\documentstyle[preprint,aps]{revtex}  % PREPRINT

\begin{document}

\title{Pairs of charged heavy-fermions from an 
SU(3)$_{L}\otimes$U(1)$_{N}$ model at $e^+ e^-$  colliders}

\author{J.\ E.\ Cieza Montalvo $^{*}$}

\address{Instituto de F\'{\i}sica, Universidade do Estado do Rio de
Janeiro, \\ Rua S\~ao Francisco Xavier 524, 20559-900 Rio de Janeiro, RJ, 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, 12228-900 S\~ao Jos\'e dos Campos, SP, Brazil}

%\date{\today}

\maketitle

                  
\begin{abstract} 
We investigate the production, backgrounds and signatures of pairs of 
charged heavy-fermions using the SU(3)$_L\otimes$U(1)$_N$ electroweak model 
in $e^+ e^-$ colliders (NLC and CLIC). We also analyze the indirect evidence 
for a boson $Z^{'}$.


PACS number: 12.60.-i,13.85.Rm,14.80.-j

%\vskip 2.5cm
%\begin{center}

\end{abstract}

%\newpage

\section{INTRODUCTION}

Although the standard electroweak model is very well successful
explaining experimental data up to order 100 GeV, there are experimental
results on the muon anomalous magnetic moment \cite{Bea01} and (solar
and atmospheric) neutrinos \cite{Aea99}, which suggest no standard
interpretation. Some other known experimental facts, such as the
proliferation of the fermion generation and their complex pattern of
masses and mixing angles, are not predicted in the framework of the
standard model. There are no theoretical explanation for the existence
of several generations and for the values of the masses. It was
established at the CERN $e^{+} e^{-}$ collider LEP that the number of
light neutrinos is three \cite{lep}. 


Many models, such as composite models \cite{af,bu1}, grand unified
theories \cite{la}, technicolor models \cite{di}, superstring-inspired
models \cite{e6}, mirror fermions \cite{maa} predict the existence of
new particles with masses around of the scale of $1$ TeV. All these
models consider the possible existence of a new generation of fermions.


Heavy-leptons are usually classified in four types: sequential leptons,
paraleptons, ortholeptons, and long-lived penetrating particle
\cite{Gea00}. In this work we will study the type of heavy leptons which
does not belong to any one mentioned above. Consequently, the existing
experimental bounds on heavy-lepton parameters do not apply to them.
The particular kind of heavy-leptons considered here are predicted 
for instance by an electroweak model based
on the SU(3)$_{C}\otimes$SU(3)$_{L}\otimes$U(1)$_{N}$ (3-3-1 for short)
semi-simple symmetry group \cite{PT93}. In this model we have only three
generations, differently as we have in  most of the
heavy-lepton models \cite{FH99,cie}. It is a chiral electroweak model
whose left-handed charged heavy-leptons, which we denote by $P_a$ $=$
$E$, $M$ and $T$, together in association with the ordinary charged
leptons and its respective neutrinos, are accommodated in SU(3)$_L$
triplets. So, we will study the production mechanism for these
heavy-exotic leptons, together with the exotic quarks and exotic
neutrinos, in $e^{-} e^{+}$ colliders such as the Next Linear Collider
(NLC) ($\sqrt{s} = 500$ GeV) and CERN Linear Collider (CLIC) ($\sqrt{s}
= 1000$ GeV). 

The outline of this paper is the following. In Sec. II we describe the
relevant features of the model. The luminosities of $\gamma \gamma$,
$\gamma Z$ and $ZZ$ for $e^{-} e^{+}$ colliders are given in Sec. III.
In Sec. IV we study the production of a pair of exotic-lepton. We
summarize the results in Sec. V.

\section{Basic facts about the 3-3-1 heavy-lepton model}   
\label{secII}    

The most interesting feature of this class of models is the occurrence
of anomaly cancellations, which is implemented only when the three
fermion families are considered together and not family by family as in
the standard model. This implies that the number of families must be a
multiple of the color number and, consequently, the 3-3-1 model suggests
a route towards the answer of the flavor question \cite{PP92}. The
model has also a great phenomenological interest since the related new
physics can be expected in a scale near of the Fermi one
\cite{MT02,CQ99}.\par Let us summarize the most relevant points of the
model (for details see Ref. \cite{PT93}). The left-handed leptons and
quarks transform under the SU(3)$_L$ gauge group as the triplets 

\begin{mathletters}  
\begin{equation}  
\psi_{aL} = \left(\begin{array}{c}  \nu_{\ell_a} \\  \ell_a^\prime \\  P^\prime_a 
\end{array}\right)_L \sim \left({\bf 3}, 0\right), \quad 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), 
\quad 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{cont} 
\end{equation}\noindent  
where $P^\prime_a$ $=$ $E^\prime$, $M^\prime$, $T^\prime$ are the new
leptons, $\ell^\prime_a$ $=$ $e^\prime$, $\mu^\prime$, $\tau^\prime$ and
$\alpha$ = 2, 3. The $J_1$ exotic quark carries $5/3$ units of
elementary electric charge while $J_2$ and $J_3$ carry $-$4/3 each. In
Eqs. (\ref{quark}) the numbers 0, 2/3 and $-$1/3 are the U(1)$_N$
charges. Each left-handed charged fermion has its right-handed
counterpart transforming as a singlet in the presence of the SU(3)$_L$
group, {\it i.e.}, 

\begin{eqnarray}  
\ell'_R \sim \left({\bf 1}, -1\right), & \qquad & P'_R \sim \left({\bf 1}, 1\right), \qquad U'_R 
\sim \left({\bf 1}, 2/3\right), \\  D'_R \sim \left({\bf 1}, -1/3\right), & \qquad & J_{1R} \sim 
\left({\bf 1}, 5/3\right), \qquad J_{2,3R}' \sim \left({\bf 1}, -4/3\right).   
\end{eqnarray}
\label{quark}
\end{mathletters}
\noindent 
We are defining $U = u, c, t$ and $D = d, s, b$. In order to avoid
anomalies, one of the quark families must transforms in a different way
with respect to the two others. In Eqs. (\ref{quark}) all the primed
fields are linear combinations of the mass eigenstates. The charge
operator is defined by 

\begin{equation}  
\frac{Q}{e} = \frac{1}{2}\left(\lambda_3 - \sqrt{3}\lambda_8\right) + N,  
\label{op} 
\end{equation}  
where the $\lambda$'s are the usual Gell-Mann matrices. We notice,
however, that since $Q_{\alpha L}$ in Eqs. (\ref{cont}) are in
anti-triplet representation of SU(3)$_L$, the anti-triplet representation
of the Gell-Mann matrices must also be used in Eq. (\ref{op}) in order
to get the correct electric charge for the quarks of the second and
third generations.\par The three Higgs scalar triplets

\begin{equation}
\eta = \left(\begin{array}{c} \eta^0 \\  \eta_1^- \\  \eta_2^+ \end{array}\right) \sim \left({\bf 3}, 0\right), \quad \rho = \left(\begin{array}{c} \rho^+ \\  \rho^0 \\  \rho^{++}
\end{array}\right) \sim \left({\bf 3}, 1\right), \quad \chi =
\left(\begin{array}{c} \chi^- \\
\chi^{--} \\ \chi^0 \end{array}\right) \sim \left({\bf 3}, -1\right),
\label{higgs}
\end{equation}
generate the fermion and gauge boson masses in the model. The neutral
scalar fields develop the vacuum expectation values (VEVs)
$\langle\eta^0\rangle = v_\eta$, $\langle\rho^0\rangle = v_\rho$ and
$\langle\chi^0\rangle = v_\chi$, with $v_\eta^2 + v_\rho^2 = v_W^2 =
(246 {\rm GeV})^2$. Neutrinos can get their masses from the $\eta^0$
scalar. A detailed scheme for Majorana mass generation for the neutrinos
in this model is given in Ref. \cite{OY99}.\par The pattern of symmetry
breaking is \[ \mbox{SU(3)}_L \otimes\mbox{U(1)}_N
\stackrel{\langle\chi\rangle}{\longmapsto}\mbox{SU(2)}_L\otimes\mbox{U(1
)}_Y \stackrel{\langle\eta, \rho\rangle}{\longmapsto}\mbox{U(1)}_{\rm
em}\] and so, we can expect $v_\chi \gg v_\eta, v_\rho$. The $\eta$ and
$\rho$ scalar triplets give masses to the ordinary fermions and gauge
bosons, while the $\chi$ scalar triplet gives masses to the new fermions
and new gauge bosons.\par Due to the transformation properties of the
fermion and Higgs fields under SU(3)$_L$ [see Eqs. (\ref{quark}) and
(\ref{higgs})] the Yukawa interactions in the model are 

\begin{mathletters} 
\begin{eqnarray}
{\cal L}_\ell^Y & = & -G_{ab}\bar\psi_{aL}\ell^\prime_{bR}\rho - G^\prime_{ab}\bar\psi_{aL}^\prime P^\prime_{bR}\chi + \mbox{H. c.}, 
\label{yl} \\ 
{\cal L}_q^Y & = & \sum_a\left[\bar Q_{1L}\left(G_{1a}U^\prime_{a R}\eta + \tilde G_{1a}D^\prime_{a R}\rho\right) + \sum_\alpha\bar  Q_{\alpha L}\left(F_{\alpha a}U^\prime_{aR}\rho^* +  \tilde F_{\alpha a}D^\prime_{aR}\eta^*\right)\right] + \cr &&  
\sum_{\alpha\beta}F^J_{\alpha\beta}\bar Q_{\alpha L}J^\prime_{\beta R}\chi^* + G^J\bar  Q_{1L}J_{1R}\chi +  \mbox{H. c.}  \nonumber  \\  
\label{yq} 
\end{eqnarray}\label{weak}\end{mathletters}\noindent
The $G$'s, $F$'s and $\tilde F$'s are Yukawa coupling constants with $a,
b = 1, 2, 3$ and $\alpha, \beta = 2, 3$. The interaction eigenstates
which appear in Eqs. (\ref{weak}) can be transformed in the
corresponding physical eigenstates by appropriated rotations. However,
since the cross section calculation imply summation on flavors (see Sec.
\ref{secIV}) and the rotation matrix must be unitary, the mixing
parameters have not essential effect for our purpose here. So,
thereafter we suppress the primes notation for the interactions
eigenstates.\par The gauge bosons consist of an octet $W^i_\mu$ $\left(i
= 1, \dots, 8\right)$ associated with SU(3)$_L$ and a singlet $B_\mu$
associated with U(1)$_N$. The covariant derivatives are

\begin{equation}
{\cal D}_\mu\varphi_a = \partial_\mu\varphi_a+ i\frac{g}{2}\left(\vec W_\mu.\vec\lambda\right)^b_a\varphi_b + ig^\prime N_\varphi\varphi_aB_\mu,
\end{equation}
where $\varphi = \eta, \rho, \chi$. The model predicts single charged $\left(V^\pm\right)$, double charged $\left(U^{\pm\pm}\right)$ vector bileptons and a new neutral gauge boson $\left(Z^\prime\right)$ in addition to the charged standard gauge bosons $W^\pm$ and the neutral standard $Z$. We take from Ref. \cite{MT02} the trilinear interactions of the $Z^\prime\left(k_1\right)$ with the $V\left(k_2\right)^\pm$ and $U^{\pm\pm}\left(k_3\right)$, in the usual notation that all the quadrimoments are incoming in the vertex,
\begin{equation}
{\cal V}_{\lambda\mu\nu} = -i\frac{g}{2}\sqrt{\frac{3}{1 + 3t_W^2}}\left[\left(k_1 - k_2\right)_\lambda g_{\mu\nu} + \left(k_2 - k_3\right)_\mu g_{\nu\lambda} + \left(k_3 - k_1\right)_\nu g_{\lambda\mu}\right]
\end{equation}
where

\begin{equation}  
t_W^2 = \frac{\sin^2{\theta_W}}{1 - 4\sin^2{\theta_W}}.  
\label{tw}
\end{equation} 
The relevant neutral vector current interactions are    

\begin{mathletters}
\begin{eqnarray}
{\cal L}_{Z} & = & -\frac{g}{2\cos\theta_W}\left[a_L\left(f\right)\overline{f}\gamma^\mu\left(1 -  \gamma_5\right)f + a_R\left(f\right)\overline{f}\gamma^\mu\left(1 - \gamma_5\right)f\right]Z_\mu, 
\label{lz}\\
{\cal L}_{Z^\prime} & = & -\frac{g}{2\cos\theta_W}\left[a^\prime_L\left(f\right)\overline{f}\gamma^\mu\left(1 - \gamma_5\right)f + a^\prime_R\left(f\right)\overline{f}\gamma^\mu\left(1 - \gamma_5\right)f\right]Z^\prime_\mu, 
\label{lzl}\\
{\cal L}_{AP} & = & -e\bar P_a\gamma^\mu P_aA_\mu, \\
{\cal L}_{ZP} & = & -g\sin{\theta_W}\tan{\theta_W}\bar P_a\gamma^\mu P_aZ_\mu,
\label{lagrb}\\
{\cal L}_{Z^\prime P} & = &  -\frac{g\tan{\theta_W}}{2\sqrt{3}t_W}\bar P_a\gamma^\mu\left[3t_W^2 - 1 + \left(3t_W^2 + 1\right)\gamma_5\right]P_aZ^\prime_\mu,
\label{lagrc}\\
{\cal L}_{Zq} & = & -\frac{g}{4\cos{\theta_W}}\sum_a\bar q_a\gamma^\mu\left(v^a + a^a\gamma^5\right)q_aZ_\mu,
\label{lagrd}\\
{\cal L}_{Z^\prime q} & = & -\frac{g}{4\cos{\theta_W}}\sum_a\bar q_a\gamma^\mu\left(v^{\prime a} 
+ a^{\prime a}\gamma^5\right)q_aZ^\prime_\mu,
\label{lagre}
\end{eqnarray}
\label{lagr}
\end{mathletters}\noindent 
where $\theta_W$ is the Weinberg mixing angle, $f$ is any fermion and
$q_a$ is any quark \cite{PT93,PP92}. The coefficients in Eqs.
(\ref{lz}), (\ref{lzl}), (\ref{lagrd}) and (\ref{lagre}) are 

\begin{mathletters}  
\begin{eqnarray}  
a_L\left(\nu_a^\prime\right) = \frac{1}{2}, \quad a_R\left(\nu_a\right) = 0, \quad a^\prime_L\left(\nu_a^\prime\right) & = & \frac{1}{2}\sqrt{\frac{1 - 4\sin^2\theta_W}{3}}, \quad a^\prime_R\left(\nu_a^\prime\right) = 0, \\
a_L\left(e_a^\prime\right) = -\frac{1}{2} + \sin\theta_W, \quad a_R\left(e_a^\prime\right) = \sin\theta_W, & \quad & a^\prime_L\left(e^\prime_a\right) = a^\prime_L\left(\nu^\prime_a\right), \quad a^\prime_R\left(e^\prime_a\right) = -\frac{\sin\theta_W}{2a^\prime_L\left(\nu^\prime_a\right)} \\
a_L\left(E^\prime_a\right) = a_R\left(E^\prime_a\right) = -\sin\theta_W, \quad a^\prime_L\left(E^\prime_a\right) & = & -\sqrt{\frac{1 - 4\sin\theta_W}{3}}, \quad a^\prime_R\left(E^\prime_a\right) = -a^\prime_R\left(e^\prime_a\right) \\
v^U = \frac{3 + 4t_W^2}{f\left(t_W\right)}, \quad v^D = -\frac{3 + 8t_W^2}{f\left(t_W\right)}, & \quad & -a^U = a^D = 1, \quad  v^{\prime u} = -\frac{1 + 
8t_W^2}{f\left(t_W\right)}, \\  
v^{\prime c} = v^{\prime t} = \frac{1 - 
2t_W^2}{f\left(t_W\right)}, \quad v^{\prime d} = -\frac{1 + 2t_W^2}{f\left(t_W\right)}, & \quad &
v^{\prime s} =  v^{\prime b} = \frac{f\left(t_W\right)}{\sqrt(3)}, \quad a^{\prime u} =  
\frac{1}{f\left(t_W\right)}, \\ 
a^{\prime c} = a^{\prime t} = -\frac{1 + 6t_W^2}{f\left(t_W\right)}, \quad a^{\prime d} = 
-a^{\prime c}, & \quad & a^{\prime s} = a^{\prime b} = -a^{\prime u} \quad v^{\prime J_1} = 
\frac{2\left(1 - 7t_W^2\right)}{f\left(t_W\right)}, \\ 
v^{\prime J_2} = v^{\prime J_3} = 
-\frac{2\left(1 - 5t_W^2\right)}{f\left(t_W\right)}, & \quad &
a^{\prime J_2} = a^{\prime J_2} = a^{\prime J_2} = a^{\prime J_1} = -\frac{2\left(1 + 
3t_W^2\right)}{f\left(t_W\right)},
\end{eqnarray} 
\label{coef}
\end{mathletters} 
with $f^2\left(t_W\right) = 3\left(1 + 4t_W^2\right)$. As we have commented in
the introduction and by inspection of Eqs. (\ref{quark}), (\ref{lagrb}) and
(\ref{lagrc}), we conclude that the heavy-leptons $P_a$ belong to
another class of exotic particles differently of the heavy-lepton
classes usually considered in the literature. Thus, the present
experimental limits do not apply directly to them \cite{Gea00} (see also
Ref. \cite{PT93}). Therefore, the 3-3-1 heavy-leptons phenomenology
deserves more detailed studies.\par 

%%%%%%%%%%%%%%%%%%%%%%%%%

\section{LUMINOSITIES}    

Let us now analyze the case of elastic $e^{-} e^{+}$ scattering. The $\gamma
\gamma$ differential luminosity is given by 

\begin{eqnarray}
\left (\frac{d \rm L^{el}}{d\tau} \right )_{\gamma \gamma/\ell \ell}  =  \int_{\tau}^{1} \frac{dx_{1}}{x_{1}} f_{\gamma/\ell} (x_{1})    
f_{\gamma/\ell} (x_{2} = \tau/x_{1})   \; ,
\end{eqnarray}  
where $\tau = x_{1} x_{2}$ and $f_{\gamma/\ell} (x)$ is the effective
photon approximation for the photon into the lepton, which is defined by

\begin{eqnarray}
f_{\gamma/\ell} (x) = \frac{\alpha}{2\pi} \frac{1+ (1- x)^{2}}{x} \ln \frac{s}{4 m_{e}^{2}} \; ,
\nonumber 
\end{eqnarray} 
where $x$ is the longitudinal momentum fraction of the lepton carried off
by the photon, $s$ is the center-of-mass energy of the $e^{-} e^{+}$
pair and $m_{e}$ is the electron mass. 


The $ZZ$ differential luminosity for elastic $e^{-} e^{+}$ scattering is given by 

\begin{eqnarray}
\left (\frac{d \rm L^{el}}{d\tau} \right )_{Z Z/\ell \ell}  = \int_{\tau}^{1} \frac{dx_{1}}{x_{1}} f_{Z/\ell} (x_{1})    
f_{Z/\ell} (x_{2} = \tau/x_{1})   \; ,
\nonumber
\end{eqnarray} 
where $f_{Z/\ell}(x)$ is the distribution function for finding a boson $Z$
of transverse and longitudinal helicities in a fermion with energy
$\sqrt{s}$ in the limit $\sqrt{s} \ge 2M_{Z}$ and which has the
following forms

\begin{eqnarray}
f^{\pm T}_{Z/\ell} (x) = \frac{\alpha}{4 \pi x \sin^{2}{\theta_{W}} \cos^{2}{\theta_{W}}} \left [ \left ( g_{V}^{\ell} \mp g_{A}^{\ell} \right )^{2} +  \left ( g_{V}^{\ell} \pm g_{A}^{\ell} \right )^{2} (1- x)^{2} \right ] \ln \frac{s}{M_{Z}^{2}} \; ,
\nonumber 
\end{eqnarray}  

\begin{eqnarray}
f^{L}_{Z/\ell} (x) = \frac{\alpha}{\pi \sin^{2}{\theta_{W}} \cos^{2}{\theta_{W}}} \left [ \left ( g_{V}^{\ell} \right )^{2} +  \left( g_{A}^{\ell} \right )^{2} \right ]  \frac{1- x}{x}  \; ,
\nonumber 
\end{eqnarray}  
where the $g_{V}^{\ell}$ and $g_{A}^{\ell}$ are the vector and axial-vector coupling,
respectively.


{}For the $Z \gamma$ differential luminosity for elastic $e^{-} e^{+}$ scattering we have 
the expression

\[
\left (\frac{d \rm L^{el}}{d\tau} \right )_{Z \gamma /\ell \ell}  = \int_{\tau}^{1} \frac{dx_{1}}{x_{1}} f_{Z/\ell} (x_{1}) f_{\gamma /\ell} (x_{2} = \tau /x_{1}) 
\]


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{CROSS SECTION PRODUCTION}

\label{secIV}

\subsection{$e^{-} e^{+} \rightarrow P^{-} P^{+}$}

Pair production of exotic particles is, to a very good approximation, a
model independent process, since it proceeds through a well known
electroweak interaction. This production mechanism can be studied
through the analysis of the reactions $e^{-} e^{+} \rightarrow P^{-}
P^{+}$, provided that there is enough available energy ($\sqrt{s} \geq
2M_{P}$). We will analyze the following processes for pair production of
exotic heavy leptons: $e^{-} e^{+} \rightarrow P^{-} P^{+}$, $e^{-}
e^{+} \rightarrow \gamma \gamma \rightarrow P^{-} P^{+}$, $e^{-} e^{+}
\rightarrow Z \gamma \rightarrow P^{-} P^{+}$ and $e^{-} e^{+}
\rightarrow Z Z \rightarrow P^{-} P^{+}$, the first process take place
through the exchange of a photon, a boson $Z^{0}$ and ${Z^{0}}^{'}$ in
the $s$ channel, while the others processes take place through the
exchange of heavy lepton in the $t$ and $u$ channel.


Using the interactions Lagrangians (8a),(8b) and (8c), it is easy to
evaluate the cross section for the process $e^{+} e^{-} \rightarrow
P^{+} P^{-}$, involving a neutral current, from which we obtain:

\begin{eqnarray} 
\left (\frac{d \sigma}{d\cos \theta} \right )_{P^+P^-} =  &&\frac{\beta 
\alpha^{2} \pi} {s^{3}} \Biggl \{ \left[ 2 s M_P^{2}
+ \left(M_P^{2} - t\right)^{2} + \left(M_P^{2} - u\right)^{2}  \right]  
\nonumber \\
&&+ \frac{1}{2 \sin^{2} \theta_{W} \cos^{2} \theta_{W} \left(s -
M_{Z,Z'}^{2} + i M_{Z,Z'} \Gamma_{Z,Z'}\right)} 
\left[ 2s M_P^{2} {g^\prime}_{V}^{PP} g_{V}^{\ell}\right. \nonumber \\ 
&& \left. + {g^\prime}_{V}^{PP} g_{V}^{\ell} \left[\left(M_P^{2} - t\right)^{2} + 
\left(M_P^{2} - u\right)^{2}\right] + {g^\prime}_{A}^{PP} g_{A}^{\ell} \left( \left(M_P^{2} - u\right)^{2} \right.\right. \nonumber \\ && \left.\left. -\left(M_P^{2} - t\right)^{2}\right) \right]  \Biggr \} + \frac{\beta \pi \alpha^{2}}{16 \cos^{4} \theta_{W} \sin^{4} \theta_{W}} \frac{1}{s \left(s - M_{Z,Z'}^{2} + i M_{Z,Z'} \Gamma_{Z,Z'}\right)^{2}} \nonumber \\
&& \times\Biggl \{\left[\left({g^\prime}_V^{PP}\right)^2 + 
\left({g^\prime}_{A}^{PP}\right)^{2}\right]\left[\left(g_V^\ell \right)^2 +  
\left(g_{A}^{\ell}\right)^{2}\right] \left[\left(M_P^{2} - u\right)^{2}+\left(M_P^{2} - t\right)^{2} \right]  \nonumber \\
&& + 2 s M_{P}^{2} \left[\left({g^\prime}_{V}^{PP}\right)^{2} - 
\left({g^\prime}_{A}^{PP}\right)^{2}\right]\left[\left(g_{V}^{\ell}\right)^{2} + 
\left(g_{A}^\ell \right)^{2}\right]   \nonumber   \\
&& + 4{g^\prime}_{V}^{PP}{g^\prime}_{A}^{PP} g_{V}^{\ell} g_{A}^{\ell} 
\left[\left(M_P^{2} - u\right)^{2} - \left(M_P^{2} - t\right)^{2}\right] \Biggr 
\}  \nonumber   \\
&&+ \frac{\beta \pi \alpha^{2}}{8 \sin^{4} \theta_{W} \cos^{4} 
\theta_{W} s\left(s- M_{Z}^{2} + i M_{Z} \Gamma_{Z}\right) \left(s- M_{Z'}^{2}+ i M_{Z'} \Gamma_{Z'}\right)}   \nonumber    \\
&&\Biggl \{2sM_{P}^{2} \left(g_{V}^{\ell} + g_{A}^{\ell}\right) 
\left(g_{V}^{PP}{g^\prime}_{V}^{PP} - g_{A}^{PP}{g^\prime}_{A}^{PP}\right) 
+ \left(M_{P}^{2}- t\right)^{2}  \left[  \left( \left(g_{V}^{\ell}\right)^{2} + 
\left(g_{A}^{\ell}\right)^{2}\right)\right.  \nonumber   \\  
&& \left. g_{V}^{PP}{g^\prime}_{V}^{PP} + g_{A}^{PP}{g^\prime}_{A}^{PP} - 2g_{V}^{\ell} g_{A}^{\ell} g_{V}^{PP}{g^\prime}_{A}^{PP} - 
2g_{V}^{\ell}g_{A}^{\ell}g_{A}^{PP}{g^\prime}_{V}^{PP}\right]  \nonumber \\
&& + \left(M_{P}^{2}- u\right)^{2}  \left[  \left( \left(g_{V}^{\ell}\right)^{2} + 
\left(g_{A}^{\ell}\right)^{2}\right) 
\left(g_{V}^{PP}{g^\prime}_{V}^{PP} + g_{A}^{PP}{g^\prime}_{A}^{PP}\right)\right.  \nonumber \\
&& \left. + 2g_{V}^{\ell} g_{A}^{\ell} g_{V}^{PP}{g^\prime}_{A}^{PP} + 2 
g_{V}^{\ell}g_{A}^{\ell} g_{A}^{PP}{g^\prime}_{V}^{PP}\right] \Biggr\},
\end{eqnarray}\\
where 

\[
g_{V,A}^{PP} = \frac{a_L \pm a_R}{2}, \qquad {g^\prime}_{V,A}^{PP} = 
\frac{a^\prime_{L} \pm a^\prime_{R}}{2}.
\]
The primes $\left(^\prime\right)$ is for the case when we take a boson
$Z'$, $\Gamma_{Z,Z'}$ are the total width of the boson Z and $Z'$
\cite{MT02}, $\beta = \sqrt{1 - 4 M_P^{2}/s}$ is the velocity of the
heavy-lepton in the c. m. of the process, $\alpha$ is the fine structure
constant, which we take equal to $\alpha =1/128$, $g^{\ell}_{V, A}$ are
the standard coupling constants, $M_{Z}$ is the mass of the $Z$ boson,
$\sqrt{s}$ is the center of mass energy of the $e^{-} e^{+}$ system, $t
= M_{P}^{2} - (1 - \beta \cos \theta)s/2$ and $u = M_{P}^{2} - (1 +
\beta \cos \theta)s/2$, where $\theta$ is the angle between the
heavy-lepton and the incident electron, in the c. m. frame. For
$Z^\prime$ boson we take $M_{Z^\prime} = \left(0.6 - 3\right)$ TeV,
since $M_{Z^\prime}$ is proportional to the VEV $v_\chi$ \cite{PP92,FR92}.
For the standard model parameters we assume PDG values, {\it i. e.},
$M_Z = 91.02$ GeV, $\sin^2{\theta_W} = 0.2315$ and $M_W = 80.33$ GeV
\cite{Gea00}. In Figs. 1 and 2, we show the cross sections $\sigma
(e^{-} e^{+} \rightarrow P^{-} P^{+})$ for the NLC and CLIC.  \par


Another way to produce a pair of heavy exotic leptons is through the
elastic reactions of the type $e^{-} e^{+} \rightarrow \gamma \gamma
\rightarrow P^{-} P^{+}$, $e^{-} e^{+} \rightarrow Z \gamma \rightarrow
P^{-} P^{+}$ and $e^{-} e^{+} \rightarrow Z Z \rightarrow P^{-} P^{+}$ 
These three processes take place through the exchange of the exotic lepton
in the $t$ and $u$ channels. So the cross section for the production of
a pair of $P^{-} P^{+}$ in the $e^{-} e^{+}$ collision can be obtained by
convoluting the cross section for the subprocesses $\gamma \gamma
\rightarrow P^{-} P^{+}$, $Z \gamma \rightarrow P^{-} P^{+}$ and $Z Z
\rightarrow P^{-} P^{+}$, with the two photon, Z$\gamma$ and ZZ
luminosities in these collisions, that is

\[
\sigma = \int_{\tau_{min}}^{1} \frac{d \it{L}}{d \tau} d\tau \ \hat{\sigma} (\hat{s} = x_{1} x_{2} s) = \int_{\tau_{min}}^{1}  \int_{\ln{\sqrt(\tau)}}^{-\ln{\sqrt(\tau)}} \frac{dx_{1}}{x_{1}} f_{V/\ell} (x_{1}) f_{V/\ell} (x_{2}) \int \frac{d \hat{\sigma}}{dcos} dcos 
\]
where $V = \gamma, Z$. The subprocess cross section for two photon
$P^{-} P^{+}$ production via elastic collisions of electron-positron is 

\begin{eqnarray} 
\left (\frac{d \sigma}{d\cos \theta} \right )_{\gamma \gamma} = &&\frac{\beta \alpha^{2} \pi}{s} \bigl [\frac{1}{(t - M_{P}^{2})^{2}} (-M_{P}^{4}- 3 M_{P}^{2} t - M_{P}^{2} u + tu)  \nonumber \\
&&+ \frac{1}{(u - M_{P}^{2})^{2}} (-M_{P}^{4}- M_{P}^{2} t - 3 M_{P}^{2} u + tu) \nonumber  \\ 
&&+ \frac{2}{(t - M_{P}^{2}) (u - M_{P}^{2})} (-2 M_{P}^{4} - M_{P}^{2} t - M_{P}^{2} u ) \bigr ]  \; ,
\end{eqnarray}
where $M_{P}$ is the mass of the exotic lepton, $\hat{t} = M_{P}^{2}-
\frac{\hat{s}}{2} (1- \beta cos \theta)$ and $\hat{u} = M_{P}^{2}-
\frac{\hat{s}}{2} (1+ \beta cos \theta)$ refer to the exchanged momenta
squared, corresponding to the direct and crossed diagrams for the two
photon, with $\beta$ being the $P$ velocity in the subprocess c.m. and
$\theta$ its angle with respect to the incident electron in this frame. 


The contribution of the subprocesses cross sections for $Z \gamma$ and $Z
Z$ luminosities to the total cross section can be shown to be very small,
therefore  
we do not
show here the explicit calculation for them, but we present their results
in Fig. $3$ for the CLIC.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{$e^{-} e^{+} \rightarrow Q \bar{Q}$}

The production of exotic quarks was already studied by both of the
authors \cite{cie,tonpe}, so that in this subsection we will restrict its
study
through the analysis of the reaction $e^{-} e^{+} \rightarrow Q
\bar{Q}$ in the case that there is enough available energy ($\sqrt{s} \ge
2 M_{Q}$). This process takes place through the exchange of a photon, Z
and a $Z^{'}$ in the s channel.


Using the interaction Lagrangians given in Sec. II, we can evaluate the
cross section involving a neutral current to obtain

\begin{eqnarray} 
\left (\frac{d \sigma}{d\cos \theta} \right )_{Q \bar{Q}} =  &&\frac{N_{c} \beta_{Q} 
\alpha^{2} \pi} {s^{3}} \Biggl \{ \left[ c_{q}^{2} (2 s M_Q^{2}
+ \left(M_Q^{2} - t\right)^{2} + \left(M_Q^{2} - u\right)^{2} )  \right]  
\nonumber \\
&&+ \frac{c_{q}}{2 \sin^{2} \theta_{W} \cos^{2} \theta_{W} \left(s -
M_{Z,Z'}^{2} + i M_{Z,Z'} \Gamma_{Z,Z'}\right)} 
\left[ 2s M_Q^{2} {g^\prime}_{V}^{Q \bar{Q}} g_{V}^{\ell}\right. \nonumber \\ 
&& \left. + {g^\prime}_{V}^{Q \bar{Q}} g_{V}^{\ell} \left[\left(M_Q^{2} - t\right)^{2} + 
\left(M_Q^{2} - u\right)^{2}\right] + {g^\prime}_{A}^{Q \bar{Q}} g_{A}^{\ell} \left( \left(M_Q^{2} - u\right)^{2} \right.\right. \nonumber \\ && \left.\left. -\left(M_Q^{2} - t\right)^{2}\right) \right]  \Biggr \} + \frac{\beta \pi \alpha^{2}}{16 \cos^{4} \theta_{W} \sin^{4} \theta_{W}} \frac{1}{s \left(s - M_{Z,Z'}^{2} + i M_{Z,Z'} \Gamma_{Z,Z'}\right)^{2}} \nonumber \\
&& \times\Biggl \{\left[\left({g^\prime}_V^{Q \bar{Q}}\right)^2 + 
\left({g^\prime}_{A}^{Q \bar{Q}}\right)^{2}\right]\left[\left(g_V^\ell \right)^2 +  
\left(g_{A}^{\ell}\right)^{2}\right] \left[\left(M_Q^{2} - u\right)^{2}+\left(M_Q^{2} - t\right)^{2} \right]  \nonumber \\
&& + 2 s M_{Q}^{2} \left[\left({g^\prime}_{V}^{Q \bar{Q}}\right)^{2} - 
\left({g^\prime}_{A}^{Q \bar{Q}}\right)^{2}\right]\left[\left(g_{V}^{\ell}\right)^{2} + 
\left(g_{A}^\ell \right)^{2}\right]   \nonumber   \\
&& + 4{g^\prime}_{V}^{Q \bar{Q}}{g^\prime}_{A}^{Q \bar{Q}} g_{V}^{\ell} g_{A}^{\ell} 
\left[\left(M_Q^{2} - u\right)^{2} - \left(M_Q^{2} - t\right)^{2}\right] \Biggr 
\}  \nonumber   \\
&&+ \frac{\beta \pi \alpha^{2}}{8 \sin^{4} \theta_{W} \cos^{4} 
\theta_{W} s\left(s- M_{Z}^{2} + i M_{Z} \Gamma_{Z}\right) \left(s- M_{Z'}^{2}+ i M_{Z'} \Gamma_{Z'}\right)}   \nonumber    \\
&&\Biggl \{2sM_{Q}^{2} \left(g_{V}^{\ell} + g_{A}^{\ell}\right) 
\left(g_{V}^{Q \bar{Q}}{g^\prime}_{V}^{Q \bar{Q}} - g_{A}^{Q \bar{Q}}{g^\prime}_{A}^{Q \bar{Q}}\right) 
+ \left(M_{Q}^{2}- t\right)^{2}  \left[  \left( \left(g_{V}^{\ell}\right)^{2} + 
\left(g_{A}^{\ell}\right)^{2}\right)\right.  \nonumber   \\  
&& \left. g_{V}^{Q \bar{Q}}{g^\prime}_{V}^{Q \bar{Q}} + g_{A}^{Q \bar{Q}}{g^\prime}_{A}^{Q \bar{Q}} - 2g_{V}^{\ell} g_{A}^{\ell} g_{V}^{Q \bar{Q}}{g^\prime}_{A}^{Q \bar{Q}} - 
2g_{V}^{\ell}g_{A}^{\ell}g_{A}^{Q \bar{Q}}{g^\prime}_{V}^{Q \bar{Q}}\right]  \nonumber \\
&& + \left(M_{Q}^{2}- u\right)^{2}  \left[  \left( \left(g_{V}^{\ell}\right)^{2} + 
\left(g_{A}^{\ell}\right)^{2}\right) 
\left(g_{V}^{Q \bar{Q}}{g^\prime}_{V}^{Q \bar{Q}} + g_{A}^{Q \bar{Q}}{g^\prime}_{A}^{Q \bar{Q}}\right)\right.  \nonumber \\
&& \left. + 2g_{V}^{\ell} g_{A}^{\ell} g_{V}^{Q \bar{Q}}{g^\prime}_{A}^{Q \bar{Q}} + 2 
g_{V}^{\ell}g_{A}^{\ell} g_{A}^{Q \bar{Q}}{g^\prime}_{V}^{Q \bar{Q}}\right] \Biggr\},
\end{eqnarray}
where $\beta_{Q} = \sqrt{1- 4 M_{Q}^{2}/s}$ is the velocity of the
exotic-quark in the c.m. of the process, $c_{q}$ is the charge of the
quark, $Q$ is the exotic quark and $\bar{Q}$ the exotic antiquark,
$\sqrt{s}$ is the center of mass energy of the $e^{-} e^{+}$ system, $t
= M_{Q}^{2} - \frac{s}{2} (1 - \beta \cos \theta)$ and {} $u = M_{Q}^{2}
- \frac{s}{2} (1 + \beta \cos \theta)$, where $\theta$ is the angle
between the exotic quark and the incident eletron, in the c.m. frame and
the couplings $g_{V}^{QQ}$ and $g_{A}^{QQ}$ are given in Sec. II.


In order to analyze the indirect evidence for a boson
$Z^{'}$, we compute the production of the quarks as in the standard model as in the $3-3-1$ model.  As a result we find that in the $3-3-1$ model at high
energies  there will be many more dijets than
expected in the scope of the standard model. In Fig. $4$, we show the
result for the
cross section $\sigma(e^{-} e^{+} \rightarrow q \bar{q} (Q \bar{Q}))$ as
a function of center of mass energy for different values of the boson
mass $M_{Z^{'}}$. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{$e^{-} e^{+} \rightarrow N_{1} N_{2}$}

Given the recent indications of the existence of massive 
neutrinos, in this section we also will study them. Some examples of
the relevance of
their study are the deficit of
solar electron neutrinos whose flux falls below that predicted by the
standard solar model \cite{bahc}, the neutrino oscillations, where the
electron neutrinos partially convert to muon neutrinos within the
interior of the sun \cite{mikh} and the need for explications concerning
hot dark matter in cosmology \cite{cald}. 

We study the production of massive neutrinos through the analysis of the
reaction $e^{-} e^{+} \rightarrow N_{1} N_{2}$. This process takes place
through the exchange of the bosons $Z, Z^{'}$ in the $s$ channel. Using
the interactions Lagrangian given by Eqs. (8a) and (8b), we evaluate the 
cross section, obtaining

\begin{eqnarray} 
\left (\frac{d \sigma}{d\cos \theta} \right )_{N_{1} N_{2}} = &&\frac{\beta_{N} \alpha^{2} \pi}{32 s \sin^{4} \theta_{W} \cos^{4} \theta_{W}} \bigl [\frac{1}{\left(s -M_{Z,Z'}^{2} + i M_{Z,Z'} \Gamma_{Z,Z'}\right)} \bigl (2 M_{N}^{4} \left ({g_{V}^{\ell}}^{2} + {g_{A}^{\ell}}^{2} \right )  \nonumber  \\
&&- 2 M_{N}^{2} t \left (g_{V}^{\ell}- g_{A}^{\ell} \right)^{2}  
- 2 M_{N}^{2} u \left (g_{V}^{\ell} + g_{A}^{\ell} \right )^{2} + t^{2} \left (g_{V}^{\ell}- g_{A}^{\ell} \right)^{2} + u^{2} \left (g_{V}^{\ell} + g_{A}^{\ell} \right )^{2}  \bigr )  \nonumber \\
&& +\frac{2}{\left(s- M_{Z}^{2} + i M_{Z} \Gamma_{Z}\right) \left(s- M_{Z'}^{2}+ i M_{Z'} \Gamma_{Z'}\right)} \bigl (2 M_{N}^{4} (g_{V}^{\ell} g_{V}^{\ell'} + g_{A}^{\ell} g_{A}^{\ell'})  \nonumber  \\
&&- 2 M_{N}^{2} g_{V}^{\ell} (g_{V}^{\ell'}- g_{A}^{\ell'}) (t - u)+ 2 M_{N}^{2} g_{A}^{\ell} t  (g_{V}^{\ell'}- g_{A}^{\ell'}) - 2 M_{N}^{2} g_{A}^{\ell} u (g_{V}^{\ell'}+ g_{A}^{\ell'})  \nonumber  \\
&&+ t^{2} (g_{V}^{\ell'}- g_{A}^{\ell'}) (g_{V}^{\ell}- g_{A}^{\ell}) + u^{2} (g_{V}^{\ell'}+ g_{A}^{\ell'}) (g_{V}^{\ell}+ g_{A}^{\ell}) \bigr ]  \; ,
\end{eqnarray}
where $\beta_{N} = \sqrt{1- 4 M_{N}^{2}/s}$ is the velocity of
exotic-neutrino in the c.m. of the process and $M_{Z,Z'}$ is the mass of
the boson $Z(Z')$. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{RESULTS AND CONCLUSIONS}

In the following we present the cross section for the process
$e^{+} e^{-} \rightarrow P^{+} P^{-}, (\bar{Q} Q), (N_{1} N_{2})$ for
the NLC and CLIC. In all calculations we take $\sin^{2} {\theta_W} =
0.2315$, $M_Z = 91.188$ GeV, and the mass of the heavy exotic lepton
equal to $200$ GeV.


In Fig. $1$, we show the cross section $\sigma (e^{-} e^{+}
\rightarrow P^{-} P^{+})$ as a function of $M_{P}$. Taking into account
that the expected integrated luminosity for the NLC, will be of order of
$6 \times 10^{4}$ pb$^{-1}$/yr, there will a total of: $\simeq 2.5 \times
10^{4}$ heavy exotic leptons pairs produced per year, considering
$M_{Z}^{'} = 1200$ GeV, while for the $M_{Z'} = 2000$ GeV the production
will be of order of $2.2 \times 10^{4}$. 
 

In Fig. $2$, taking into account that the integrated luminosity for the
CLIC will be of order of $2 \times 10^{5}$ pb$^{-1}$/yr, then the
statistics that we can expect for this collider is a little larger. So
for the process $e^{-} e^{+} \rightarrow P^{-} P^{+}$, considering the
mass of the boson $Z^{'}$ equal to $1200$ GeV, we will have a total of
$\simeq 2 \times 10^{5}$ lepton pairs produced per year, while for
$M_{Z'}= 2000$ GeV will be $\simeq 3 \times 10^{4}$, respectively. For
both Figs. 1 and 2 we considered $M_{J_{1}} = 300$ GeV, $M_{J_{2}} = 400$ GeV,
$M_{J_{3}} = 600$ GeV and $M_{V} = 800$ GeV. 


In Fig. $3$, we show the pair production of exotic heavy leptons through
the elastic reactions, so the statistics that we can expect for the CLIC
collider, for photon-photon $P^{-} P^{+}$ production, will be of order
of $\simeq 2 \times 10^{4}$ lepton pairs produced per year, while for
the $Z \gamma$ will be of $\simeq 200$ events per year and for the $ZZ$
will be very small. It should be noted that here was taken only the
transverse helicity of the boson Z, since the longitudinal one gives a
small contribution.


In Fig. $4$, we compare the standard cross section $\sigma (e^{-} e^{+}
\rightarrow q \bar{q})$ \ with the production cross section $\sigma
(e^{-} e^{+} \rightarrow q \bar{q} + Q \bar{Q})$, when the $3-3-1$ model
is applied. We see from these results that using the $3-3-1$ model we
will have more dijets than using the standard model at high energies.
This figure was obtained imposing the cut $|\cos \theta| < 0.95$ and
assuming three bosons $Z'$ with masses equal to $800$, $1200$ and $2000$
GeV, respectively. This figure still show the resonance peaks
associated with the boson $Z'$. We have also considered for this figure
$M_{J_{1}} = 200$ GeV, $M_{J_{2}} = 220$ GeV, $M_{J_{3}} = 245$ GeV, whose
masses would be accessible to the NLC. 


In Figs. $5$ and $6$ we show the cross sections for the production of
exotic quarks $e^{+} e^{-} \rightarrow \bar{Q} Q$, in the colliders NLC
and CLIC. We see from these results that we can expect for the first
collider a total of $\simeq 3.6 \times 10^{5}$ heavy quark pairs
produced per year, considering the mass of the boson $Z^{'}$ equal to
$1.2$ and $2$ TeV. We see that the cross section for both masses of the
boson $Z^{'}$ is not different one from another. For the second
collider, the CLIC, we expect a total of $\simeq 2.4 \times
10^{6}$ exotic quarks for the mass of the boson $Z^{'}$ equal to $1.2$
TeV, while for $M_{Z}^{'} = 2$ TeV we obtain $4 \times 10^{5}$ events
per year. Here the cross sections are different one from another, which
is not the case for the NLC, this is due to the propagator, that for the
CLIC is larger than for the NLC.


In Figs. $7$ and $8$ we show the cross sections for the production of
exotic neutrinos, $e^{+} e^{-} \rightarrow N_{1} N_{2}$, in the
colliders NLC and CLIC. We see from these results that we can expect, in
the NLC, a total of around $1.5 \times 10^{3}$ heavy neutrinos pairs
produced per year for the mass of the boson $Z^{'}$ equal to $1.2$ TeV,
while for the mass equal to $M_{Z}^{'}= 2$ TeV, the total of events is
$1.3 \times 10^{3}$. We see that the cross sections are nearly equal. We
also have that the CLIC can produce a total of $2 \times 10^{4}$ pairs
of exotic neutrinos for the mass of the boson $Z^{'}$ equal to $1200$
GeV, while for $M_{Z}^{'} = 2$ TeV the number of events will be $5.8
\times 10^{3}$. The discrepancy between these cross sections in both
colliders has the same reason as above. Here, for both figures, we considered
$M_{J_{1}} = 300$ GeV, $M_{J_{2}} = 400$ GeV, $M_{J_{3}} = 600$ GeV and
$M_{V} = 800$ GeV. 


The main background for the signal, $e^{-} e^{+} \rightarrow P^{-} P^{+}
\rightarrow \bar{\nu} \bar{u} J_{1} \ (\nu u \bar{J_{1}} )$, can be found
e.g. in Ref. \cite{MT02}. The backgrounds for the signal,
$e^{-} e^{+} \rightarrow Q \bar{Q}
\rightarrow q \ell^{-} \ell^{-} (\bar{q} \ell^{+} \ell^{+})$, are shown
in \cite{tonpe}, and the backgrounds for heavy neutrinos are determined in
\cite{sim}. Here it is to remark that even so a detailed simulation of 
Monte
Carlo must be done in all cases to extract the signal from the background,
due to the possibility of production of additional jets,
the balances of energy that may occur if the missing energy may be
averaged out, and other small backgrounds, for example, for the signal
$q \ell^{-} \ell^{-} (\bar{q} \ell^{+} \ell^{+})$,
like  HZZ, WZZ and $q \bar{q} ZZ$.


In summary, we have shown in this work that in the context of the 3-3-1
model the signatures for heavy-fermions can be significant  in both the NLC 
and in the CLIC colliders. Our study indicates the possibility of obtaining a
clear signal of these new particles with a satisfactory number of
events.


\acknowledgements 

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 (Processo No.
99/07956-3) for full financial support. The authors are also
in debt with R. O. Ramos for a careful
reading of the manuscript. 



%^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^



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\newpage

\begin{center}
FIGURE CAPTIONS
\end{center}

%\vspace{0.5cm}   
%\begin{figure}  

{\bf Figure 1}: Total cross section for the process $e^{-} e^{+} \rightarrow P^-P^+$ as a 
function of $M_{P}$ at $\sqrt{s} = 500$ GeV: (a) $M_{Z'} = 1200$ GeV (solid line), and  (b) 2000 GeV (dashed line).  

%\label{fig:1}  
%\end{figure}  
%\begin{figure}  

{\bf Figure 2}: Total cross section for the process $e^{-} e^{+} \rightarrow P^-P^+$ as a 
function of $M_{P}$ at $\sqrt{s} = 1000$ GeV: (a) $M_{Z'} = 1200$ GeV (solid line), and  (b) $M_{Z'} = 2000$ GeV (dashed line).  

%\label{fig:2}  
%\end{figure}  
%\begin{figure}  

{\bf Figure 3}: Total cross section for the process $e^{-} e^{+} \rightarrow P^{-} P^{+}$ as a function of $M_{P}$ at $\sqrt{s} = 1000$ GeV for different elastic production mechanisms: (a) $\gamma \gamma$ (dot-dot-dashed line); (b) $Z\gamma$ (dashed line); (c) $Z Z$ (solid). 

%\label{fig:3}  
%\end{figure}  
%\begin{figure}  

{\bf Figure 4}: Total cross section versus the total c.m. energy $\sqrt{s}$ for the following masses of the gauge boson $Z^{'}$ (a) $M_{Z'} = 800$ GeV (dashed line), (b) $M_{Z'} = 1200$ GeV (dot-dot-dashed line), (c) $M_{Z'} = 2000$ GeV (dot-dashed line), (d) standard model (solid line). 
%\label{fig:4}  
%\end{figure}  
%\begin{figure}  

{\bf Figure 5}: Total cross section for the process $e^{-} e^{+} \rightarrow Q \bar{Q}$ as a function of $M_{Q}$ at $\sqrt{s} = 500$ GeV: (a) $M_{Z'} = 1200$ GeV (dot-dashed line), (b)$M_{Z'} = 2000$ GeV (solid line). 

%\label{fig:5}
%\end{figure}  
%\begin{figure}  

{\bf Figure 6}: Total cross section for the process $e^{-} e^{+} \rightarrow Q \bar{Q}$ as a function of $M_{Q}$ at $\sqrt{s} = 1000$ GeV: (a) $M_{Z'} = 1200$ GeV (dot-dashed line), (b)$M_{Z'} = 2000$ GeV (solid line). 

%\label{fig:6}
%\end{figure}  
%\begin{figure}  

{\bf Figure 7}: Total cross section for the process $e^{-} e^{+} \rightarrow N_{1} N_{2}$ as a function of $M_{N}$ at $\sqrt{s} = 500$ GeV: (a) $M_{Z'} = 1200$ GeV (dot-dashed line), (b)$M_{Z'} = 2000$ GeV (solid line). 

%\label{fig:7}
%\end{figure}  
%\begin{figure}  

{\bf Figure 8}: Total cross section for the process $e^{-} e^{+} \rightarrow N_{1} N_{2}$ as a function of $M_{N}$ at $\sqrt{s} = 1000$ GeV: (a) $M_{Z'} = 1200$ GeV (dot-dashed line), (b)$M_{Z'} = 2000$ GeV (solid line). 

%\label{fig:8}  
%\end{figure}  

\end{document}



