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\flushbottom
\preprint{\vbox{  
\hbox{IFT-P.014/2001}
\hbox{ LPM 01-25}  
\hbox
\hbox{March 2001} }}  
 
\begin{document}
\draft 
\title{Charginos and Neutralinos Production at 3-3-1 Supersymmetric Model in 
$e^-e^-$ Scattering}
\author{M. Capdequi-Peyran\`ere and M. C. Rodriguez\footnote{Permanent 
Address:Instituto de F\'\i sica  Te\'orica Universidade  Estadual Paulista }} 
\address{
Physique Math\'ematique et Th\'eorique, CNRS-UMR 5825 \\
Universit\'e Montpellier II, F-34095, Montpellier Cedex 5. } 
\maketitle
\begin{abstract}  

The goal of this article is to derive the Feynman rules involving charginos, 
neutralinos, double charged gauge bosons and sleptons in a 3-3-1 
supersymmetric model. Using these Feynman rules we will calculate the 
production of a double charged chargino with a neutralino and also the 
production of a pair of single charged charginos , both in an electron-
electron $e^-e^-$ process.

\pacs{PACS   numbers: 12.60.-i %Models beyond the standard model
12.60.Jv %Supersymmetric Model
13.85.Lg %Total cross section
} 

\end{abstract}

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

The Standard Model is exceedingly successful in describing leptons, 
quarks and their interactions. Nevertheless, the Standard Model is 
not considered as the ultimate theory since neither the fundamental 
parameters, masses and couplings, nor the symmetry pattern are 
predicted. Even though many aspects of the Standard Model are experimentally 
supported to a very accuracy, the embedding of the model into a 
more general framework is to be expected. The possibility 
of a gauge symmetry based on the following symmetry 
$SU(3)_{C} \otimes SU(3)_{L} \otimes U(1)_{N}$ (3-3-1) \cite{331} 
is particularly interesting, because it explains some fundamental questions 
that are eluded in the Standard Model \cite{331susy}. 

Recently was proposed the supersymmetric 
extension of the 3-3-1 model \cite{331susy}; in this kind of model the supersymmetry 
is naturally broken at a TeV scale, and the lightest scalar boson has an upper limit, 
which is  124.5 GeV for a given set of parameters at the tree level.  We must 
remember that in MSSM the bound using radiative corrections to the mass of the 
lightest Higgs scalar is 130 GeV \cite{radl}. On another hand no direct observation of a 
Higgs boson has been made yet and current direct searches constraint its mass to 
$M_h>113$ GeV \cite{exp1} in the minimal Standard Model.

Linear colliders would be most versatile tools in experimental high 
energy physics. A large international effort is currently under way 
to study the technical feasibility and physics possibilities of
linear $e^+ e^-$ colliders in the TeV range. A number of designs 
have already been proposed (NLC, JLC, TESLA, CLIC, VLEPP, ...) and 
several workshops have recently been devoted to the subject. Not only they can provide 
$e^+ e^-$ collisions and high luminosities, but also very energetic beams of real photons. 
One could thus exploit $\gamma \gamma$, $e^- \gamma$ and  even $e^- e^-$ 
collisions for physics studies.


These exciting prospects have prompted a growing number of theoretical 
studies devoted to the investigation of the physics potential of such 
new $e^-e^-$ accelerator experiments. From the experimental point of view such 
a machine would provide relevant new possibilities, including the use of 
highly polarized beams or the production of high energy electron beam. 
Of course, in the realm of the Standard Model this option is not particularly 
interesting because mainly M\o ller scattering and bremsstrahlung events are 
to be observed. However, it is just for that reason that $e^-e^-$ collisions 
can provide crucial information on exotic processes, in particular on 
processes involving lepton and/or fermion number violation. Therefore, new 
perspectives emerge in detecting new physics beyond the Standard Model in 
processes having non-zero initial electric charge (and non-zero lepton number) 
like in electron-electron $(e^-e^-)$ process.



In this paper we will explicitely work out two $e^-e^-$ processes and we will show 
on an example a difference between MSSM and the 3-3-1 supersymmetric 
model. This paper is organized as follows. The model is  outlined in section 
\ref{sec1},  where leptons, sleptons, scalars, higgsinos, gauge bosons and 
gauginos are defined. We derive the mass spectrum in section \ref{sec2}, while 
the interactions are get in section \ref{sec3}. In order to give some 
examples, we derive the Feynman rules in the section \ref{sec4} while the 
examples are worked out in section \ref{sec5}. Our conclusions are given in 
\ref{sec:con}. 


The lagrangian is given in Appendix 
\ref{sec:lagrangian}, while in Appendix \ref{c} we show all non diagonal mass 
matrices of the charginos and neutralinos. The procedure to write two-components 
spinors in terms of four-components spinors is given in Appendix \ref{2t4}. 
The differential cross sections of the processes we have calculated are given in 
the Appendix \ref{a6}.

\section{Particle Content}
\label{sec1}

First of all, let us outline the model, following the notation given in 
\cite{331susy}. We are writing here the particle content that we are using in 
this study, so it means we are not going to discuss quarks and squarks.

The leptons and sleptons are given by
\begin{equation} 
   L_{l} = 
\left( \begin{array}{ccc} \nu_{l}& 
                  l &
                  l^{c}          
\end{array} \right)^t_{L}, \,\
   \tilde{L}_{l} = 
      \left( \begin{array}{ccc} \tilde{ \nu}_{l} & 
                  \tilde{l} &
                  \tilde{l}^{c}          \end{array} \right)^t_{L}, 
\,\ l= e, \mu , \tau .
\label{sl}
\end{equation}

The scalars of our theory are 
\begin{eqnarray} 
\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), \quad
S = 
      \left( \begin{array}{ccc} 
\sigma^{0}_{1}& \frac{h^{+}_{2}}{ \sqrt{2}}& \frac{h^{-}_{1}}{ \sqrt{2}} \\ 
\frac{h^{+}_{2}}{ \sqrt{2}}& H^{++}_{1}& \frac{\sigma^{0}_{2}}{ \sqrt{2}} \\
\frac{h^{-}_{1}}{ \sqrt{2}}& \frac{\sigma^{0}_{2}}{ \sqrt{2}}&  H^{--}_{2} 
\end{array} \right), 
\label{defscalars} 
\end{eqnarray}
the higgsinos, the superpartners of the scalars, are defined as
\begin{eqnarray} 
\tilde{ \eta} &=& 
      \left( \begin{array}{c} \tilde{ \eta}^{0} \\ 
                  \tilde{ \eta}^{-}_{1} \\
                  \tilde{ \eta}^{+}_{2}          \end{array} \right),\quad 
\tilde{ \rho} = 
      \left( \begin{array}{c} \tilde{ \rho}^{+} \\ 
                  \tilde{ \rho}^{0} \\
                  \tilde{ \rho}^{++}          \end{array} \right),\quad
\tilde{ \chi} = 
      \left( \begin{array}{c} \tilde{ \chi}^{-} \\ 
                  \tilde{ \chi}^{--} \\
                  \tilde{ \chi}^{0}          \end{array} \right), \quad 
\tilde{S} = 
      \left( \begin{array}{ccc} 
\tilde{\sigma}^{0}_{1}& \frac{ \tilde{h}^{+}_{2}}{ \sqrt{2}}& 
\frac{ \tilde{h}^{-}_{1}}{\sqrt{2}} \\ 
\frac{ \tilde{h}^{+}_{2}}{ \sqrt{2}}& \tilde{H}^{++}_{1}& 
\frac{ \tilde{\sigma}^{0}_{2}}{ \sqrt{2}} \\
\frac{ \tilde{h}^{-}_{1}}{ \sqrt{2}}& 
\frac{\tilde{\sigma}^{0}_{2}}{ \sqrt{2}}&  \tilde{H}^{--}_{2}        
\end{array} \right).
\label{shs}
\end{eqnarray}
To cancel chiral anomalies generated by these higgsinos we have to add the 
followings fields $\tilde{ \eta}^{\prime}$, $\tilde{ \rho}^{\prime}$, 
$\tilde{ \chi}^{\prime}$ and $\tilde{S}^{\prime}$ and their respectives scalars, see 
\cite{331susy}.


When one breaks the 3-3-1 symmetry to the $SU(3)_{C} \otimes U(1)_{EM}$, the 
scalars get the following vacuum expectation values (VEVs):
\begin{eqnarray} 
< \eta > &=& 
      \left( \begin{array}{c} v \\ 
                  0 \\
                  0          \end{array} \right),\quad 
< \rho > = 
      \left( \begin{array}{c} 0 \\ 
                  u \\
                  0          \end{array} \right),\quad 
< \chi > = 
      \left( \begin{array}{c} 0 \\ 
                  0 \\
                  w          \end{array} \right), \quad
< S > = 
      \left( \begin{array}{ccc} 
0& 0& 0 \\ 
0& 0& \frac{z}{ \sqrt{2}} \\
0& \frac{z}{ \sqrt{2}}&  0
\end{array} \right), 
\label{vev1} 
\end{eqnarray}
and similar expressions to the prime fields, where $v=v_{\eta}/ \sqrt{2}$, $u=v_{\rho}/ \sqrt{2}$, 
$w=v_{\chi}/ \sqrt{2}$, $z=v_{\sigma_2}/ \sqrt{2}$, $v^{\prime}=
v_{\eta^{\prime}}/ \sqrt{2}$, $u^{\prime}=v_{\rho^{\prime}}/ \sqrt{2}$, 
$w^{\prime}=v_{\chi^{\prime}}/ \sqrt{2}$, and $z^{\prime}=v_{\sigma^{\prime}_2}
/ \sqrt{2}$.


In the 3-3-1 supersymmetric model the bosons of the symmetry 
$SU(3)_L$ are $V^a$ and their superpartners, the gauginos, are $\lambda^a_A$, 
with $a=1, \cdots,8$. For the $U(1)_N$ symmetry, we have the boson $V$ and its 
superpartner is written as $\lambda_B$(that we call gaugino too).

In the usual 3-3-1 model \cite{331} the gauge bosons are defined as
\begin{eqnarray}
W^{ \pm}_{m}(x)&=&-\frac{1}{\sqrt{2}}(V^{1}_{m}(x) \mp i V^{2}_{m}(x)),
\,\
V^{ \pm}_{m}(x)=-\frac{1}{\sqrt{2}}(V^{4}_{m}(x) \pm i V^{5}_{m}(x)), 
\nonumber \\
U^{\pm \pm}_{m}(x) &=&- \frac{1}{\sqrt{2}}(V^{6}_{m}(x) \pm i V^{7}_{m}(x)), 
\,\
A_{m}(x) = \frac{1}{\sqrt{1+4t^{2}}}
\left[ (V^{3}_{m}(x)- \sqrt{3}V^{8}_{m}(x))t+V_{m} \right], 
\nonumber \\
Z^{0}_{m}(x)  &=&- \frac{1}{\sqrt{1+4t^{2}}}
\left[ \sqrt{1+3t^{2}}V^{3}_{m}(x)+ 
\frac{ \sqrt{3}t^{2}}{\sqrt{1+3t^{2}}}V^{8}_{m}(x)-
\frac{t}{\sqrt{1+3t^{2}}}V_{m}(x) \right], \nonumber \\
Z^{\prime 0}_{m}(x) &=& \frac{1}{\sqrt{1+3t^{2}}}
(V^{8}_{m}(x)+ \sqrt{3}tV_{m}(x)),
\label{defbosons}
\end{eqnarray}
where $t \equiv \tan \theta = \frac{g^{ \prime}}{g}$ and $g^{\prime}$ and $g$ 
are the gauge coupling constants of $U(1)$ and $SU(3)$, 
respectively. We can define the charged gauginos, in analogy with the gauge 
bosons, in the following way
\begin{eqnarray}
\lambda_{W}^{\pm}(x)&=& -\frac{1}{\sqrt{2}}(\lambda^{1}_{A}(x) \mp i 
\lambda^{2}_{A}(x)), \,\
\lambda_{V}^{\pm}(x)=-\frac{1}{\sqrt{2}}(\lambda^{4}_{A}(x) \pm i 
\lambda^{5}_{A}(x)),\nonumber \\
\lambda_{U}^{\pm \pm}(x) &=&-\frac{1}{\sqrt{2}}(\lambda^{6}_{A}(x) \pm i 
\lambda^{7}_{A}(x)).
\label{defgauginos}
\end{eqnarray}


\section{Mass Spectrum}
\label{sec2}

The higgsino mass term comes from ${\cal L}_{HMT}$, see Eq.(\ref{compsup}), 
the mass of gauginos is given by ${\cal L}^{\mbox{gaugino}}_{GMT}$, see 
Eq.(\ref{smt}) and the mixing term between gauginos and higgsinos is given by 
${\cal L}^{\mbox{scalar}}_{H \tilde{H} \tilde{V}}$, Eq.(\ref{mix1}). After the
mixture the physical states are double charged charginos, charginos and
neutralinos. Now we will discuss these states. 


\subsection{Double Charged Chargino}

Doing the calculation  we obtain the mass matrix of the double charged 
chargino in the following \footnote{Here we are considerating the 
conservation of ${\cal F}$ then $f_{2},f^{\prime}_{2}=0$ \cite{331susy}.}
way
\begin{eqnarray}
{\cal L}^{\mbox{double}}_{\mbox{mass}}&=&- m_{\lambda} 
\lambda^{--}_{U} \lambda^{++}_{U}-
\frac{\mu_{\rho}}{2} \tilde{\rho}^{ \prime --} \tilde{\rho}^{++}- 
\frac{\mu_{\chi}}{2} \tilde{\chi}^{--} \tilde{\chi}^{ \prime ++}- 
\frac{\mu_{S}}{2} ( 
\tilde{H}_{1}^{ \prime --} \tilde{H}_{1}^{++}+ 
\tilde{H}_{2}^{--} \tilde{H}_{2}^{ \prime ++}) \nonumber \\
&-&ig \left( u \tilde{\rho}^{++}-w^{\prime} \tilde{\chi}^{ \prime ++}- 
\frac{z}{\sqrt{2}} \tilde{H}^{++}_{1}+ \frac{z^{\prime}}{\sqrt{2}} 
\tilde{H}^{ \prime ++}_{2} \right) \lambda^{--}_U- ig \left( w 
\tilde{\chi}^{--}-u^{\prime} \tilde{\rho}^{ \prime --}- 
\frac{z}{\sqrt{2}} \tilde{H}^{--}_{2}+ \frac{z^{\prime}}{\sqrt{2}} 
\tilde{H}^{\prime --}_{1} \right) \lambda^{++}_U \nonumber \\
&+& \frac{f_{1}v}{3} \tilde{\chi}^{--} \tilde{\rho}^{++}-
f_{3}u \tilde{\chi}^{--} \tilde{H}_{1}^{++}- \sqrt{2}f_{3}z 
\tilde{\chi}^{--} \tilde{\rho}^{++}-f_{3}w \tilde{H}_{2}^{--} 
\tilde{\rho}^{++} \nonumber \\
&+& \frac{f^{\prime}_{1}v^{\prime}}{3} \tilde{\chi}^{ \prime ++} \tilde{\rho}^{ \prime --}-
f^{\prime}_{3}u^{\prime} \tilde{\chi}^{ \prime ++} \tilde{H}_{1}^{ \prime --}- 
\sqrt{2}f^{\prime}_{3}z^{\prime} \tilde{\chi}^{ \prime ++} \tilde{\rho}^{ \prime --}-
f^{\prime}_{3}w^{\prime} \tilde{H}_{2}^{ \prime ++} \tilde{\rho}^{ \prime --}+hc,
\label{dc1}
\end{eqnarray}
which can be writen in analogy with the MSSM\footnote{In the 
Appendix 
\ref{c} we show all non diagonal mass matrices of the charginos and 
neutralinos},
see Appendix \ref{c1}, as follows
\begin{eqnarray}
{\cal L}^{\mbox{double}}_{\mbox{mass}}&=&- \frac{1}{2} \left( \Psi^{\pm \pm} 
\right)^{t} Y^{\pm \pm} \Psi^{\pm \pm}+hc.
\label{y++}
\end{eqnarray}



The double chargino mass matrix is diagonalized using two rotation matrices, 
$A$ and $B$, defined by
\begin{eqnarray}
\tilde{ \chi}^{++}_{i}=A_{ij} \Psi^{++}_{j}, \,\
\tilde{ \chi}^{--}_{i}=B_{ij} \Psi^{--}_{j}, \,\ i,j=1, \cdots , 5. 
\label{2dc}
\end{eqnarray}
where $A$ and $B$ are unitarity matrices choosen such that
\begin{equation}
M_{DCM}=B^{*}TA^{-1}.
\end{equation}
To determine $A$ and $B$, we note that
\begin{equation}
M^{2}_{DCM}=AT^{t} \cdot TA^{-1}=B^{*}T \cdot T^{t}(B^{*})^{-1},
\end{equation}
which means that $A$ diagonalizes $T^{t} \cdot T$ while $B$ diagonalizes 
$T \cdot T^{t}$.


We define the following Dirac spinors to represent the mass 
eigenstates:
\begin{eqnarray}
\Psi(\tilde{ \chi}^{++}_{i})= \left( \begin{array}{cc}
             \tilde{ \chi}^{++}_{i} &
	     \bar{ \tilde{\chi}}^{--}_{i} 
\end{array} \right)^{t}, \,\
\Psi^{c}(\tilde{ \chi}^{--}_{i})= \left( \begin{array}{cc}
             \tilde{ \chi}^{--}_{i} &
	     \bar{ \tilde{\chi}}^{++}_{i} 
\end{array} \right)^{t},
\label{emassdou}
\end{eqnarray}
where $\tilde{ \chi}^{++}_{i}$ is the particle and $\tilde{ \chi}^{--}_{i}$ 
is the anti-particle( we are using the same notation as in \cite{mssm}). 



\subsection{Charged Chargino}

We can write the mass matrix of the charged 
chargino in the following way
\begin{eqnarray}
{\cal L}^{\mbox{unique}}_{\mbox{mass}}&=&- m_{\lambda} \left( 
\lambda^{-}_{V} \lambda^{+}_{V}+ \lambda^{-}_{W} \lambda^{+}_{W} \right)
- \frac{\mu_{\eta}}{2} \left( \tilde{\eta}_1^{-} \tilde{\eta}_{1}^{ \prime +}+ 
\tilde{\eta}_2^{ \prime -} \tilde{\eta}_{2}^{+} \right)
-\frac{\mu_{\rho}}{2} \tilde{\rho}^{ \prime -} \tilde{\rho}^{+}- 
\frac{\mu_{\chi}}{2} \tilde{\chi}^{-} \tilde{\chi}^{ \prime +} \nonumber \\
&-& \frac{\mu_{S}}{2} \left( 
\tilde{h}_{1}^{-} \tilde{h}_{1}^{ \prime +}+ \tilde{h}_{2}^{ \prime -} \tilde{h}_{2}^{+} \right) 
+ ig \left( v \tilde{\eta}_2^{+}-w^{ \prime} \tilde{\chi}^{ \prime +}- \frac{z}{2} \tilde{h}_{2}^{+} 
\right) \lambda^{-}_{V}+ig 
\left( w \tilde{\chi}^{-}-v^{ \prime} \tilde{\eta}^{ \prime -}_{2}+ 
\frac{z}{2} \tilde{h}_{2}^{ \prime -} \right) \lambda^{+}_{V} \nonumber \\
&-& ig \left( u \tilde{\rho}^{+}-v^{ \prime} \tilde{\eta}^{ \prime +}_{1}+ 
\frac{z}{2} \tilde{h}_{1}^{ \prime +} \right) \lambda^{-}_{W}-ig 
\left( v \tilde{\eta}_{1}^{-}-u^{ \prime} \tilde{\rho}^{ \prime -}- \frac{z}{2} 
\tilde{h}_{1}^{-} \right) \lambda^{+}_{W}- 
\frac{f_{1}u}{3} \tilde{\chi}^{-} \tilde{\eta}_{2}^{+}+
\frac{f_{1}w}{3} \tilde{\eta}_{1}^{-} \tilde{\rho}^{+} \nonumber \\ &-&  
\frac{f^{\prime}_{1}u^{ \prime}}{3} \tilde{\eta}_{2}^{ \prime -} \tilde{\chi}^{\prime +}+
\frac{f^{\prime}_{1}w^{ \prime}}{3} \tilde{\rho}^{\prime -} \tilde{\eta}_{1}^{\prime +}- 
\frac{f_{3}u}{\sqrt{2}} \tilde{\chi}^{-} \tilde{h}_{2}^{+}-
\frac{f_{3}w}{\sqrt{2}} \tilde{h}_{1}^{-} \tilde{\rho}^{+}- 
\frac{f^{\prime}_{3}u^{\prime}}{\sqrt{2}} \tilde{\chi}^{ \prime +} \tilde{h}_{2}^{ \prime -}-
\frac{f^{\prime}_{3}w^{\prime}}{\sqrt{2}} \tilde{h}_{1}^{ \prime +} \tilde{\rho}^{ \prime -}+
hc,
\label{uc1}
\end{eqnarray}
but Eq.(\ref{uc1}), see Appendix \ref{c2}, takes the form
\begin{eqnarray}
{\cal L}^{\mbox{unique}}_{\mbox{mass}}&=&- \frac{1}{2} \left( \Psi^{\pm} 
\right)^{t} Y^{\pm} \Psi^{\pm}+hc.
\label{y-}
\end{eqnarray}


The chargino mass matrix is diagonalized using two rotation matrices, 
$D$ and $E$, defined by
\begin{eqnarray}
\tilde{ \chi}^{+}_{i}=D_{ij} \Psi^{+}_{j}, \,\
\tilde{ \chi}^{-}_{i}=E_{ij} \Psi^{-}_{j}, \,\ i,j=1, \cdots , 8.
\label{2sc} 
\end{eqnarray}
then we can write the diagonal mass matrix( $D$ and $E$ are 
unitary) as
\begin{equation}
M_{SCM}=E^{*}XD^{-1}.
\end{equation}




As in the previous subsection we will define the following Dirac spinors:
\begin{eqnarray}
\Psi(\tilde{ \chi}^{+}_{i})= \left( \begin{array}{cc}
             \tilde{ \chi}^{+}_{i} &
	     \bar{ \tilde{\chi}}^{-}_{i} 
\end{array} \right)^{t}, \,\
\Psi^{c}(\tilde{ \chi}^{-}_{i})= \left( \begin{array}{cc}
             \tilde{ \chi}^{-}_{i} &
	     \bar{ \tilde{\chi}}^{+}_{i} 
\end{array} \right)^{t},
\label{emasssim}
\end{eqnarray}
where $\tilde{ \chi}^{+}_{i}$ is the particle and $\tilde{ \chi}^{-}_{i}$ 
is the anti-particle. This structure is the same as is the chargino sector of 
the MSSM \cite{mssm}.



\subsection{Neutralino}

For the neutralino case we have
\begin{eqnarray}
{\cal L}^{\mbox{neutralino}}_{\mbox{mass}}&=&- \frac{m_{\lambda}}{2} \left( 
\lambda^3_A 
\lambda^3_A+ \lambda^8_A \lambda^8_A \right)- \frac{m^{\prime}}{2} 
\lambda_B \lambda_B- \frac{\mu_{\eta}}{2} \tilde{\eta}^{0} \tilde{\eta}^{\prime 0}
-\frac{\mu_{\rho}}{2} \tilde{\rho}^{0} \tilde{\rho}^{\prime 0}- 
\frac{\mu_{\chi}}{2} \tilde{\chi}^{0} \tilde{\chi}^{\prime 0} \nonumber \\ 
&-& \frac{\mu_{S}}{2} \left( 
\tilde{\sigma}_{1}^{ \prime 0} \tilde{\sigma}_{1}^{0}+ 
\tilde{\sigma}_{2}^{ \prime 0} \tilde{\sigma}_{2}^{0} \right)- 
\frac{ig^{\prime}}{\sqrt{2}}(w \tilde{\chi}^{0}-u \tilde{\rho}^{0}+
u^{\prime} \tilde{\rho}^{\prime 0}-w^{\prime} \tilde{\chi}^{\prime 0}) 
\lambda_B
\nonumber \\
&-& \frac{ig}{\sqrt{2}} \left( u \tilde{\rho}^{0}- v \tilde{\eta}^{0}+ 
v^{\prime} \tilde{\eta}^{\prime 0}-u^{\prime} \tilde{\rho}^{\prime 0} 
+ \frac{z}{2} \tilde{\sigma}_{2}^{0}- 
\frac{z^{\prime}}{2} \tilde{\sigma}_{2}^{\prime 0} \right) \lambda^3_A 
\nonumber \\
&-& \frac{ig}{\sqrt{6}} \left( 2w \tilde{\chi}^{0}-u \tilde{\rho}^{0}-v 
\tilde{\eta}^{0}+
u^{\prime} \tilde{\rho}^{\prime 0}+v^{\prime} \tilde{\eta}^{\prime 0}-
2w^{\prime} \tilde{\chi}^{\prime 0}+ \frac{z}{2} \tilde{\sigma}_{2}^{0}- 
\frac{z^{\prime}}{2} \tilde{\sigma}_{2}^{\prime 0} \right) \lambda^8_A 
\nonumber \\
&+& \frac{f_{1}}{3} \left( u \tilde{\chi}^{0} \tilde{\eta}^{0}-
v \tilde{\chi}^{0} \tilde{\rho}^{0}-w \tilde{\eta}^{0} \tilde{\rho}^{0} 
\right)+\frac{f^{\prime}_{1}}{3} \left( 
u^{\prime} \tilde{\chi}^{\prime 0} \tilde{\eta}^{\prime 0}-
v^{\prime} \tilde{\chi}^{\prime 0} \tilde{\rho}^{\prime 0}-w^{\prime} 
\tilde{\eta}^{\prime 0} \tilde{\rho}^{\prime 0} \right) 
\nonumber \\
&-& \frac{f_{3}}{3} \left( u \tilde{\chi}^{0} \tilde{\sigma}_{2}^{0}+
w \tilde{\rho}^{0} \tilde{\sigma}_{2}^{0}+2z \tilde{\chi}^{0} 
\tilde{\rho}^{0} \right)-
\frac{f^{\prime}_{3}}{3} \left( u^{\prime} 
\tilde{\chi}^{\prime 0} \tilde{\sigma}_{2}^{\prime 0}+
w^{\prime} \tilde{\rho}^{\prime 0} \tilde{\sigma}_{2}^{\prime 0}+
2z^{\prime} \tilde{\chi}^{\prime 0} \tilde{\rho}^{\prime 0} \right)+hc
\label{neu1}
\end{eqnarray}
Eq.(\ref{neu1}) can be put in the following form, see Appendix \ref{c3}
\begin{eqnarray}
{\cal L}^{\mbox{neutralino}}_{\mbox{mass}}&=&- \frac{1}{2} \left( \Psi^{0}
\right)^{t} Y^{0} \Psi^{0}+hc.
\label{y0}
\end{eqnarray}

The neutralino mass matrix is diagonalized by a $8 \times 8$ rotation matrix 
$N$, a unitary matrix satisfying
\begin{equation}
M_{NMD}=N^{*}Y^{0}N^{-1},
\end{equation}
and the mass eigenstates are
\begin{eqnarray}
\tilde{ \chi}^{0}_{i}&=&N_{ij} \Psi^{0}_{j}, \,\ j=1, \cdots ,8.
\label{emasneu}
\end{eqnarray}



We can define the following Majorana spinor to represent the mass eigenstates
\begin{eqnarray}
\Psi(\tilde{ \chi}^{0}_{i})&=& \left( \begin{array}{cc}
             \tilde{ \chi}^{0}_{i} &
	     \bar{ \tilde{\chi}}^{0}_{i} 
\end{array} \right)^{t}.
\label{emassneu} 
\end{eqnarray}

In Supersymmetric Left-Right models, there are double charged 
higgsinos too \cite{LR}, and in both models the diagonalization of the 
charginos and neutralinos sectors is numerically performed.

\subsection{Sleptons}

We can write the slepton mass term as
\begin{eqnarray}
{\cal L}^{ \mbox{slepton}}_{ \mbox{mass}}&=& 
{\cal L}^{ \mbox{slep}}_{ \mbox{mass}}+{\cal L}^{\mbox{slepton}}_F+
{\cal L}^{\mbox{slepton}}_D \nonumber \\ 
&=&- \tilde{m}^{2}_{\tilde{ \nu}} 
\tilde{ \nu}^*_l \tilde{ \nu}_l- \left( \tilde{m}^2_L+ 
\frac{4 m^2_l}{9} \right) \tilde{l}^* \tilde{l}- \left( 
\tilde{m}^2_R+
\frac{4 m^2_l}{9}  \right) \tilde{l}^{c *} \tilde{l}^c+ 
\tilde{m}^2_{LR}( \tilde{l} \tilde{l}^{c}- \tilde{l}^{c *} \tilde{l}^*),
\label{smt3}  
\end{eqnarray}
where
\begin{eqnarray}
\tilde{m}^{2}_{\tilde{ \nu}}&=&m^2_{\tilde{ \nu}}+ \frac{z^2+2v^2-u^2-w^2}{6}, 
\nonumber \\
\tilde{m}^2_L&=&m^2_l+\lambda^{2}_{2}v^{2}+ 
\frac{z^2-2(w^2-v^2-u^2)}{12}, \nonumber \\
\tilde{m}^2_R&=&m^2_{l^c}+\lambda^{2}_{2}v^{2}+ 
\frac{z^2-2(u^2-2w^2-v^2)}{12}, 
\nonumber \\
\tilde{m}^2_{LR}&=& \frac{9 m_{l}}{4} \left[ \left( \frac{ \mu_S}{\sqrt{2}}+ 
\frac{f_3uw}{9z} \right) - \frac{4}{9}\zeta_0z \right] .
\end{eqnarray}

Performing the diagonalization we find the following states
\begin{eqnarray}
&&\left( \begin{array}{c}
\tilde{l}^-_1 \\
\tilde{l}^-_2
\end{array} \right)= \left( \begin{array}{cc}
\cos \theta_f& \sin \theta_f \\
- \sin \theta_f& \cos \theta_f
\end{array} \right) \left( \begin{array}{c}
\tilde{l} \\
\tilde{l}^{c *}
\end{array}\right), \,\
\left( \begin{array}{cc}
\tilde{l}^+_1& \tilde{l}^+_2
\end{array} \right)=
\left( \begin{array}{cc}
\tilde{l}^{*}& \tilde{l}^{c}
\end{array}\right) 
\left( \begin{array}{cc}
\cos \theta_f&- \sin \theta_f \\
\sin \theta_f& \cos \theta_f
\end{array} \right),
\label{defslepmass}
\end{eqnarray}
where the mixing angle is given by:
\begin{equation}
\tan 2 \theta_f= \frac{2 \tilde{m}^2_{LR}}{ \tilde{m}^2_L- \tilde{m}^2_R},
\label{mixingangleslep}
\end{equation}
and the masses of these states are:
\begin{equation}
m^2_{1,2}= \frac{4 m^2_l}{9}+ \frac{1}{2} \left[ \tilde{m}^2_L+ 
\tilde{m}^2_R \pm \sqrt{ ( \tilde{m}^2_L- \tilde{m}^2_R)^2+4 \tilde{m}^4_{LR}}
\right].
\end{equation}

Substituing the diagonal sleptons states, given in Eq.(\ref{defslepmass}), in 
the Eq.(\ref{smt3}), we obtain the following diagonal lagrangian
\begin{equation}
{\cal L}^{ \mbox{slepton}}_{ \mbox{mass}}=- \tilde{m}_{\tilde{ \nu}} 
\tilde{ \nu}^*_l \tilde{ \nu}_l- m^2_1 \tilde{l}^+_1 \tilde{l}^-_1-
m^2_2 \tilde{l}^+_2 \tilde{l}^-_2.
\end{equation}
We wish to stress that the slepton sector of this model is the same as in the 
MSSM \cite{mssm}.

\subsection{Neutral Gauge Boson}


The neutral gauge boson mass is given by
\begin{eqnarray}
{\cal L}^{\mbox{mass}}_{\mbox{neutral}}&=& \left( \begin{array}{ccc}
V_{3m}& V_{8m}& V_{m} \end{array} \right) M^2 \left( \begin{array}{c}
V_{3}^{m} \\
V_{8}^{m} \\
V^{m} 
\end{array} \right),
\label{numerar}
\end{eqnarray}
where
\begin{eqnarray}
M^2= \frac{g^2}{2} \left( \begin{array}{ccc}
(v^2+u^2+z^2+4z_{1}^2)& \frac{1}{ \sqrt{3}}(v^2-u^2+z^2+4z_{1}^2)& 
-2tu^2 \\
\frac{1}{ \sqrt{3}}(v^2-u^2+z^2+4z_{1}^2)& \frac{1}{3}(v^2+u^2+4w^2+z^2+
4z_{1}^2)& \frac{2t}{ \sqrt{3}}(u^2+2w^2) \\
-2tu^2& \frac{2t}{ \sqrt{3}}(u^2+2w^2)& 4t^2 (u^2+w^2)
\end{array} \right), 
\label{numerar}
\end{eqnarray}
where $t=g/g^{ \prime}$.

In the approximation that $w^2 \gg v^2,u^2,z^2$, the masses of the neutral 
gauge bosons are: $0$, 
$M_Z^2$ and $M_{Z^{ \prime}}^2$, and the masses are given by
\begin{eqnarray}
M_Z^2 \approx \frac{1}{2} \frac{g^2+4g^{ \prime 2}}{g^2+3g^{ \prime 2}}
(v^2+u^2+z^2+v^{\prime 2}+u^{\prime 2}+z^{\prime 2}), \,\
M_{Z^{ \prime}}^2 \approx \frac{1}{3}(g^{2}+3g^{\prime 2})(2w^2+2w^{\prime 2}).
\label{zmass}
\end{eqnarray}
Using $M_W$ given in Eq.(\ref{wmass}) and $M_Z$ in Eq.(\ref{zmass}) we get
 the following relation:
\begin{eqnarray}
\frac{M^2_Z}{M^2_W}= \frac{1+4t^2}{1+3t^2}=\frac{1}{1- \sin^2 \theta_W},
\label{rel1}
\end{eqnarray}
therefore we obtain
\begin{equation}
t^2= \frac{ \sin^2 \theta_W}{1-4 \sin^2 \theta_W}.
\label{tdef}
\end{equation}

\subsection{Charged Gauge Boson}

The gauge mass term is given by
${\cal L}^{\mbox{scalar}}_{HHVV}$, see Eq.(\ref{mix1}), which we can divided 
in ${\cal L}^{\mbox{mass}}_{\mbox{charged}}$ and 
${\cal L}^{\mbox{mass}}_{\mbox{neutral}}$.

The charged gauge boson mass is
\begin{eqnarray}
{\cal L}^{\mbox{mass}}_{\mbox{charged}}
&=& M^2_{W} W^{-}_{m}W^{+ m}+ M^2_{V} V^{-}_{m} V^{+ m}+ 
M^2_U U^{--}_{m}U^{++ m},
\label{bgmcc}
\end{eqnarray}
where 
\begin{eqnarray}
M^2_{U}&=& \frac{g^2}{4}(v^2_{ \rho}+v^2_{ \chi}+4v^2_{ \sigma_{2}}+
v^2_{ \rho^{\prime}}+v^2_{ \chi^{\prime}}+4v^2_{ \sigma^{\prime}_{2}}), 
\nonumber \\ 
M^2_{W}&=& \frac{g^2}{4}(v^2_{ \eta}+v^2_{ \rho}+2v^2_{ \sigma_{2}}+
v^2_{ \eta^{\prime}}+v^2_{ \rho^{\prime}}+2v^2_{ \sigma^{\prime}_{2}}), \nonumber \\
M^2_{V}&=& \frac{g^2}{4}(v^2_{ \eta}+v^2_{ \chi}+2v^2_{ \sigma_{2}}+
v^2_{ \eta^{\prime}}+v^2_{ \chi^{\prime}}+2v^2_{ \sigma^{\prime}_{2}}).
\label{wmass}
\end{eqnarray}
We want to mention that the 
gauge boson sector is exactly the same as in the non-supersymmetric 
3-3-1 model \cite{331}.



\section{Interactions}
\label{sec3}

In the previous section we have analysed the physical spectrum of the model. 
Now we are going to get interactions between these particles.

The procedure to write two component spinors in terms of four components spinors is 
given in Appendix \ref{2t4}.

\subsection{Lepton Gauge Boson Interaction}



We will define the following Dirac spinors for the leptons
\begin{eqnarray}
\Psi(l)&=& \left( \begin{array}{cc}
             l &
	     \bar{l}^{c} 
\end{array} \right)^{t}, \,\
\Psi^c(l)= \left( \begin{array}{cc}
             l^c &
	     \bar{l} 
\end{array} \right)^{t}, \,\
\Psi( \nu_{l})= \left( \begin{array}{cc}
             \nu_{l} &
	     0 
\end{array} \right)^{t}.
\label{defspoinors}
\end{eqnarray}


Due to Eqs.(\ref{defspoinors}), 
we can rewrite Eq.(\ref{lepbos}) as follows
\begin{eqnarray}
{\cal L}^{lep}_{llV}&=&- \frac{g}{ \sqrt{2}} \left( \bar{ \Psi^{c}}(l) 
\gamma^{m}L \Psi(l)U^{++}_{m}+ \bar{ \Psi^{c}}(l)
\gamma^{m}L \Psi( \nu_{l})V^{+}_{m}+ \bar{ \Psi}( \nu_{l}) 
\gamma^{m}L \Psi(l)W^{+}_{m}+hc \right)) \nonumber \\
&-& \frac{gt}{ \sqrt{1+4t^{2}}} \bar{ \Psi}(l) \gamma^{m} \Psi(l)A_{m}
- \frac{g}{2} \sqrt{ \frac{1+4t^{2}}{1+3t^{2}}} 
\bar{ \Psi}( \nu_{l}) \gamma^{m}L \Psi( \nu_{l}) \left[ Z_{m}-
\frac{1}{ \sqrt{3}} \frac{1}{ \sqrt{1+4t^{2}}} Z^{ \prime}_{m} \right]
\nonumber \\
&-& \frac{g}{4} \sqrt{ \frac{1+4t^{2}}{1+3t^{2}}} 
\left[ \left( \frac{-1}{1+4t^{2}} \bar{ \Psi}(l) \gamma^{m} \Psi(l)
+ \bar{ \Psi}(l) \gamma^{m} \gamma^{5} \Psi(l) \right)Z_{m} \right. \nonumber\\
&+& \left. \left(
\frac{- \sqrt{3}}{ \sqrt{1+4t^{2}}} \bar{ \Psi}(l) \gamma^{m} \Psi(l)-
\frac{1}{ \sqrt{3}} \frac{1}{ \sqrt{1+4t^{2}}} 
\bar{ \Psi}(l) \gamma^{m} \gamma^{5} \Psi(l) \right) Z^{ \prime}_{m} \right]= 
{\cal L}^{NC}_{llV}+ {\cal L}^{CC}_{llV},
\label{lepbosint}
\end{eqnarray}
where $t$ is defined in Eq.(\ref{tdef}).

Let us define also the following parameters
\begin{eqnarray}
e&=& \frac{g \sin \theta}{ \sqrt{1+3 \sin^{2} \theta}}, \,\
h(t)=1+4t^{2}, \,\
v_{l}=- \frac{1}{h(t)}, \,\
a_{l}=1 ,\,\
v^{ \prime}_{l}=- \frac{ \sqrt{3}}{ \sqrt{h(t)}}, \,\
a^{ \prime}_{l}= \frac{v^{ \prime}_{l}}{3}, \,\ g^2=\frac{8G_F M^2_W}{\sqrt2}.
\label{pardef}
\end{eqnarray}
Then we can now rewrite the neutral part of 
Eq.(\ref{lepbosint}):
\begin{eqnarray}
{\cal L}^{NC}_{llV}&=&-e \bar{ \Psi}(l) \gamma^{m} \Psi(l)A_{m}-
\frac{g}{2} \frac{M_{Z}}{M_{W}}
\bar{ \Psi}( \nu_{l}) \gamma^{m}L \Psi( \nu_{l}) \left[ Z_{m}-
\frac{1}{ \sqrt{3}} \frac{1}{ \sqrt{h(t)}} Z^{ \prime}_{m} \right] \nonumber \\
&-& \frac{g}{4} \frac{M_{Z}}{M_{W}} \left[ \bar{ \Psi}(l) 
\gamma^{m}
(v_{l}+a_{l} \gamma^{5}) \Psi(l) Z_{m}+ \bar{ \Psi}(l) \gamma^{m}
(v^{ \prime}_{l}+a^{ \prime}_{l} \gamma^{5}) \Psi(l) Z^{ \prime}_{m} \right],
\label{lagr1}
 \end{eqnarray}
The charged part of 
Eq.(\ref{lepbosint}) is
\begin{eqnarray}
{\cal L}^{CC}_{llV}&=&- \frac{g}{ \sqrt{2}} \left( \bar{ \Psi^{c}}(l) 
\gamma^{m}L \Psi(l)U^{++}_{m}+ \bar{ \Psi^{c}}(l)
\gamma^{m}L \Psi( \nu_{l})V^{+}_{m}+ \bar{ \Psi}( \nu_{l}) 
\gamma^{m}L \Psi(l)W^{+}_{m}+hc \right) ,
\label{lagr2}
\end{eqnarray}
where $g$ is defined in Eq.(\ref{pardef}). The lagrangian in the Eqs.(\ref{lagr1},
\ref{lagr2})
 is the same that it was shown in Ref~\cite{331}.


\subsection{Chargino(Neutralino) Bilepton $U^{--}$ Interaction}

Using Eq.(\ref{w4ssc}) in the Eqs(\ref{tutty}) and (\ref{mix1}), we 
can write the interaction between the double charged vector with the 
double charged chargino and the neutralino, in the following way
\begin{eqnarray}
{\cal L}_{U \tilde{\chi}^{++} \tilde{\chi}^{0}}&=&- \frac{g}{2} \left \{ \left[
\bar{ \tilde{W}}_{3}L \gamma^{m}R \tilde{U}- \bar{ \tilde{U}}^{c}L \gamma^{m}R 
\tilde{W}_{3}+ \sqrt{3} \left( \bar{ \tilde{U}}^{c}L \gamma^{m}R \tilde{W}_{8}-
\bar{ \tilde{W}}_{8}L \gamma^{m}R \tilde{U} \right) \right. \right. \nonumber 
\\
&+& \left. \left. \sqrt{2} \left(
\bar{ \tilde{T}}^{0}_{5}L \gamma^{m}R \tilde{T}^{++}_{2}+
\bar{ \tilde{T}}^{0}_{4}L \gamma^{m}R \tilde{T}^{++}_{1}+
\bar{ \tilde{T}}^{c ++}_{2}L \gamma^{m}R \tilde{T}^{0}_{6}+ 
\bar{ \tilde{T}}^{c ++}_{1}L \gamma^{m}R \tilde{T}^{0}_{3}
\right) \right. \right. \nonumber 
\\
&+& \left. \left.
\bar{ \tilde{S}}^{c ++}_{2}L \gamma^{m}R \tilde{S}^{0}_{4}+
\bar{ \tilde{S}}^{c ++}_{1}L \gamma^{m}R \tilde{S}^{0}_{3}+
\bar{ \tilde{S}}^{0}_{4}L \gamma^{m}R \tilde{S}^{++}_{1}+
\bar{ \tilde{S}}^{0}_{3}L \gamma^{m}R \tilde{S}^{++}_{2} \right]U^{--}_{m}+hc 
\right \} \nonumber \\
&=&- \frac{g}{2} \left \{ \left[
\left( (N^{*}_{i1}- \sqrt{3}N^{*}_{i2})B_{j1}+ \sqrt{2}(N^{*}_{i8}B_{j3}+
N^{*}_{i7}B_{j2})+N^{*}_{i12}B_{j5}+N^{*}_{i13}B_{j4}  \right) 
\bar{ \Psi}( \tilde{ \chi}^{0}_{i})L 
\gamma^{m}R \Psi( \tilde{ \chi}^{++}_{j}) \right. \right. \nonumber \\
&+& \left. \left. \left( A^{*}_{i1}( \sqrt{3}N_{j2}-N_{j1})+ 
\sqrt{2}(A^{*}_{i3}N_{j9}+A^{*}_{i2}N_{j6})+ A^{*}_{i5}N_{j13}+A^{*}_{i4}N_{j12} 
\right) 
\bar{ \Psi}^{c}( \tilde{ \chi}^{--}_{i})L \gamma^{m}R 
\Psi( \tilde{ \chi}^{0}_{j}) \right]U^{--}_{m}+ hc \right \}.
\label{basemassa1}
\end{eqnarray}
 

In a similar way we get for the charginos
\begin{eqnarray}
{\cal L}_{U \tilde{\chi}^{+} \tilde{\chi}^{+}}&=&- \frac{g}{\sqrt{2}} \left[ 
\left( \bar{ \tilde{W}}^{c}L \gamma^{m}R \tilde{V}-
\bar{ \tilde{V}}^{c}L \gamma^{m}R \tilde{W}+
\bar{ \tilde{T}}^{c +}_{2}L \gamma^{m}R \tilde{T}^{+}_{1}+
\bar{ \tilde{T}}^{c +}_{1}L \gamma^{m}R \tilde{T}^{+}_{2} \right. \right. 
\nonumber \\
&+& \left. \left.
\frac{1}{2} \bar{ \tilde{S}}^{c +}_{1}L \gamma^{m}R \tilde{S}^{+}_{2}+
\frac{1}{2} \bar{ \tilde{S}}^{c +}_{2}L \gamma^{m}R \tilde{S}^{+}_{1}
\right)U^{--}_{m}+hc \right] \nonumber \\
&=& \frac{g}{\sqrt{2}} \left[
\left( D^{*}_{i1}E_{j2}-D^{*}_{i2}E_{j1}+D^{*}_{i4}E_{j3}+D^{*}_{i3}E_{j4} 
\frac{1}{2}D^{*}_{i7}E_{j8}+\frac{1}{2}D^{*}_{i8}E_{j7} 
\right) \bar{ \Psi}^{c}( \tilde{ \chi}^{-}_{i})L \gamma^{m}R 
\Psi( \tilde{ \chi}^{+}_{j}) U^{--}_{m}+hc \right].
\label{basemassa2}
\end{eqnarray}

\subsection{Lepton Slepton Chargino(Neutralino) Interaction}

The interaction between lepton slepton chargino(neutralino) is given by 
\begin{eqnarray}
{\cal L}_{l \tilde{l} \tilde{\chi}}&=& \left\{ -g \cos \theta_{f} A_{i} 
\bar{ \Psi}( \tilde{\chi}^{0}_{i})L \Psi(l)  
- g \sin \theta_{f} A_{i1} \bar{\Psi}(l)R \Psi^{c}( \tilde{\chi}^{--}_{i}) 
+2 \lambda_{3} \sin \theta_{f} \left[
A^{*}_{i5}\bar{\Psi}^{c}(\tilde{\chi}^{--}_{i})L \Psi^{c}(l)+
N^{*}_{i8}\bar{\Psi}(\tilde{\chi}^{0}_{i})L \Psi(l) 
\right] \right\} \tilde{l}^{+}_{1} \nonumber \\
&+& \left\{ g \sin \theta_{f} A^{*}_{i} 
\bar{ \Psi}( \tilde{\chi}^{0}_{i})L \Psi(l)  
- g \cos \theta_{f} A_{i1} \bar{\Psi}(l)R \Psi^{c}( \tilde{\chi}^{--}_{i}) 
+2 \lambda_3  \cos \theta_{f} \left[
A^{*}_{i5}\bar{\Psi}^{c}(\tilde{\chi}^{--}_{i})L \Psi^{c}(l)+
N^{*}_{i8}\bar{\Psi}(\tilde{\chi}^{0}_{i})L \Psi(l) \right] \right\} 
\tilde{l}^{+}_{2} \nonumber \\
&+&
\left\{ -g  
\frac{2 \lambda_{3}}{\sqrt{2}} \left[
E^{*}_{i8}\bar{\Psi}(\tilde{\chi}^{+}_{i})L \Psi(l)+
D^{*}_{i7}\bar{\Psi}^{c}(\tilde{\chi}^{-}_{i})L \Psi^{c}(l)+
\sqrt{2} N^{*}_{i7}\bar{\Psi}(\tilde{\chi}^{0}_{i})L \Psi(\nu_{l})
\right] \right\} \tilde{\nu_{l}}+hc,
\label{sleplepchar}
\end{eqnarray}
where $A_{i}= \left( \frac{N^{*}_{i1}}{ \sqrt{2}}+ 
\frac{N^{*}_{i2}}{ \sqrt{6}} \right)$.

\section{Feynman Rules}
\label{sec4}

In the previous section we got the interaction lagrangians, mainly 
the following vertices:
\begin{itemize}
\item Lepton-Lepton-Gauge Boson,
\item Chargino-Chargino(Neutralino)-Gauge Boson,
\item Slepton-Lepton-Chargino(Neutralino).
\end{itemize}
In the Table \ref{t1} we give the Feynman rules for the interactions mentioned 
above.



In Table \ref{t1} we have defined the following operators:
\begin{eqnarray}
O^{1}_{ij}=A^{*}_{i1}(\sqrt{3}N_{j2}-N_{j1})+\sqrt{2}(A^{*}_{i3}N_{j9}+
A^{*}_{i2}N_{j6})+ A^{*}_{i5}N_{j13}+A^{*}_{i4}N_{j12};\\
O^{2}_{ij}=- \left( D^{*}_{i1}E_{j2}-D^{*}_{i2}E_{j1}+D^{*}_{i4}E_{j3}+
D^{*}_{i3}E_{j4} \frac{1}{2}D^{*}_{i7}E_{j8}+\frac{1}{2}D^{*}_{i8}E_{j7} 
\right).
\end{eqnarray}

Because there are neutralinos, charginos, sleptons and sneutrinos in the MSSM, 
we show in Table \ref{t2} two different solutions got in mSUGRA \cite{desy}. 
To derive the total and differential cross sections, we will assume that the 
mass of the lightest particle of this model coincides with the mass parameter 
coming from MSSM and shown in Table \ref{t2}.


\section{Applications}
\label{sec5}

In this section, we will perform the calculation of differential and total 
cross sections of the lightest chargino, double charged chargino and 
neutralinos, in $e^-e^-$ scattering. First we present the chargino production. 

In the two subsections below, we neglect the electron mass and we assume that 
electrons have energy $E/2$. We will consider that $P_1$ and $P_2$ are the 
four momenta of the incoming electrons while $K_1$ and $K_2$ are the four momenta 
of 
the outgoing particles.

\subsection{ Charginos Production $e^{-}e^{-} \to \tilde{ \chi}^{-} 
\tilde{ \chi}^{-}$}

Considering the Table \ref{t1}, we have 5 diagrams at the tree level 
corresponding to this process, see Fig.(\ref{process2}).
The first and the third diagrams that appear in Fig.(\ref{process2}) exist 
in the MSSM, but the other ones are  new contributions coming from the 3-3-1 
supersymmetric model.



We have calculated the differential cross section, see Eq.(\ref{dif1}), and 
the
total cross section, and we have displayed several plots of the total cross 
section with $M_U \geq 0.5$TeV and $\sqrt{s}=0.5,1.0$ and $2.0$TeV.

The plots show that outside the $U$ resonance, the total cross section is of 
order of pb, like in the MSSM \cite{cp}. This result is diplayed in  
Figs.(\ref{plot1},\ref{plot2},\ref{plot3}).


\subsection{ Double Chargino and Neutralino Production 
$e^{-}e^{-} \to \tilde{ \chi}^{--} \tilde{ \chi}^{0}$}

Considering the Table \ref{t1}, we have at the tree level 9 diagrams 
for this process, see Fig.(\ref{process1}). In Supersymmetric Left-Right 
model \cite{LR} the production of double charged higgsino occurs via a 
selectron exchange in t-channel, like in the third diagram of 
Fig.(\ref{process1}).



As in the previous subsection, we have calculated the differential cross 
section, see Eq.(\ref{dif2}), and the total cross section. We have done 
several plots
using $M_{\chi^{0}}$, $M_{\tilde{l}_1}$ and $M_{\tilde{l}_2}$, given in 
Table \ref{t2}, and $M_U=0.5$TeV. Some of our results are show in 
Figs.(\ref{plot4},\ref{plot5},\ref{plot6}).

\section{Conclusions}
\label{sec:con}

Because of low level of Standard Model backgrounds, the total cross section 
$\sigma \approx 10^{-3}nb$ at $\sqrt{s}=500GeV$ \cite{assi3}, $e^-e^-$ 
collisions are a good reaction for discovering and investigating new physics 
at linear colliders.

We have shown in this work that the production of single charginos have more 
contributions in this model than in the MSSM. Although in the MSSM 
the chargino pairs can be only produced through $e^-e^-$ collisions by sneutrino exchanges 
in u and t channels , in the 3-3-1 supersymmetric model we have also 
a s channel contribution due to the exchange of a bilepton $U^{--}$. This new 
contribution induces a peak at $\sqrt{s} \simeq M_U$, where $M_U$ 
is the bilepton mass, and gives an clean signal. Near the resonance the 
dominant term in the total cross 
section is given by $|O_{2}|^2(m^4_{\tilde{\chi}^+}- 8sm^2_{\tilde{\chi}^+}+
4s^2)$ which is a clear signal coming from the bilepton $U$ contribution. 
The total cross section outside the $U$'s resonance has 
the same order of magnitude than the cross section in the MSSM, as we should 
expected, because in this case we don't have an enhancement due to the $s$- 
channel contribution.

It was shown too that in this model we have double charged charginos, that is 
impossible in the MSSM framework. Therefore it is a very useful way to 
distinguish the 3-3-1 supersymmetric model from the MSSM, and also from the 
usual 3-3-1 model because in this case we don't have double charged leptons.
We have considerated the double chargino mass in the range from $700 \leq 
M_{\tilde{\chi}^{++}} \leq 800$ GeV, and we could get cross section of the 
order of pb outside the $U$ ressonance, while in the bilepton ressonance we 
have an enhance in the cross section. We believe that these new states can be 
discovered, if they really exist, in the linear colliders as (NLC, JLC, 
TESLA, CLIC, VLEPP, ...).

\acknowledgments 
This research was financed  by Funda\c c\~ao de Amparo \`a Pesquisa do 
Estado de S\~ao Paulo (M.C.R.). One of us (M.C.R.) would like to thank 
the Laboratoire de Physique Math\'ematique et Th\'eorique (Montpellier) for it 
kind hospitality and also G. Moultaka for useful discussions.

\appendix


\section{Lagrangian}
\label{sec:lagrangian}

With the fields introduced in section \ref{sec1}, we can built the following 
lagrangian \cite{331susy}
\begin{eqnarray} 
   {\cal L}_{331S} &=& {\cal L}_{SUSY} + {\cal L}_{\mbox{soft}} \,\ , 
\end{eqnarray}
where
\begin{eqnarray} 
{\cal L}_{SUSY} &=&   {\cal L}_{\mbox{Lepton}}+ {\cal L}_{\mbox{Quarks}}+ 
{\cal L}_{\mbox{Gauge}}+ {\cal L}_{\mbox{Scalar}}, 
\end{eqnarray}
is the supersymmetric part while ${\cal L}_{\mbox{soft}}$ breaks supersymmetry. 
Now we are going to present all the lagrangian of the model, except the quark 
part, because we are not considering them in this study.

\subsection{Lepton Lagrangian}
\label{a1}

In the ${\cal L}_{Lepton}$ term we have the interaction between leptons and 
gauge bosons in component are given by
\begin{eqnarray}
{\cal L}^{lep}_{llV}&=&\frac{g}{2} \bar{L}\bar\sigma^m\lambda^a LV^a_m,
\label{lepbos}
\end{eqnarray}
where $\lambda^a$ are the usual Gell-Mann Matrices.The next part, we are 
interested here, is the interaction between lepton-slepton-gaugino is 
given by the following term
\begin{eqnarray}
{\cal L}^{lep}_{l \tilde{l} \tilde{V}}=- \frac{ig}{ \sqrt{2}}
( \bar{L}\lambda^a\tilde{L}\bar{\lambda}^a_{A}- 
\bar{\tilde{L}}\lambda^aL\lambda^a_{A}).
\label{llchar}
\end{eqnarray}


\subsection{Gauge Lagrangian}
\label{a3}

The part we are interested in ${\cal L}_{Gauge}$ is the interaction between 
higgsino and gauge boson, that is:
\begin{eqnarray}
{\cal L}^{\mbox{gauge}}_{ \lambda \lambda V}=-igf^{abc}\bar{\lambda}^{a}_{A}
\lambda^{b}_{A} \sigma^{m}V^{c}_{m},
\label{tutty}
\end{eqnarray}
$f^{abc}$ are the structure constants of the gauge group $SU(3)$.

\subsection{Scalar Lagrangian}
\label{a2}


While in the scalar sector we have the following three terms, considering all 
triplets and anti-sextet:
\begin{eqnarray}
{\cal L}^{\mbox{scalar}}_{H \tilde{H} \tilde{V}}&=&- \frac{ig}{ \sqrt{2}} 
\left[
\bar{\tilde{\eta}}\lambda^a\eta\bar{\lambda}^a_{A} 
- \bar{\eta}\lambda^a\tilde{\eta}\lambda^a_{A}+
\bar{\tilde{\rho}}\lambda^a\rho\bar{\lambda}^a_{A} 
- \bar{\rho}\lambda^a\tilde{\rho}\lambda^a_{A}+
\bar{\tilde{\chi}}\lambda^a\chi\bar{\lambda}^a_{A} 
- \bar{\chi}\lambda^a\tilde{\chi}\lambda^a_{A}+
\bar{\tilde{S}}\lambda^aS\bar{\lambda}^a_{A} \right. \nonumber \\ 
&-& \left. \bar{S}\lambda^a\tilde{S}\lambda^a_{A} \right]V^{a}_{m}-
\frac{ig^{ \prime}}{ \sqrt{2}} \left[  
\bar{\tilde{\rho}}\rho\bar{\lambda}_{B} 
-\bar{\rho}\tilde{\rho}\lambda_{B}-
\bar{\tilde{\chi}}\chi\bar{\lambda}_{B} 
+\bar{\chi}\tilde{\chi}\lambda_{B} \right]V_{m}, \nonumber \\
{\cal L}^{\mbox{scalar}}_{ \tilde{H} \tilde{H}V}&=& \frac{g}{2} \left[ 
\bar{\tilde{\eta}}\bar\sigma^m\lambda^a \tilde{\eta}+ 
\bar{\tilde{\rho}}\bar\sigma^m\lambda^a \tilde{\rho}+
\bar{\tilde{\chi}}\bar\sigma^m\lambda^a \tilde{\chi}+ 
\bar{\tilde{S}}\bar\sigma^m\lambda^a \tilde{S}  \right]V^{a}_{m}+ 
\frac{g^{ \prime}}{2} \left[ \bar{\tilde{\rho}}\bar{\sigma}^m\tilde{\rho}-
\bar{\tilde{\chi}}\bar{\sigma}^m\tilde{\chi} \right]V_{m}, \nonumber \\
{\cal L}^{\mbox{scalar}}_{HHVV}&=& \frac{1}{4} \left[  
g^2V_m^aV^{bm}\bar{\eta}\lambda^{a}\lambda^{b}\eta+
g^2V_m^aV^{bm}\bar{\rho}\lambda^{a}\lambda^{b}\rho+
g^2V_m^aV^{bm}\bar{\chi}\lambda^{a}\lambda^{b}\chi \right. \nonumber \\
&+& \left.
g^2V_m^aV^{bm} 
\left( \lambda^{a}_{ik} \bar{S}_{kj}+ \lambda^{a}_{jk} \bar{S}_{ki} \right)
\left( \lambda^{a}_{ik}S_{kj}+ \lambda^{a}_{jk}S_{ki} \right) 
\right. \nonumber \\ &+& \left.
g^{\prime 2}V^m V_m \bar{\rho}\rho + g^{\prime 2}V^m V_m \bar{\chi}\chi +
2gg^\prime V^a_m V^m(\bar{\rho}\lambda^a\rho)-2 gg^\prime V^a_m V^m(\bar{\chi}
\lambda^a\chi) \right].
\label{mix1}
\end{eqnarray}


\subsection{Superpotential}
\label{a4}

\begin{eqnarray}
%%% Higgsino Mass Term %%%%%%%%%%%
{\cal L}_{HMT}&=&- 
\frac{\mu_{ \eta}}{2} \tilde{ \eta}_i \tilde{ \eta}^{*}_i
-\frac{\mu_{ \rho}}{2} \tilde{ \rho}_i \tilde{ \rho}^{*}_i-
\frac{\mu_{ \chi}}{2} \tilde{ \chi}_i \tilde{ \chi}^{*}_i-
\frac{\mu_{S}}{2} \tilde{S}_{ij} \tilde{S}^{*}_{ji}+hc. \nonumber \\
{\cal L}_{F}&=& \frac{1}{3}[
3 \lambda_{1} \epsilon F_L \tilde{L} \tilde{L}+
\lambda_{2} \epsilon ( 2F_{L}\eta + F_{ \eta}\tilde{L}) \tilde{L}+
\lambda_{3} ( 2F_{L}S + F_{S}\tilde{L}) \tilde{L} \nonumber \\
&+&
f_{1} \epsilon (F_{ \rho} \chi \eta+ \rho F_{ \chi} \eta+ \rho \chi F_{ \eta})
+f_{2}(2 F_{ \eta} \eta S+ \eta \eta F_{S})+
f_{3}( F_{ \rho} \chi S+ \rho F_{ \chi} S+ \rho \chi F_{S})+hc  \\
{\cal L}^{W3}_{H \tilde{H} \tilde{H}}&=&- \frac{1}{3}[ 
f_{1} \epsilon ( \tilde{ \rho} \tilde{ \chi} \eta+ \rho \tilde{ \chi} 
\tilde{ \eta}+ \tilde{ \rho} \chi \tilde{ \eta})+
f_{2}( \tilde{ \eta} \tilde{ \eta} S+ \eta \tilde{ \eta} \tilde{S}+ 
\tilde{ \eta} \eta \tilde{S})+
f_{3}( \tilde{ \rho} \tilde{ \chi}S+ \rho \tilde{ \chi} \tilde{S}+ 
\tilde{ \rho} \chi \tilde{S}) \nonumber \\
&+&
f^{\prime}_{1} \epsilon(\tilde{\rho}^{\prime}\tilde{\chi}^{\prime}\eta^{\prime}
+ \rho^{\prime} \tilde{ \chi}^{\prime} \tilde{ \eta}^{\prime}+ 
\tilde{ \rho}^{\prime} \chi^{\prime} \tilde{ \eta}^{\prime})+
f^{\prime}_{2}( 
\tilde{ \eta}^{\prime} \tilde{ \eta}^{\prime} S^{\prime}+ 
\eta^{\prime} \tilde{ \eta}^{\prime} \tilde{S}^{\prime}+ 
\tilde{ \eta}^{\prime} \eta^{\prime} \tilde{S}^{\prime}) \nonumber \\
&+& 
f^{\prime}_{3}( 
\tilde{ \rho}^{\prime} \tilde{ \chi}^{\prime}S^{\prime}+ 
\rho^{\prime} \tilde{ \chi}^{\prime} \tilde{S}^{\prime}+ 
\tilde{ \rho}^{\prime} \chi^{\prime} \tilde{S}^{\prime})
] +hc.
\label{compsup}
\end{eqnarray}

\subsection{Soft Term}
\label{a5}

\begin{eqnarray}
{\cal L}^{\mbox{gaugino}}_{GMT}&=&- \frac{1}{2} \left[ 
m_{ \lambda} \sum_{a=1}^{8} 
\left( \lambda^{a}_{A} \lambda^{a}_{A} \right) 
+m^{ \prime} \lambda_{B} \lambda_{B}+hc \right], \nonumber \\
{\cal L}^{\mbox{soft}}_{\mbox{scalar}}&=&
-m^2_{\eta}\bar{\eta}\eta-m^2_{\rho}\bar{\rho}\rho-m^2_{\chi}\bar{\chi}\chi
-m^2_{S}\bar{S}S+
(k_1\epsilon_{ijk}\rho_i\chi_j\eta_k+k_2\eta_i\eta_j \bar{S}_{ij} \nonumber \\
&+&k_3\chi_i\rho_j \bar{S}_{ij}+hc), \nonumber \\ 
{\cal L}_{SMT}&=&-m_{L}^{2} \tilde{L}^{\dagger} \tilde{L}+ 
\zeta_{0} \sum_{i=1}^{3} \sum_{j=1}^{3} \left( \tilde{L}_{i} \tilde{L}_{j} 
S_{ij}+ \bar{ \tilde{L}}_{i} \bar{ \tilde{L}}_{j} S^{*}_{ij} \right).
\label{smt}
\end{eqnarray}



\section{Non Diagonal Mass Matrix}
\label{c}

In this Appendix we display all non diagonal mass matrix of the charginos and 
neutralinos. 

\subsection{Double Charged Chargino}
\label{c1}

Introducing the notation
\begin{eqnarray}
\psi^{++}= \left( \begin{array}{rrrrr}
-i \lambda^{++}_{U}&
\tilde{\rho}^{++}&
\tilde{\chi}^{ \prime ++}&
\tilde{H}_1^{++}&
\tilde{H}_2^{ \prime ++}
\end{array}
\right)^t, \,\
\psi^{--}= \left( \begin{array}{rrrrr}
-i \lambda^{--}_{U}&
\tilde{\rho}^{ \prime --}&
\tilde{\chi}^{--}&
\tilde{H}_1^{ \prime --}&
\tilde{H}_2^{--}
\end{array}
\right)^t, \nonumber
\end{eqnarray}
and
\begin{equation}
\Psi^{\pm \pm}= \left( \begin{array}{rr}
\psi^{++} &
\psi^{--}
\end{array}
\right)^{t},
\end{equation}
we can write Eq.(\ref{dc1}) as follows
\begin{eqnarray}
{\cal L}^{\mbox{double}}_{\mbox{mass}}&=&- \frac{1}{2} \left( \Psi^{\pm \pm} 
\right)^{t} Y^{\pm \pm} \Psi^{\pm \pm}+hc,
\end{eqnarray}
where
\begin{equation}
Y^{\pm \pm}= \left( \begin{array}{cc}
0  & T^{t} \\
T  & 0
\end{array}
\right),
\end{equation}
with
\begin{equation}
T= \left( \begin{array}{ccccc}
-m_{\lambda}& -gu& gw^{\prime}& \frac{gz}{\sqrt{2}}& - \frac{gz^{\prime}}{\sqrt{2}} \\
gu^{\prime}& \frac{\mu_{\rho}}{2}&- 
\left( \frac{f_{1}^{\prime}v^{\prime}}{3}- \sqrt{2}f_{3}^{\prime}z^{\prime} \right)& 0& 
f_{3}^{\prime}w^{\prime} \\
-gw&- \left( \frac{f_{1}v}{3}- \sqrt{2}f_{3}z \right)& \frac{\mu_{\chi}}{2}& f_{3}u& 0 \\
- \frac{gz^{\prime}}{\sqrt{2}}& 0& f_{3}^{\prime}u^{\prime}& \frac{\mu_{S}}{2}& 0 \\
\frac{gz}{\sqrt{2}}& f_{3}w& 0& 0& \frac{\mu_{S}}{2}
\end{array}
\right).
\end{equation}

The matrix $Y^{\pm \pm}$ in Eq.(\ref{y++}) satisfy the 
following relation
\begin{eqnarray}
\det (Y^{\pm \pm}- \lambda I)= \det \left[ \left( \begin{array}{cc}
- \lambda & T^{t} \\
T  &- \lambda 
\end{array} \right) \right]= \det( \lambda^{2}-T^{t} \cdot T),
\label{propmat1}
\end{eqnarray}
so we only have to calculate $T^{t} \cdot T$ to obtain eigenvalues. 
Since $T^{t} \cdot T$ is a symmetric matrix, $\lambda^2$ must be real, and 
positive because $Y^{\pm \pm}$ is also symmetric.

\subsection{Charged Chargino}
\label{c2}


Introducing the notation
\begin{eqnarray}
\psi^{+}= \left( \begin{array}{rrrrrrrr}
-i \lambda^{+}_{W}&
-i \lambda^{+}_{V}&
\tilde{\eta}_{1}^{ \prime +}&
\tilde{\eta}_{2}^{+}&
\tilde{\rho}^{+}&
\tilde{\chi}^{ \prime +}&
\tilde{h}_{1}^{ \prime +}&
\tilde{h}_{2}^{+}
\end{array}
\right)^t, \,\
\psi^{-}= \left( \begin{array}{rrrrrrrr}
-i \lambda^{-}_{W}&
-i \lambda^{-}_{V}&
\tilde{\eta}_{1}^{-}&
\tilde{\eta}_{2}^{ \prime -}&
\tilde{\rho}^{ \prime -}&
\tilde{\chi}^{-}&
\tilde{h}_{1}^{-}&
\tilde{h}_{2}^{ \prime -}
\end{array}
\right)^t, \nonumber
\end{eqnarray}
and
\begin{equation}
\Psi^{\pm}= \left( \begin{array}{rr}
\psi^{+} &
\psi^{-}
\end{array}
\right)^{t},
\end{equation}
Eq.(\ref{uc1}) takes the form
\begin{eqnarray}
{\cal L}^{\mbox{unique}}_{\mbox{mass}}&=&- \frac{1}{2} \left( \Psi^{\pm} 
\right)^{t} Y^{\pm} \Psi^{\pm}+hc,
\end{eqnarray}
where
\begin{equation}
Y^{\pm}= \left( \begin{array}{cc}
0  & X^{t} \\
X  & 0
\end{array}
\right),
\end{equation}
with
\begin{equation}
X= \left( \begin{array}{cccccccc}
- m_{\lambda}& 0& gv^{\prime}& 0&- gu& 0&- \frac{gz^{\prime}}{2}& 0 \\
0&- m_{\lambda}& 0&- gv& 0& gw^{\prime}& 0&- \frac{gz}{2}  \\
-gv& 0& \frac{\mu_{\eta}}{2}& 0&- \frac{f_{1}w}{3}& 0& 0& 0 \\
0& gv^{\prime}& 0& \frac{\mu_{\eta}}{2}& 0& \frac{f^{\prime}_{1}u^{\prime}}{3}& 0& 0 \\
gu^{\prime}& 0&- \frac{f^{\prime}_{1}w^{\prime}}{3}& 0& \frac{\mu_{\rho}}{2}& 0& 0& 0 \\
0&- gw& 0& \frac{f_{1}u}{3}& 0& \frac{\mu_{\chi}}{2}& 0& 0 \\
- \frac{gz}{2}& 0& 0& \frac{f_{3}w}{\sqrt{2}}& & 0& \frac{\mu_{S}}{2}& 0 \\
0 & \frac{gz^{ \prime}}{2}& 0& 0& \frac{f^{ \prime}_{3}u^{ \prime}}{\sqrt{2}}& 0& 0& 
\frac{\mu_{S}}{2}
\end{array}
\right).
\end{equation}

We can show that our matrix $X$ satisfy the same properties than $T$ ( 
Eq.(\ref{propmat1})). 


\subsection{Neutralinos}
\label{c3}

Introducing the notation
\begin{eqnarray}
\Psi^{0}= \left( \begin{array}{rrrrrrrrrrrrr}
i \lambda^{3}_{A}&
i \lambda^{8}_{A}&
i \lambda_{B}&
\tilde{\eta}^{0}&
\tilde{\eta}^{ \prime 0}&
\tilde{\rho}^{0}&
\tilde{\rho}^{ \prime 0}&
\tilde{\chi}^{0}&
\tilde{\chi}^{ \prime 0}&
\tilde{\sigma}_{1}^{0}&
\tilde{\sigma}_{1}^{ \prime 0}&
\tilde{\sigma}_{2}^{0}&
\tilde{\sigma}_{2}^{ \prime 0}
\end{array}
\right)^t, \nonumber 
\end{eqnarray}
Eq.(\ref{neu1}) takes the following form:
\begin{eqnarray}
{\cal L}^{\mbox{neutralino}}_{\mbox{mass}}&=&- \frac{1}{2} \left( \Psi^{0} 
\right)^{t} Y^{0} \Psi^{0}+hc,
\end{eqnarray}
where
\begin{equation}
Y^{0}= \left( \begin{array}{ccccccccccccc}
-m_{\lambda}& 0& 0&- \frac{gv}{\sqrt{2}}& \frac{gv^{\prime}}{\sqrt{2}}& \frac{gu}{\sqrt{2}}&- 
\frac{gu^{\prime}}{\sqrt{2}}& 0& 0& 0& 0& \frac{gz}{2 \sqrt{2}}&- 
\frac{gz^{\prime}}{2 \sqrt{2}} \\
0&- m_{\lambda}& 0&- \frac{gv}{\sqrt{6}}& \frac{gv^{\prime}}{\sqrt{6}}&- 
 \frac{gu}{\sqrt{6}}& \frac{gu^{\prime}}{\sqrt{6}}& 
\frac{2}{\sqrt{6}} gw&- \frac{2}{\sqrt{6}}gw^{\prime}& 0& 0& \frac{gz}{2 \sqrt{6}}&- 
\frac{gz^{\prime}}{2 \sqrt{6}}\\
0& 0&- m^{\prime}& 0& 0&- \frac{g^{\prime}u}{\sqrt{2}}& 
\frac{g^{\prime}u^{\prime}}{\sqrt{2}}&  
\frac{g^{\prime}w}{\sqrt{2}}&- \frac{g^{\prime}w^{\prime}}{\sqrt{2}}& 0 & 0& 0& 0 \\
- \frac{gv}{\sqrt{2}}&- \frac{gv}{\sqrt{6}}& 0& 0& \frac{\mu_{\eta}}{2}& 
\frac{f_{1}w}{3}& 0&- \frac{f_{1}u}{3}& 0& 0& 0& 0& 0 \\
\frac{gv^{\prime}}{\sqrt{2}}& \frac{gv^{\prime}}{\sqrt{6}}& 0& \frac{\mu_{\eta}}{2}& 
0& 0& \frac{f^{\prime}_{1}w^{\prime}}{3}& 0&- \frac{f^{\prime}_{1}u^{\prime}}{3}& 
0& 0& 0& 0 \\
\frac{gu}{\sqrt{2}}&- \frac{gu}{\sqrt{6}}&- \frac{g^{\prime}u}{\sqrt{2}}& 
\frac{f_{1}w}{3}& 0& 0& \frac{ \mu_{\rho}}{2}& \frac{f_{1}v}{3}+\sqrt{2}f_{3}z 
& 0& 0& 0& \frac{f_{3}w}{\sqrt{2}}& 0 \\
- \frac{g^{\prime}u^{\prime}}{\sqrt{2}}& \frac{g^{\prime}u^{\prime}}{\sqrt{6}}& 
\frac{g^{\prime}u^{\prime}}{\sqrt{2}}& 0& \frac{f^{\prime}_{1}w^{\prime}}{3}& 
\frac{ \mu_{\rho}}{2}& 0& 0& \frac{f^{\prime}_{1}v^{\prime}}{3}+
\sqrt{2}f^{\prime}_{3}z^{\prime}& 0& 0& 0&- 
\frac{f^{\prime}_{3}w^{\prime}}{\sqrt{2}} \\
0& \frac{2}{\sqrt{6}} gw& \frac{g^{\prime}w}{\sqrt{2}}&- \frac{f_{1}u}{3}& 0& 
\frac{f_{1}v}{3}+\sqrt{2}f_{3}z& 0& 0& 
\frac{ \mu_{\chi}}{2}& 0& 0& \frac{f_{3}u}{\sqrt{2}}& 0 \\
0&- \frac{2}{\sqrt{6}} gw^{\prime}&- \frac{g^{\prime}w^{\prime}}{\sqrt{2}}& 0&- 
\frac{f^{\prime}_{1}u^{\prime}}{3}& 0& \frac{f^{\prime}_{1}v^{\prime}}{3}+
\sqrt{2}f^{\prime}_{3}z^{\prime}& \frac{ \mu_{\chi}}{2}& 0& 0& 
0& 0& \frac{f^{\prime}_{3}u^{\prime}}{\sqrt{2}} \\
0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{ \mu_{S}}{2}& 0&  0 \\
0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{ \mu_{S}}{2}& 0& 0&  0 \\
\frac{gz^{\prime}}{2 \sqrt{2}}& \frac{gz}{2 \sqrt{6}}& 0& 0& 0& \frac{f_{3}w}{\sqrt{2}}& 0& 
\frac{f_{3}u}{\sqrt{2}}& 0& 0& 0& 0& \frac{ \mu_{S}}{2} \\
- \frac{gz^{\prime}}{2 \sqrt{2}}&- \frac{gz^{\prime}}{2 \sqrt{6}}& 0& 0& 0& 0&- 
\frac{f^{\prime}_{3}w^{\prime}}{\sqrt{2}}& 0& \frac{f^{\prime}_{3}u^{\prime}}{\sqrt{2}}& 
0& 0& \frac{ \mu_{S}}{2}& 0
\end{array}
\right).
\end{equation}

\section{Connection between the Two- and Four- Component Spinors}
\label{2t4}

In this Appendix we show the procedure to write two-components spinors in 
terms of four-components spinors.

\subsection{Weak Eigenstates}

The weak interaction eigenstates are:
\begin{eqnarray}
\tilde{U}&=& \left( \begin{array}{c}
                -i \lambda^{++}_{U} \\
                i \bar{ \lambda}^{--}_{U}
\end{array} \right), \,\  
\tilde{U}^{c}= \left( \begin{array}{c}
                 -i \lambda^{--}_{U} \\
                i \bar{ \lambda}^{++}_{U} 
\end{array} \right), \,\
\tilde{T}^{++}_{1}= \left( \begin{array}{c}
                 \tilde{\rho}^{++} \\
		 \bar{ \tilde{\rho}}^{ \prime --}
\end{array} \right), \,\
\tilde{T}^{c ++}_{1}= \left( \begin{array}{c}
                   \tilde{\rho}^{ \prime --}   \\
                   \bar{ \tilde{\rho}}^{++} 
\end{array} \right), \nonumber \\
\tilde{T}^{++}_{2}&=& \left( \begin{array}{c}
                 \tilde{\chi}^{ \prime ++} \\
                 \bar{ \tilde{\chi}}^{--}
\end{array} \right), \,\
\tilde{T}^{c ++}_{2}= \left( \begin{array}{c}
                    \tilde{\chi}^{--}  \\
                   \bar{ \tilde{\chi}}^{ \prime ++}  
\end{array} \right), \,\
\tilde{S}^{++}_{1}= \left( \begin{array}{c}
                 \tilde{H}_{1}^{++}  \\
                  \bar{ \tilde{H}}_{1}^{ \prime --}
\end{array} \right), \,\
\tilde{S}^{c ++}_{1}= \left( \begin{array}{c}
                  \tilde{H}_{1}^{ \prime --} \\
                  \bar{ \tilde{H}}_{1}^{++}
\end{array} \right), \nonumber \\
\tilde{S}^{++}_{2}&=& \left( \begin{array}{c}
                  \tilde{H}_{2}^{ \prime ++} \\
                  \bar{ \tilde{H}}_{2}^{--}
\end{array} \right), \,\
\tilde{S}^{c ++}_{2}= \left( \begin{array}{c}
                  \tilde{H}_{2}^{--} \\
                  \bar{ \tilde{H}}_{2}^{ \prime ++}
\end{array} \right), \,\
\tilde{W}= \left( \begin{array}{c}
              -i \lambda^{+}_{W} \\
                i \bar{ \lambda}^{-}_{W}  
\end{array} \right), \,\
\tilde{W}^{c}= \left( \begin{array}{c}
                 -i \lambda^{-}_{W} \\
                i \bar{ \lambda}^{+}_{W}
\end{array} \right), \nonumber \\
\tilde{V}&=& \left( \begin{array}{c}
                -i \lambda^{+}_{V} \\
                i \bar{ \lambda}^{-}_{V}
\end{array} \right), \,\
\tilde{V}^{c}= \left( \begin{array}{c}
                 -i \lambda^{-}_{V} \\
                i \bar{ \lambda}^{+}_{V}
\end{array} \right), \,\ 
\tilde{T}^{+}_{1}= \left( \begin{array}{c}
                     \tilde{\eta}_{1}^{ \prime +}   \\
                      \bar{ \tilde{\eta}}_{1}^{-} 
\end{array} \right), \,\
\tilde{T}^{c +}_{1}= \left( \begin{array}{c}
                       \tilde{\eta}_{1}^{-} \\
                       \bar{ \tilde{\eta}}_{1}^{ \prime +}
\end{array} \right), \nonumber \\
\tilde{T}^{+}_{2}&=& \left( \begin{array}{c}
                       \tilde{\eta}_{2}^{+} \\
			\bar{ \tilde{\eta}}_{2}^{ \prime -}
\end{array} \right), \,\
\tilde{T}^{c +}_{2}= \left( \begin{array}{c}
                       \tilde{\eta}_{2}^{ \prime -} \\
		       \bar{ \tilde{\eta}}_{2}^{+}
\end{array} \right), \,\
\tilde{T}^{+}_{3}= \left( \begin{array}{c}
                      \tilde{\rho}^{+}  \\
                      \bar{ \tilde{\rho}}^{ \prime -} 
\end{array} \right), \,\
\tilde{T}^{c +}_{3}= \left( \begin{array}{c}
                       \tilde{\rho}^{ \prime -} \\
                       \bar{ \tilde{\rho}}^{+}
\end{array} \right), \nonumber \\
\tilde{T}^{+}_{4}&=& \left( \begin{array}{c}
                       \tilde{\chi}^{ \prime +} \\
                       \bar{ \tilde{\chi}}^{-}
\end{array} \right), \,\
\tilde{T}^{c +}_{4}= \left( \begin{array}{c}
                       \tilde{\chi}^{-} \\
                       \bar{ \tilde{\chi}}^{ \prime +}
\end{array} \right), \,\
\tilde{S}^{+}_{1}= \left( \begin{array}{c}
                     \tilde{h}_{1}^{-}   \\
                      \bar{ \tilde{h}}_{1}^{ \prime +} 
\end{array} \right), \,\
\tilde{S}^{c +}_{1}= \left( \begin{array}{c}
                       \tilde{h}_{1}^{ \prime +} \\
                       \bar{ \tilde{h}}_{1}^{-}
\end{array} \right), \nonumber \\
\tilde{S}^{+}_{2}&=& \left( \begin{array}{c}
                       \tilde{h}_{2}^{+} \\
                       \bar{ \tilde{h}}_{2}^{ \prime -}
\end{array} \right), \,\
\tilde{S}^{c +}_{2}= \left( \begin{array}{c}
                       \tilde{h}_{2}^{ \prime -} \\
                       \bar{ \tilde{h}}_{2}^{+}
\end{array} \right), \,\
\tilde{W}_{3}= \left( \begin{array}{c}
                 -i \lambda^{3}_{A} \\
                i \bar{ \lambda}^{3}_{A}  
\end{array} \right), \,\
\tilde{W}_{8}= \left( \begin{array}{c}
                 -i \lambda^{8}_{A} \\
                i \bar{ \lambda}^{8}_{A}
\end{array} \right), \,\
\tilde{B}= \left( \begin{array}{c}
             -i \lambda_{B} \\
                i \bar{ \lambda}_{B}
\end{array} \right), \nonumber \\
\tilde{T}^{0}_{1}&=& \left( \begin{array}{c}
                       \tilde{\eta}^{0} \\
                        \bar{ \tilde{\eta}}^{0}
\end{array} \right), \,\
\tilde{T}^{0}_{2}= \left( \begin{array}{c}
                     \tilde{\eta}^{\prime 0} \\
                     \bar{ \tilde{\eta}}^{\prime 0} 
\end{array} \right), \,\
\tilde{T}^{0}_{3}= \left( \begin{array}{c}
                     \tilde{\rho}^{0} \\
                     \bar{ \tilde{\rho}}^{0} 
\end{array} \right), \,\ 
\tilde{T}^{0}_{4}= \left( \begin{array}{c}
                     \tilde{\rho}^{\prime 0} \\
                     \bar{ \tilde{\rho}}^{\prime 0} 
\end{array} \right), \,\
\tilde{T}^{0}_{5}= \left( \begin{array}{c}
                     \tilde{\chi}^{0} \\
                     \bar{ \tilde{\chi}}^{0} 
\end{array} \right), \nonumber \\
\tilde{T}^{0}_{6}&=& \left( \begin{array}{c}
                      \tilde{\chi}^{\prime 0} \\
                     \bar{ \tilde{\chi}}^{\prime 0} 
\end{array} \right), \,\
\tilde{S}^{0}_{1}= \left( \begin{array}{c}
                       \tilde{\sigma}_{1}^{0} \\
                       \bar{ \tilde{\sigma}}_{1}^{0}
\end{array} \right), \,\
\tilde{S}^{0}_{2}= \left( \begin{array}{c}
                     \tilde{\sigma}_{1}^{ \prime 0} \\
                      \bar{ \tilde{\sigma}}_{1}^{ \prime 0}
\end{array} \right), \nonumber \\
\tilde{S}^{0}_{3}&=& \left( \begin{array}{c}
                       \tilde{\sigma}_{2}^{0} \\
                       \bar{ \tilde{\sigma}}_{2}^{0}
\end{array} \right), \,\
\tilde{S}^{0}_{4}= \left( \begin{array}{c}
                     \tilde{\sigma}_{2}^{ \prime 0} \\
                     \bar{ \tilde{\sigma}}_{2}^{ \prime 0} 
\end{array} \right).
\label{weakdef} 
\end{eqnarray}

With the states defined in Eqs.(\ref{weakdef}) we get the following identities, 
that allow us to write the two-component spinors in terms of (four-component) 
weak eigenstates
\begin{eqnarray}
\lambda^{--}_{U} \sigma^{m} \bar{ \lambda}^{3}_{A}&=&-
\bar{ \tilde{U}}L \gamma^{m}R \tilde{W}_{3}, \,\
\lambda^{++}_{U} \sigma^{m} \bar{ \lambda}^{3}_{A}=-
\bar{ \tilde{U}}^{c}L \gamma^{m}R \tilde{W}_{3},  \nonumber \\
\lambda^{8}_{A} \sigma^{m} \bar{ \lambda}^{++}_{U}&=&-
\bar{ \tilde{W}}_{8}L \gamma^{m}R \tilde{U}^{c}, \,\
\lambda^{++}_{U} \sigma^{m} \bar{ \lambda}^{8}_{A}=-
\bar{ \tilde{U}}^{c}L \gamma^{m}R \tilde{W}_{8},  \nonumber \\
%%%%% Higgsinos double-charge in triplets %%%%%%%%%%%
\tilde{ \chi}^{\prime ++} \sigma^{m} \bar{ \tilde{ \chi}}^{\prime 0}&=&-
\bar{ \tilde{T}}^{c++}_{3}L \gamma^{m}R \tilde{T}^{0}_{6}, \,\
\tilde{ \rho}^{++} \sigma^{m} \bar{ \tilde{ \rho}}^{0}=-
\bar{ \tilde{T}}^{c ++}_{1}L \gamma^{m}R \tilde{T}^{0}_{3}, \nonumber \\
\tilde{ \chi}^{0} \sigma^{m} \bar{ \tilde{ \chi}}^{--}&=&-
\bar{ \tilde{T}}^{0}_{5}L \gamma^{m}R \tilde{T}^{++}_{2}, \,\
\tilde{ \rho}^{\prime 0} \sigma^{m} \bar{ \tilde{ \rho}}^{\prime --}=-
\bar{ \tilde{T}}^{0}_{4}L \gamma^{m}R \tilde{T}^{++}_{1}, \nonumber \\
%%%%% Higgsinos double-charge in sextet %%%%%%%%%%%
\tilde{ \sigma}^{\prime 0}_{2} \sigma^{m} \bar{ \tilde{H}}^{\prime --}_{1}&=&-
\bar{ \tilde{S}}^{0}_{4}L \gamma^{m}R \tilde{S}^{++}_{1}, \,\
\tilde{H}^{++}_{1} \sigma^{m} \bar{ \tilde{ \sigma}}^{0}_{2}=-
\bar{ \tilde{S}}^{c ++}_{1}L \gamma^{m}R \tilde{S}^{0}_{3}, \nonumber \\
\tilde{H}^{\prime ++}_{2} \sigma^{m} \bar{ \tilde{ \sigma}}^{\prime 0}_{2} &=&-
\bar{ \tilde{S}}^{c ++}_{2}L \gamma^{m}R \tilde{S}^{0}_{4}, \,\
\tilde{ \sigma}^{0}_{2} \sigma^{m} \bar{ \tilde{H}}^{--}_{2}=-
\bar{ \tilde{S}}^{0}_{3}L \gamma^{m}R \tilde{S}^{++}_{2}, \nonumber \\
%%%%% Gauginos with one charge %%%%%%%%%%%%%
\lambda^{-}_{V} \sigma^{m} \bar{ \lambda}^{+}_{W}&=&-
\bar{ \tilde{V}}L\gamma^{m}R \tilde{W}^{c}, \,\
\lambda^{+}_{W} \sigma^{m} \bar{ \lambda}^{-}_{V}=-
\bar{ \tilde{W}}^{c}L \gamma^{m}R \tilde{V},  \nonumber \\
%%%%% Higgsinos charge %%%%%%%%%%%
\tilde{ \eta}^{+}_{2} \sigma^{m} \bar{ \tilde{ \eta}}^{-}_{1}&=&-
\bar{ \tilde{T}}^{c +}_{2}L \gamma^{m}R \tilde{T}^{+}_{1}, \,\
\tilde{h}^{\prime +}_{1} \sigma^{m} \bar{ \tilde{h}}^{\prime -}_{2}=-
\bar{ \tilde{S}}^{c +}_{1}L \gamma^{m}R \tilde{S}^{+}_{2}, \nonumber \\
\tilde{ \eta}^{\prime +}_{1} \sigma^{m} \bar{ \tilde{ \eta}}^{\prime -}_{2}&=&-
\bar{ \tilde{T}}^{c +}_{1}L \gamma^{m}R \tilde{T}^{+}_{2}, \,\
\tilde{h}^{+}_{2} \sigma^{m} \bar{ \tilde{h}}^{-}_{1}=-
\bar{ \tilde{S}}^{c +}_{2}L \gamma^{m}R \tilde{S}^{+}_{1}.
\label{w4ssc}
\end{eqnarray}

\subsection{Mass Eigenstates}

From Eq.(\ref{2dc}, \ref{2sc}, \ref{emasneu}), we can write the inverse 
transformation as:
\begin{eqnarray}
\Psi^{++}_{k}&=&A^{*}_{ik} \tilde{ \chi}^{++}_{i}, \,\
\Psi^{--}_{k}=B^{*}_{ik} \tilde{ \chi}^{--}_{i}, \,\
\Psi^{+}_{k}=D^{*}_{ik} \tilde{ \chi}^{+}_{i}, \,\
\Psi^{-}_{k}=E^{*}_{ik} \tilde{ \chi}^{-}_{i}, \nonumber \\
\Psi^{0}_{k}&=&N^{*}_{ik} \tilde{ \chi}^{0}_{i}.
\label{inv1}
\end{eqnarray}
 Equations(\ref{w4ssc}) above do not involve physical particles. From 
Eqs.(\ref{emassdou}) and (\ref{emassneu}) we can show the following
\begin{eqnarray}
\tilde{ \chi}^{++}_{i}&=&L \Psi( \tilde{ \chi}^{++}_{i}), \,\
\bar{ \tilde{ \chi}}^{++}_{i}=R \Psi^{c}( \tilde{\chi}^{--}_{i}), \,\
\bar{ \tilde{ \chi}}^{--}_{i}=R \Psi( \tilde{ \chi}^{++}_{i}), \,\
\tilde{ \chi}^{--}_{i}=L \Psi^{c}( \tilde{ \chi}^{--}_{i}), \nonumber \\
\tilde{ \chi}^{0}_{i}&=&L \Psi( \tilde{ \chi}^{0}_{i}), \,\
\bar{ \tilde{ \chi}}^{0}_{i}=R \Psi( \tilde{ \chi}^{0}_{i}), \nonumber \\
\tilde{ \chi}^{+}_{i}&=&L \Psi( \tilde{ \chi}^{+}_{i}), \,\
\bar{ \tilde{ \chi}}^{+}_{i}=R \Psi^{c}( \tilde{\chi}^{-}_{i}), \,\
\bar{ \tilde{ \chi}}^{-}_{i}=R \Psi( \tilde{ \chi}^{+}_{i}), \,\
\tilde{ \chi}^{-}_{i}=L \Psi^{c}( \tilde{ \chi}^{-}_{i}).
\label{4sss}
\end{eqnarray}


\section{Differential Cross Section}
\label{a6}

In this Appendix we calculate the differential cross section to the processes 
we have studied in section \ref{sec5}.

\subsection{$e^-e^- \to \tilde{\chi}^- \tilde{\chi}^-$( Charginos productions)}

\begin{eqnarray}
\frac{d \sigma}{d \Omega}(e^-e^- \to \tilde{\chi}^- \tilde{\chi}^-)= 
\frac{1}{128 \pi^2s} \sqrt{\frac{s}{\frac{s}{4}-m^2_{\tilde{\chi}^+}}}
|{\cal M}_T|^2,
\label{dif1}
\end{eqnarray}
where
\begin{eqnarray}
|{\cal M}_T|^2&=&|D_{i7}|^4 \left( \frac{1}{(t-m^2_{\tilde{\nu}})^2}+ 
\frac{1}{(u-m^2_{\tilde{\nu}})^2} \right) \left\{  2m^4_{\tilde{\chi}^+}+ 
2(E-2m^2_{\tilde{\chi}^+})m^2_{\tilde{\chi}^+} \right. \nonumber \\
&+& \left. 2 \left[ 
(m^2_{\tilde{\chi}^+}-\frac{s}{2})^2+s \left( \frac{s}{4}-
m^2_{\tilde{\chi}^+} \right) \cos \theta \right] \right\} \nonumber \\
&-& \frac{2g^2(s-M^2_U)O_{2}|D_{i7}|^2}{(s-M^2_U)^2+(\Gamma_UM_U)^2} \left[  
m^4_{\tilde{\chi}^+}\left( \frac{1}{(t-m^2_{\tilde{\nu}})}+ 
\frac{1}{(u-m^2_{\tilde{\nu}})} \right) \right. \nonumber \\
&+& \left. (2m^2_{\tilde{\chi}^+}+s)
\left( \frac{u}{(t-m^2_{\tilde{\nu}})}+ 
\frac{t}{(u-m^2_{\tilde{\nu}})} \right) \right], \nonumber \\
&+& \frac{g^4|O_{2}|^2}{(s-M^2_U)^2+(\Gamma_UM_U)^2}(m^4_{\tilde{\chi}^+}- 
8sm^2_{\tilde{\chi}^+}+4s^2)+ 
\frac{4s|D_{i7}|^2}{(t-m^2_{\tilde{\nu}})(u-m^2_{\tilde{\nu}})}. \nonumber
\end{eqnarray}

\subsection{$e^-e^- \to \tilde{\chi}^{--} \tilde{\chi}^0$( Double charged 
Neutralinos production)}
\begin{eqnarray}
\frac{d \sigma}{d \Omega}(e^-e^- \to \tilde{\chi}^{--} \tilde{\chi}^0)= 
\frac{1}{64 \pi^2s} \sqrt{\frac{s}{E^2_{\chi^{++}}-m^2_{\chi^{++}}}}
|{\cal M}_T|^2,
\label{dif2}
\end{eqnarray}
where
\begin{eqnarray}
|{\cal M}_T|^2&=& \left( 
\frac{X^2_1 \cos^2 \theta_f}{(u-m^2_{\tilde{l}_2})^2}+
\frac{X^2_2 \cos^2 \theta_f}{(t-m^2_{\tilde{l}_2})^2}+
\frac{X^2_3 \sin^2 \theta_f}{(t-m^2_{\tilde{l}_1})^2}+
\frac{X^2_4 \sin^2 \theta_f}{(u-m^2_{\tilde{l}_1})^2} \right. \nonumber \\
&+& \left.
\frac{X_1X_4 \sin \theta_f \cos \theta_f}{(u-m^2_{\tilde{l}_2})(u-m^2_{\tilde{l
}_1})}+
\frac{X_2X_3 \sin \theta_f \cos \theta_f}{(t-m^2_{\tilde{l}_2})(t-m^2_{\tilde{l
}_1})}
\right) \nonumber \\
&\cdot& [2(m^4_{\chi^{++}}+m^4_{\chi^{0}})+s(m^2_{\chi^{++}}+m^2_{\chi^{0}}-s)+
2t(t-2m^2_{\chi^{++}})+2u(u-2m^2_{\chi^{0}})] \nonumber \\
&+&2m^4_{\chi^{++}}m^4_{\chi^{0}}O^1 \cos \theta_f \left( 
\frac{X_1}{(u-m^2_{\tilde{l}_2})^2[(s-M^2_U)^2+(\Gamma_UM_U)^2]} \right. 
\nonumber \\
&+& \left. 
\frac{X_2}{(t-m^2_{\tilde{l}_2})^2[(s-M^2_U)^2+(\Gamma_UM_U)^2]} 
\right) \nonumber \\
&+&
2m^4_{\chi^{++}}m^4_{\chi^{0}}O^1 \sin \theta_f \left( 
\frac{X_3}{(t-m^2_{\tilde{l}_1})^2[(s-M^2_U)^2+(\Gamma_UM_U)^2]} \right. 
\nonumber \\
&+& \left. 
\frac{X_4}{(u-m^2_{\tilde{l}_1})^2[(s-M^2_U)^2+(\Gamma_UM_U)^2]} 
\right) \nonumber \\
&+& \frac{O^1}{(s-M^2_U)^2+(\Gamma_UM_U)^2}[2(m^4_{\chi^{++}}+m^2_{\chi^{0}})+ 
4(2(m^2_{\chi^{++}}+m^2_{\chi^{0}}))s+s^2], \nonumber 
\end{eqnarray}
where
\begin{eqnarray}
X_1&=&\frac{2}{\sqrt{2}}A_{i5}\lambda_3N_{i8}, \,\
X_2=\frac{2}{\sqrt{2}}A_{i1}gN_{i8}, \nonumber \\
X_3&=&A_{i1}g \left( 
\frac{N_{i1}}{\sqrt{2}}+\frac{N_{i2}}{\sqrt{6}} \right), \,\
X_4=A_{i5}\lambda_3\left( 
\frac{N_{i1}}{\sqrt{2}}+\frac{N_{i2}}{\sqrt{6}} \right). 
\end{eqnarray}

\begin{references}  
\bibitem{331} F. Pisano and V. Pleitez {\sl Phys. Rev.}{\bf D46}, 410, (1992);
P. H. Frampton, {\sl Phys. Rev. Lett.} {\bf 69}, 2889, (1992); 
R. Foot, O. F. Hernandez, F. Pisano and V. Pleitez,
{\sl Phys. Rev.}{\bf D47}, 4158, (1993).
\bibitem{tec} Some earlier reviews of technicolor theories are \\ 
E. Farhi and L. Suskind, {\sl Phys. Rep.}{\bf 74}, 277 (1981); \\  
R. Kaul, {\sl Rev. Mod. Phys.}{\bf 55}, 449 (1983).
\bibitem{331susy} J. C. Montero, V. Pleitez and M. C. Rodriguez, .
\bibitem{radl} H. E. Haber, {\sl Eur. Phys. J.}{\bf C 15}, 817, (2000).
\bibitem{exp1} The Lep working group for Higgs boson searches, L3 Note 253.
\bibitem{mssm} H. E. Haber and G. L. Kane, Phys. Rep. {\bf117}, 75 (1985).
\bibitem{LR} K. Huitu, J. Maalampi and M. Raidal, {\sl Nucl. Phys. }{\bf B420} 
449 (1994).
\bibitem{jj} P. Jain and S. D. Joglekar, Phys. Lett. {\bf B407}, 151 (1997).
\bibitem{desy} DESY/ECFA 1998/99 SUSY Working Group see \\
http://www.desy.de/conferences/ecfa-desy-lc98.html
\bibitem{cp} F. Cuypers, G. J. van Oldenborgh and R. R\"uckl, Nucl. Phys. 
{\bf B49}, 128 (1993).
\bibitem{wb} {\it Supersymmetry and Supergravity}, J. Wess and J. Bagger,
{\sl 2nd edition, Princeton University Press, Princeton NJ}, (1992).
\bibitem{assi3} J. C. Montero, V. Pleitez and M. C. Rodriguez, 
{\sl Int. J. Mod. Phys. A16}, 1147, (2001). 
\end{references}

\begin{figure}[ht]
\begin{center}
\vglue -0.009cm
\mbox{\epsfig{file=graf.eps,width=0.3\textwidth,angle=0}}       
\end{center}
\caption{Feynman diagrams for the process 
$e^-e^- \to \tilde{ \chi}^{-} \tilde{ \chi}^{-}$.}
\label{process2}
\end{figure}

\begin{figure}[ht]
\begin{center}
\vglue -0.009cm
\mbox{\epsfig{file=cgraf.eps,width=0.3\textwidth,angle=90}}       
\end{center}
\caption{Total Cross Section $e^-e^- \to \tilde{ \chi}^{-} \tilde{ \chi}^{-}$ 
at $\sqrt{s}=0.5$TeV and $O_{2}=1$, $A_{i1}=10^{-1}$.}
\label{plot1}
\end{figure}

\begin{figure}[ht]
\begin{center}
\vglue -0.009cm
\mbox{\epsfig{file=graf3.eps,width=0.3\textwidth,angle=90}}       
\end{center}
\caption{Total Cross Section $e^-e^- \to \tilde{ \chi}^{-} \tilde{ \chi}^{-}$ 
at $\sqrt{s}=1.0$TeV and $D_{i7}=10^{-1}$, $O_{2}=10^{-1}$.}
\label{plot2}
\end{figure}


\begin{figure}[ht]
\begin{center}
\vglue -0.009cm
\mbox{\epsfig{file=graf2.eps,width=0.3\textwidth,angle=90}}       
\end{center}
\caption{Total Cross Section $e^-e^- \to \tilde{ \chi}^{-} \tilde{ \chi}^{-}$ 
at $\sqrt{s}=2.0$TeV and $D_{i7}=1$, $O_{2}=10^{-1}$.}
\label{plot3}
\end{figure}

\begin{figure}[ht]
\begin{center}
\vglue -0.009cm
\mbox{\epsfig{file=graf1.eps,width=0.3\textwidth,angle=0}}       
\end{center}
\caption{Feynman diagrams for the process 
$e^-e^- \to \tilde{ \chi}^{--} \tilde{ \chi}^{0}$, and $i=1,2$.}
\label{process1}
\end{figure}

\begin{figure}[ht]
\begin{center}
\vglue -0.009cm
\mbox{\epsfig{file=dgraf.eps,width=0.3\textwidth,angle=90}}       
\end{center}
\caption{Total Cross Section $e^-e^- \to \tilde{ \chi}^{--} \tilde{ \chi}^{0}$ 
at $\sqrt{s}=0.5$TeV and \\
$X_1 \cos \theta_f=X_2 \cos \theta_f=
X_3 \sin \theta_f=X_4 \sin \theta_f=10^{-1}$, $O^1=10^{-2}$.}
\label{plot4}
\end{figure}

\begin{figure}[ht]
\begin{center}
\vglue -0.009cm
\mbox{\epsfig{file=dgraf1.eps,width=0.3\textwidth,angle=90}}       
\end{center}
\caption{Total Cross Section $e^-e^- \to \tilde{ \chi}^{--} \tilde{ \chi}^{0}$ 
at $\sqrt{s}=1.0$TeV and \\
$X_1 \cos \theta_f=X_2 \cos \theta_f=
X_3 \sin \theta_f=X_4 \sin \theta_f=10^{-1}$, $O^1=10^{-1}$.}
\label{plot5}
\end{figure}

\begin{figure}[ht]
\begin{center}
\vglue -0.009cm
\mbox{\epsfig{file=dgraf2.eps,width=0.3\textwidth,angle=90}}       
\end{center}
\caption{Total Cross Section $e^-e^- \to \tilde{ \chi}^{--} \tilde{ \chi}^{0}$ 
at $\sqrt{s}=2.0$TeV and \\
$X_1 \cos \theta_f=X_2 \cos \theta_f=
X_3 \sin \theta_f=X_4 \sin \theta_f=10^{-2}$, $O^1=10^{-2}$.}
\label{plot6}
\end{figure}

\begin{table}
\begin{center}
\begin{tabular}{|c|c|} 
Vertices & Feynman rules   \\ \hline 
$l^-l^-U^{--}$ & $- \frac{ig}{\sqrt{2}} C \gamma^{m}L$ \\ \hline 
$\tilde{ \chi}^{--}_{j} \tilde{ \chi}^{0}_{i}U^{--}$ & 
$\frac{ig}{2} O^{1}_{ij}C \gamma^{m}R$ \\ \hline
$\tilde{ \chi}^{-}_{i} \tilde{ \chi}^{-}_{j}U^{--}$ &
$\frac{ig}{2}O^{2}_{ij}C \gamma^{m}R$ \\ \hline
$\tilde{l}^{-}_{1}l^{-} \tilde{\chi}^{--}_{i}$ & 
$- 2i \lambda_3A_{i5} \sin \theta_{f}R$ \\ \hline
$\tilde{l}^{-}_{2}l^{-} \tilde{\chi}^{--}_{i}$ &
$- 2i \lambda_3A_{i5} \cos \theta_{f}R$ \\ \hline
$\tilde{l}^{-}_{1}l^{-} \tilde{\chi}^{0}_{i}$ &
$i \left[ g \left( 
\frac{N_{i1}}{\sqrt{2}}+ \frac{N_{i2}}{\sqrt{6}} \right) \cos \theta_{f}R- 
\lambda_{3} \frac{2}{\sqrt{2}} \sin \theta_{f}N_{i8}R \right]$ \\ \hline
$\tilde{l}^{-}_{2}l^{-} \tilde{\chi}^{0}_{i}$ &
$i \left[ g \left( 
\frac{N_{i1}}{\sqrt{2}}+ \frac{N_{i2}}{\sqrt{6}} \right) \sin \theta_{f}R+ 
\lambda_{3} \frac{2}{\sqrt{2}} \cos \theta_{f}N_{i8}R \right]$ \\ \hline
$\tilde{l}^{+}_{1}l^{-} \tilde{\chi}^{0}_{i}$ &
$i \left[ g \left(
\frac{N^*_{i1}}{\sqrt{2}}+ \frac{N^*_{i2}}{\sqrt{6}} \right) \cos \theta_{f}L- 
\lambda_{3} \frac{2}{\sqrt{2}} \sin \theta_{f}N^*_{i8}L \right]$ \\ \hline
$\tilde{l}^{+}_{2}l^{-} \tilde{\chi}^{0}_{i}$ &
$i \left[ g \left( 
\frac{N^*_{i1}}{\sqrt{2}}+ \frac{N^*_{i2}}{\sqrt{6}} \right) \sin \theta_{f}L+ 
\lambda_{3} \frac{2}{\sqrt{2}} \cos \theta_{f}N^*_{i8}L \right]$ \\ \hline
$\tilde{l}^{+}_{1}l^{-} \tilde{\chi}^{--}_{i}$ &
$-igA_{i1} \sin \theta_{f}RC$ \\ \hline
$\tilde{l}^{+}_{2}l^{-} \tilde{\chi}^{--}_{i}$ &
$-igA_{i1} \cos \theta_{f}RC$ \\ \hline
$\tilde{\nu_{l}}l^{-} \tilde{\chi}^{-}_{i}$ &
$-i \lambda_{3} \frac{2}{\sqrt{2}}D^{*}_{i7}L$ \\ \hline
$\tilde{\nu_{l}}^{*}l^{-} \tilde{\chi}^{-}_{i}$ &
$-i \lambda_{3} \frac{2}{\sqrt{2}}D_{i7}R$ \\ \hline
\end{tabular}
\end{center}
\caption{Feynman rules derived from Eqs.
(\ref{lagr2},\ref{basemassa1},\ref{basemassa2},\ref{sleplepchar}).}
\label{t1}
\end{table}

\begin{table}
\begin{center}
\begin{tabular} {|c|c|c|}
$\tilde{m}$ [GeV]    & \mbox{RR1} : $\tan\beta=3$ & 
\mbox{RR2} : $\tan\beta=30$ \\ \hline 
$\tilde{\chi}^\pm_1$ & 128                 & 132   \\ \hline
$\tilde{\chi}^0_1$   & 70                  & 72    \\ \hline
$\tilde{e}_L^-$	     & 176		   & 217   \\ \hline
$\tilde{e}_R^-$	     & 132		   & 183   \\ \hline
$\tilde{\nu}$        & 166                 & 206   \\ \hline 
\end{tabular}
\end{center}
\caption{Mass values of the ligthest chargino and neutralino, sleptons and of
the sneutrino at electro weak scale corresponding to the mSUGRA solution.}
\label{t2}
\end{table}



\end{document}


