\frac{m_\varphi^4}{2}\hspace{0.05 cm} 
\ln\frac{m_\varphi^4}{\sigma^4}\right]+3\sum_{i=1}^3{\mu_i^4}\hspace{0.05
cm}\ln\frac{\mu_i^4}{\sigma^4}\right). 
\label{V1TCOR}
\end{eqnarray}
\noindent
Where $v_\theta^2(d/dv_\theta^2)=v_\eta^2(d/dv_\eta^2)+v_\rho^2(d/dv_\rho^2)
+v_\chi^2(d/dv_\chi^2)$, and $m^2_\varphi\equiv m^2_\varphi(v^2_\theta)$
denotes the mass of the physical field $\varphi$. 
We  pointed out that this derivative has to be done before the $\mu_{i}^2$
elimination.  The sum over $\varphi(phys)$ means that we are disregarding the
nonphysical fields in $V_{1\rm (min)}$ for the reason we have mentioned above.
Hence, the mass dimension functions $m_\varphi^2$ which appear in
Eq.~(\ref{V1TCOR}), take in this point the value of the mass of the respective
particle $\varphi$.  
 
As we said before, by consistency of the approach, the vanishing or the small
value of the contribution to the cosmological constant should also
occur for the order $\hbar$ which arise from radiative corrections. So, we need
to look for conditions, over the mass parameters, that would make
$V^{(1)}_{\rm 331(min)}$ to give us a zero (in fact an extremely small and
positive cosmological term) contribution to the minimum of the 
potential. Of course, the radiative corrections will be inside of the validity
domain of the perturbation theory, only if
$\frac{a(\lambda_i)}{64\pi^2}\ln\frac{m_i^2}{\sigma^2}<1$. Where $\lambda_i$ is
the largest coupling constant of the electroweak sector~\cite{SH89}. It can be
shown, using the parameters found before, that this is the case for the 3-3-1
model.  

The main contribution for $V_{331(mim)}^{(1)}$ comes from the heaviest
particles. They are the vector bosons
$U^{++}$, $V^{+}$ and $Z^{\prime}$, the exotic quarks $J$,
$j_1$ and $j_2$, the top quark and the Higgs scalars. 
The explicit form of the first order correction in Eq.~(\ref{V1TCOR}) is
troublesome, but we present in the Fig.~\ref{fig1} the surface satisfying the
condition $V^{(1)}_{1(mim)} = 0$ as a function of the mass of the exotic quarks
and assuming the tree level parameters given above and with $m_V = m_U = 1$ TeV.
For a numerical example we can take $m_J = 1.098$ TeV and $m_{j_1} = m_{j_2} =
1$ TeV. Of course, there exist other values for the quark masses which satisfy
the condition of flat potential. This shows that in the 3-3-1 model context,
unlike the other electroweak models we have considered above, the values of the
parameters stay in a reasonable range even when we impose the measured value of
the cosmological constant.  Although we have consider the version of the model
with only three scalar triplets it is easy to convince ourselves that the same
will happen if we add a scalar sextet to the model since in  case there are at
least two trilinear terms and for this reason it seems obvious that similar
results should be obtained in this extension of the model. Details will be given
elsewhere.
 
\section{Conclusions}
\label{sec:con}

We conclude that, under reasonable hypotheses, in some gauge models stringent
bounds on symmetry breaking parameters come from the vacuum 
structure but this is not the case for 3-3-1 models. Of course, from the 
phenomenological point of view this is not an argument against any electroweak 
model, however it is certainly a virtue of 3-3-1 models and probably other
models with complicated Higgs scalar sectors to have a flat potential and still
a realistic mass spectra for all the particles that are present in the model.
Our starting point was that the vacuum energy density receives 
a contribution from the minimum of the Higgs potential of the 
respective electroweak model. It is already known that the standard 
model is not compatible with such a small and positive value of the cosmological
constant unless we chose an appropriate value for $V_0\equiv V(\Phi=0)$.
However, it implies that a different value for $V_0$ is needed in any of its
extensions. For this reason we assumed above that for any electroweak model
$V_0=0$.  
Next we showed that the non supersymmetric two  Higgs doublet model gets
values rather small for the dimensionless coupling constants of the Higgs
potential $\left(\sim 10^{-44}\right)$ leading, as in the standard electroweak
model, to a non realistic scalar mass
spectrum.  On the other hand, the minimal supersymmetric model gives a negative
value for $\Lambda$, contrary to the recently observed data. 
The 3-3-1 models, even with only three triplets, give realistic and restrictive
constraints on the coupling constants maintaining the masses in the scalar
sector compatible with phenomenology. The key ingredient which allows the 3-3-1
model to accommodate an extremely flat potential i.e., a tiny and 
positive contribution to the cosmological constant, is the trilinear term
present in the Higgs potential in Eq.~(\ref{vt}) which modifies the 
$\lambda\phi^4$ form of the scalar potential. It is then possible to have an
extremely flat potential having still agreement with other phenomenological
aspects. It is interesting that no one of the proposed solutions to the
cosmological constant problem using scalar fields introduce trilinear
interactions~\cite{vilenkin}. In the 3-3-1 model these interactions are
necessary in order to have the correct number of Goldstone bosons i.e., to
obtain the breakdown of the correct gauge symmetry.
We stress that other gauge models have not yet been analyzed under this point of
view. 

Loop corrections are not important when we compare several models among
themselves. For instance, the two doublet model 
without supersymmetry in Sec.~\ref{subsec:2h} has values for the $\beta$
parameters (the parameters of the quartic terms in the potential) 
of the order of $10^{-44}$. 
Even if loop corrections were considered 
they will not affect the main issue of the model: it implies
too small masses for the physical scalars of the model.  Hence, higher order
corrections are not important in comparing it, for example, with the 3-3-1
models in Sec.~\ref{subsec:331a}. The same occurs with the MSSM model considered
in Sec.~\ref{subsec:mssm} which also gives a negative $\Lambda$. 
However, loop corrections are important for a model which, already at the tree
level gives a flat potential and a realistic mass spectra in the Higgs scalar
sector as in the case of the 3-3-1 model. 
One possibility is that the effective potential must be extremely flat 
order by order. However, since the loop correction depends on the masses of
the other particles (fermion and vector boson) any fine tune implies constraints
on the masses of these particles too. In particular we have shown in the figure
that if the masses of the vector bosons are of the order of 1 TeV, the flatness
condition of the effective potential at the 1 loop level implies constraints on
the masses of the exotic quarks of the model which are compatible with what 
is expected on phenomenological grounds~\cite{das}. 
Hence, we have shown that in the 3-3-1 model radiative corrections do not spoil
the fine tuning at least at the 1 loop level. 

Finally we would like to remember that flat and realistic scalar potential may
be usefulness in inflation scenarios~\cite{ag}.


\acknowledgements  
This work was supported by Funda\c{c}\~ao de Amparo \`a Pesquisa
do Estado de S\~ao Paulo (FAPESP), Conselho Nacional de 
Ci\^encia e Tecnologia (CNPq) and by Programa de Apoio a
N\'ucleos de Excel\^encia (PRONEX). 
One of us (M. D. T.) would like to thank the Instituto de F\'\i sica Te\'orica 
of the UNESP for the use of its facilities and the Funda\c c\~ao de Amparo 
\`a Pesquisa no Estado de S\~ao Paulo (Process No. 99/07956-3) for full 
financial support.  Programa de Apoio a
N\'ucleos de Excel\^encia (PRONEX). The authors would like to thanks G. E. A. 
Matsas for useful discussions. 


\newpage

\begin{appendix}

\section{The 3-3-1 Model.}
\label{a1}

The fermion representation content is of the model is
\begin{mathletters}
\begin{eqnarray} 
\psi_{aL} & = & \left(\begin{array}{c}\nu_a \\ l^\prime_a \\ {l^\prime}^c_a
\end{array}\right)_L \sim \left({\bf 3}, 0\right),  
\label{lep} \\ 
Q_{1L} & = & \left(\begin{array}{c} u^\prime_1 \\ d^\prime_1 \\ J_1
\end{array}\right)_L \sim \left({\bf 3}, \frac{2}{3}\right), 
\label{quark1} \qquad Q_{\alpha L} = \left(\begin{array}{c} J^\prime_\alpha \\
u^\prime_\alpha \\ d^\prime_\alpha \end{array}\right)_L \sim \left({\bf 3}^*,
-\frac{1}{3}\right),\label{quark2} 
\end{eqnarray}
\noindent
where $l^\prime_a = e^\prime, \mu^\prime, \tau^\prime$, $\alpha$ = 2,
3~\cite{PT93,331}. The primed fields are symmetry eigenstates. The left-handed
quark fields have their right-handed counterparts transforming as singlets of
$SU(3)_L$ group, {\it i. e.}, 
\begin{equation} 
U'_R \sim \left({\bf 1}, 2/3\right), \quad D'_R \sim \left({\bf 1}, -1/3\right),
\quad  
J_{1R} \sim \left({\bf 1}, 5/3\right), \quad J_{2,3R}' \sim \left({\bf 1}, -
4/3\right). 
\label{quarkr}\end{equation}\label{cont}
\end{mathletters}\noindent
In Eqs. (\ref{cont}) the numbers 0, 2/3, $-$1/3, 5/3 and $-$4/3 are the U(1)$_N$
charges. Here we are defining $U^\prime_R = u^\prime_R, c^\prime_R, t^\prime_R$
and $D^\prime_R = d^\prime_R, s^\prime_R, b^\prime_R$. In order to avoid
anomalies one of the quark families must transform in a different way with
respect to others. \par 
In the gauge boson sector the single charged $\left(V^\pm\right)$ and  double
charged $\left(U^{\pm\pm}\right)$ vector bileptons, together with a new neutral
gauge boson $Z^{0\prime}$, complete the particle spectrum with the charged
$W^\pm$ and the neutral $Z^0$  gauge bosons from the SM.\par 
The content of the scalar sector is the three triplets of the Eqs. (\ref{trip})
and, in principle, one additional sextet. However, since the sextet is
introduced to gives mass to the electron and break the mass degeneration between
the muon and the tau leptons \cite{331}, we can consider here that the its
contribution is negligible. There is also versions of the model in that the
sextet is not necessary \cite{PT93}. The physical particle spectrum comes from
the three triplets is four neutral Higgs ($H^0_1$, $H^0_2$, $H^0_3$ and $h^0$),
four single charged ($H_1^\pm$ and $H_2^\pm$) and two double charged scalar
bosons ($H^{\pm\pm}$). 

\section{One loop correction to the potential}
\label{a2}
%novo
Here we show how to obtain the formula used to get the radiative quantum
corrections to the potential. 
In one loop approximation, the whole $\hbar$ contribution for the effective
potential can be extracted from the quadratic part of the Lagrangian
\cite{jackiw}. It is, for computational convenience, chosen a gauge fixing of
$R_\xi$ type, see Ref.~\cite{SH89}. The basic integral that must be computed,
using dimensional regularization, is given by~\cite{miran} 

\begin{eqnarray}
V^{(1)}&=&-\frac{i\hbar}{2}\int \frac{d^4k}{(2\pi)^4} \ln \hspace{0.1
cm}\det \hspace{0.1 cm} D_{\alpha\beta} (k,m)\nonumber\\ \nonumber\\ 
&=&-\frac{\hbar\sigma^{2\epsilon}}{2} \int \frac{d^Dk}{(2\pi)^D}
\ln(k^2-m^2)\nonumber\\\nonumber\\ 
&=&\frac{-i\hbar\sigma^{2\epsilon}}{2(4\pi)^{\frac{D}{2}}}\Gamma(-D/2)\hspace{0.1
cm}m^D\nonumber\\\nonumber\\ 
&=&-\frac{\hbar}{32\pi^2}m^4 \left[\frac{4\pi\sigma^2}{m^2}
\right]^\epsilon\Gamma(-2+\epsilon)\nonumber\\\nonumber\\ 
&=&-\frac{1\hbar}{64\pi^2}m^4\left
[\frac{1}{\epsilon}+\frac{3}{2}-\gamma+\ln4\pi-\ln\frac{m^2}{\sigma^2}\right],  
\end{eqnarray}  
where $\sigma$ is a  mass scale parameter introduced to keep the correct dimension
for $V^{(1)}$ when the integral is extended to others dimensions. And
$\epsilon=\frac{4-D}{2}$. It was also the usual expansion for the $\Gamma$
function and taken the limit $D\rightarrow 4$, where it was possible.   

The formula above must be multiplied by the number of the degrees of freedom of
the respective field, including a minus one factor for fermionic fields, which
gives a nonzero contribution to the effective potential. This numbers are, 6 
for a charged vectorial boson, 3 for a neutral vectorial boson, 2 for a charged
scalar, 1 for a neutral scalar and -4 for a fermion. The mass parameter, $m$,
in $V^{(1)}$ are functions of the shifts in the real components of the neutral scalars fields which get a vacuum
expectation value \cite{colemanweinberg,jackiw}. We call these shifts by $u_\varphi\equiv(u_\eta,u_\rho,u_\chi)$, and are to be identified at the end of the computation with their respective fields. Renormalization
guarantees that we can absorb the infinities through  redefinitions of the
arbitrary parameters of the model. We use a renormalization prescription based
in a modified minimal subtraction of the kind $\overline{MS}$~\cite{bardeen} in
the sense that we will keep only the logarithmic term as loop corrections. Thus
the whole one loop corrections is summarized in 

\begin{eqnarray}
V^{(1)}=\frac{\hbar}{64\pi^2}\sum^ N_i n_i m^4_i(u_\varphi) 
\ln\frac{m^2_i(u_\varphi)}{\sigma^2}, 
\end{eqnarray}
where $n_i$ stands for the number related to the degrees of freedom, as said
above and the sum is over the N  physical and nonphysical fields of the theory.     

\end{appendix}

%\newpage
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(1975).  
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\narrowtext

\vglue 0.01cm
\begin{figure}[ht]
\begin{center}
%\centering\leavevmode
%\epsfxsize=\hsize
%\epsfbox{mmnu2.eps}
\vglue -0.009cm
\mbox{\epsfig{file=ccfig1.eps,width=0.7\textwidth,angle=270}}       
%\epsfxsize=430pt \epsffile{mex99f1.eps}}
\end{center}
\vglue 2cm
\caption{ In this figure we show the surface
of flat potential as a function of the exotic quark masses. We have defined
$\alpha = m_{J}/v_\chi$, $\kappa = m_{j_1}/v_\chi$ and $\theta =
m_{j_2}/v_\chi$.}
\label{fig1}
%
\end{figure}
\end{document}

