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%                Neutrinos and Supersymmetry                         %
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%                file=book03c8.tex                                   %
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%                Last changed 23/8/2003 by JCR                       %
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\setcounter{chapter}{7}
\chapter{Phenomenology of Bilinear R Parity Violation}
\label{BRpVphenomenology}

\section{Introduction}

In this chapter we will discuss the phenomenology of the Bilinear
R-parity Violation (BRpV) model. We will see that low energy
supersymmetry with bilinear breaking of R-parity leads to a weak-scale
seesaw mechanism for the atmospheric neutrino scale and a radiative
mechanism for the solar neutrino scale. From this analysis we will
discover that the BRpV parameters have to be at most $\mathcal{O}(1\
\hbox{GeV})$, therefore small compared with the electroweak
scale. This in turn implies that from the point of view of production
of supersymmetric particles the model will not differ from the
MSSM. However, due to the R-parity breaking, the otherwise Lightest
Supersymmetric Particle (LSP) will decay given origin to decay modes
that are quite different from the MSSM. We should therefore 
look at the decays to distinguish between the
models. In fact, we will discover that the model has
striking implications for collider searches of supersymmetric
particles. We will discuss the decays of the lightest
SUSY particle in the cases where this particle is either the
neutralino or the stau as well as the decays of the stop and show how
these decays can be correlated to the neutrino properties.



\section{Neutrino Properties and BRpV Parameters}

\subsection{The Atmospheric Neutrino Anomaly}

The BRPV model produces a hierarchical mass spectrum for almost all
choices of parameters. The largest mass can be estimated by the tree
level value as shown in \Fig{cascais2000fig3}. As the figure shows,
correct $\Delta m^2_{atm}$ can be easily obtained by an appropriate
choice of $| \vec \Lambda|$. The mass scale for the solar neutrinos is
generated a 1--loop level and therefore depends in a complicated way
in the model parameters. This is shown in \Fig{allmasses} where
we have fixed the SUSY parameters. The parameter $\epsilon^2/|\vec \Lambda|$ 
is the most important in determining the solar mass scale, but some
other parameters also play a role~\cite{romao:1999up,hirsch:2000ef}.


\begin{figure}[htb]
\begin{center}
\begin{picture}(9,6)
\put(0,0){\includegraphics[width=9cm]{cascais2000fig3.ps}}
\end{picture}
\end{center}
\vspace{-5mm}
\caption{The atmospheric $\Delta m^2$ as function of 
$|\vec \Lambda |/(\sqrt{M_2}\mu)$}
\label{cascais2000fig3}
\end{figure}



\begin{figure}[htb]
\begin{center}
\begin{picture}(9,6)
\put(0,0){\includegraphics[width=9cm]{cascais2000fig1.ps}}
\end{picture}
\end{center}
\vspace{-5mm}
\caption{Neutrino masses as a function of $\epsilon^2/|\vec \Lambda|$}
\label{allmasses}
\end{figure}


\ni
Now we turn to the discussion of the mixing angles. As it can be seen
from Fig.~\ref{allmasses} if
$\epsilon^2/|\vec \Lambda| \ll 100$ then the one loop corrections are
not larger then the tree level results and the flavour composition of
the 3rd mass eigenstate is approximately given by
%
\beq
U_{\alpha 3}\approx\Lambda_{\alpha}/|\vec \Lambda |
\eeq
%
As the atmospheric and reactor neutrino data tell us that
$\nu_{\mu}\ra \nu_{\tau}$ oscillations are preferred over 
$\nu_{\mu}\ra \nu_e$, we conclude that  
%
\begin{equation}
\label{eq:AtmosphericConditions}
\Lambda_e \ll \Lambda_{\mu} \simeq \Lambda_{\tau}
\end{equation}
%
are required for BRPV to fit the the data. This is sown in
\Fig{val2000fig56}. On the left panel we plot $U_{e3}^2$ constrained
from the CHOOZ experiment to be small, while on the left panel the
atmospheric angle is plotted. It is clear that the conditions in
Eq.~(\ref{eq:AtmosphericConditions}) will be in agreement with the
experimental data. In practice $\Lambda_e$ does not need to be too
small. In fact $5 \Lambda_e \simeq \Lambda_{\mu} \simeq \Lambda_{\tau}$
will be enough.

\begin{figure}[htb]
\centering
\begin{tabular}{cc}
\includegraphics[width=65mm,height=55mm]{val2000fig6.ps}
&
\includegraphics[width=65mm,height=55mm]{val2000fig5.ps}
\end{tabular}
\vspace{-5mm}
\caption{ a) $U^2_{e3}$ as a function of 
$|\Lambda_e|/\sqrt{\Lambda^2_{\mu}+\Lambda^2_{\tau}}$.
b) Atmospheric angle as a function of 
$|\Lambda_{\mu}|/\sqrt{\Lambda^2_{\mu}+\Lambda^2_{\tau}}$.}
\label{val2000fig56}
\end{figure}


\subsection{The Solar Neutrino Anomaly}


For the solar angle the situation is more complicated and there are
two cases to consider~\cite{romao:1999up,hirsch:2000ef}. With the
usual SUGRA assumptions, ratios of $\epsilon_i/\epsilon_j$ fix the
ratios of $\Lambda_i/\Lambda_j$.  Since atmospheric and reactor data
tell us that $\Lambda_e \ll \Lambda_{\mu},\Lambda_{\tau}$ in this case
only the small angle solution can be obtained in the BRpV model as
shown in Fig.~(\ref{val2000fig78} a).  However it can be
shown~\cite{romao:1999up,hirsch:2000ef} that even a tiny deviation of
from universality of the soft parameters at the GUT scale relaxes this
constraint. In this case the ratio $\epsilon_i/\epsilon_j$ is not
constrained and also large angle solutions can be obtained as shown in
Fig.~(\ref{val2000fig78} b).



\begin{figure}[htb]
\centering
\begin{tabular}{cc}
\includegraphics[width=65mm,height=55mm]{val2000fig7.ps}
&
\includegraphics[width=65mm,height=55mm]{cascais2000fig8.ps}
\end{tabular}
\vspace{-2mm}
\caption{The solar angle as function of:
a)$|\Lambda_e|/\sqrt{\Lambda^2_{\mu}+\Lambda^2_{\tau}}$;
b)$\epsilon_e/\sqrt{\epsilon^2_{\mu}+\epsilon^2_{\tau}}$}
\label{val2000fig78}
\end{figure}

\subsection{The BRpV parameters}

From Fig.~\ref{cascais2000fig3} one can get that the atmospheric
neutrino data requires  $|\Lambda| \simeq 0.1 \hbox{GeV}^2$. From the
definition of the $\Lambda_i$, Eq.~(\ref{eq:lambdai}), we can conclude
that if we do not want excessive fine tunning then the BRpV parameters
$\epsilon_i$ have to obbey,
%
\begin{equation}
  \label{eq:epsilonLimits}
  |\epsilon_i| \leq 1 \hbox{GeV}
\end{equation}
%
As we will show below this in turn will imply that R-parity violation
will never be seen in the production, only in the decay.


\section{Supersymmetric Particle Production}

small RPV leads to small single supersymmetric particle production

standard pair production

BARTL ,porod

\section{Supersymmetric Particle Decays}

\subsection{Charginos \& neutralinos}

porod +++ 

\subsection{h, sleptons, squarks}

porod +++ 

\section{NEW}

\section{Neutralino decays}
\label{sec:NeutralinoDecays}

\subsection{Approximate Formulas for neutralino couplings}
\label{sec:approx}
%
\begin{figure}[t]
 \setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(140,80)
\put(-35,-112){\mbox{\epsfig{figure=FeynGraphs1.eps,height=29.7cm,width=21.cm}}}
\end{picture}
\end{center}
\caption[]{ Feynman graphs for the decay 
  $\chiz{1} \to \nu_i \, l^-_j \, l^+_k$.}
\label{fig:graphs}
\end{figure}

\begin{figure}[ht] %[h] Please don't erase the h option
 \setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(140,80)
\put(-35,-112){\mbox{\epsfig{
               figure=FeynGraphs2.eps,height=29.7cm,width=21.cm}}}
\end{picture}
\end{center}
\caption[]{ Generic Feynman graphs for semi-leptonic neutralino decays.}
\label{fig:graphs1}
\end{figure}
%
\begin{figure}[ht] %[h] Please don't erase the h option
 \setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(140,30)
\put(-35,-140){\mbox{\epsfig{
               figure=FeynGraphs3.eps,height=29.7cm,width=21.cm}}}
\end{picture}
\end{center}
\caption[]{ Generic Feynman graphs for invisible neutralino decays.}
\label{fig:graphs2}
\end{figure}
The set of Feynman diagrams involved in neutralino decays is indicated
in Figs.~\ref{fig:graphs}, \ref{fig:graphs1} and \ref{fig:graphs2}.
%
Most of the relevant couplings involved have been given in appendix B
of ref.~\cite{hirsch:2000ef} and the remaining ones will be included in
appendix B of the present paper. Even though these are sufficient for
our calculation of neutralino production and decay properties, it is
very useful to have approximate formulas for the neutralino couplings,
since this allows some qualitative understanding of the correlations
we are going to discuss.
%
To achieve this we make use of the expansions for the neutralino mass
matrix and also a corresponding one for the charginos as given in
\cite{hirsch:1998kc}\footnote{Note that one has to reverse the sign of the
  $\epsilon_i$ in \cite{hirsch:1998kc} to be consistent with our present
  notation.}.  For this purpose we will confine ourselves to the
tree-level neutralino/neutrino mass matrix and we refer to
\sect{sec:predictions} for a short discussion of the necessary changes
once the 1-loop corrections to the mass matrix are included.  However,
we have used exact numerical diagonalizations and loop effects in the
calculation of all resulting physical quantities presented in Secs. 3
and 4.

One class of decays which is important are those involving a
$W$-boson, either virtual or real. The $\chiz{1}$-$W^\pm$-$l_i$
couplings are approximatively given by:
%
\bea
\label{eq:coupWRChi}
O^{cnw}_{Ri1} &=& \frac{g h_E^{ii} v_D}{2 \mathrm{Det}_+} 
      \left[ \frac{g v_D N_{12} + M_2 N_{14}}{\mu} \epsilon_i \right.
     \nonumber  \\
     && \hspace{1.2cm} \left.
           + g \frac{ \left(2 \mu^2 + g^2 v_D v_U \right) N_{12}
                    + \left(\mu + M_2 \right) g v_U N_{14} }
                    {2 \mu \mathrm{Det}_+} \Lambda_i
             \right] \\
O^{cnw}_{Li1} &=& \frac{g \Lambda_i}{\sqrt{2}}
     \left[ - \frac{g' M_2 \mu }{2 \mathrm{Det}_0} N_{11} 
       + g \left( \frac{1}{\mathrm{Det}_+} 
                + \frac{M_1 \mu}{2 \mathrm{Det}_0} \right) N_{12} \right. \no
       & & \hspace{7mm} - \frac{v_U}{2} \left( 
                   \frac{g^2 M_1 + {g'}^2 M_2}{2 \mathrm{Det}_0}
                   + \frac{g^2}{\mu \mathrm{Det}_+} \right) N_{13} % \no
        + \left. \frac{v_D (g^2 M_1 + {g'}^2 M_2)}{4 \mathrm{Det}_0}
                 N_{14} \right] % \no
\label{eq:CoupWLChi}
\eea
%
Here $\mathrm{Det}_+$ and $\mathrm{Det}_0$ denote the determinant of
the MSSM chargino and neutralino mass matrix, respectively.  $N_{ij}$
are the elements of the mixing matrix which diagonalizes the MSSM
neutralino mass matrix.

For the coupling $Z$-$\chiz{1}$-$\nu_i$ we find
%
\begin{eqnarray}
\label{eq:coupZnu}
O^{nnz}_{L\chi^0_1 \nu_1} &=& O^{nnz}_{L\chi^0_1 \nu_2} = 0 \\
O^{nnz}_{L\chi^0_1 \nu_3} &=& \left(
        \frac{g \left(g M_1  N_{12}-  g' M_2 N_{11} \right) \mu}
             {4 \cos \theta_W \mathrm{Det}_0} 
      + \frac{g \left(g^2 M_1 + {g'}^2 M_2\right) v_D N_{14}}
             {4 \cos \theta_W \mathrm{Det}_0}  \right)
        |\vec{\Lambda}| \, \, . 
\end{eqnarray}
%
As already mentioned, the tree-level the states $\nu_1$ and $\nu_2$
are not well defined.  Therefore one has to consider the complete
1-loop mass matrix as it will be done in the numerical part in
sections 3 and 4. However, as one cannot detect single neutrino
flavours, in experiments one observes the decay of $\tilde \chi^0_1
\to X + \nu_i$ summing over all neutrinos $\nu_i$. Therefore, for the
$Z$-mediated decay the interesting quantity is $\sum_{i=1,3}
|{O^{nnz}_{L\chi^0_1 \nu_i}}|^2$ and, in contrast to the individual
$\chi^0_1 \to Z \, \nu_i$ decay rates, this only gets small radiative
corrections.


For the coupling $\chiz{1}$-$\nu_i$-($S^0_1 \simeq h^0$) we get
%
\begin{eqnarray}
\label{eq:coupSnuA}
O^{nnh}_{111} &=& E_{\chiz{1}} 
           \left( \sina \, c_2 \, c_4 \, c_6 
    \frac{- \epsilon_e \left( \Lambda_\mu^2 + \Lambda_\tau^2 \right)
          + \Lambda_e \left( \epsilon_\mu  \Lambda_\mu
                           + \epsilon_\tau \Lambda_\tau \right)}
         {\mu \sqrt{ \Lambda_e^2 +  \Lambda_\tau^2} |\vec{\Lambda}|}
        \right. \nonumber \\
 &&  \hspace{1cm} + \left.
    \frac{- s_2 \left( \Lambda_\mu^2 + \Lambda_\tau^2 \right)
          + \Lambda_e \left( s_4  \Lambda_\mu
                           + s_6 \Lambda_\tau \right)}
         { \sqrt{ \Lambda_e^2 +  \Lambda_\tau^2} |\vec{\Lambda}|} 
        \right) \\
%
O^{nnh}_{121} &=& E_{\chiz{1}} 
           \left( \sina \, c_2 \, c_4 \, c_6 
    \frac{ \epsilon_\tau \Lambda_\mu - \epsilon_\mu \Lambda_\tau }
         {\mu \sqrt{ \Lambda_e^2 +  \Lambda_\tau^2} } 
    + \frac{ s_6 \Lambda_\mu - s_4 \Lambda_\tau }
           { \sqrt{ \Lambda_e^2 +  \Lambda_\tau^2} } \right)  \\
%
O^{nnh}_{131} &=& E_{\chiz{1}}
        \left(\sina \, c_2 \, c_4 \, c_6
           \frac{(\vec{\epsilon}.\vec{\Lambda})}{|\vec{\Lambda}|}
         + \frac{(\vec{s}.\vec{\Lambda})}{|\vec{\Lambda}|} 
        \right)
       - D_{\chiz{1}} \, c_2 \, c_4 \, c_6 |\vec{\Lambda}|
\label{eq:coupSnu}
\end{eqnarray}
with
\begin{eqnarray}
\vec{s} &=& (s_2,s_4,s_6) \hspace{0.2cm}, \\
E_{\chiz{1}} &=& \frac{(g' N_{11} - g N_{12})}{2} \hspace{0.2cm}, \mathrm{and} \\
D_{\chiz{1}} &=&  \frac{\left( g^2 M_1 + {g'}^2 M_2 \right)
       \left[ \left( \cosa  \,v_D + \sina  \, v_U \right)
              \left(g' N_{11} - g N_{12} \right)
            + 2 \mu \left(\sina N_{13} - \cosa N_{14} \right) 
       \right]}{8 \mathrm{Det}_0} \no
\end{eqnarray}

The quantities $s_i$ and $c_i$ are parts of the mixing matrix for the neutral
scalars, which is discussed in section XXX.  

For the couplings $\tilde d_{Li} - d_i - \nu_j$ one finds
%
\begin{eqnarray}
\label{eq:CoupSdDChi1}
 O^{dnL}_{Li1} &=& h^{ii}_D 
    \frac{- \epsilon_e \left( \Lambda_\mu^2 + \Lambda_\tau^2 \right)
          + \Lambda_e \left( \epsilon_\mu  \Lambda_\mu
                           + \epsilon_\tau \Lambda_\tau \right)}
         {\mu \sqrt{ \Lambda_e^2 +  \Lambda_\tau^2} |\vec{\Lambda}|} \\
 O^{dnL}_{Ri1} &=& 0 \\
 O^{dnL}_{Li2} &=& h^{ii}_D 
    \frac{ \epsilon_\tau \Lambda_\mu -  \epsilon_\mu \Lambda_\tau }
         {\mu \sqrt{ \Lambda_e^2 +  \Lambda_\tau^2} } \\
 O^{dnL}_{Ri2} &=& 0 \\
 O^{dnL}_{Li3} &=& h^{ii}_D \left(
    G_{\chiz{1}} |\vec{\Lambda}|
   - \frac{(\vec{\epsilon},\vec{\Lambda})}{\mu |\vec{\Lambda}|} \right) \\
 O^{dnL}_{Ri3} &=& H_{\chiz{1}} |\vec{\Lambda}|
\label{eq:CoupSdDChi2}
\end{eqnarray}
%
with $G_{\chiz{1}} = (g^2 M_1 + {g'}^2 M_2) v_U / (4 \mathrm{Det}_0)$
and $H_{\chiz{1}} = (3 g^2 M_1 + {g'}^2 M_2) \mu / (6 \sqrt{2}
\mathrm{Det}_0)$.  For the couplings $\tilde d_{Ri} - d_i - \nu_j$ one
finds that $O^{dnR}_{Rij} = O^{dnL}_{Lij}$ and
$O^{dnR}_{Lij} = O^{dnL}_{Rij}$ as above but
with $H_{\chiz{1}} \to {g'}^2 M_2 \mu / (3 \sqrt{2} \mathrm{Det}_0)$.

One can obtain the couplings between $\tilde l_{Li}$-$l_i$-$\nu_j$ by
replacing $h_D \to h_E$ and $H_{\chiz{1}} \to (g^2 M_1 + {g'}^2 M_2)
\mu / (2 \sqrt{2} \mathrm{Det}_0)$ in the above equations.  For the
case of $\tilde l_{Ri}$-$l_i$-$\nu_j$ one finds the couplings by replacing
$h_D \to h_E$ and $H_{\chiz{1}} \to {g'}^2 M_2 \mu / ( \sqrt{2}
\mathrm{Det}_0)$.

For the couplings $\tilde u_{Li} - u_i - \nu_j$ one finds
%
\begin{eqnarray}
  O^{unL}_{Li1} &=& O^{unL}_{Ri1} =
  O^{unL}_{Li2}  =  O^{unL}_{Ri2} = 0 \\
  O^{unL}_{Li3} &=& - h_U^{ii} I_{\chiz{1}} |\vec{\Lambda}| \\
 O^{unR}_{Li3} &=& J_{\chiz{1}} |\vec{\Lambda}|
\end{eqnarray}
%
with $I_{\chiz{1}} = (g^2 M_1 + {g'}^2 M_2) v_D / (2 \mathrm{Det}_0)$
and $J_{\chiz{1}} = (- 3 g^2 M_1 + {g'}^2 M_2) \mu / (6 \sqrt{2}
\mathrm{Det}_0)$.  For the couplings $\tilde u_{Ri}$-$u_i$-$\nu_j$ one
finds that $O^{unR}_{Rij} = O^{unL}_{Lij}$ and
$O^{unR}_{Lij} = O^{unL}_{Rij}$ as above but
with $J_{\chiz{1}} \to - \sqrt{2} {g'}^2 M_2 \mu / (3
\mathrm{Det}_0)$.

For the couplings $\tilde u_j - d_k - l_i$ one finds
%
\begin{eqnarray}
 \label{eq:char1}
 C^{\tilde u}_{Lk l_i} &=& h_D^{kk} R^{\tilde u}_{j1} \left( 
          \frac{\epsilon_i}{\mu}
        + \frac{g^2 v_U}{2 \mu} \frac{\Lambda_i}{ \mathrm{Det}_+}
                   \right)  \\
 C^{\tilde u}_{R l_i} &=& \frac{h_E^{ii} v_D}{\sqrt{2} \mathrm{Det}_+}
     \left\{ 
     \left( \frac{g^2 v_D R^{\tilde u}_{j1}}{\sqrt{2} }
           + h_U^{kk} M_2 R^{\tilde u}_{j2} \right) 
            \frac{\epsilon_i}{\mu} \right. \nonumber \\
    &&\hspace{16mm} + \left.
          \left[ \frac{g^2 \mu  R^{\tilde u}_{j1}}{\sqrt{2}}
                 \left(1 + \frac{g^2 v_D v_U}{2 \mu^2} \right)
                    + \frac{g^2 h_U^{kk} v_U R^{\tilde u}_{j2}}{2}
                 \left(1 + \frac{M_2}{\mu} \right)  
               \right] \frac{\Lambda_i}{\mathrm{Det}_+} \right\}
\end{eqnarray}

For the couplings $\tilde d_j - u_k - l_i$ one finds
%
\begin{eqnarray}
 C^{\tilde d}_{Lk l_i} &=& \frac{h_E^{ii} h_U^{kk} v_D
                  R^{\tilde d}_{j1}}{\sqrt{2}\mathrm{Det}_+}
        \left[ M_2 \frac{\epsilon_i}{\mu}
        + \frac{g^2 v_U }{2}
          \left( 1 + \frac{M_2}{\mu} \right)
          \frac{\Lambda_i}{ \mathrm{Det}_+}
                   \right]  \\
 C^{\tilde d}_{Rk l_i} &=& h_D^{kk} R^{\tilde d}_{j2} \frac{\epsilon_i}{\mu}
          + \left( \frac{g^2 R^{\tilde d}_{j1}}{\sqrt{2}} 
                 + \frac{g^2 h_D^{kk} v_U R^{\tilde d}_{j2}}{2 \mu} \right)
             \frac{\Lambda_i}{\mathrm{Det}_+}
 \label{eq:char4}
\end{eqnarray}
In \eq{eq:char1} - (\ref{eq:char4}) we have assumed that there is no
generation mixing between the squarks implying that $j=1,2$.

Data from reactor experiments~\cite{apollonio:1999ae,boehm:1999gk} indicate that the mixing
element $U_{e3}$ must be small~\cite{gonzalez-garcia:2000sq}. This
implies that $|\Lambda_e| \ll |\Lambda_{2,3}|$.  In the limit
$\Lambda_e / \Lambda_{2,3}\to 0$ some of the above formulas simplify
to
%
\begin{eqnarray}
\label{eq:simple1}
O^{nnh}_{111} &=& - E_{\chiz{1}} 
         \left( \frac{ c_2 \, c_4 \, c_6 \, \sina \, \epsilon_e }{\mu } + s_2 
         \right) \\
%
O^{nnh}_{121} &=& E_{\chiz{1}} 
           \left( \sina \, c_2 \, c_4 \, c_6 
    \frac{ \epsilon_\tau \Lambda_\mu - \epsilon_\mu \Lambda_\tau }
         {\mu |\Lambda_\tau| } 
    + \frac{ s_6 \Lambda_\mu - s_4 \Lambda_\tau }{|\Lambda_\tau|} \right)  \\
%
O^{nnh}_{131} &=& E_{\chiz{1}}
        \left(\sina  \,c_2  \, c_4 \, c_6
    \frac{ \epsilon_\mu \Lambda_\mu + \epsilon_\tau \Lambda_\tau }
         {\mu \sqrt{\Lambda^2_2+\Lambda^2_3} } 
    + \frac{ s_4 \Lambda_\mu + s_6 \Lambda_\tau }{\sqrt{\Lambda^2_2+\Lambda^2_3}}
           \right) 
       - D_{\chiz{1}} \, c_2 \, c_4 \, c_6 \sqrt{\Lambda^2_2+\Lambda^2_3} \\
O^{dnL}_{Li\nu_1} &=& O^{dnR}_{Ri\nu_1}
                          = -  \frac{h_D^{ii} \epsilon_e}{\mu} \\
O^{dnL}_{Li\nu_2} &=& O^{dnR}_{Ri\nu_2}
  =  \frac{h_D^{ii} \left( \epsilon_\mu \Lambda_\tau- \epsilon_\tau \Lambda_\mu\right) }
          {\mu |\Lambda_\tau|} \\
O^{dnL}_{Li\nu_3} &=& O^{dnR}_{Ri\nu_3}
  =  h_D^{ii} \left( G_{\chiz{1}} \sqrt{\Lambda_\mu^2 + \Lambda^2_\tau}
         - \frac{\epsilon_\mu \Lambda_\mu+ \epsilon_\tau \Lambda_\tau}
          {\mu \sqrt{\Lambda_\mu^2 + \Lambda_\tau^2}} \right) 
\label{eq:simple6}
\end{eqnarray}
%
Later on we will also use the so-called sign-condition~\cite{hirsch:2000ef},
defined by
%
\bea
\frac{\epsilon_\mu}{\epsilon_\tau} \frac{\Lambda_\mu}{\Lambda_\tau} < 0 \, \, .
\eea 
%
Its origin can be traced back to the above \eq{eq:simple1} -
(\ref{eq:simple6}). Assuming $\epsilon_\mu \simeq \epsilon_\tau$ as
indicated by unification and $|\Lambda_\mu| \simeq |\Lambda_\tau|$ as
required by the atmospheric neutrino problem one sees easily from the
above equations that either the $\epsilon$ part\footnote{The $\Lambda$
  parts lead only to a renormalization of the heaviest neutrino state
  whereas the $\epsilon$ part gives mass to the lighter neutrinos.}
of the couplings to the second or the third neutrino state is very
small depending on the relative sign between $\Lambda_\mu$ and
$\Lambda_\tau$.  If $\Lambda_\mu \simeq -\Lambda_\tau$ one can show, after a
lengthy calculation \cite{diaz:2003as}, that the resulting effective
neutrino mixing matrix is given by
%
\bea
V_{\nu,\mathrm{loop}}=
\left(\begin{array}{ccc}
  \cos\theta_{12} &  -\sin\theta_{12} & 0 \\
  \sin\theta_{12} &  \cos\theta_{12}  & 0 \\
                0 & 0 &               1 
\end{array}\right) \times V_{\nu,\mathrm{tree}}
\eea
%
with nearly unchanged $\theta_{13}$ and $\theta_{23}$. In contrast, if
this sign condition is not fulfilled the $\theta_{13}$ and
$\theta_{23}$ angles get large corrections. One sees therefore that if
the sign condition is satisfied the atmospheric and solar neutrino
features decouple: the atmospheric is mainly tree-level physics, while
the solar neutrino anomaly is accounted for by genuine loop physics
in a simple factorizable way. Thus the sign condition is helpful to
get a better control on the parameters for the solar neutrino problem.


\subsection{Neutralino Production and Decays}
\label{sec:decays}

\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}(150,90)
\put(-3,-50){\mbox{\epsfig{figure=tree_app.ps,height=18.7cm,width=7.cm}}}
\put(-2,86){\makebox(0,0)[bl]{{ a)}}}
\put(3,85){\makebox(0,0)[bl]{{\small $\Delta m^2_{atm}$}}}
\put(77,-3){\makebox(0,0)[br]{
             {$10^5 |{\vec \Lambda}|/(\sqrt{M_2} \mu)$ $[GeV]$}}}
%
\put(82,0){\mbox{\epsfig{figure=LengthMchi.eps,height=8.7cm,width=7.cm}}}
\put(82,87){\makebox(0,0)[bl]{{ b)}}}
\put(87,85){\makebox(0,0)[bl]{{\small $L(\chiz{1})$~[cm]}}}
\put(158,-3){\makebox(0,0)[br]{{ $m_{\chiz{1}}$~[GeV]}}}
\end{picture}
\caption[]{ a) $\Delta m^2_{atm}$ and  b) neutralino decay length.}
\label{fig:TestAtmMass}
\end{figure}

\begin{figure}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(80,80)
\put(0,0){\mbox{\epsfig{figure=Prod11LC1TeV.eps,height=7.7cm,width=7.cm}}}
\put(2,76){\makebox(0,0)[bl]
            {{\small $\sigma(e^+ e^- \to \chiz{1} \chiz{1})$~[fb]}}}
\put(71,-3){\makebox(0,0)[br]{{$\mchiz{1}$~[GeV]}}}
\end{picture}
\end{center}
\caption[]{Production cross section for the process
$\sigma(e^+ e^- \to \chiz{1} \chiz{1})$ as a function of $\mchiz{1}$ at a
Linear Collider with 1~TeV c.m.s energy. ISR-corrections are included.}
\label{fig:Prod1TeV}
\end{figure}

In this section we will discuss the production and the decay modes of
the lightest neutralino $\chiz{1}$.  In order to reduce the numbers of
parameters we have performed the calculations in the framework of a
minimal SUGRA version of bilinearly \rp SUSY model. Unless noted
otherwise the parameters have been varied in the following ranges:
%
$M_2$ and $|\mu|$ from 0 to 1 TeV, $m_0$ [0.2 TeV, 1.0 TeV], $A_0/m_0$
and $B_0/m_0$ [-3,3] and $\tan\beta$ [2.5,10], and for the \rp
parameters, $|\Lambda_\mu/\sqrt{\Lambda_e^2+\Lambda^2_\tau}|= 0.4-2$,
$\epsilon_\mu/\epsilon_\tau = 0.8-1.25$, $|\Lambda_e/\Lambda_\tau|=
0.025-2$, $\epsilon_e/\epsilon_\tau = 0.015-2$ and $|\Lambda| =
0.05-0.2$ GeV$^2$. They were subsequently tested for consistency with
the minimization (tadpole) conditions of the Higgs potential as well
as for phenomenological constraints from supersymmetric particle
searches. Moreover, they were checked to provide a solution to both
solar and atmospheric neutrino problems. For the case of the solar
neutrino anomaly we have accepted points which give either one of the
large mixing angle solutions or the small mixing angle MSW solution.

We have seen in \eq{eq:mnutree} that the atmospheric scale is
proportional $|\vec{\Lambda}|^2/ \mathrm{Det}(\mchiz{})$. As has been
shown in \cite{romao:1999up,hirsch:2000ef} this statement remains valid after
inclusion of 1-loop corrections provided that
$|\vec{\epsilon}|^2/|\vec{\Lambda}| < 1$ implying that 1-loop
corrections to the heaviest neutrino mass remain small. As we have
seen in section (\ref{sec:approx}), most of the couplings are
proportional to $|\vec{\Lambda}|/ \sqrt{\mathrm{Det}(\mchiz{}})$
and/or $\epsilon_i / \mu$.
%
Although $|\vec{\Lambda}|/(\sqrt{M_2}\mu)$ has to be small in order to
account for the atmospheric mass scale (see \figz{fig:TestAtmMass}{a})
the previously discussed couplings are still large enough so that the
neutralino decays inside the detector, as can be seen in
\figz{fig:TestAtmMass}{b}.

In \fig{fig:Prod1TeV} we show the cross section $\sigma(e^+ e^- \to
\chiz{1} \chiz{1})$ in fb for $\sqrt{s} = 1$~TeV.  Assuming now that
an integrated luminosity of 1000~fb$^{-1}$ per year can be achieved at
a future linear collider (see \cite{accomando:1998wt,blair:2000gy} 
and references
therein) this implies that between $10^4$ to $10^5$ neutralino pairs
can be directly produced per year. Due to the smallness of the
R-parity violating couplings, most of the SUSY particles will decay
according to the MSSM scheme implying that there will be many more
neutralinos to study, namely from direct production as well as
resulting from cascade decays of heavier SUSY particles. From this
point of view the measurement of branching ratios as small as
$10^{-5}$ should be feasible. As we will see in what follows this
might be required in order to establish some of the correlations
between neutrino mixing angles and the resulting neutralino decay
observables, which is a characteristic feature of this class of models.
 
In this model the neutralino can decay in the following ways
%
\bea
\chiz{1} &\to& \nu_i \, \nu_j \, \nu_k \\
         &\to& \nu_i \, q \, \bar{q} \\
         &\to& \nu_i \, l^+_j \, l^-_k \\
         &\to& l^\pm_i \, q \, \bar{q}' \\
         &\to& \nu_i \, \gamma
\eea
%
In the following we will discuss these possibilities in detail except
$\chiz{1} \to \nu_i \, \gamma$ because its branching ratio is always
below $10^{-7}$.

In the following discussion we have always computed the complete
three-body decay widths even in cases where $\mchiz{1}$ has been
larger than one of the exchanged particle masses, so that two-body
channels are open. This has turned out to be necessary because there
are parameter combinations where the couplings to the lightest
exchanged particle are $O(10)$ smaller than the coupling to one of the
heavier particles, implying that the graph containing the heavy
particle cannot be neglected with respect to the lighter particle
contribution.  An example is the case of $Z$-boson and
$S^0_1$-mediated gaugino-like neutralino decays discussed later on.
Here $S^0_1$ denotes the lightest neutral scalar.
%
In addition we want to be sure not to miss possibly important
interference effects as there are several graphs which contribute to a
given process. A typical example is the process $\chiz{1} \to \nu_i \,
l^-_j \, l^+_k$ where 26 contributions exist, as can be seen from the
generic diagrams  shown in \fig{fig:graphs}.

\begin{figure}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(80,80)
\put(0,-1){\mbox{\epsfig{
     figure=BrInvMChi.eps,height=7.7cm,width=7.cm}}}
\put(5,75){\makebox(0,0)[bl]{{\small
        Br(${\tilde \chi}^0_1 \to \sum_{i,j,k} \nu_i \nu_j \nu_k$)}}}
\put(70,-3){\makebox(0,0)[br]{{$m_{{\tilde \chi}^0_1}$~[GeV]}}}
\end{picture}
\end{center}
\caption[]{Invisible neutralino branching ratio summing over all neutrinos.}
\label{fig:chiInv}
\end{figure}

The first important question to be answered is how large the invisible
neutralino decay modes to neutrinos can be. This is important to
ensure that sufficient many neutralino decays can be observed. As can
be seen from \fig{fig:chiInv} the invisible branching ratio never
exceeds 10\%.  
%
The main reason for this behaviour can be found in the fact that 
for the SUGRA motivated scenario under consideration the couplings 
of the lightest neutralino to the Z-boson are suppressed.
%
This and the comparison with other couplings will be discussed in some
detail later on.

\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}(150,75)
\put(0,0){\mbox{\epsfig{
          figure=BrNbbMChi.eps,height=7.cm,width=5.cm}}}
\put(0,68){\makebox(0,0)[bl]{{\small
                  a)   Br(${\tilde \chi}^0_1 \to b \bar{b} \sum_i \nu_i$)}}}
\put(50,-3){\makebox(0,0)[br]{{$m_{{\tilde \chi}^0_1}$~[GeV]}}}
%
\put(54,0){\mbox{\epsfig{
        figure=BrNqqMChi.eps,height=7.cm,width=5.cm}}}
\put(54,68){\makebox(0,0)[bl]{{\small
     b) Br(${\tilde \chi}^0_1 \to  \sum_{q=u,d,s} q \bar{q}  \sum_i \nu_i$)}}}
\put(104,-3){\makebox(0,0)[br]{{$m_{{\tilde \chi}^0_1}$~[GeV]}}}
%
\put(108,0){\mbox{\epsfig{
        figure=BrNccMChi.eps, height=7.cm,width=5.cm}}}
\put(108,68){\makebox(0,0)[bl]{{\small
                 c)  Br(${\tilde \chi}^0_1 \to c \bar{c} \sum_i \nu_i$)}}}
\put(158,-3){\makebox(0,0)[br]{{$m_{{\tilde \chi}^0_1}$~[GeV]}}}
\end{picture}
\caption[]{Neutralino branching ratios for the decays into $q \bar{q} \nu_i$
           final states summing over all neutrinos. }
\label{fig:chiQQnu}
\end{figure}
%
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}(150,75)
\put(0,0){\mbox{\epsfig{
          figure=BrEqqMChi.eps,height=7.cm,width=5.cm}}}
\put(0,68){\makebox(0,0)[bl]{{\small
                  a)   Br(${\tilde \chi}^0_1 \to e^\pm \sum \bar{q} q'$)}}}
\put(50,-3){\makebox(0,0)[br]{{$m_{{\tilde \chi}^0_1}$~[GeV]}}}
%
\put(54,0){\mbox{\epsfig{
        figure=BrMuqqMChi.eps,height=7.cm,width=5.cm}}}
\put(54,68){\makebox(0,0)[bl]{{\small
     b) Br(${\tilde \chi}^0_1 \to \mu^\pm \sum \bar{q} q'$)}}}
\put(104,-3){\makebox(0,0)[br]{{$m_{{\tilde \chi}^0_1}$~[GeV]}}}
%
\put(108,0){\mbox{\epsfig{
        figure=BrTauqqMChi.eps, height=7.cm,width=5.cm}}}
\put(108,68){\makebox(0,0)[bl]{{\small
                 c)  Br(${\tilde \chi}^0_1 \to \tau^\pm \sum \bar{q} q'$)}}}
\put(158,-3){\makebox(0,0)[br]{{$m_{{\tilde \chi}^0_1}$~[GeV]}}}
\end{picture}
\caption[]{Neutralino branching ratios for the decays into
           $l^\pm q'\bar{q}$ final states summing over all $q' \bar{q}$
           combinations.}
\label{fig:chiLQQp}
\end{figure}
%
The mainly ``visible'' nature of the lightest neutralino decay,
together with the short neutralino decay path discussed above,
suggests the observability of neutralino-decay-induced events at
collider experiments and this should stimulate dedicated detector
studies.

In \fig{fig:chiQQnu} we show the branching ratios for the decays into
$q \bar{q} \nu_i$. Here we single out the $b$-quark
(\figz{fig:chiQQnu}{a}) and the $c$-quark (\figz{fig:chiQQnu}{c})
because in these cases flavour detection is possible. One can clearly
see that for $\mchiz{1} \lsim 1.1 \, m_W$ the decay into $b \bar{b}
\nu_i$ can be the dominant one. The reason is that the scalar
contributions stemming from $S^0_j$, $P^0_j$ and/or $\tilde b_k$ can
be rather large. This can be understood with the help of
\eq{eq:CoupSdDChi1}-(\ref{eq:CoupSdDChi2}) where terms proportional to
$h_D \epsilon_i / \mu$ appear. This kind of terms is absent in the
corresponding couplings for the u-type squarks implying that the
branching ratio for $c \bar{c} \nu_i$ is rather small as can be seen
in \figz{fig:chiQQnu}{c}. One can see in \figz{fig:chiQQnu}{b} and
\figz{fig:chiQQnu}{c} a pronounced 'hole' around 80-100 GeV. It occurs
because for $\mchiz{1} > m_W$ the $W$ becomes on-shell implying a
reduction for these decays. This is compensated as the $Z$ becomes
on-shell.

\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}(150,160)
\put(0,79){\mbox{\epsfig{
             figure=BrEEMChi.eps,height=7.cm,width=5.cm}}}
\put(0,148){\makebox(0,0)[bl]{{\small
                  a) Br(${\tilde \chi}^0_1 \to e^- e^+ \sum_i \nu_i$)}}}
\put(50,77){\makebox(0,0)[br]{{$m_{{\tilde \chi}^0_1}$~[GeV]}}}
%
\put(54,79){\mbox{\epsfig{
         figure=BrEMuMChi.eps,height=7cm,width=5.cm}}}
\put(54,148){\makebox(0,0)[bl]{{\small
                 b) Br(${\tilde \chi}^0_1 \to e^\pm \mu^\mp \sum_i \nu_i$)}}}
\put(104,77){\makebox(0,0)[br]{{$m_{{\tilde \chi}^0_1}$~[GeV]}}}
%
\put(108,79){\mbox{\epsfig{
         figure=BrETauMChi.eps,height=7cm,width=5.cm}}}
\put(108,148){\makebox(0,0)[bl]{{\small
                 c) Br(${\tilde \chi}^0_1 \to e^\pm \tau^\mp \sum_i \nu_i$)}}}
\put(158,77){\makebox(0,0)[br]{{$m_{{\tilde \chi}^0_1}$~[GeV]}}}
%
\put(0,0){\mbox{\epsfig{
          figure=BrMuMuMChi.eps,height=7.cm,width=5.cm}}}
\put(0,68){\makebox(0,0)[bl]{{\small
                  d)   Br(${\tilde \chi}^0_1 \to  \mu^- \mu^+ \sum_i \nu_i$)}}}
\put(50,-3){\makebox(0,0)[br]{{$m_{{\tilde \chi}^0_1}$~[GeV]}}}
%
\put(54,0){\mbox{\epsfig{
        figure=BrMuTauMChi.eps,height=7.cm,width=5.cm}}}
\put(54,68){\makebox(0,0)[bl]{{\small
     e) Br(${\tilde \chi}^0_1 \to  \mu^\pm \tau^\mp \sum_i \nu_i$)}}}
\put(104,-3){\makebox(0,0)[br]{{$m_{{\tilde \chi}^0_1}$~[GeV]}}}
%
\put(108,0){\mbox{\epsfig{
        figure=BrTauTauMChi.eps, height=7.cm,width=5.cm}}}
\put(108,68){\makebox(0,0)[bl]{{\small
           f)  Br(${\tilde \chi}^0_1 \to  \tau^- \tau^+ \sum_i \nu_i$)}}}
\put(158,-3){\makebox(0,0)[br]{{$m_{{\tilde \chi}^0_1}$~[GeV]}}}
\end{picture}
\caption[]{Neutralino branching ratios for the decays into various lepton
           final states summing over all neutrinos.}
\label{fig:chiLLnu}
\end{figure}

The semi-leptonic branching ratios into charged leptons are shown in
\fig{fig:chiLQQp}. The decays into $\mu$ and $\tau$ are particularly
important because, as we will see in \sect{sec:correlations}, they
will give a measure of the atmospheric neutrino angle. Note that these
branching ratios are larger than $10^{-4}$ and in most cases larger
than $10^{-3}$, implying that there should be sufficient statistics
for investigations. In case of the $e$ final state it might happen
that one can only give an upper bound on this branching ratio.  This
is just a result of the reactor neutrino bound~\cite{apollonio:1999ae,boehm:1999gk}.
%
Note that due to the Majorana nature of the neutralino one expects in
large regions of the parameter space several events with same sign
di-leptons and four jets.

In \fig{fig:chiLLnu} the fully leptonic branching ratios are shown.
One can clearly see a difference between the branching ratios into
channels containing different charged leptons of the same flavour,
i.e.  $\tau^- \tau^+$ versus $\mu^- \mu^+$ and $e^- e^+$.
%
This difference is due to the importance of the $S^0_1$ state which
corresponds mainly to the lightest Higgs boson $h^0$ of the MSSM. We
have found that for gaugino-like $\chiz{1}$ the \rp couplings $S^0_1 -
\chiz{1} - \nu_i$ are in general larger than the corresponding $Z^0 -
\chiz{1} - \nu_i$ couplings.  This can be understood by inspecting the
formulas given in \eq{eq:coupZnu} - (\ref{eq:coupSnu}) in
\sect{sec:approx}, in particular the parts proportional to
$\epsilon_k$ in \eq{eq:coupSnuA} -- (\ref{eq:coupSnu}). Other reasons
for having ``non-universal'' $\tau^- \tau^+$, $\mu^- \mu^+$ and $e^-
e^+$ couplings are the graphs containing $W$ or charged sleptons as
exchanged particle (see \fig{fig:graphs}). From \eq{eq:coupWRChi} one
can see that the coupling $O^{cnw}_{Ri1}$ is proportional to
$h_E^{ii}$ implying that they only play a role if a $\tau$ is present
in the final state.

Notice also that the largeness of the branching ratios for neutralino
decays into lepton-flavour-violating channels can be simply understood
from the importance of $W^\pm$ and $S^\pm_n$ contributions present in
\fig{fig:graphs} \footnote{The charged scalars are a mixture of the
  charged Higgs bosons and the charged sleptons, and in particular the
  later are the important ones}.

\subsection{Probing Neutrino mixing via Neutralino Decays}
\label{sec:correlations}

In this section we will demonstrate that neutralino decay branching
ratios are strongly correlated with neutrino mixing angles. We will
consider two cases: 1) The situation before supersymmetry is
discovered. In this case we demonstrate that neutrino physics implies
predictions for neutralino decays which will be tested at future
colliders.   
2) The situation when the spectrum is known to the 1\% level or better
as could, for example, be achieved at a future linear collider
\cite{accomando:1998wt,martyn:1999tc}. In this case our model allows for several
consistency checks between neutrino physics (probed by underground and
reactor experiments~\cite{fukuda:1998tw,fukuda:1998ub,fukuda:1999pp,fukuda:1998ah,cleveland:1998nv,davis:1994jw,abdurashitov:1999zd,hampel:1998xg,apollonio:1999ae,boehm:1999gk}) and neutralino
physics.
%
Moreover, some neutralino decay observables are sensitive to which of
the possible solutions to the solar neutrino problem is the one
realized, i.e. they can discriminate large angles solutions from the
small angle MSW solution.
%
% For displaying these correlations we extend the ranges of the R-parity
% violating parameters beyond the regions indicated by the neutrino
% experiments, given in \sect{sec:decays}.

\subsubsection{Before SUSY is discovered}
\label{sec:predictions}
 
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(80,80)
\put(0,1){\mbox{\epsfig{figure=TestCoupling.eps,height=7.3cm,width=7.cm}}}
\put(0,76){\makebox(0,0)[bl]{{\small
        $O^{cnw}_{L31,approx.}/O^{cnw}_{L31}$}}}
\put(70,-3){\makebox(0,0)[br]{{$10^7 \, O^{cnw}_{L31}$}}}
\end{picture}
\end{center}
\caption[]{Approximated coupling $O^{cnw}_{L31,approx.}$ using formula 
           \eq{eq:CoupWLChi} 
           divided by the exact calculated coupling as a function of
           the exact calculated coupling. The bright (dark) points are for 
           $\mu > (<) 0$.}
\label{fig:TestCoupling}
\end{figure}

Let us first consider the situation before SUSY is discovered.  Before
working out the predictions for neutralino decays we would like to
point out a fact concerning the 1-loop corrected neutrino/neutralino
mass matrix.  It has been noticed in \cite{hirsch:2000ef} that the sign of
the $\mu$ parameter determines to some extent how large the absolute
radiative corrections are (see Fig.~5 of \cite{hirsch:2000ef})~\footnote{The
  important information is the relative sign between $\mu$ and the
  gaugino mass parameters $M_{1,2}$. Since in \cite{hirsch:2000ef} as well as
  here we assume that $M_{1,2}>0$ then the absolute sign of $\mu$
  becomes relevant. }.  The reason is that depending on this sign the
interference between the 1-loop graphs containing gauginos and the
1-loop graphs containing Higgsinos is constructive or destructive.

This fact has of course implications on whether the approximate
couplings presented in \sect{sec:approx} remain valid after the 1-loop
corrections are taken into account~\footnote{Of course the couplings
  involving $\nu_1$ and/or $\nu_2$ are exceptional ones, as the angle
  between these states is only meaningful after performing 1-loop
  corrections are included.}.  A typical example is shown in
\fig{fig:TestCoupling} where the approximated coupling
$O^{cnw}_{L31,approx.}$ divided by the coupling $O^{cnw}_{L31}$ as a
function of $O^{cnw}_{L31}$. One clearly sees that for $\mu <0$ the
tree-level result~\cite{mukhopadhyaya:1998xj,chun:1998ub,choi:1999tq} is a good approximation to
within 20\%, but for $\mu >0$ it can be off by a factor up to 5 in
some extreme cases where a constructive interference between gaugino
and Higgsino loops takes place.
%
We have checked that the same is true for the other couplings
involving either the charged leptons and/or $\nu_3$.

\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}(150,80)
\put(0,1){\mbox{\epsfig{
               figure=LLBrMuTauQQpTanAtm2.eps,height=7.3cm,width=7.cm}}}
\put(0,75){\makebox(0,0)[bl]{{\small a) Br($\mu q q'$)/Br($\tau q q'$)}}}
\put(70,-3){\makebox(0,0)[br]{{$\tan^2(\theta_{atm})$}}}
%
\put(88,1){\mbox{\epsfig{figure=LLBrMuTauQQpTanAtm2A.eps,
                               height=7.3cm,width=7.cm}}}
\put(85,75){\makebox(0,0)[bl]{{\small b) Br($\mu q q'$)/Br($\tau q q'$)}}}
\put(158,-3){\makebox(0,0)[br]{{$\tan^2(\theta_{atm})$}}}
\end{picture}
\caption[]{Testing the atmospheric angle.
           In case of the dark (bright) points  $\mu<(>)0$. 
           In the second figure we have taken 
           only those points with $|\sin 2 \theta_{\tilde b}| > 0.1$.}
\label{fig:TestAngleAtm}
\end{figure}
%
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}(150,80)
\put(0,1){\mbox{\epsfig{
               figure=LLBrTauQQpEMuNuTanAtm2.eps,height=7.3cm,width=7.cm}}}
\put(0,75){\makebox(0,0)[bl]{{\small
        a) Br($e^\pm \mu^\mp \sum_i \nu_i$)/Br($\tau q q'$)}}}
\put(70,-3){\makebox(0,0)[br]{{$\tan^2(\theta_{atm})$}}}
%
\put(88,1){\mbox{\epsfig{figure=LLBrTauQQpEMuNuTanAtm2A.eps,
                               height=7.3cm,width=7.cm}}}
\put(85,75){\makebox(0,0)[bl]{{\small
             b) Br($e^\pm \mu^\mp \sum_i \nu_i$)/Br($\tau q q'$)}}}
\put(158,-3){\makebox(0,0)[br]{{$\tan^2(\theta_{atm})$}}}
\end{picture}
\caption[]{Testing the atmospheric angle.
           In case of the dark (bright) points  $\mu<(>)0$. 
           In the second figure we have taken 
           only those points with $|\sin 2 \theta_{\tilde b}| > 0.1$.}
\label{fig:TestAngleAtmA}
\end{figure}
%
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}(150,80)
\put(0,1){\mbox{\epsfig{
               figure=LLBrTauQQpMuMuNuTanAtm2.eps,height=7.3cm,width=7.cm}}}
\put(0,75){\makebox(0,0)[bl]{{\small
        a) Br($\mu^- \mu^+ \sum_i \nu_i$)/Br($\tau q q'$)}}}
\put(70,-3){\makebox(0,0)[br]{{$\tan^2(\theta_{atm})$}}}
%
\put(88,1){\mbox{\epsfig{figure=LLBrTauQQpMuMuNuTanAtm2A.eps,
                               height=7.3cm,width=7.cm}}}
\put(85,75){\makebox(0,0)[bl]{{\small
             b) Br($\mu^- \mu^+ \sum_i \nu_i$)/Br($\tau q q'$)}}}
\put(158,-3){\makebox(0,0)[br]{{$\tan^2(\theta_{atm})$}}}
\end{picture}
\caption[]{Testing the atmospheric angle.
           In case of the dark (bright) points  $\mu<(>)0$. 
           In the second figure we have taken 
           only those points with $|\sin 2 \theta_{\tilde b}| > 0.1$.}
\label{fig:TestAngleAtmB}
\end{figure}

As can be seen from the discussion in \sect{sec:approx} the
approximate formulas depend on the SUSY parameters, in particular on
the parameters of the MSSM chargino/neutralino sector. However, one
can see that the ratios of neutralino partial decay widths or of its
branching ratios is rather insensitive to the MSSM parameters. As has
been pointed out in \cite{hirsch:2000ef} the atmospheric angle depends on the
ratio of $\Lambda_\mu / \Lambda_\tau$. This ratio (at tree level) can
be obtained by taking the ratio $O^{cnw}_{L21} / O^{cnw}_{L31}$. This
leads immediately to the idea that the semi-leptonic branching ratios
into $\mu^\pm q \bar{q}'$ and $\tau^\pm q \bar{q}'$ should be related
to the atmospheric angle.  This is clearly demonstrated in
\fig{fig:TestAngleAtm} where we show the ratio of the corresponding
branching ratios as a function of $\tan^2(\theta_{atm})$. One sees
that present data imply that this ratio should be $O(1)$.
%
In particular, the relative yield of muons and taus will specify
whether or not the solution to the atmospheric neutrino anomaly occurs
for parameter choices in the ``normal'' range or in the ``dark-side'',
i.e. $\tan^2(\theta_{atm}) <1$ or $\tan^2(\theta_{atm})
>1$~\cite{gonzalez-garcia:2000cm,degouvea:2000cq}.

The observed width of the band simply expresses the residual SUSY
parameter dependence, which comes partly from the 1-loop calculated
mass matrix and partly from the different contributions to these
decays.  If for some reason $|\sin 2 \theta_{\tilde b}| > 0.1$ the
dependence on the parameters in the 1-loop calculation is considerably
reduced because the sbottom/bottom loop dominates.  This leads to a
stronger correlation as seen in \figz{fig:TestAngleAtm}{b}.  The fact
that for $\mu > 0$ the band is wider is a consequence of the
discussion in the previous paragraph.

In \fig{fig:TestAngleAtmA} and \fig{fig:TestAngleAtmB} we show two
additional ratios which exhibit also a correlation with $\tan^2
\theta_{atm}$: Br$(\chiz{1} \to e^\pm \mu^\mp \sum_i \nu_i$) /
Br$(\chiz{1} \to \tau^\pm q \bar{q}'$) and Br$(\chiz{1} \to \mu^-
\mu^+ \sum_i \nu_i$) / Br$(\chiz{1} \to \tau^\pm q \bar{q}'$).  The
(nearly) maximal mixing of atmospheric neutrinos implies that several
other ratios of branching ratios are also fixed to within one order of
magnitude,
see Table 1 and \fig{fig:TestAngleSolPred}. 
% which will depend to some extent on the solution of the
% solar neutrino problem and will be thus discussed later.
%
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}(150,80)
\put(0,1){\mbox{\epsfig{figure=LLBrETauQQpUe32.eps,height=7.3cm,width=7.cm}}}
\put(0,76){\makebox(0,0)[bl]{{ a)}}}
\put(5,75){\makebox(0,0)[bl]{{\small Br($e q q'$)/Br($\tau q q'$)}}}
\put(70,-3){\makebox(0,0)[br]{{$U^2_{e3}$}}}
%
\put(88,1){\mbox{\epsfig{figure=LLBrETauQQpUe32A.eps,
                               height=7.3cm,width=7.cm}}}
\put(84,76){\makebox(0,0)[bl]{{ b)}}}
\put(89,75){\makebox(0,0)[bl]{{\small Br($e q q'$)/Br($\tau q q'$)}}}
\put(158,-3){\makebox(0,0)[br]{{$U^2_{e3}$}}}
\end{picture}
\caption[]{Testing the Chooz angle.
           In case of the dark (bright) points  $\mu<(>)0$. 
            In b) we have taken 
           only those points with $|\sin 2 \theta_{\tilde b}| > 0.1$.}
\label{fig:TestChooz}
\end{figure}

In this model the so-called Chooz-angle is given by $|\Lambda_e /
\Lambda_\tau|$~\cite{hirsch:2000ef} where we already have used the fact that
the atmospheric data implies $|\Lambda_\mu| \simeq |\Lambda_\tau|$.
The same discussion as in the previous paragraph is valid. This leads
automatically to the correlation between Br($\chiz{1} \to e^\pm q
q'$)/Br($\chiz{1} \to \tau^\pm q q'$) and $U^2_{e3}$ which is shown in
\fig{fig:TestChooz}. For $U^2_{e3} < 0.01$ the correlation is less
stringent because it implies that the tree level couplings have to be
rather small and therefore loop corrections are more important. Note
that existing reactor data~\cite{apollonio:1999ae,boehm:1999gk} give the constraint on
$U^2_{e3} \lsim 0.05$ at 90\% CL~\cite{gonzalez-garcia:2000sq}. This in
turn implies an upper bound of $\sim0.2$ on this ratio of branching
ratios.

The discussion of the solar angle is more involved. As illustrated in
\fig{fig:angles} this angle is strongly correlated with
$\epsilon_e/\epsilon_\mu$ ratio.
%
\begin{figure}%2
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(80,70)
\put(0,-5){\mbox{\epsfig{figure=TanSol2Eps12.eps,height=7.7cm,width=7.7cm}}}
\put(2,73){\mbox{$\tan^2 \theta_{sol}$}}
\put(71,-3){\mbox{$\epsilon_e/\epsilon_\mu$}}
\end{picture}
\end{center}
%
\caption{The solar mixing angle as a function of $\epsilon_e/\epsilon_\mu$.
         \label{fig:angles}}
\end{figure}
%
In order to get information on the $\epsilon_i$ from neutralino decays
one must take into account that, as already mentioned, the solar angle
acquires a meaning only once the complete 1-loop corrections to the
mass matrix have been included. For an easier understanding we
focus on leptonic decays of the type
$\chiz{1} \to l^+_i l^-_j \nu_k$ with $i\ne j$ which depend on the
$\chiz{1}$-$W$-$l_{j,i}$ and $W$-$l_{i,j}$-$\nu_k$ couplings.
%
The way a correlation appears is non-trivial. To understand it note
that the couplings $W$-$l_i$-$\nu_j$ depend on the neutrino mixing,
since one must use the {\sl mass eigenstates} for the calculation of
the partial decay widths and not the electroweak eigenstates. In
addition the $\epsilon_i$ enter via the $\nu_j$-$S_k^\pm$-$l_i$ and
$\chiz{1}$-$S_k^\pm$-$l_i$ couplings.
%
Remarkably, despite the non-trivial way the $\epsilon_i$ parameters
enter here, one still has some residual correlation with $\epsilon_i$
ratios, as displayed in  \fig{fig:TestAngleSol}.
%
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}(150,80)
\put(0,1){\mbox{\epsfig{figure=LLBrETauMuTauSol1.eps,height=7.3cm,width=7.cm}}}
\put(0,76){\makebox(0,0)[bl]{{ a)}}}
\put(5,75){\makebox(0,0)[bl]{{\small Br($e \tau \nu_i$)/Br($\mu \tau \nu_i$)}}}
\put(70,-3){\makebox(0,0)[br]{{$\tan^2(\theta_{sol})$}}}
%
\put(88,1){\mbox{\epsfig{figure=LLBrETauMuTauSol2.eps,
                               height=7.3cm,width=7.cm}}}
\put(84,76){\makebox(0,0)[bl]{{ b)}}}
\put(89,75){\makebox(0,0)[bl]{{\small Br($e\tau \nu_i$)/Br($\mu \tau \nu_i$)}}}
\put(158,-3){\makebox(0,0)[br]{{$\tan^2(\theta_{sol})$}}}
\end{picture}
\caption[]{Testing the solar angle.
           In case of the dark (bright) points  $\mu<(>)0$. 
           In a) we have 
           taken $\epsilon_\mu \Lambda_\mu /(\epsilon_\tau \Lambda_\tau)$
     $> 0$. and in b) $\epsilon_\mu \Lambda_\mu /(\epsilon_\tau \Lambda_\tau)$
           $< 0$ }
\label{fig:TestAngleSol}
\end{figure}
%
This figure shows that, although one does not get a strong correlation
in this case, one can still derive lower and upper bounds depending on
$\tan^2(\theta_{sol})$. For the favored
case~\cite{gonzalez-garcia:2000sq} of the large mixing angle solution
one finds that Br($\chiz{1} \to e \tau \nu_i$)/Br($\chiz{1} \to \mu \tau
\nu_i$) is determined to be one to within an order of magnitude.
%
For the general bilinear \rp model the spread in
\fig{fig:TestAngleSol} is due to the lack of knowledge of SUSY
parameters. As will be shown in the next subsection, a much stronger
correlation appears once the SUSY parameters get determined.
%

In \tab{tab:TestSolar} we list upper and lower bounds on several
ratios of branching ratios which are required by the consistency of
the model. The values in the table are hardly dependent on the
solution for the solar neutrino problem.
%
\begin{table}
\label{tab:TestSolar}
\caption[]{Ratio of branching ratios as required by the consistency of
the model. In Br($q \bar{q} \sum_i \nu_i$) we have summed over $u$, $d$, 
and $s$.
Also in case of $\nu_i$ we have summed over all neutrinos.\\[-0.2cm]}
\begin{center}
\begin{tabular}{|l|c|c|} \hline
  Ratio & lower bound & upper bound  \\ \hline 
Br($ q \bar{q} \nu_i$) / Br($ c \bar{c} \nu_i$) &
                                                     2.5 & 6 \\
Br($ q \bar{q} \nu_i$) / Br($\mu^\pm q \bar{q}'$) &
                                                     0.1 & 3.5 \\
Br($ q \bar{q} \nu_i$) / Br($\tau^\pm q \bar{q}'$) &
                                                     0.1 & 3.5 \\
Br($ q \bar{q} \nu_i$) / Br($e^+ e^- \nu_i$) &
                                                       5 & 35 \\
Br($ q \bar{q} \nu_i$) / Br($e^\pm \mu^\mp \nu_i$) &
                                                    0.3 & 9.5 \\
Br($ q \bar{q} \nu_i$)/ Br($\mu^+ \mu^-  \nu_i$) & 
                                                     0.3 & 9 \\
Br($\mu^\pm q \bar{q}'$) / Br($\tau^\pm q \bar{q}'$) &
                             0.5 & 3 \\
Br($\mu^\pm q \bar{q}'$) / Br($\mu^+ \mu^- \nu_i$) &
                             1 & 5  \\
Br($\tau^\pm q \bar{q}'$) / Br($\mu^+ \mu^- \nu_i$) &
                             0.5 & 6.5  \\
Br($e^\pm \mu^\mp \nu_i$) / Br($\mu^+ \mu^- \nu_i$) &
                             0.4 & 1.6  \\
\hline
%
\end{tabular}
\end{center}
\end{table}
%
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(150,80)
\put(0,-1){\mbox{\epsfig{figure=BrRanges.eps,height=8cm,width=16.cm}}}
\end{picture}
\end{center}
\caption[]{Predicted ranges for the ratios of various branching ratios.
  The dark stripes are the ranges if one of the large mixing solutions
  (LMA, LOW or just-so) is realized in nature, the bright stripes are
  if SMA is realized in nature. The various ratios are given in the text. }
\label{fig:TestAngleSolPred}
\end{figure}

The values in \tab{tab:TestSolar} can be viewed as important consistency checks
of our model. However, one can also have observables which are able to
discriminate between large and small angle solution of the solar
neutrino problem.
%
In \fig{fig:TestAngleSolPred} we show how several ratios of neutralino
decay branching ratios can be used to discriminate between large and
small angle solution of the solar neutrino problem.
%
The numbers in  \fig{fig:TestAngleSolPred} correspond to the 
following branching ratios: 
%
1 \dots Br($ q \bar{q} \nu_i$) / Br($e^\pm \tau^\mp \nu_i$),
2 \dots Br($b \bar{b} \nu_i$) / Br($\mu^\pm \tau^\mp \nu_i$),
3 \dots Br($b \bar{b} \nu_i$) / Br($\tau^- \tau^+ \nu_i$),
4 \dots Br($e^\pm q \bar{q}'$) / Br($\mu^\pm q \bar{q}'$),
5 \dots Br($e^\pm q \bar{q}'$) / Br($\tau^\pm q \bar{q}'$),
6 \dots Br($e^\pm q \bar{q}'$) / Br($e^\pm \mu^\mp \nu_i$),
7 \dots Br($\mu^\pm q \bar{q}'$) / Br($e^\pm \mu^\mp \nu_i$),
8 \dots Br($\mu^\pm q \bar{q}'$) / Br($e^\pm \tau^\mp \nu_i$),
9 \dots Br($\tau^\pm q \bar{q}'$) / Br($e^\pm \mu^\mp \nu_i$),
10 \dots Br($\tau^\pm q \bar{q}'$) / Br($e^\pm \tau^\mp \nu_i$),
11 \dots Br($e^\pm \mu^\mp \nu_i$) / Br($e^\pm \tau^\mp \nu_i$),
12 \dots Br($e^\pm \tau^\mp \nu_i$) / Br($\mu^+ \mu^- \nu_i$), and 
13 \dots Br($\mu^\pm \tau^\mp \nu_i$) / Br($\tau^+ \tau^- \nu_i$).
%
In Br($q \bar{q} \sum_i \nu_i$) we have summed over $u$, $d$, and $s$.
Also for the case of $\nu_i$ we have summed over all neutrinos.

\subsubsection{After the  SUSY spectrum is measured}
 
In the previous section we have discussed the predictions which can be
established between neutralino decay branching ratios and neutrino
mixing angles before the first SUSY particle is discovered. 
%
Let us assume now that the entire spectrum has been measured with some
precision, e.g. at a future Linear Collider
\cite{accomando:1998wt,martyn:1999tc}.  
%
As a typical example we discuss the point $M_2=120$~GeV,
$\mu=500$~GeV, $\tan \beta=5$, setting all scalar mass parameters to
$500$~GeV, and also the A-parameter is assumed to be equal for all
sfermions $A = -500$~GeV.  Note, that we have taken $\mu$ positive to
be conservative, as this corresponds to a 'worst-case' scenario."
%
There are at least two parameters which need to be measured precisely:
$\tan \beta$ and $|\sin 2 \theta_{\tilde b}|$ because the 1-loop mass
matrix is dominated by the sbottom/bottom loop if at least one of
these parameters is large.


\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}(150,75)
\put(0,-1){\mbox{\epsfig{
          figure=BrMuQQpTauQQpTanAtmsq.eps,height=6.8cm,width=5.cm}}}
\put(0,68){\makebox(0,0)[bl]{{\small
                  a)   Br($\mu^\pm q \bar{q}'$)/Br($\tau^\pm q \bar{q}'$)}}}
\put(50,-3){\makebox(0,0)[br]{{$\tan^2 \theta_{atm}$}}}
%
\put(54,-2){\mbox{\epsfig{
        figure=BrEMuNuTauQQpTanAtmsq.eps,height=7.cm,width=5.cm}}}
\put(54,68){\makebox(0,0)[bl]{{\small
             b)   Br($e^\pm \mu^\mp \sum_i \nu_i$)/Br($\tau^\pm q \bar{q}'$)}}}
\put(104,-3){\makebox(0,0)[br]{{$\tan^2 \theta_{atm}$}}}
%
\put(108,-1){\mbox{\epsfig{
        figure=BrMuQQpTauQQpTanAtmsq.eps, height=6.8cm,width=5.cm}}}
\put(108,68){\makebox(0,0)[bl]{{\small
             c)   Br($\mu^+ \mu^- \sum_i \nu_i$)/Br($\tau^\pm q \bar{q}'$)}}}
\put(158,-3){\makebox(0,0)[br]{{$\tan^2 \theta_{atm}$}}}
\end{picture}
\caption[]{Correlations between $\tan^2 \theta_{atm}$ and ratios of
           branching ratio for the parameter point specified in the text
           assuming that $10^5$ neutralino decays have been measured.
           The bands correspond to an 1-$\sigma$ error.}
\label{fig:PostDictAtm}
\end{figure}

\begin{figure}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(80,80)
\put(0,0){\mbox{\epsfig{figure=BrEQQpTauQQpUe3sq.eps,height=7.7cm,width=7.cm}}}
\put(2,76){\makebox(0,0)[bl]{{\small
                Br($e^\pm q \bar{q}'$)/Br($\tau^\pm q \bar{q}'$)}}}
\put(71,-3){\makebox(0,0)[br]{{$U^2_{e3}$}}}
\end{picture}
\end{center}
\caption[]{Correlation between $U^2_{e3}$ and the ratio
           Br($e^\pm q \bar{q}'$)/Br($\tau^\pm q \bar{q}'$) 
           for the parameter point specified in the text
           assuming that $10^5$ neutralino decays have been measured.
           The band corresponds to an 1-$\sigma$ error.}
\label{fig:ProsDictUe3}
\end{figure}

\begin{figure}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(80,80)
\put(0,0){\mbox{\epsfig{figure=BrETauMuTauTanSolsq.eps,
                                     height=7.5cm,width=7.cm}}}
\put(2,76){\makebox(0,0)[bl]{{\small
     Br($e^\pm \tau^\mp \sum_i \nu_i$)/Br($\mu^\pm \tau^\mp \sum_i \nu_i$)}}}
\put(71,-3){\makebox(0,0)[br]{{$\tan^2 \theta_{sol}$}}}
\end{picture}
\end{center}
\caption[]{Correlation between $\tan^2 \theta_{sol}$ and the ratio
         Br($e^\pm \tau^\mp \sum_i \nu_i$)/Br($\mu^\pm \tau^\mp \sum_i \nu_i$)
           for the parameter point specified in the text
           assuming that $10^5$ neutralino decays have been measured.
           The band corresponds to an 1-$\sigma$ error.}
\label{fig:ProsDictSol}
\end{figure}

In \fig{fig:PostDictAtm} --- \fig{fig:ProsDictSol} the same
relationships as discussed above are displayed assuming that the
particle spectrum and the corresponding mixing angles are known to the
1\% level or better. In addition we have assumed that $10^5$
neutralino decays have been identified and measured.  Taking at the
moment only the statistical error this translates to a relative error
on the branching ratio Br$(X)$ of the form $1/\sqrt{10^5
  \mathrm{Br}(X)}$. It is clear from these figures that there exist
excellent correlations between the ratio of various branchings and
$\tan^2 \theta_{atm}$ as well as the parameter $U^2_{e3}$ probed in
reactor experiments. For the solar angle we observe a strong
dependence on $\tan^2 \theta_{sol}$ for the case of large mixing angle
solutions (LMA, LOW or vacuum) of the solar neutrino problem. 
%
For the small mixing angle MSW solution, even though the dependence on
$\tan^2 \theta_{sol}$ becomes, unfortunately, rather weak, the ratio
of branching ratios for Br($e^\pm \tau^\mp \sum_i \nu_i$)/Br($\mu^\pm
\tau^\mp \sum_i \nu_i$) is predicted with good accuracy for any
$\tan^2 \theta_{sol} \lsim 0.1$.




\section{stau  decays}


  Supersymmetry with bilinear R-parity violation provides a predictive
  framework for neutrino masses and mixings in agreement with current
  neutrino oscillation data.  The model leads to striking signals at
  future colliders through the R-parity violating decays of the
  lightest supersymmetric particle. Here we study charged scalar
  lepton decays and demonstrate that if the scalar tau is the LSP (i)
  it will decay within the detector, despite the smallness of the
  neutrino masses, (ii) the relative ratio of branching ratios
  $Br({\tilde \tau}_1 \to e \sum \nu_i)/ Br({\tilde \tau}_1 \to \mu
  \sum \nu_i)$ is predicted from the measured solar neutrino angle,
  and (iii) scalar muon and scalar electron decays will allow to test
  the consistency of the model.  Thus, bilinear R-parity breaking SUSY
  will be testable at future colliders also in the case where the LSP
  is not the neutralino.




\subsection{The charged scalar mass matrix}
\label{sec:charged-scalar-mass}

For the decays of the charged sleptons it is necessary to calculate the 
mixings between neutrinos and neutralinos, charginos and charged 
leptons, as well as the charged scalar mixing. Since the various 
mass matrices can be found in \cite{hirsch:2000ef,romao:1999up}, 
we will discuss only the 
charged scalar mass matrix in this section. 

With R-parity broken by the bilinear terms in Eq. (\ref{eq:WRPV}) the
left-handed and right-handed charged sleptons mix with the charged
Higgs of the MSSM, resulting in an ($8 \times 8$) mass matrix for
charged scalars. As in the MSSM this matrix contains the Goldstone
boson, providing the mass of the W-boson after electro-weak symmetry
breaking. One can rotate away the Goldstone mode from this mass
matrix, using the following rotation matrix

\begin{equation}
\label{eq:Rhat}
{\hat R} = 
\left[
\begin{array}{cccccccc}
\frac{v_D}{w_3} & -\frac{v_U}{w_3} & \frac{v_1}{w_3} & 
\frac{v_2}{w_3} &\frac{v_3}{w_3} & 0 & 0 & 0\cr
\frac{v_U}{w_0} & \frac{v_D}{w_0} & 0 & 
0 & 0 & 0 & 0 & 0\cr
- \frac{v_1 v_D}{w_0 w_1} & \frac{v_1 v_U}{w_0 w_1} & \frac{w_0}{w_1} & 
0 & 0 & 0 & 0 & 0\cr
- \frac{v_2 v_D}{w_1 w_2} & \frac{v_2 v_U}{w_1 w_2} & 
-\frac{v_2 v_1}{w_1 w_2} & \frac{w_1}{w_2} & 0 & 0 & 0 & 0\cr
- \frac{v_3 v_D}{w_2 w_3} & \frac{v_3 v_U}{w_2 w_3} & 
-\frac{v_3 v_1}{w_2 w_3} & -\frac{v_2 v_3}{w_2 w_3} & 
\frac{w_2}{w_3} & 0 & 0 & 0 \cr
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \cr
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \cr
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \cr
\end{array}
\right]
\end{equation}
where,

\begin{eqnarray}
  \label{eq:shorthands}
  w_0 = \sqrt{v_D^2+v_U^2} \\
w_1 = \sqrt{v_1^2+v_D^2+v_U^2} \\
w_2 = \sqrt{v_1^2+v_2^2+v_D^2+v_U^2} \\
w_3 = \sqrt{v_1^2+v_2^2+v_3^2+v_D^2+v_U^2} 
\end{eqnarray}

This matrix has the property that
\begin{eqnarray}
  \label{eq:rot}
{\hat R} M_{S^{\pm}}^2 {\hat R}^T = \left[  
\begin{array}{cc}  
0 & {\vec 0}^T \cr
{\vec 0} & M_{S_7^{\pm}}^2 \cr
\end{array}
\right]
\end{eqnarray}
where $M_{S_7^{\pm}}^2$ is a ($7\times 7$) matrix and the zeroes  
in the first row and first column correspond to the (massless) Goldstone 
state in $\xi = 0$ gauge.

We divide the remaining $M_{S_7^{\pm}}^2$ into two parts,
\begin{equation}
\label{eq:splitMSC}
M_{S_7^{\pm}}^2 = (M_{S_7^{\pm}}^2)^{(0)} +(M_{S_7^{\pm}}^2)^{(1)}
\end{equation}
where $(M_{S_7^{\pm}}^2)^{(0)}$ [$(M_{S_7^{\pm}}^2)^{(1)}$] contains 
only R-parity conserving [R-parity violating] terms. 
Note that in the following we assume for simplicity that there is no 
inter-generational mixing among the charged sleptons. This is motivated 
by existing constraints from flavour changing neutral currents
\cite{gabbiani:1996hi} 
and is consistent with the minimal SUGRA scenario of the MSSM, which we 
will use in the numerical part of this paper. With this assumption  
also the branching ratio $\mu \rightarrow e \gamma$ is small
\cite{carvalho:2002bq}  
in the bilinear model in agreement with experimental data.
The R-parity conserving part of $M_{S_7^{\pm}}^2$ is given by
\begin{equation}
\label{eq:ChSc0}
(M_{S_7^{\pm}}^2)^{(0)}=
\left[
\begin{array}{ccccccc}
m_{H^{\pm}}^2 & \cdot &\cdot &\cdot &\cdot &\cdot &\cdot \cr
0 & {\hat m}_{L_1}^2 & \cdot & \cdot& \cdot& \cdot  & \cdot \cr
0 & 0 & {\hat m}_{L_2}^2 & \cdot& \cdot& \cdot  & \cdot \cr
0 & 0 & 0 & {\hat m}_{L_3}^2 & \cdot& \cdot  & \cdot \cr
0 & {\hat m}_{LR1}^2 & 0 & 0 & {\hat m}_{R_1}^2 & \cdot  & \cdot \cr
0 & 0 & {\hat m}_{LR2}^2 & 0 & 0 & {\hat m}_{R_2}^2   & \cdot \cr
0 & 0 & 0 & {\hat m}_{LR3}^2 & 0 & 0 & {\hat m}_{R_3}^2  \cr
\end{array}
\right]
\end{equation}
where the dots indicate that the matrix is symmetric and 
\begin{equation}
\label{eq:defhmpm}
m_{H^{\pm}}^2 = m_A^2 + \frac{g^2 v_{R_P}^2}{4}
\end{equation}
\begin{equation}
\label{eq:defhmli}
{\hat m}_{L_i}^2 = m_{L_i}^2 - (g^2 - {g'}^2)\frac{v_{R_P}^2}{8}c_{2\beta} 
                 + \frac{1}{2}(h_{i}^E)^2 v_D^2
\end{equation}
\begin{equation}
\label{eq:defhmri}
{\hat m}_{R_i}^2 = m_{R_i}^2 - {g'}^2\frac{v_{R_P}^2}{4}c_{2\beta} 
                 + \frac{1}{2}(h_{i}^E)^2 v_D^2
\end{equation}
\begin{equation}
\label{eq:defhmlri}
{\hat m}_{LRi}^2 = + \frac{1}{\sqrt{2}}(h_{i}^E)(A_i v_D - \mu v_U)
\end{equation}
with $v_{R_P}^2=v_U^2+v_D^2$. $m_A^2$ is the MSSM pseudoscalar Higgs
mass parameter $m_A^2 = (\mu B)/(s_{\beta}c_{\beta})$, $h_i^E$ and
$A_i$ are the Yukawa couplings and soft breaking trilinear parameters
of the charged lepton of generation $i$, $\mu$ is the Higgsino mixing
parameter characterizing the superpotential, and $c_{2\beta} =
\cos(2\beta)$, where $\beta$ is defined in the usual way as $\tan\beta
= v_U/v_D$.  The R-parity violating part of $M_{S_7^{\pm}}^2$
can be written as

\begin{equation}
\label{eq:ChSc1}
(M_{S_7^{\pm}}^2)^{(1)} =
\left[
\begin{array}{ccc}
\Delta m_{H^{\pm}}^2 & ({\vec X}_{HL})^T & ({\vec X}_{HR})^T \cr
{\vec X}_{HL} & M_{LL}^{2(1)} & (M_{LR}^{2(1)})^T \cr
{\vec X}_{HR} & M_{LR}^{2(1)} & M_{RR}^{2(1)} 
\end{array} 
\right] .
\end{equation}
The Higgs mass correction and the Higgs-Slepton mixing terms in eq.
(\ref{eq:ChSc1}) are

\begin{equation}
\label{eq:DeltamHp}
\Delta m_{H^{\pm}}^2 = \sum {\Big\{ }
                   (\frac{v_i}{v_D})^2 {\bar m}_{\tilde \nu_i}^2
                        \frac{c_{\beta}^4}{s_{\beta}^2}
                     -  \epsilon_i \mu \frac{v_i}{v_D} 
                        \frac{c_{2\beta}}{s_{\beta}^2}
                     + \frac{g^2}{4} v_i^2 c_{2\beta}
                     + \frac{1}{2}  (h_i^E v_i)^2 s_{\beta}^2{\Big\} }
\end{equation}
\begin{equation}
\label{eq:DelMHLi}
(X_{HL})_i = \frac{v_i}{v_D} {\bar m}_{\tilde \nu_i}^2
                        \frac{c_{\beta}^2}{s_{\beta}}
                     - \mu \epsilon_i \frac{1}{s_{\beta}}
             +\frac{1}{2} (g^2 - (h_i^E)^2) v_D v_i s_{\beta}
\end{equation}
\begin{equation}
\label{eq:DelMHRi}
(X_{HR})_i = - \frac{1}{\sqrt{2}}h_i^E v_i (A_is_{\beta} + \mu c_{\beta})
             - \frac{1}{\sqrt{2}}h_i^E \epsilon_i v_D \frac{1}{c_{\beta}}
\end{equation}

$M_{LL}^{2(1)}$ can be written as,
\begin{equation}\label{DefMLL21}
M_{LL}^{2(1)}= 
\left[
\begin{array}{ccc}
\Delta m_{L_1}^2 & (X_{LL})_{12} & (X_{LL})_{13} \cr
(X_{LL})_{12} & \Delta m_{L_2}^2 & (X_{LL})_{23} \cr
(X_{LL})_{13} & (X_{LL})_{23} &  \Delta m_{L_3}^2
\end{array}
\right]
\end{equation}
with the diagonal terms given by
\begin{equation}
\label{eq:DefMLLii}
\Delta m_{L_i}^2 = (\frac{v_i}{v_D})^2 {\bar m}_{\tilde \nu_i}^2 c_{\beta}^2 +
 \epsilon_i^2 + \frac{1}{2}  (g^2 + (h_i^E)^2) v_i^2 c_{\beta}^2
                  + \frac{1}{8} (g^{\prime 2} - g^2) \sum v_i^2
\end{equation}
whereas the off-diagonals are
\begin{eqnarray}
  \label{eq:xllij}
(X_{LL})_{12} = \epsilon_1 \epsilon_2 
        &+& (\frac{v_1}{v_D}) (\frac{v_2}{v_D})m_{L_2}^2 c_{\beta}^2 \\ \nn
        &+& v_1 v_2 \Big[\frac{1}{4}(g^2 + (h_2^E)^2) -
                          \frac{1}{8} (g^2 -  g^{\prime 2}) c_{2\beta} +
                          \frac{1}{4}(h_2^E)^2 c_{2\beta}\Big] \\
(X_{LL})_{13} = \epsilon_1 \epsilon_3 
        &+& (\frac{v_1}{v_D}) (\frac{v_3}{v_D})m_{L_3}^2 c_{\beta}^2 \\ \nn
        &+& v_1 v_3 \Big[\frac{1}{4}(g^2 + (h_3^E)^2) -
                          \frac{1}{8} (g^2 -  g^{\prime 2}) c_{2\beta} +
                          \frac{1}{4}(h_3^E)^2 c_{2\beta}\Big] \\
(X_{LL})_{23} = \epsilon_2 \epsilon_3 
        &+& (\frac{v_2}{v_D}) (\frac{v_3}{v_D})m_{L_3}^2 c_{\beta}^2 \\ \nn
        &+& v_2 v_3 \Big[\frac{1}{4}(g^2 + (h_3^E)^2) -
                          \frac{1}{8} (g^2 -  g^{\prime 2}) c_{2\beta} +
                          \frac{1}{4}(h_3^E)^2 c_{2\beta}\Big]
\end{eqnarray}
Similarly for $M_{RR}^{2(1)}$, 
\begin{equation}
\label{eq:defmrrii}
\Delta m_{R_i}^2 = \frac{1}{2} (h_i^E)^2 v_i^2 -
                    \frac{1}{4}  g^{\prime 2} \sum v_i^2
\end{equation}
and
\begin{equation}\label{defmrrij}
(X_{RR})_{ij}=\frac{1}{2} (h_i^E)(h_j^E) v_i v_j
\end{equation}
Finally, the matrix
$M_{LR}^{2(1)}$ has the following peculiar structure,

\begin{equation}\label{DefMLR21}
M_{LR}^{2(1)}= 
\left[
\begin{array}{ccc}
(X_{LR})_{11} & 0 & 0 \cr
(X_{LR})_{12} & (X_{LR})_{22} & 0 \cr
(X_{LR})_{13} & (X_{LR})_{23} & (X_{LR})_{33}
\end{array}
\right]
\end{equation}
where
\begin{equation}\label{DefXLRii}
(X_{LR})_{ii} = - \frac{1}{2\sqrt{2}}(h_i^E)(\frac{v_i}{v_D})^2 c_{\beta} 
                  v_D \Big[ \mu s_{\beta} - A_i c_{\beta} \Big]
\end{equation}
\begin{equation}\label{DefXLRij}
(X_{LR})_{ij} = - \frac{1}{\sqrt{2}}(h_i^E)(\frac{v_i}{v_D})
                   (\frac{v_j}{v_D}) c_{\beta} v_D 
                  \Big[ \mu s_{\beta} - A_i c_{\beta} \Big]
\end{equation}
In the above equations we have used the following abbreviation
%
\begin{equation}\label{defmsnu}
{\bar m}_{\tilde \nu_i}^2 = m_{L_i}^2 + \frac{1}{8} (g^2 +  g^{\prime 2})
                                                    (v_D^2-v_U^2).
\end{equation}
With the definitions outlined above, once can easily derive approximate 
expressions for the mixing between the charged Higgs and the charged 
sleptons induced by the R-parity breaking parameters. These are given 
by
\begin{equation}\label{Lmix}
\sin\theta_{HL_i} \simeq \frac{X_{HL,i}}{(m^2_{H^{\pm}}-m^2_{L_i})},
\end{equation}
\begin{equation}\label{Rmix}
\sin\theta_{HR_i} \simeq \frac{X_{HR,i}}{(m^2_{H^{\pm}}-m^2_{R_i})}.
\end{equation}
Note that one expects $\sin\theta_{HR_i} \sim h^E_i
\sin\theta_{HL_i}$, i.e. the mixing between right-handed sleptons and
the Higgs should be typically much smaller than the left-handed
Higgs-slepton mixing.

Finally, the R-parity conserving mixing between left-handed and 
right-handed sleptons is approximately given by 

\begin{equation}\label{LRmix}
\sin 2\theta_{\tilde l_i} \simeq \frac{2 {\hat m}_{LRi}^2}
                       {{\hat m}_{Li}^2-{\hat m}_{Ri}^2}.
\end{equation}


\subsection{Formulas for two-body decays}
\label{sec:formulas-two-body}

Charged scalar leptons lighter than all other supersymmetric particles
will decay through R-parity violating couplings. Possible final states
are either $l_j\nu_k$ or $q{\bar q}'$. For right-handed charged
sleptons (${\tilde l}_{Ri}$) the former by far dominates over the
hadronic decay mode, since the mixing between ${\tilde l}_{Ri}$ and
the charged Higgs is small, as explained above.

In the limit $(m_{f_j},m_{\nu_k}) \ll m_{{\tilde f}_i}$ one has the
simple formula for the two-body decays ${\tilde f}_i \rightarrow f_j +
\nu_k$,
\begin{equation}\label{width}
\Gamma_{{\tilde f}_if_j\nu_k} = \frac{m_{{\tilde f}_i}}{16\pi} 
\Big[(O^{cns}_{Lf_j\nu_k{\tilde f}_i})^2 + 
(O^{cns}_{Rf_j\nu_k{\tilde f}_i})^2\Big]
\end{equation}
Exact expressions for these couplings can be found, for
example, in ref.~\cite{hirsch:2000ef,romao:1999up}. 
Even though in the results presented
in this paper we have always calculated the couplings appearing in
eq. (\ref{width}) exactly using our numerical code, it is instructive to
consider an approximate diagonalization procedure for the various mass
matrices.  This method is based on the fact that
neutrino masses are much smaller than all other particle masses in the
theory and therefore one expects that the bilinear R-parity breaking
parameters are (somewhat) smaller than the corresponding MSSM
parameters. For the charged scalar mass matrix all necessary
definitions have been given above, for details for the corresponding
procedure for neutralino and chargino mass matrices we refer to
\cite{hirsch:2000ef,romao:1999up,hirsch:1998kc,nowakowski:1996dx}.

For the case where $i \ne j$ for ${\tilde l}_{Ri} \rightarrow l_j \sum
\nu_k$ one finds
%
\begin{eqnarray}
\label{eq:SlLNu}
\sum_k \Big[(O^{cns}_{L l_j\nu_k{\tilde l}_{i}})^2 + 
(O^{cns}_{Rl_j \nu_k {\tilde l}_{i}})^2 \Big] &=& 
(-h^E_{l_i}c_{\tilde l_i}\frac{\epsilon_j}{\mu}- (gs_{\tilde l_i}y_1+
h^E_{l_i}c_{\tilde l_i} y_2) \Lambda_{j})^2 \\ \nn
& + & (h^E_{l_j})^2 (s_{\beta} \sin\theta_{HR_i}-c_{\beta}^2 
 s_{\tilde l_i} {\tilde v}_i)^2   
\end{eqnarray}
\begin{eqnarray}
\label{eq:SimSlLNu}
&\simeq & 
(c_{\tilde l_i}h^E_{l_i}\frac{\epsilon_j}{\mu})^2
\end{eqnarray}
Here $c_{\tilde l_i} \equiv \cos(\theta_{\tilde l_i})$ and $s_{\tilde
  l_i} \equiv \sin(\theta_{\tilde l_i})$ where $\theta_{\tilde l_i}$
is the left-right mixing angle for ${\tilde l_i}$, $\sin\theta_{HR_i}$
characterizes the charged Higgs-(right-handed)-Slepton mixing and
${\vec \Lambda}$ is given by
\begin{equation}
\label{deflam}
\Lambda_i = \epsilon_i v_D + \mu v_i.
\end{equation}
The quantities $y_1$ and $y_2$ are defined as

\begin{eqnarray}\label{eq:defy}
y_1 = \frac{g}{\sqrt{2}{\rm Det}M_{\chi^{\pm}}} \\
y_2 =- \frac{g^2 v_U}{2\mu{\rm Det}M_{\chi^{\pm}}}
\end{eqnarray}
with ${\rm Det}M_{\chi^{\pm}}$ being the determinant of the MSSM 
chargino mass matrix.

While eq. (\ref{eq:SlLNu}) above keeps all R-parity breaking paramters in
the expansion up to second order, eq. (\ref{eq:SimSlLNu}) should be valid
in the parameter region in which the 1-loop neutrino masses are
smaller than the tree-level contribution.

For the case $i=j$ the corresponding formulas are rather cumbersome
and therefore of limited utility, except for the case ${\tilde l} =
\te$.  Here, since $h_e \ll 1$ one can simplify the couplings to,

\begin{equation}\label{SeENu}
\sum_k \Big[(O^{cns}_{Le{\nu_k}{\tilde e}})^2 + 
(O^{cns}_{Re{\nu_k}{\tilde e}})^2 \Big] 
\simeq 2  g^{\prime 2} x_1^2 |{\vec \Lambda}|^2 
\end{equation}
The parameter ${\vec \Lambda}$ has been defined above and $x_1$ is
given by
\begin{equation}
\label{defx1}
x_1 = \frac{g' M_2 \mu}{2 {\rm Det}M_{\chi^0}}
\end{equation}
with ${\rm Det}M_{\chi^0}$ being the determinant of the MSSM neutralino 
mass matrix and $M_2$ the soft SUSY breaking $SU(2)$ mass parameter.

From eq. (\ref{eq:SimSlLNu}) one expects that various ratios of branching
ratios should contain rather precise information on ratios of the
bilinear R-parity breaking parameters, for example, $Br({\tilde
  \tau}_1 \rightarrow e \sum \nu_i)/ Br({\tilde \tau}_1 \rightarrow
\mu \sum \nu_i) \simeq (\epsilon_1/\epsilon_2)^2$. We will discuss
this important point in more detail in the next section.

\subsection{Slepton production and decays}
\label{sec:slept-prod-decays}

In this section we will discuss charged slepton production and decay
modes. In order to reduce the number of parameters, the numerical
calculations were performed in the mSUGRA version of the MSSM. Unless
noted otherwise, we have scanned the parameters in the following
ranges: $M_2$ from [0,1.2] TeV, $|\mu|$ from [0,2.5] TeV, $m_0$ in the
range [0,0.5] TeV, $A_0/m_0$ and $B_0/m_0$ [-3,3] and $\tan\beta$
[2.5,10].  All randomly generated points were subsequently tested for
consistency with the minimization (tadpole) conditions of the Higgs
potential as well as for phenomenological constraints from
supersymmetric particle searches. In addition, we selected points in
which at least one of the charged sleptons was lighter than the
lightest neutralino, and thus the LSP. This latter cut prefers
strongly $m_0 <<M_2$.

R-parity violating parameters were chosen in such a way
\cite{hirsch:2000ef,romao:1999up}
that the neutrino masses and mixing angles are approximately
consistent with the experimental data. A good ``fit'' to the data
would require: a) $\Lambda_{\mu} \simeq \Lambda_{\tau}$, in order to
account for a nearly maximal $\nu_{\mu} \rightarrow \nu_{\tau}$
angle in atmospheric oscillations; b) $\Lambda_{e}
< \Lambda_{\tau}$, to fulfil the constraints from $\nu_e$-oscillation
searches at reactors~\cite{apollonio:1999ae}; c) $|\vec \Lambda|
\simeq [0.05,2]$ GeV$^2$, for the atmospheric neutrino mass
scale; d) $\epsilon_1 \simeq \epsilon_2$, to have
a large angle in solar oscillations; and e)
$|{\vec \epsilon}|^2/|\vec \Lambda| \simeq [0.1,10]$, for the solar
mass scale.

In order to investigate the dependence of our results on the
assumptions about the R-parity violating parameters, we construct
three different sets of points. Set1 was calculated to give an
approximate ``fit'' to the neutrino data, as described above. Set2 is
similar to Set1, except that $\epsilon_1/\epsilon_2$ has been varied
in a wider range ([0.1,10]), so as to cover both large and small solar
angles~\footnote{Although at the moment the small angle solar solution
  is ruled out by a careful analysis of the solar
  data~\cite{Maltoni:2002ni}, it does not cost us much additional
  effort to keep this option in mind.}. 
The last set, called Set3 in the following, is again similar to Set1,
except that $\epsilon_2/\epsilon_3$, which is hardly constrained by
neutrino data, is varied in the interval $\epsilon_2/\epsilon_3 \simeq
0.1-2$.

In supersymmetric models in which the scalar leptons have a common
soft SUSY breaking mass parameter at some high scale ($m_0$ in mSUGRA)
the renormalization group evolution leads to some splitting between the
scalar taus and the $\te$ and $\tm$ states at the weak scale. While
the lightest mass eigenstate in the charged slepton sector is usually
mainly the $\st_R$, the eigenvalues for $\te_R$ and $\tm_R$ are not
much heavier, such that also $\te_R$ and $\tm_R$ decay mainly via
R-parity violating two-body decays. In our numerical calculation we
therefore not only consider the decays of $\st_R$, but also those of
$\te_R$ and $\tm_R$. These decays can provide information on the
R-parity violating parameters not accesible in $\st_R$ decays and
allow for additional cross checks of the consistency of the model.
%
This is true especially for the case of lepton flavor violating
slepton decays since from eq.~\ref{eq:SimSlLNu} one expects them to
be directly correlated with the BRpV parameters $\epsilon_i$.
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(80,70)
\put(0,0){\mbox{\epsfig{figure=ProductionNf4.eps,height=7.7cm,width=7.cm}}}
\put(2,76){\makebox(0,0)[bl]
            {{\small $\sigma(e^+ e^- \to {\tilde l}{\tilde l})$~[fb]}}}
\put(71,-3){\makebox(0,0)[br]{{$m_{\tilde l_1}$ [GeV]}}}
\put(20,67){\mbox{${\tilde e}_R$}}
\put(20,56){\mbox{${\tilde \mu}_R$}}
\put(20,48){\mbox{${\tilde \tau}_R$}}
\end{picture}
\end{center}
\caption[]{$e^+ e^- \to {\tilde l}{\tilde l}$ production 
  cross section as a function of $m_{\tilde l}$ at a Linear Collider
  with 0.8 TeV c.m.s energy.  From top to bottom: ${\tilde e}$ (dark,
  on colour printers blue), ${\tilde \mu}$ (light shaded, green) and
  ${\tilde \tau}$ (dark shaded, red).}
\label{fig:Prod0.8TeV}
\end{figure}


\begin{figure}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(80,50)
\put(0,-30){\mbox{\epsfig{figure=LengthNf4b.eps,height=7.7cm,width=7.cm}}}
\put(2,46){\makebox(0,0)[bl]
            {{\small $L({\tilde l})$~[cm]}}}
\put(71,-33){\makebox(0,0)[br]{{$m_{\tilde l}$ [GeV]}}}
\end{picture}
\end{center}
\vskip28mm
\caption[]{Charged slepton decay length 
  as a function of $m_{\tilde l}$ at a linear collider with 0.8 TeV
  c.m.s.  energy. From top to bottom: ${\tilde e}$ (dark, on color
  printers blue), ${\tilde \mu}$ (light shaded, green) and ${\tilde
    \tau}$ (dark shaded, red).}
\label{fig:DecayLength}
\end{figure}

For the calculation of the cross section we have adapted the
formulas given in \cite{Blochinger:2002zw} to the bilinear model
taking into account correctly all mixing effects in the numerical
calculation. 
In \fig{fig:Prod0.8TeV} we show the cross section $\sigma(e^+ e^- \to
{\tilde l}{\tilde l})$ in [fb] for $\sqrt{s}=0.8$ TeV as a
function of the charged scalar mass, for ${\tilde e}$, ${\tilde \mu}$
and ${\tilde \tau}$, respectively. Assuming an integrated luminosity
of 1000 fb$^{-1}$ per year can be achieved at a future linear collider
\cite{accomando:1998wt,aguilar-saavedra:2001rg} this implies that around $10^4$ 
scalar muons and 
scalar taus can be directly produced per year. For scalar electrons
one expects between $10^4$ and $10^5$ produced pairs per year. Since
the three R-parity violating two-body decay channels of the
right-handed sleptons nearly add up to 100 \%, one can expect that
individual branching ratios will be measured to an accuracy of 1 \% if
they occur with similar strength.

At the LHC the direct production of right--sleptons is small. As a
result, they will be produced mainly in cascade decays. The relative
$\te_R$, $\tm_R$ and $\st_R$ yields will depend on the details of the
cascade decays involved. Let us consider for simplicity the case where
the cascade decays of the coloured particles end up in the lightest
neutralino as in the MSSM.  Beside the kinematics, the resulting
number of $\te_R$, $\tm_R$ and $\st_R$ arising from these decays
depends on the nature of the lightest neutralino. When this is mainly
bino-like, one expects that it decays dominantly into an equal number
of $\te_R$, $\tm_R$ and $\st_R$'s. As a result the number of
right-sleptons roughly equal to the number of neutralinos. Also in
case of a wino--like neutralino the amount of $\te_R$, $\tm_R$ and
$\st_R$ will be equal. However, in this case the main lightest
neutralino decay mode will be to a $W$--boson and a charged lepton,
leaving fewer sleptons to be studied. However, as discussed in
\cite{porod:2000hv,mukhopadhyaya:1998xj,chun:1998ub,choi:1999tq}, in this case the neutralino decay modes
can be used to probe the large atmospheric neutrino angle. For the case
where the lightest neutralino is higgsino--like it will decay into a
$W$--boson and a charged lepton, or into a $Z$--boson and a neutrino,
similar to the wino case. However for large $\tan\beta$ the decay into
$\st_R$ will again be important, even for higgsino--like neutralinos.

In \fig{fig:DecayLength} we show the charged scalar leptons decay
length ($\te$, $\tm$ and $\st$, from top to bottom) as a function of
the scalar lepton masses for Set3. Very similar results hold for the
other sets which are therefore not shown. All decay lengths are small
compared to typical detector sizes, despite the smallness of the
neutrino masses.  The three generations of sleptons decay with quite
different decay lengths and thus it should be possible to separate the
different generations experimentally at a future linear collider.
Note that the ratio of the decay lengths $L(\st)/L(\tm)$ is
approximately given by $(h_{\mu}/h_{\tau})^2$.

As mentioned in the previous section, one expects that ratios of
branching ratios of various charged slepton decays contain rather
precise information on ratios of the bilinear parameters $\epsilon_i$.
That this is indeed the case is shown in \fig{BrSet4a} for the data of
Set2 and in \fig{BreSet4b} and \fig{BrSet4b} for the data of Set3.

\begin{figure}
  \begin{center}
    \includegraphics[clip,height=70mm]{Set4aBrStauemu.ps}    
  \end{center}

\vskip -10mm
\begin{rotate}{90}
$Br({\tilde \tau}_1 \rightarrow e \sum \nu_i)/
Br({\tilde \tau}_1 \rightarrow \mu \sum \nu_i)$
\end{rotate}

\vskip5mm
\hskip110mm
$(\epsilon_1/\epsilon_2)^2$

\vskip10mm
\begin{center}
  \includegraphics[height=70mm]{Set4aBrSmuetau.ps}  
\end{center}


\vskip-10mm
\begin{rotate}{90}
$Br({\tilde \mu}_1 \rightarrow e \sum \nu_i)/
Br({\tilde \mu}_1 \rightarrow \tau \sum \nu_i)$
\end{rotate}

\vskip5mm
\hskip110mm
$(\epsilon_1/\epsilon_3)^2$

\vskip10mm
\caption[]{Ratios of branching ratios for scalar tau  decays (top panel)
  versus $(\epsilon_1/\epsilon_2)^2$, and scalar muon decays (bottom
  panel) versus $(\epsilon_1/\epsilon_3)^2$ for Set2.}
\label{BrSet4a}
\end{figure}



\begin{figure}
  \begin{center}
    \includegraphics[height=70mm]{Set4bBrSemutau.ps}    
  \end{center}


\vskip-20mm
\begin{rotate}{90}
$Br({\tilde e}_1 \rightarrow \mu \sum \nu_i)/
Br({\tilde e}_1 \rightarrow \tau \sum \nu_i)$
\end{rotate}

\vskip10mm
\hskip110mm
$(\epsilon_2/\epsilon_3)^2$

\vskip5mm

\caption[]{Ratios of branching ratios for scalar electron  decays 
versus $(\epsilon_2/\epsilon_3)^2$ for Set3.}
\label{BreSet4b}
\end{figure}



\begin{figure}

  \begin{center}
    \includegraphics[height=70mm]{Set4bBrStauemu-v1.ps}    
  \end{center}


\vskip-20mm
\begin{rotate}{90}
$Br({\tilde \tau}_1 \rightarrow e \sum \nu_i)/
Br({\tilde \tau}_1 \rightarrow \mu \sum \nu_i)$
\end{rotate}

\vskip15mm
\hskip110mm
$(\epsilon_1/\epsilon_2)^2$

\begin{center}
  \includegraphics[height=70mm]{Set4bBrSmuetau.ps}
\end{center}


\vskip-20mm
\begin{rotate}{90}
$Br({\tilde \mu}_1 \rightarrow e \sum \nu_i)/
Br({\tilde \mu}_1 \rightarrow \tau \sum \nu_i)$
\end{rotate}

\vskip15mm
\hskip110mm
$(\epsilon_1/\epsilon_3)^2$

\vskip5mm

\caption[]{Ratios of branching ratios for scalar tau (top panel) decays 
  versus $(\epsilon_1/\epsilon_2)^2$ and scalar muon decays (bottom
  panel) versus $(\epsilon_1/\epsilon_3)^2$ for Set3.}
\label{BrSet4b}
\end{figure}

As can be seen from these figures, the ratio of charged slepton
branching ratios are correlated with the ratios of the corresponding
BRpV parameters $\epsilon_i$, following very closely the expectation
from Eq. \ref{eq:SimSlLNu}, nearly insensitive to variation of the
other parameters.  Recall, that all the points were generated through
a rather generous scan over the mSUGRA parameters.  Ratios of
$\epsilon_i$'s should therefore be very precisely measurable.
Moreover, since only two of the three ratios of $\epsilon_i$'s are
independent it is possible to derive the following
\texttt{prediction}:

\smallskip
\centerline{$Br({\tilde \tau}_1 \rightarrow e \sum \nu_i)/
Br({\tilde \tau}_1 \rightarrow \mu \sum \nu_i)$ : 
$Br({\tilde \mu}_1 \rightarrow e \sum \nu_i)/
Br({\tilde \mu}_1 \rightarrow \tau \sum \nu_i)$ $\simeq$}
 
\centerline{$\simeq$ $Br({\tilde e}_1 \rightarrow \mu \sum \nu_i)/
Br({\tilde e}_1 \rightarrow \tau \sum \nu_i)$ }
\smallskip

\noindent
which provides an important cross check of the validity of our
bilinear R-parity model. Any significant departure from this equality
would be a clear sign that the bilinear model is incomplete.

\begin{figure}
  \begin{center}
    \includegraphics[height=70mm]{Set4aBrStauemuVtsol.ps}    
  \end{center}


\vskip-10mm
\begin{rotate}{90}
$Br({\tilde \tau}_1 \rightarrow e \sum \nu_i)/
Br({\tilde \tau}_1 \rightarrow \mu \sum \nu_i)$
\end{rotate}

\vskip5mm
\hskip110mm
$\tan^2\theta_{\odot}$

\vskip10mm

\begin{center}
  \includegraphics[height=70mm]{Set4aBrStauemuVtsolx90.ps}  
\end{center}


\vskip-20mm
\begin{rotate}{90}
$Br({\tilde \tau}_1 \rightarrow e \sum \nu_i)/
Br({\tilde \tau}_1 \rightarrow \mu \sum \nu_i)$
\end{rotate}

\vskip15mm
\hskip110mm
$\tan^2\theta_{\odot}$

\vskip0mm

\caption[]{Ratios of branching ratios for scalar tau decays 
  versus $\tan^2\theta_{\odot}$ for Set2. The top panel shown all data
  points, the bottom one refers only to data points with
  $\epsilon_2/\epsilon_3$ restricted to the range [0.9,1.1].}
\label{BrSolar}
\end{figure}

As mentioned in the introduction current 
solar neutrino data prefer a large angle solution (LMA). In the BRpV
model the solar angle is mainly determined by the ratio
$\epsilon_1/\epsilon_2$ \cite{hirsch:2000ef,romao:1999up}.  
A measured solar angle
therefore leads to a prediction for $Br({\tilde \tau}_1 \rightarrow e
\sum \nu_i)/ Br({\tilde \tau}_1 \rightarrow \mu \sum \nu_i)$, as shown
in \fig{BrSolar} for the data of Set2. With the current limits on
$\tan^2\theta_{\odot}$, which are $0.25 < \tan^2\theta_{\odot} < 0.83$
for the preferred LMA-MSW solution to the solar neutrino problem
\cite{Maltoni:2002ni} at 3 $\sigma$ \CL, one can currently predict
that this ratio in the BRpV model must be in the range $[0.09,1.8]$.
Additional input on $\epsilon_2/\epsilon_3$, for example
$\epsilon_2/\epsilon_3\simeq 1$ to within 10 \% would sharpen the
predicted value to $[0.15,1.1]$.  Obviously, also a more precise
measurement of the solar angle will lead to a tighter prediction in
the future. In this context it is worth noting that KamLAND
 should be able to fix the solar angle to within $\sim$
30 \%, if LMA is indeed the correct solution to the solar neutrino
problem.

Up to now we have discussed only \texttt{ratios} of R-parity violating
parameters, but charged scalar lepton decays allow, in principle, also
to gain information on \texttt{absolute values} of these parameters,
as relevant, e.~g. to fix the scale of neutrino masses determined
through the analysis of current solar and atmospheric
data~\cite{Maltoni:2002ni}.  However, such a measurement would require
at least some information on MSSM parameters which is at the moment 
unavailable.

\begin{figure}

  \begin{center}
    \includegraphics[height=70mm]{GamStauTot.ps}    
  \end{center}


\vskip-50mm
\begin{rotate}{90}
$\Gamma({\tilde \tau_1})$ [eV]
\end{rotate}

\vskip48mm
\hskip73mm
$(|\epsilon|/\mu)^2 m_{\tilde \tau_1}$ [GeV]

\begin{center}
  \includegraphics[height=70mm]{GamSmuTot.ps}  
\end{center}


\vskip-50mm
\begin{rotate}{90}
$\Gamma({\tilde \mu_1})$ [eV]
\end{rotate}

\vskip48mm
\hskip73mm
$(|\epsilon|/\mu)^2 m_{\tilde \mu_1}$ [GeV]

\caption[]{Total widths in [eV] for scalar tau decays (top) 
and scalar muon decays (bottom) for the data of Set1 versus 
$(|\epsilon|/\mu)^2 m_{\tilde l}$. Once $\mu$ and $m_{\tilde l}$ 
are measured, the widths provide information on the absolute 
value of $|\epsilon| \equiv |{\vec \epsilon}|$. Note that Set1 
fixes $\epsilon_2/\epsilon_3 \simeq 1$. In general, this ratio 
must be known with some accuracy, before a value for $|\epsilon|$ 
can be derived from the widths.}
\label{TotWidth}
\end{figure}

In \fig{TotWidth} we show the total widths in [eV] for scalar tau
decays (top panel) and scalar muon decays (bottom panel) for the data
of Set1 displayed versus $(|\epsilon|/\mu)^2 m_{\tilde l}$. Once $\mu$
and $m_{\tilde l}$ have been measured with some accuracy, one can
determine the absolute value of $|\epsilon|$ from this measurement,
provided $\epsilon_2/\epsilon_3$ is known (for example, from the ratio
$Br({\tilde e}_1 \rightarrow \mu \sum \nu_i)/ Br({\tilde e}_1
\rightarrow \tau \sum \nu_i)$).

In a similar way, the decay width of the scalar electron contains
information on $|{\vec \Lambda}|$, as is demonstrated in \fig{SeWidth}. A
priori knowledge on $\epsilon_2/\epsilon_3$ leads to a tighter
correlation, as can be seen from the comparison of the results for
Set2 and Set3.

To deduce the value of $|{\vec \Lambda}|$ from this measurement one
needs the parameter combination $x_1$, as defined in Eq.
(\ref{defx1}).  It contains the MSSM parameters $M_1$, $M_2$, $\mu$
and $\tan\beta$, which could be determined, for example, if at least
some of the neutralino and chargino eigenstates are accesible at the
LHC or a possible linear collider.

\begin{figure}

  \begin{center}
    \includegraphics[height=70mm]{GamSeTot4a.ps}    
  \end{center}


\vskip-50mm
\begin{rotate}{90}
$\Gamma({\tilde e_1})$ [meV]
\end{rotate}

\vskip48mm
\hskip73mm
$c \cdot m_{\tilde e_1}$ [GeV]

\begin{center}
  \includegraphics[height=70mm]{GamSeTot4b.ps}  
\end{center}



\vskip-50mm
\begin{rotate}{90}
$\Gamma({\tilde e_1})$ [meV]
\end{rotate}

\vskip48mm
\hskip73mm
$c \cdot m_{\tilde e_1}$ [GeV]

\caption[]{Charged scalar electron total decay widths in [meV], the 
  top panel refers to Set2 while the bottom one is for Set3. The plots
  are versus $c \cdot m_{\tilde c_1}$, where $c=\frac{1}{8\pi} (g')^2
  x_1 |\Lambda|^2$.}
\label{SeWidth}
\end{figure}


\subsection{Conclusions}
\label{sec:conclusions}

Supersymmetric models with bilinear R-parity breaking provide a
simple, testable framework for neutrino masses and mixings in
agreement with current solar, atmospheric and reactor neutrino
oscillation data. The model is testable at future colliders if the
neutralino is the LSP, as was shown previously, as well as in the
alternative case where one of the charged scalar leptons is the LSP,
as we have demonstrated here.

The measured neutrino mixing angles fix certain ratios of the bilinear
R-parity breaking parameters and, therefore, lead to well-defined
predictions for the ratio of branching ratios of certain slepton decay
modes, which should be easily measurable at a future collider such as
a high energy linear collider. Our main result is shown in
\fig{BrSolar}, where we display $Br({\tilde \tau}_1 \to e \sum \nu_i)/
Br({\tilde \tau}_1 \rightarrow \mu \sum \nu_i)$ versus the solar
neutrino angle, $\tan^2\theta_{\odot}$.

We have also shown how charged scalar lepton decays allow the
determination of other parameters of our model, thus providing a
definite test that bilinear R-parity breaking SUSY is the origin of
neutrino masses.

\section{stop  decays}
  
  We discuss the phenomenology of the lightest stop in models where
  R-parity is broken by bilinear superpotential terms. In this class
  of models we consider scenarios where the R-parity breaking two-body
  decay ${\tilde t}_1 \to \tau^+ \, b $ competes with the leading
  three-body decays such as ${\tilde t}_1 \to W^+ \, b \, {\tilde
    \chi}^0_1$.  We demonstrate that the R--parity violating decay can



  be sizable and in some parts of the parameter space even the
  dominant one.  Moreover we discuss the expectations for ${\tilde
    t}_1 \to \mu^+ \, b $ and ${\tilde t}_1 \to e^+ \, b $. The recent
  results from solar and atmospheric neutrinos suggest that these are
  as important as the $\tau^+ b$ mode.  The ${\tilde t}_1 \to l^+ \,
  b$ decays are of particular interest for hadron colliders, as they
  may allow a full mass reconstruction of the lighter stop.  Moreover
  these decay modes allow cross checks on the neutrino mixing angle
  involved in the solar neutrino puzzle complementary to those
  possible using neutralino decays. For the so--called small mixing
  angle or SMA solution ${\tilde t}_1 \to e^+ \, b$ should be
  negligible, while for the large mixing angle type solutions
  \texttt{all} ${\tilde t}_1 \to l^+ \, b$ decays should have
  comparable magnitude.


\subsection{Numerical Analyses}
\label{sec:numerical-analisis}

In this section we present our numerical results for the branching
ratios of the lighter stop $\tilde t_1$. Here we consider scenarios
where all two-body decays induced at tree-level are kinematically
forbidden except the $b \, l^+$ decays.  Before going into detail it
is useful to have some approximate formulas at
hand~\cite{diaz:1999ge}:
%
\begin{eqnarray}
  \Gamma(\tilde t_1\to b\,\tau) \approx\,&
  \frac{g^2 |U_{32}|^2 h_b^2 \cos^2_{\theta_{\tilde t}} m_{\tilde t_1}}{16\pi} 
  \approx
  \frac{g^2 |\epsilon_3|^2 h_b^2 \cos^2_{\theta_{\tilde t}} m_{\tilde t_1}}{16\pi |\mu|^2} 
  \label{eq:6} \\
  \Gamma(\tilde t_1 \to c\tilde\chi_1^0)\approx\,& F h_b^4(
  \delta_{m_0^2}\cos\theta_{\tilde t}-\delta_A\sin\theta_{\tilde t})^2
  f_L^2 %\nonumber\\&&
   m_{\tilde t_1} 
  \left(
    1-\frac{m_{\tilde \chi_1^0}^2}{m_{\tilde t_1}^2}
  \right)^2\,
\label{eq:7} ,
\end{eqnarray}
where $F = \frac{g^2}{16\pi}\left(
 \log(m_{GUT}/m_Z) K_{cb}K_{tb}/16 \pi^2 \right)^2
 \sim 6 \times 10^{-7}$,
$f_L = \sqrt2(\tan\theta_W\,N_{11}+3N_{12}) / 6$ and
the parameter $\delta_{m_0^2}$ is given by
%
\begin{equation}
\delta_{m_0^2}=\frac{M_Q^2+M_D^2+m_{H_1}^2+A_b^2}{m_{\tilde c_L}^2-m_{\tilde
t_L}^2}
\end{equation}
%
For the minimal SUGRA models one finds $\delta_{m_0^2}={\cal O}(1)$ which is
basically independent of the initial conditions due to the $m_0$
dependence both in the numerator and in the denominator. Finally we
have
\begin{equation}
\delta_A=\frac{m_t(A_b+\frac12A_t)}{m_{\tilde c_L}^2-m_{\tilde t_L}^2}
\end{equation}
%
The complete formulas are given in \cite{bartl:1996gz,diaz:1999ge},
while for the three--body decays they are given in the Appendix
of~\cite{restrepo:2001me}. They reduce to the ones given
in~\cite{porod:1997at,porod:1998yp} for vanishing R-parity violation
parameters.

We have fixed the parameters as in \cite{porod:1998yp} to avoid colour
breaking minima, while in the top squark sector we have used
$m_{\tilde t_1}$, $\cos \theta_{\tilde t}$, $\tan \beta$, and $\mu$ as
input parameters.  For the sbottom sector we have fixed $M_{\tilde Q},
M_{\tilde D}$ and $A_b$ as input parameters whereas for the charged
scalars we took $m_{P^0_2}$, $M_{\widetilde E}, M_{\tilde L}$, and
$A_\tau$ as input~\footnote{ $m_{P^0_2}$ plays here the same role as
  $m_{A^0}$ in the MSSM case \cite{porod:1998yp}.}.  In addition we
have chosen the R-parity violating parameters $\epsilon_3$ and $v_3$
in such a way that the heaviest neutrino mass is fixed with the help
of Eq.~YYY.
%
For simplicity, we have also assumed that the soft SUSY breaking
parameters are equal for all generations.

In order to get a feeling for minimal size of branching ratios that
can be measured let us first shortly discuss the expected size for the
direct production of light stops at future colliders.
%
One expects for example at the LHC a production cross section of $\sim
35$~pb for 220 GeV stop mass.  Therefore, once the full luminosity has
been reached, one has to expect approximately 3.5 $10^6$ events per
year.
%%
The corresponding stop production cross section at a future $e^+ e^-$
linear collider of $800$ c.m.s.~energy is of ${\cal O}(10-100 \mathrm{fb})$
\cite{bartl:1997wt}.  For an integrated luminosity of 500 fb$^{-1}$
per year one can expect ${\cal O}(10^4)$ stop pairs per year.  This implies
that branching ratio as low as $10^{-3}$ can in principle be measured.

We consider first the simplest case of one generation model which, as
already mentioned, is sufficient to describe the relative importance
of the ${\tilde t}_1 \to \tau^+ \, b $ decay mode relative to the
possible 3-body decay modes
\begin{eqnarray}
{\tilde t}_1 &\to& W^+ \, b \, {\tilde \chi}^0_1 \nonumber\\
{\tilde t}_1 &\to& S^+_k \, b \, {\tilde \chi}^0_1 \nonumber\\
{\tilde t}_1 &\to& S^+_k \, b \, \nu_3 \nonumber\\
\label{eq:8}
{\tilde t}_1 &\to& b \, S^0_i \, \tau^+ \,,  \\
{\tilde t}_1 &\to& b \, P^0_j \, \tau^+ \, \nonumber\\
{\tilde t}_1 &\to& b \, {\tilde l}^+_i \, \nu_l \,, \nonumber\\
{\tilde t}_1 &\to& b \, {\tilde \nu}_l \, l^+ \hspace{1cm} (l=e,\mu) \, .
\nonumber
\end{eqnarray}
In general the important final states are those that conserve
R-parity. For example, for the case of decays involving $S^+_k$ the
most important are those in which the scalars mainly a stau.  Due to
the fact that the existing bounds on the MSSM sneutrinos are below 100
GeV there exists the possibility that the sneutrino has nearly the
same mass as one of the Higgs boson.  Similarly it could be that the
charged MSSM boson has nearly the same mass as one of the staus. This
implies large mixing effects even for small R-parity breaking
parameters \cite{decampos:1995av,akeroyd:1998iq,porod:2000pw}.  We
therefore have used the complete formulas for the 3-body decay modes
which are presented in the Appendix. The latter include R-parity
violating decays such as $\tilde t_1 \to W^+ b \nu_3$. In addition to
the above mentioned decays there is also ${\tilde t}_1 \to b \, Z^0 \,
\tau^+$.  This decay mode is kinematically suppressed compared to
${\tilde t}_1 \to b \, \tau^+$ and there is no possible enhancement
due to a mixing with an R-parity conserving final state.  Therefore it
can be safely neglected.

In Fig.~\ref{fig:1} we show the branching ratios for the $\tilde t_1$
as a function of $\cos \theta_{\tilde t}$ in different scenarios.  In
order to calculate the partial width for the decay $\tilde t_1\to
c\tilde\chi_1^0$ we have taken the formula given in ref.
\cite{hikasa:1987db}. According to the analysis performed
in~\cite{diaz:1999ge}, where the full calculation was done in the
mSUGRA scenario, the result obtained with the present approximation
should be taken as one upper bound. This implies also that the shown
branching ratio for $\tilde t_1 \to b \tau^+$ can be viewed as a lower
bound. 
%%
The parameters and physical quantities used in Fig.~\ref{fig:1} are
given in Tab.~\ref{tab:1}. For the case of Fig.~\ref{fig:1}(a) we have
fixed in addition the R-parity violating parameters such that $m_{\nu}
\approx 1$~eV. With this choice of parameters $S^0_1$, $S^0_3$,
$P^0_3$, and $S^-_4$ are mainly the MSSM Higgs-bosons whereas $S^0_2$,
$P^0_2$, $S^-_2$, and $S^-_3$ are mainly the MSSM sleptons of the
third generation.
%
In the plot we show the various branching ratios of the lighter stop
summing up those branching ratios for the decays into sleptons that
give the same final state, for example:
%
\begin{equation}
{\tilde t}_1 \to b \, \nu_e \, {\tilde e}^+_L \,
          \to \, b \, e^+ \, \nu_e \, {\tilde \chi}^0_1 \;, \hspace{5mm}
{\tilde t}_1 \to b \, e^+ \, {\tilde \nu}_e \,
          \to \, b \, e^+ \, \nu_e \, {\tilde \chi}^0_1 \, .
\end{equation}
The branching ratios for decays into $\tilde{\mu}_{L}$ or
$\tilde{\nu}_{\mu}$ are practically the same as those into
$\tilde{e}_{L}$ or $\tilde{\nu}_{e}$.  Note that the energy spectrum
of the leptons in the final will be somewhat different depending on
whether the scalar in the intermediate step is charged or neutral.
This offers in principle the possibility of determining the branching
ratios of the different decay chains even if the final state topology
is common.  Note, that states containing scalars or neutralinos will
lead to additional jet and/or lepton multiplicities absent in the
MSSM.

\begin{figure}[htp]
 \setlength{\unitlength}{1mm}
 \begin{picture}(142,160)
 \put(5,175){\mbox{{\bf a)}}}
 \put(0,90){\includegraphics[height=8.5cm,width=8.5cm]{full1ev.eps}}
 \put(75,175){\mbox{{\bf b)}}}
 \put(70,90){\includegraphics[height=8.5cm,width=8.5cm]{partial1ev.eps}}
 \put(5,85){\mbox{{\bf c)}}}
 \put(0,0){\includegraphics[height=8.5cm,width=8.5cm]{partialatm.eps}}
 \put(75,85){\mbox{{\bf d)}}}
 \put(70,0){\includegraphics[height=8.5cm,width=8.5cm]{partialatmtb3.eps}}
\end{picture}
    \caption{Branching ratios for the $\tilde t_1$ as a function of 
     $\cos \theta_{\tilde t}$ for different scenarios. We have fixed in
    a) $m_{\nu_3}=1\,$eV,
    b) $m_{\nu_3}=1\,$eV, $M_{\tilde
        E}>225\,$GeV, $M_{\tilde L}>225\,$GeV,
    c) $m_{\nu_3}=0.06\,$eV, $M_{\tilde E}>225\,$GeV,
      $M_{\tilde L}>225\,$GeV,
    d) Branching ratios for the $\tilde t_1$ as a function of $\cos
      \theta_{\tilde t}$ for $\tan \beta = 3$. $m_{\nu_3}=0.06\,$eV,
      $M_{\tilde E}>225\,$GeV, $M_{\tilde L}>225\,$GeV.
All the other inputs are given in Table~\protect{\ref{tab:1}}.
 }
\label{fig:1}
\end{figure}

In Fig.~\ref{fig:1}(b) the slepton mass parameters are chosen such
that decays into scalars are kinematically forbidden.  Here we display
the channels ${\tilde t}_1 \to b \, W^+ {\tilde \chi}^0_1$, ${\tilde
  t}_1 \to b\, \tau^+$ and ${\tilde t}_1 \to c \, {\tilde \chi}^0_1$.
The remaining modes, such as ${\tilde t}_1 \to b \, S^0_1 \, \tau^+$,
turn out to be completely negligible.  In both cases, with and without
sleptons in the final state, one can see that in general the three
body mode ${\tilde t}_1 \to b \, W^+ {\tilde \chi}^0_1$, dominates
except for a somewhat narrow range of negative $\cos \theta_{\tilde
  t}$.  However, the branching ratio for $\tilde t_1\to b\,\tau^+$ is
above 0.1\% for most values of $|\cos \theta_{\tilde t}|$ implying the
observability of this mode.  Most importantly, note that even in the
parameter ranges where the three-body decay mode is dominant, its
resulting signature is rather different from that of the MSSM due to
the fact the lightest neutralino decays into SM-fermions, leading to
enhanced jet and/or lepton multiplicities, as discussed in detail in
\cite{bartl:2000yh,porod:2000hv}. In the remaining part of this section
we assume that 3-body decays into scalars are kinematically forbidden.

\begin{table}
  \begin{center}
    \begin{tabular}{|l|lll|}\hline
      Input: & $\tan \beta = 6$ & $\mu = 500$ GeV & $M = 250$ GeV \\
      & $M_{\tilde D}=370$ GeV & $M_{\tilde Q}=340$ GeV & $A_b=150$ GeV \\
      & $M_{\tilde E}=210$ GeV & $M_{\tilde L}= 210$ GeV & $A_\tau=150$ GeV \\ 
      & $m_{{\tilde t}_1}=220$ GeV & $\cos \theta_{\tilde t}=-0.8$ &
      $m_{P^0_3}=300$ GeV \\ \hline
      Calculated & $m_{{\tilde \chi}^0_1}=122$ GeV &
      $m_{{\tilde \chi}^+_1}=234$ GeV & $m_{{\tilde \chi}^+_2}=519$ GeV \\
      &  $m_{{\tilde b}_1}=334$ GeV & $m_{{\tilde b}_2}=381$ GeV & 
      $\cos \theta_{\tilde b}=0.879$ \\
      & $m_{S^0_1}=107$ GeV & $m_{S^0_2}=200$ GeV & $m_{S^0_3}=302$ GeV \\
      &  $m_{P^0_2}=200$ GeV & $m_{P^0_3}=300$ GeV  & \\
      & $m_{S^-_2}=203$ GeV & $m_{S^-_3}=226$ GeV & $m_{S^-_4}=311$ GeV  \\
      & $m_{{\tilde e}_L}=215$ GeV &
      $m_{{\tilde \nu}_e}=m_{{\tilde \nu}_\mu }=200$ GeV  & \\ \hline
    \end{tabular}
  \end{center}
  \caption[]{Input parameters and resulting quantities used in 
Fig.~\protect{~\ref{fig:1}}.}
\label{tab:1}
\end{table}


In Fig.~\ref{fig:1}(c) the R-parity violating parameters are fixed in
such a way that the heaviest neutrino mass is in the range suggested
by the oscillation interpretation of the atmospheric neutrino anomaly
\cite{gonzalez-garcia:2000sq}.  

\begin{figure}[htp]
%  \begin{minipage}[t]{7.7cm}
 \begin{center}
  \includegraphics[height=8.0cm,width=8.5cm]{fig3rpv.eps}
 \end{center}
    \caption{\small Branching ratios for ${\tilde t}_1$ decays
  for $m_{{\tilde t}_1} = 220$~GeV, $\mu = 500$~GeV, $M =
  240$~GeV, and $m_\nu = 100$, 1 and $0.06\,$eV.
  The branching ratios are shown as a function of
  $\tan \beta$. ($\cos\theta_{\tilde t}=-0.8$)}
   \label{fig:2}
\end{figure}


In Fig.~\ref{fig:1}(d) we show the same scenario as in
Fig.~\ref{fig:1}(c) but for $\tan \beta = 3$. The branching ratio into
$b \tau$ now increases, whereas the branching ratio into $c \tilde
\chi^0_1$ decreases.  This is easily understood by inspecting
Eqs.~(\ref{eq:6}) and (\ref{eq:7}). Indeed for the $b \, \tau$ case
the partial width is proportional to $h_b^2$, whereas for
$c\tilde\chi_1^0$ it is proportional to $h_b^4$.  This implies that
the partial width for $\tilde t_1\to c\tilde\chi_1^0$ grows faster
with $\tan \beta$ than the width for $\tilde t_1\to b \, \tau$.
%
This is also demonstrated in Fig.~\ref{fig:2} where we show the $\tan
\beta$ dependence of the branching ratio for the decay of $\tilde t_1$
into $b\tau^+$ for several values of the neutrino mass. For
$m_{\nu_3}=0.06\,$eV the $B(\tilde t_1\to b\,\tau)$ is still above
$0.1\%$ if $\tan\beta$ is not too large, as favored by the explanation
of the neutrino anomalies in this model~\cite{romao:1999up}. As seen
from the figure, the $\tilde t_1 \to b\tau^+$ branching ratio is
also somewhat correlated to the $\nu_3$ mass.  Should one add a
sterile neutrino to the model~\cite{hirsch:2000xe}, then the neutrino
state $\nu_3$ could in principle be heavier than assumed above,
favoring ${\tilde t}_1 \to \tau^+ \, b $ decay mode.

Let us now turn to the general three neutrinos case. There are new
features that arise in this case, as opposed to the 1-generation case
considered so far.  In this model the solution to the present neutrino
anomalies implies that all the $\epsilon_i$ are of the same order of
magnitude~\cite{romao:1999up}.  

Two further important results of \cite{romao:1999up} are that the
atmospheric neutrino angle is controlled by the ratio $(\epsilon_2 v_d
+\mu v_2)/(\epsilon_3 v_d +\mu v_3)$ and that the solar mixing angle
is controlled by $(\epsilon_1/\epsilon_2)^2$.  One can get approximate
formulas for the decay widths ${\tilde t}_1 \to b \, e^+$ and ${\tilde
  t}_1 \to b \, \mu^+$ similar to Eq.~(\ref{eq:6}) by replacing
$\epsilon_3$ by $\epsilon_{1,2}$. This implies that {\it (i)} The
decays into $ b \, e^+$ and $ b \, \mu^+$ are as important as the
decay into $ b \, \tau^+$.  {\it (ii)} The decays ${\tilde t}_1 \to b
\, e^+$ and ${\tilde t}_1 \to b \, \mu^+$ are related with the solar
mixing angle.  Moreover, we find that $\sum_{l=e,\mu,\tau}
\Gamma(\tilde t_1 \to b \, l^+)$ in the 3-generation model is nearly
equal to $\Gamma(\tilde t_1 \to b \, \tau^+)$ in the 1-generation
model provided that $\sum_{i=1}^3 \epsilon^2_i$ is identified to
$\epsilon^2$ in the 1-generation model.

In Fig.~\ref{fig:3} we show the ratio of B$({\tilde t}_1 \to b \,
e^+)/$B$({\tilde t}_1 \to b \, \mu^+)$ versus
$(\epsilon_1/\epsilon_2)^2$ for different values of $\cos
\theta_{\tilde t}$. For definiteness we have fixed the heaviest
neutrino mass at the best-fit value indicated by the atmospheric
neutrino anomaly.
%
One can see that the dependence is nearly linear even for rather small
$\cos \theta_{\tilde t}$.  For $|\cos \theta_{\tilde t}| \lesssim
10^{-2}$ the approximation in Eq.~(\ref{eq:6}) breaks down and
additional pieces dependent on $\sin \theta_{\tilde t}$
\cite{bartl:1996gz,diaz:1999ge} become important, leading to the non-linear
dependence.  One sees from the figure that, as long as $\cos
\theta_{\tilde t} \gtrsim 10^{-2}$ there is a good degree of
correlation between the branching ratios into $B(\tilde t_1 \to
b\,e^+)$ and $B(\tilde t_1 \to b\,\mu^+)$ and the ratio
$(\epsilon_1/\epsilon_2)^2$. Thus by measuring these branchings one
will get information on the solar neutrino mixing, since $\tan^2
\theta_{sol}$ is proportional to
$(\epsilon_1/\epsilon_2)^2$~\cite{romao:1999up} which makes it a
rather important quantity.
%%
For the so--called small mixing angle or SMA solution of the solar
neutrino problem we expect ${\tilde t}_1 \to e^+ \, b$ to be
negligible. In contrast, for the large mixing angle type solutions
(LMA, LOW and QVAC, see ref.~\cite{gonzalez-garcia:2000sq} and
references therein) we expect \texttt{all} ${\tilde t}_1 \to l^+ \, b$
decays to have comparable rates.
%%
As a result in this model one can directly test the solution to the
solar neutrino problem against the lighter stop decay pattern. 
%
This is also complementary to the case of neutralino decays considered
in \cite{porod:2000hv}. In that case the sensitivity is mainly to
atmospheric mixing, as opposed to solar mixing. Testing the latter in
neutralino decays at a collider experiment requires more detailed
information on the complete spectrum to test the solar angle
\cite{porod:2000hv}. In contrast we have obtained here a rather neat
connection of stop decays with the solar neutrino physics.

%
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(70,70)
\put(-5,-5){\includegraphics[height=7.0cm,width=7.cm]{nStopEps12.eps}}
\put(0,63){\makebox(0,0)[bl]{{\small $\mbox{B}({\tilde t}_1 \to b \, e^+)/
                                    \mbox{B}({\tilde t}_1 \to b \, \mu^+) $}}}
\put(68,-8){\makebox(0,0)[br]{{\small $(\epsilon_1 / \epsilon_2)^2$}}}
\put(42,50){{\small $\geq 0.1$}}
\put(49,38){{\small $ 10^{-2}$}}
\put(52,41){\vector(0,1){6}}
\put(8,18){{\small $10^{-2}$}}
\put(52,28){{\small $10^{-3}$}}
\put(15,25){{\small $10^{-3}$}}
\put(12,7){{\small $\geq 0.1$}}
\end{picture}
\end{center}
\caption[]{Ratio of branching ratios: $\mbox{B}({\tilde t}_1 \to b e^+)/
  \mbox{B}({\tilde t}_1 \to b \mu^+) $ as a function of $(\epsilon_1 /
  \epsilon_2)^2$ for $m_{{\tilde t}_1} = 220$~GeV, $\mu = 500$~GeV, $M
  = 240$~GeV; $|\cos \theta_{\tilde t}| \geq 0.1, 0.01, 10^{-3}$,
  $m_{\nu_3} = 0.6$~eV.}
\label{fig:3}
\end{figure}
Note, that this result is much more general than the scenarios
discussed in this paper. It is of particular importance in scenarios
where only the R-parity violating decays and the decay into $\tilde
\chi^0_1 \, c$ are present \cite{bartl:1996gz,diaz:1999ge}. Similarly,
the other ratios of the final states $b \, l^+$ are proportional to
the square of the ratio of corresponding $\epsilon_i$ provided that
$\cos \theta_{\tilde t}$ is not too small.


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

We have studied the phenomenology of the lightest stop in scenarios
where R-parity violating decays such as ${\tilde t}_1 \to b \, \tau^+$
compete with three--body decays. We have found that for $m_{{\tilde
    t}_1} \lesssim 250$~GeV there are regions of parameter where
${\tilde t}_1 \to b \, \tau^+$ is an important decay mode if not the
most important one. This implies that there exists the possibility of
full stop mass reconstruction from $\tau^+ \, \tau^- \, b \, \bar{b}$
final states, favoring the prospects for its discovery. In contrast, in
the MSSM the discovery of the lightest stop might not be possible at
the LHC within this mass range.
%%
This implies that it is important to take into account this new decay
mode when designing the stop search strategies at a future $e^+ e^-$
Linear Collider.  Spontaneously and bilinearly broken R-parity
violation also imply additional leptons and/or jets in stop cascade
decays.  Looking at the three generation model the decays into
${\tilde t}_1 \to b \, l^+$ imply the possibility of probing
$\epsilon^2_1 /\epsilon^2_2$ and thus the solar mixing angle.
%%
This complements information which can be obtained using neutralino
decays. In the latter case the sensitivity is mainly to the
atmospheric mixing, as opposed to solar mixing. In this model
neutralino decays is ideal to test the atmospheric anomaly at a
collider experiment, while stop decays provide neat complementary
information on the solar mixing angle.
%%
Obtaining solar mixing information from neutralino decays would
require more detailed knowledge on the supersymmetric spectrum, since
it would be involved in the relevant loop calculations of the solar
neutrino mass scale and mixing angle.
%%
By combining the two one can probe the parameters associated with both
solar and atmospheric neutrino anomalies at collider experiments.




\begin{boxedverbatim}
gluino and Drell-Yan Charginos \& neutralino Production at Large
  Hadron Collider.

Charginos \& neutralino Production at Next Linear  Collider
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