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%********************************************
\newcommand {\stbt} {\tilde{t}_1 \to b\,\tau }
\newcommand {\stch} [1] {\tilde{t}_1 \to c\,{\tilde{\chi}^0_{#1}} }
\def\rp{R-parity }
\def\nt{\hbox{$\nu_\tau$ }}
\newcommand{\chiz} [1] {\tilde{\chi}^{0}_{#1} }
\begin{document}
\baselineskip=1.5\baselineskip
\begin{titlepage}
\begin{flushright}
hep-ph/yymmnnn\\ FTUV/99-25\\ IFIC/99-27\\ FSU-HEP-990712\\ Version of \today
\end{flushright}
\vspace*{5mm}
\begin{center} 
{\Large \bf Two-Body Decays of the Lightest Stop
in Supergravity with and without R-Parity }\\[15mm]
{\large Marco A. D\'\i az${}^1$, D. A. Restrepo${}^2$, and J. W. F. 
Valle${}^2$}\\
\hspace{3cm}\\
{\small ${}^1$Department of Physics, Florida State University,}\\ 
{\small Tallahassee, Florida 32306, USA}
\vskip.5cm
{\small ${}^2$Departamento de F\'\i sica Te\'orica, IFIC-CSIC, 
Universidad de Valencia}\\ 
{\small Burjassot, Valencia 46100, Spain}
\hspace{3cm}\\
\end{center}
\vspace{5mm}
\begin{abstract}

We study the decays of the lightest stop quark in supergravity 
models with and without R-parity. Using the simplest model with an
effective explicit bilinear breaking of R-parity and radiative
electroweak symmetry breaking we show that, below the threshold for
decays into charginos, the lightest stop decays mainly into third
generation fermions, $\stbt$ instead of the R-parity conserving mode
$\stch{1}$, even for tiny tau neutrino mass values.  Moreover we show
that, even above the threshold for decays into charginos, the decay
$\stbt$ may be dominant. We study the r\^ole played by the
universality of the boundary conditions on the soft supersymmetry
breaking terms.  This new decay mode $\stbt$ as well as the cascades
originated by the conventional $\stch{1}$ decay followed by the \rp
violating neutralino decays can provide new signatures for stop
production at LEP and the Tevatron.

\end{abstract}

\end{titlepage}

\setcounter{page}{1}

\section{Introduction}

The {\sl Minimal Supersymmetric Standard Model (MSSM)} \cite{mssm} or
its minimal supergravity (SUGRA) version~\cite{msugra} are by far the
most well studied realizations of supersymmetry. However, neither
gauge invariance nor supersymmetry requires the conservation of
R-parity. Indeed, there is considerable theoretical and
phenomenological interest in studying possible implications of
alternative scenarios~\cite{beyond} in which R-parity is
broken~\cite{aul,expl0,rpold,arca}.  The violation of R-parity could
arise explicitly \cite{expl} as a residual effect of some larger
unified theory \cite{expl0}, or spontaneously, through nonzero vacuum
expectation values (vev's) for scalar neutrinos~\cite{aul,rpold,arca}.
In realistic spontaneous R-parity breaking models there is an \21
singlet sneutrino vev characterizing the scale of R-parity violation
\cite{MASIpot3,MASI,ROMA,ZR} which is expected to be the same as the
effective supersymmetry breaking scale.

There are two generic cases of spontaneous R-parity breaking models to
consider.
%
In the absence of any additional gauge symmetry, these models lead to
the existence of a physical massless Nambu-Goldstone boson, called
majoron (J) which is {\sl the lightest SUSY particle}, massless and
therefore stable. It plays an important r\^ole in making these models
fully consistent with astrophysics and cosmology~\cite{beyond}.  If
the majoron acquires a small mass due to explicit breaking effects at
the Planck scale then it may decay into neutrinos and photons, on very
large time scales of cosmological interest, playing a possible role as
unstable dark matter~\cite{KEV}. In this case LSP decays would not be
seen in laboratory experiments~\cite{KEV} and the conventional missing
momentum signal would be kept.
%
If lepton number is part of the gauge symmetry and \rp is
spontaneously broken then there is an additional gauge boson which
gets mass via the Higgs mechanism, and there is no physical Goldstone
boson \cite{ZR}.  As in the standard case in \rp breaking models the
lightest SUSY particle (LSP) is in general a neutralino. However, it
now decays mostly into visible states, therefore diluting the missing
momentum signal and bringing in increased multiplicity events which
arise mainly from three-body decays such as
%
\beq
\label{vis}
\chiz{1} \to  f \bar{f} \nu,
\eeq
where f denotes a charged fermion. The neutralino also has the invisible
decay mode
\beq
\label{invi}
\chiz{1} \to  3 \nu.
\eeq
%
as well as
%
\beq
\label{invis}
\chiz{1} \to  \nu\,J,
\eeq
%
in the case the breaking of \rp is spontaneous \cite{MASIpot3,MASI}.
This last decay conserves R-parity since the majoron has a large R-odd
singlet sneutrino component.

Although most searches for supersymmetric particles have so far been
performed within the framework of the MSSM it has become clear that
one needs alternative studies.
%
In either of the above cases the supersymmetric (SUSY) particles need
not be produced only in pairs, and the lightest of them could decay.
The effects of \rp violation can be large enough to be experimentally
observable.

In this paper we focus on the decay modes of the lightest top squark
in supergravity models where supersymmetry is realized with R-parity
violation. In such models the lightest stop could even be the lightest
supersymmetric particle and be produced at LEP. Neither $e^+ e^-$
collider data~\cite{LEPSEARCH} nor $p\bar{p}$ data from the Tevatron
\cite{D0} preclude this possibility. This way we refine the work 
presented in ref.~\cite{stop1} first by improving the treatment of the
\rp conserving decays and also by discussing the \rp violating ones
both in the context of supergravity-type models. In contrast with
ref.~\cite{stop1} we explicitly consider the case of light tau
neutrino masses, motivated by the simplest oscillation interpretation
of the Super-Kamiokande atmospheric neutrino data.

For definiteness and simplicity we focus on supersymmetric models
where the breaking of R-parity is parametrized explicitly through a
bilinear superpotential term of the type $\ell H_u$~\cite{epsi}. The
stop can have new decay modes such as
%
\beq
\label{btau}
\stbt
\eeq
due to mixing between charged leptons and charginos.  We show that
this decay may be dominant or at least comparable to the ordinary
R-parity conserving mode 
%
\beq
\label{chic}
\stch{1},
\eeq
where $\chiz{1}$ denotes the lightest neutralino.

The paper is organized as follows. The model and an analytical
analysis of the tree--level tau neutrino in terms of SUGRA parameters
is described in section~\ref{section2}. The mass matrices are given in
section~\ref{section3} while in section 4 we present the top squark
decay widths in the minimal supergravity model with universal soft
SUSY breaking terms~\cite{martin}, MSUGRA, for short. The relevant
Feynman rules and the squark decay widths and branching ratios are
calculated in appendix~\ref{appendixb}. They are studied numerically
in section 5 and we present our conclusions in section 6.

\section{The Model}
\label{section2}

The supersymmetric Lagrangian is specified by the superpotential $W$
given by
% 
\begin{equation}  
W=\varepsilon_{ab}\left[ 
 h_U^{ij}\widehat Q_i^a\widehat U_j\widehat H_2^b 
+h_D^{ij}\widehat Q_i^b\widehat D_j\widehat H_1^a 
+h_E^{ij}\widehat L_i^b\widehat R_j\widehat H_1^a 
-\mu\widehat H_1^a\widehat H_2^b 
\right] + \varepsilon_{ab}\epsilon_i\widehat L_i^a\widehat H_2^b\,,
\label{eq:Wsuppot} 
\end{equation}
%
where $i,j=1,2,3$ are generation indices, $a,b=1,2$ are $SU(2)$
indices, and $\varepsilon$ is a completely antisymmetric $2\times2$
matrix, with $\varepsilon_{12}=1$. The symbol ``hat'' over each letter
indicates a superfield, with $\widehat Q_i$, $\widehat L_i$, $\widehat
H_1$, and $\widehat H_2$ being $SU(2)$ doublets with hypercharges
$\third$, $-1$, $-1$, and $1$ respectively, and $\widehat U$,
$\widehat D$, and $\widehat R$ being $SU(2)$ singlets with
hypercharges $-{\textstyle{4\over 3}}$, ${\textstyle{2\over 3}}$, and
$2$ respectively. The couplings $h_U$, $h_D$ and $h_E$ are $3\times 3$
Yukawa matrices, and $\mu$ and $\epsilon_i$ are parameters with units
of mass.
 
Supersymmetry breaking is parametrized by the standard set of soft
supersymmetry breaking terms 
%
\begin{eqnarray} 
V_{soft}&=& 
M_Q^{ij2}\widetilde Q^{a*}_i\widetilde Q^a_j+M_U^{ij2} 
\widetilde U^*_i\widetilde U_j+M_D^{ij2}\widetilde D^*_i 
\widetilde D_j+M_L^{ij2}\widetilde L^{a*}_i\widetilde L^a_j+ 
M_R^{ij2}\widetilde R^*_i\widetilde R_j \nonumber\\ 
&&\!\!\!\!+m_{H_1}^2 H^{a*}_1 H^a_1+m_{H_2}^2 H^{a*}_2 H^a_2\nn\\
&&\!\!\!\!- \left[\half M_3\lambda_3\lambda_3+\half M\lambda_2\lambda_2 
+\half M'\lambda_1\lambda_1+h.c.\right] 
\nn\\ 
&&\!\!\!\!+\varepsilon_{ab}\left[ 
A_U^{ij}h_U^{ij}\widetilde Q_i^a\widetilde U_j H_2^b 
+A_D^{ij}h_D^{ij}\widetilde Q_i^b\widetilde D_j H_1^a 
+A_E^{ij}h_E^{ij}\widetilde L_i^b\widetilde R_j H_1^a\right.
\nn\\ 
&&\!\!\!\!\left.-B\mu H_1^a H_2^b+B_i\epsilon_i\widetilde L_i^a H_2^b\right] 
\,,\label{eq:Vsoft}
\end{eqnarray} 

For definiteness and simplicity we assume only R-parity Violation
(RPV) in the third generation, but we do allow for Flavour Changing
Neutral Currents (FCNC) effects such as the process $\s t_1\to
c\,\s\chi_1^0$ involving the three generations of quarks.

Note that the bilinear term $\epsilon_3$ can not be rotated away,
since the rotation that eliminates it reintroduces an R--Parity
violating trilinear term, as well as a sneutrino vacuum expectation
value. 

In order to study the R--Parity violating decay mode $\s t_1\to b\,
\tau$ it is sufficient to consider the superpotential
\cite{moreBRpV,otros,javi}
%
\begin{equation} 
W=h_t\widehat Q_3\widehat U_3\widehat H_2
 +h_b\widehat Q_3\widehat D_3\widehat H_1
 +h_{\tau}\widehat L_3\widehat R_3\widehat H_1
 -\mu\widehat H_1\widehat H_2
 +\epsilon_3\widehat L_3\widehat H_2
\label{eq:Wbil}
\end{equation}
%
This amounts to neglecting the effects of RPV on the two first
families.

Such model can be described in various equivalent bases, for example
%
\begin{enumerate}
\item
one in which bilinear term and sneutrino vev are non-zero, $\epsilon_3^I
\neq 0$ and $v_3^I \neq 0$~\cite{beyond}
\item
one in which trilinear and sneutrino vev are non-zero, $\lambda_3^{II} \neq
0$ and $v^{II}_3 \neq 0$~\cite{Borzumati:1999th}
\item
the vev-less basis in which in which $\epsilon_3^{III}$ and
$\lambda_3^{III}$ are non-zero but
$v_3^{III}=0$~\cite{Borzumati:1999th,Bisset:1999nw}
\end{enumerate}
%
Note that the R-parity violating parameters can be expressed in terms
of dimension-less basis-independent alignment parameters $ \sin \xi$,
$
\sin \xi'$ and $ \sin \xi''$~\cite{javi,sacha} defined by ($X=I,II$ or $III$)
%
\begin{eqnarray}
\label{supxi}
\sin\xi&=&\frac{\epsilon_3^Xv_d^X+\mu^X v_3^X}{\mu'{v_d'}}=
\frac{v_3^{II}}{{v_d'}}=\frac{\epsilon_3^{III}}{\mu'}\\
\label{supxip}
\sin\xi'&=&\frac{\mu^X \lambda_3^X+\epsilon_3^Xh_b^X}{\mu'h_b'}=
\frac{\lambda_3^{II}}{h_b'}=\frac{\epsilon_3^I}{\mu'}\\
\label{supxipp}
\sin\xi''&=&\frac{-v_d^X\lambda_3^X+v_3^Xh_b^X}{{v_d'}h_b'}=
\frac{v_3^I}{{v_d'}}=-\frac{\lambda_3^{III}}{h_b'}
\end{eqnarray}
%
of which only two are independent because they satisfy
%
\begin{equation}
\sin\xi''=\cos\xi'\sin\xi-\sin\xi'\cos\xi
\label{sinxippp}
\end{equation}
%
and where
%
\begin{equation} 
h_b'=\sqrt{{h_b^X}^2+{\lambda_3^X}^2}\qquad
\mu'=\sqrt{{\mu^X}^2+{\epsilon_3^X}^2}\qquad
{v_d'}=\sqrt{{v_d^X}^2+{v_3^X}^2}, \quad X=I,II,\hbox{ or }III
\label{hbmuvd}
\end{equation}
%
In the limit when the \rp violating parameters vanish one recovers the
MSSM. From now on we will work in the $\lambda_3^I=0$--basis, unless
otherwise stated.  As a result we will omit the label $I$ in all the
parameters associated with this basis. We also will drop out the prime
in $h_b$. One of the advantages in working in this basis is that the
RGE's evolution does not induce the trilinear \rp violating terms
neither in the superpotential nor in the scalar potential if at the
beginning we impose universality between the parameters $B$ and
$B_3$~\cite{javi}.

It is convenient to introduce the following notation in spherical
coordinates for the vacuum expectation values (vev):
%
\begin{eqnarray} 
v_d&=&v\sin\theta\cos\beta\cr 
v_u&=&v\sin\theta\sin\beta\cr 
v_3&=&v\cos\theta 
\label{eq:vevs} 
\end{eqnarray} 
% 
which preserves the MSSM definition $\tan\beta=v_u/v_d$. In the MSSM
limit, where $\epsilon_3=v_3=0$, the angle $\theta$ is equal to
$\pi/2$. This make sense in the $\lambda_3^I=0$--basis where the usual
MSSM relation
%
\begin{equation}
m_b=%&=&\frac1{\sqrt2}\left(h_bv_d+\lambda_3v_3\right)\\%&=&
\frac1{\sqrt2}h_bv_d\\
%&=&\frac1{\sqrt2}h_b'{v_d'}\cos\xi''
\label{mb}
\end{equation}
%
holds. In this model the presence of RPV induces a mass for the tau
neutrino at the tree level~\cite{rpold,arca}, as well as radiative
masses to the the \ne and \nm. As already mentioned it is sufficient
for our present discussion of stop decays to keep only the tau
neutrino.  In order to study the \nt mass it is convenient to have an
analytical expression for \mnt in this limit. The tree level tau
neutrino mass may be expressed
as~\cite{expl,expl0,javi}~--~\cite{anomaly}
%
\begin{equation}
m_{\nu_{\tau}} \approx 
-\frac{(g^2M'+g'^2M){\mu'}^2}{
4 M M'{\mu'}^2-2(g^2M'+g'^2M){\mu'}v_u{v_d'} \cos\xi}{v_d'}^2\sin^2\xi
\label{mNeutrinoApp}
\end{equation}
%
in terms of basis-independent parameters $\mu'$, $v_d'$ and $\sin \xi$
defined in Eqs~(\ref{hbmuvd}) and (\ref{supxi}).
%
The second term in the denominator may be neglected if $M,\mu\gsim
m_Z$, as often happens in minimal supergravity models with universal
soft SUSY breaking terms~\cite{martin}.  Thus one may obtain an
estimate of the neutrino mass by keeping only the first term in the
denominator.
%
\begin{equation}
m_{\nu_{\tau}}\approx\frac{g^2}{2M}{v_d'}^2\sin^2\xi
\label{mNeutrinoAppp}
\end{equation}
where we have used $M'=Mg'^2/g^2$.  For $\sin \xi \approx 1$ one can
easily check that \mnt could be as large as the experimental upper
bound of 18 Mev~\cite{ubmnt}. However in MSUGRA models one may obtain
naturally small $\sin \xi $ values, calculable from the RGE evolution
from the unification scale down to the weak scale, because from the
minimization equations it can be written in terms $\Delta
m^2=m^2_{H_1}-m^2_{L_3}$ and $\Delta B=B_3-B$~\cite{rptalks} as
%
\begin{equation}
\sin\xi=-\cos\xi'\sin\xi'
\left(\cos\xi\frac{\Delta m^2}{{m'}_{\tilde\nu^0_{\tau}}^2}+
\frac{v_2}{v_d'}\frac{\mu'\Delta B}{{m'}_{\tilde\nu^0_{\tau}}^2}\right)
\label{sinxi}
\end{equation}
%
One may give a simplified approximate analytical expression for the
tau neutrino mass in this model by solving the renormalization group
equations for the soft mass parameters $m^2_{H_1}$, $m^2_{L_3}$, $B$,
and $B_3$ in one-step approximation.  This gives the results~\cite{rptalks}
\begin{eqnarray}
\sin\xi\left|_{\Delta m^2}\right.&\approx&-\cos\xi'\sin\xi'\cos\xi
h_b^2\left[\frac{m_{H_1}^2+M_Q^2+M_D^2+A_D^2}
{{m'}_{\tilde\nu^0_{\tau}}^2}\right]\left(\frac3{8\pi^2}
\ln\frac{M_{U}}{m_t}\right)\nn\\
&\sim&-\cos\xi'\sin\xi'\cos\xi
h_b^2\left(\frac3{8\pi^2}\ln\frac{M_{U}}{m_t}\right)
\label{sinxidm}
\end{eqnarray}
and
\begin{equation}
\sin\xi\left|_{\Delta B}\right.\approx\cos\xi'\sin\xi'
\tan\beta'
h_b^2\left[\frac{\mu'A_D}{{m'}_{\tilde\nu^0_{\tau}}^2}\right]
\left(\frac3{8\pi^2}\ln\frac{M_{U}}{m_t}\right)
\label{sinxidb}
\end{equation} 
where $\sin\xi|_{\Delta m^2}$ and $\sin\xi|_{\Delta B}$ are the two terms
contributing to $\sin\xi$ in eq.~(\ref{sinxi}). With them, we find
%
\begin{equation}
m_{\nu_{\tau}}\left|_{\mathrm{min}}\right.\sim
\frac{g^2m_b^2}{M}\left(\sin^2\xi'
h_b^2\right)\left(\frac3{8\pi^2}
\ln\frac{M_{U}}{m_t}\right)^2
\label{mNeutrinods}
\end{equation}

The minimum value for $\sin \xi' h_b$ is determined by the value $\sin
\xi'$ and that of $\tan \beta$. For $\sin \xi' \sim 1$ and relatively small 
$\tan \beta$ so that $h_t$ is perturbative, one has
%
\begin{equation}
m_{\nu_{\tau}}\gsim 10 \hbox{KeV}
\label{1mev}
\end{equation}
%
for $M \sim 1$ TeV. In order to get smaller \nt masses one need to
suppress $\sin^2 \xi'$ additionally, for example to reach one
electron-volt the required R-parity violating parameters are given in
Table~\ref{rpvev}.  These order of magnitude estimates are given in
terms of the basis--independent angles $\xi$ and $\xi'$, and in the
three basis defined before. Note that whenever the parameter has two
values, the first correspond to $\tan\beta=2$ (the lower
perturbativity limit) and the second to $\tan\beta=35$. In
Table~\ref{rpvev}, $\sin\xi$ was estimated from
Eq.~(\ref{mNeutrinoAppp}) and $\sin\xi'$ from
Eq.~(\ref{mNeutrinods}). Note that the induced RGE suppression depends
basically in the $h_b^2$ factor in Eq.~(\ref{sinxidm}) that is
$10^{-3}$ ($1$) for small (large) $\tan\beta$ and therefore the bigger
the value of $\tan\beta$ the smaller $\sin\xi'$ will be. The RPV
parameters in the several bases were estimated from
Eqs.~(\ref{supxi}--\ref{supxipp}) and~(\ref{sinxippp}). As expected,
the constraint on the trilinear RPV term is stronger than the obtained
from the one-loop induced tau neutrino mass~\cite{Rakshit:1998kd}


\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
\multicolumn{4}{|c|}{basis--independent}&
\multicolumn{4}{|c|}{Basis I: $\lambda_3^I=0$}&
\multicolumn{2}{c|}{Basis II: $\epsilon_3^{II}=0$}&
\multicolumn{3}{c|}{Basis III: $v_3^{III}=0$ }\\ \hline
\multicolumn{2}{|c|}{$\sin\xi$}&
\multicolumn{2}{c|}{$\sin\xi'$}&
\multicolumn{2}{|c|}{$\!\!\epsilon_3(\hbox{GeV})\!\!$}&
\multicolumn{2}{c|}{$\!\!v_3(\hbox{GeV})\!\!$}&
$\lambda_3^{II}$&$\!\!v_3^{II}(\hbox{GeV})\!\!$&$\lambda_3^{III}$&
\multicolumn{2}{c|}{$\!\!\epsilon_3^{III}(\hbox{GeV})\!\!$}\\ \hline
$\!\!10^{-5}\!\!$&$\!\!10^{-4}\!\!$&$\!\!10^{-2}\!\!$&
$\!\!10^{-4}\!\!$&1&$\!\!10^{-2}\!\!$&1&$\!\!10^{-3}\!\!$&
$\!\!10^{-4}\!\!$&$\!\!10^{-3}\!\!$&$\!\!10^{-4}\!\!$&
$\!\!10^{-3}\!\!$&$\!\!10^{-2}\!\!$\\ \hline
\end{tabular}
\end{center}
\caption{\small
Estimated magnitude of R-parity violating parameters required for a
tau neutrino mass in the eV range, without requiring cancellation
in $\sin\xi$ in the three bases defined before. }
\label{rpvev}
\end{table}

In \eq{mNeutrinods} we have neglect $\Delta B$ contribution with
respect to the one coming from $\Delta m^2$. It is possible, however,
that the $\Delta B$ term may be sizeable. In this case then it may
cancel the $\Delta m^2$ contribution in $\sin\xi$, leading to an
additionally suppressed neutrino mass. As we will see, however, in
SUGRA models with universal soft terms at the unification scale
($\epsilon$SUGRA for short) we do not need any substantial
cancellation in order to obtain calculable \nt masses below the
electron-volt scale.

\section{Squark Mass Matrices}
\label{section3}

The up and down-type squark mass matrices of our model have already
been given previously in ref.~\cite{Diaz:1997xc}. Here we generalize
those to the three-generation case. The mass matrix of the up squark
sector follows from the quadratic terms in the scalar potential
% 
\begin{equation} 
V_{quadratic}=\mx{cc}
\s {\bold{u}}_L^\dagger&\s{\bold{ u}}_R^\dagger
\finmx 
{\s M_U^2}\mx{c}
\s{\bold{ u}}_L\\
\s{\bold{ u}}_R
\finmx+\cdots
\end{equation} 
given by 
\beq
{\s M_U^2}=\mx{cc}
{M_Q^2}+\frac12v_u^2{h_U}{h_U}^\dagger+\Delta_{UL}&
\frac1{\sqrt2}v_u{A_U^h}-\frac1{\sqrt2}(\mu v_d-\epsilon_3v_3)
{h_U}\\
\frac1{\sqrt2}v_u{A_U^h}^\dagger-\frac1{\sqrt2}(\mu v_d-\epsilon_3v_3)
{h_U}^\dagger &
{M_U^2}+\frac12v_u^2{h_U}^\dagger{h_U}+\Delta_{UR}
\finmx
\eeq
where $\Delta_{UL}=\frac18\big(g^2-\frac13{g'}^2\big)\big(v_d^2-v_u^2
+v_3^2\big)\bold{1}$ and $\Delta_{UR}=\frac16
{g'}^2(v_d^2-v_u^2+v_3^2)\bold{1}$ are the splitting in the squark
mass spectrum produced by electro-weak symmetry breaking, and
${A_U^h}_{ij}\equiv A_U^{ij}h_U^{ij}$. The eigenvalues of ${\s M_U^2}$
are
\begin{equation}
\textrm{diag}\,\{m_{\s u_1},m_{\s u_2},\ldots,m_{\s u_6}\}=\mx{cc}
\Gamma_{UL}& 
\Gamma_{UR}
\finmx
{\s M_U^2}
\mx{c}
\Gamma_{UL}^\dagger \\
\Gamma_{UR}^\dagger
\finmx
\end{equation}
This way the six weak-eigenstate fields $\s u_{iL}$ and $\s u_{iR}$
($i=1,2,3$) combine into six up-type mass eigenstate squarks $\s u_k$
as follows: $\s u_{iL}=\Gamma^{\dagger ik}_{UL}\s u_k=\Gamma^{*
ki}_{UL}\s u_k$, $\s u_{iR}=\Gamma^{\dagger ik}_{UR}\s u_k=\Gamma^{*
ki}_{UR}\s u_k$.

For completeness, we also give the mass matrix of the down squark
sector. The quadratic scalar potential includes
% 
\begin{equation} 
V_{quadratic}=\mx{cc}
\s{\bold{d}}_L^*&\s{\bold{d}}_R^*
\finmx 
{\s M_D^2}\mx{c}
\s{\bold{d}}_L\\
\s{\bold{d}}_R
\finmx+\cdots
\end{equation} 
given by 
\beq
{\s M_D^2}=\mx{cc}
{M_Q^2}+\frac12v_d^2{h_D}{h_D}^\dagger+\Delta_{DL}&
\frac1{\sqrt2}v_d{A_D^h}-\frac1{\sqrt2}\mu v_u
{h_D}\\
\frac1{\sqrt2}v_d{A_D^h}^\dagger -\frac1{\sqrt2}\mu v_u
{h_D}^\dagger  &
{M_D^2}+\frac12v_d^2{h_D}^\dagger{h_D}+\Delta_{DR}
\finmx
\eeq
where $\Delta_{DL}=-\frac18\big(g^2+\frac13{g'}^2\big)\big(v_d^2-v_u^2
+v_3^2\big)\bold{1}$, $\Delta_{DR}=-\frac1{12}
{g'}^2(v_d^2-v_u^2+v_3^2)\bold{1}$, and
${A_D^h}_{ij}\equiv A_D^{ij}h_D^{ij}$. The eigenvalues of ${\s M_D^2}$
are 
\begin{equation}
\textrm{diag}\,\{m_{\s d_1},m_{\s d_2},\ldots,m_{d_6}\}=\mx{cc}
\Gamma_{DL}& 
\Gamma_{DR}
\finmx
{\s M_D^2}
\mx{c}
\Gamma_{DL}^\dagger \\
\Gamma_{DR}^\dagger
\finmx
\end{equation}
One is left with six mass-eigenstate down squarks fields $\s d_{k}$
related to $\s d_{iL}$ and $\s d_{iR}$ fields as follows: $\s
d_{iL}=\Gamma^{\dagger ik}_{DL}\s d_k=\Gamma^{* ki}_{DL}\s d_k$, $\s
d_{iR}=\Gamma^{\dagger ik}_{DR}\s d_k=\Gamma^{* ki}_{UR}\s d_k$.

For the Higgs-slepton part of the quadratic scalar potential in the
one generation case of the Bilinear R-parity Violating (BRpV) model, see refs.~\cite{v3cha} and
\cite{epsi}.

Of particular interest to us is the chargino/tau mass matrix. For our
present purposes it is sufficient to have the form of this matrix for
one generation, which is given by
%
\begin{equation}
{\bf M_C}=\left[\matrix{
M & {\textstyle{1\over{\sqrt{2}}}}gv_2 & 0 \cr
{\textstyle{1\over{\sqrt{2}}}}gv_d & \mu & 
-{\textstyle{1\over{\sqrt{2}}}}h_{\tau}v_3 \cr
{\textstyle{1\over{\sqrt{2}}}}gv_3 & -\epsilon_3 &
{\textstyle{1\over{\sqrt{2}}}}h_{\tau}v_d}\right]
\label{eq:ChaM6x6}
\end{equation}
%
This form is common to all models with spontaneous breaking of
R-parity, as well as in the simplest truncation of these models
provided by the BRpV model.  We note that the chargino sector
decouples from the tau sector in the limit $\epsilon_3=v_3=0$.  As in
the MSSM, the chargino mass matrix is diagonalized by two rotation
matrices $\bf U$ and $\bf V$
%
\begin{equation}
{\bf U}^*{\bf M_C}{\bf V}^{-1}=\left[\matrix{
m_{\s\chi^{\pm}_1} & 0 & 0 \cr
0 & m_{\s\chi^{\pm}_2} & 0 \cr
0 & 0 & m_{\tau}}\right]\,.
\label{eq:ChaM3x3}
\end{equation}
%
The lightest eigenstate of this mass matrix must be the tau lepton
($\tau^{\pm}$) and so the mass is constrained to be
$1.77705^{+0.29}_{-0.26}$ GeV.  To obtain this the tau Yukawa coupling
becomes a function of the parameters in the mass matrix, and the full
expression is given in \cite{v3cha}. The composition of the tau is
given by
%
\begin{equation}
\tau^+_R=V_{3j}\psi_j^+, \qquad \tau^-_L=U_{3j}\psi_j^- 
\label{taucomp}
\end{equation}
%
where $\psi^{+T}=(-i\lambda^+,\widetilde H_2^1,\tau_R^{0+})$ and
$\psi^{-T}=(-i\lambda^-,\widetilde H_1^2,\tau_L^{0-})$. The two
component Weyl spinors $\tau_R^{0-}$ and $\tau_L^{0+}$ are weak
eigenstates and, similarly, the two component Weyl spinors $\tau^+_R$
and $\tau^-_L$ are mass eigenstates.  It follows easily from
eq.~(\ref{eq:ChaM3x3}) that the matrix ${\bf{M_CM_C^T}}$ is diagonalized
by ${\bf U}$ and the matrix ${\bf{M_C^TM_C}}$ is diagonalized by ${\bf V}$.

The soft SUSY breaking parameters at the electroweak scale needed for
the evaluation of the mass matrices and couplings are calculated by
solving the renormalization group equations (RGE's) of the MSSM and
imposing the radiative electroweak symmetry breaking condition. Taking
the quark masses, the CKM matrix and $\tan\beta$ as inputs, we first
solve one-loop RGE's for the gauge and Yukawa coupling constants to
calculate the values at the unification scale, where we assume
universal soft supersymmetry breaking boundary conditions. Then we
solve the RGE's for all MSSM parameters downward with universal initial
conditions for each set of soft SUSY breaking parameters.  We include
all generation mixing in the RGE's for Yukawa coupling constants as
well as soft SUSY breaking parameters. Next, we evaluate the Higgs
potential at the $m_t$ scale including the one-loop corrections
induced by the Yukawa coupling constants of the third generation. The
requirement of the radiative electroweak symmetry breaking fixes the
magnitude of the SUSY Higgs mass parameter $\mu$ and the soft SUSY
breaking parameters $B$ and $B_3$. At this stage we also choose
$\epsilon_3$ and $v_3$ in order to get a sufficiently light tau
neutrino.  At this point, all RPV parameters at the electroweak scale
are determined as functions of the input parameters $(\tan\beta,\,
m_0,\, A_0,m_{1/2}, \hbox{sign}(\mu), \epsilon_3)$.  Notice that due
to the third minimization condition one can solve for $v_3$ as a
function of $\epsilon_3$.  Iteration is required because $\mu$ and
$\epsilon_3$ are inputs to evaluate the loop corrected minimum.
Having determined all parameters at the electroweak scale, we obtain
the masses and the mixings of all the SUSY particles by diagonalizing
the corresponding mass matrices.

We scan the soft SUSY breaking parameter space in the range 
\begin{eqnarray}
&m_0&\le 700 \,\hbox{GeV}\nn \\
100<&m_{1/2}&\le 400 \,\hbox{GeV}\nn \\ 
&|A_0|&\le 1000 \,\hbox{GeV},\nn \\
&|\epsilon_3|&<200 \,\hbox{GeV}\nn \\
1.8<&\tan\beta&<60 \nn
\end{eqnarray}
the previous range on $\tan\beta$ guarantee that both $h_t$ and $h_b$
will be perturbative. For the CKM matrix, we use the convention of the
Particle Data Group, taking $V_{us}=0.2205$, $V_{cb}=0.041$,
$|V_{ub}/V_{cb}|=0.08$ and $\delta=0$.

The resulting region of lightest stop and chargino masses is displayed
in Fig.~\ref{par-sp}. Neglecting the three body decays, we find that
in Region I of the $m_{{\tilde t}_1}$--$m_{{\tilde\chi}_1^+}$ plane,
$BR(\s t_1\to c\,\s\chi_1^0)+ BR(\s t_1\to b\,\tau)\approx 1$. In
Region II $BR(\s t_1\to b\,\tau)+BR(\s t_1\to b\,\s\chi_i^+)\approx 1$
(i=1,2).  In Region III $BR(\s t_1\to b\,\tau)+BR(\s t_1\to
b+\s\chi_i^+)+BR(\s t_1\to t\,\nu_\tau)\approx 1$ (i=1,2), while in
region IV $BR(\s t_1\to b\,\tau)+BR(\s t_1\to b\,\s\chi_i^+)+BR(\s
t_1\to t\,\nu_\tau)+BR(\s t_1\to t\,\s\chi_j^0)
\approx 1$ $(j=1,\ldots,4)$. Note that in each region 
the exact equality to 1 is reached when the FCNC processes are fully
included.

\begin{figure}
\centerline{\protect\hbox{
\psfig{file=paramspace.eps,height=8cm}}}
\caption{\small 
Kinematical regions in the $m_{{\tilde t}_1}$--$m_{{\tilde\chi}_1^+}$
plane.  From left to right: Region I
$m_{\tilde t_1} < m_{{\tilde\chi}_1^+}+m_b$; 
Region II $m_{{\tilde\chi}_1^+}+m_b<m_{{\tilde t}_1}<m_t$; 
Region III $m_t<m_{{\tilde t}_1}<m_{{\tilde\chi}_1^0}+m_t$; 
and region IV $m_{{\tilde t}_1}>m_{{\tilde\chi}_1^0}+m_t$ }
\label{par-sp}
\end{figure}

In Appendix~\ref{appendixb} we give the Feynman rules for all vertices
involving squarks, quarks and charginos or neutralinos, as well as the
two--body squark decay widths, for squarks of all three generations.
These equations reduce to the expressions found in ref.~\cite{BMP}
provided one identifies $\Gamma_{UL}^{33}=\cos\theta_{\s t}$ and
$\Gamma_{UR}^{33}=\sin\theta_{\s t}$.  They also generalize the
results for the BRpV model to the three generation case. 

\section{Two-Body Decays of the Lightest Stop in MSUGRA}

In the region I of Fig.~\ref{par-sp} the main $\s t_1$ decay channel
is expected to be the loop--induced and flavour--changing $\s t \to
c\,\s\chi_1^0$~\cite{HikKob,baer,porod}. The FCNC processes in the
MSSM in general involve a very large number of input
parameters. Following common practice, we prefer to perform the
phenomenological study of flavour changing processes in the framework
of a supergravity theory with universal supersymmetry breaking.  The
simplest description of FCNC processes in \rp conserving minimal SUGRA
models uses the so-called one-step approximation.  Here we start by
reproducing the standard calculation for $\s t \to c\,\s\chi_1^0$ as
in~\cite{HikKob}. To do this consider only the effect of the third
generation Yukawa coupling. From our general Eq.~(\ref{tause}) we have
for $\s t_1= \s u_l$
%
\begin{equation}
\Gamma(\s t_1\to c\,\s\chi_1^0)
\approx \frac{g^2}{8\pi} \left(\Gamma_{UL13}\right)^2
\left[
\fracs23 \sin\theta_W N'_{11}+\left(\fracs12-\fracs23\sin^2\theta_W\right)
\frac{N'_{12}}{\cos\theta_W}\right]^2
m_{\s t_1}
\left(1-\frac{m_{\s \chi_1^0}^2}{m_{\s t_1}^2}\right)^2
\label{gtn2cos}
\end{equation}
with
%
\begin{equation}
\Gamma_{UL13} =
\frac{\Delta_L\cos\theta_{\s t}-\Delta_R\sin\theta_{\s t}}
{m_{\s c_L}^2-m_{\s t_L}^2}
\end{equation}
%
and by using the one--step approximation\footnote{In~\cite{HikKob} the
factor $\frac12 A_t$ is missing.}
%
\begin{eqnarray}
\Delta_L&=&(\s M_U^2)_{23}\approx (M_Q^2)_{23}\approx-
\frac{t_U}{16\pi^2}K_{cb}K_{tb}h_b^2(M_Q^2+M_D^2+m_{H_1}^2+A_b^2)\\
\Delta_R&=&(\s M_U^2)_{26}\approx (A_U^h)_{23}\approx-
\frac{t_U}{16\pi^2}K_{cb}K_{tb}h_b^2m_t(A_b+\frac12A_t)
\end{eqnarray}
%
with $t_U=\ln(M_G/m_t)$. So, in the one--step approximation we have
%
\begin{eqnarray}
\label{gtn2coshb}
\Gamma(\s t_1\to c\,\s\chi_1^0)&\approx&\frac{g^2}{16\pi}\left(
\frac{t_U}{16\pi^2}K_{cb}K_{tb}\right)^2 h_b^4(
\delta_{m_0^2}\cos\theta_{\s t}-\delta_A\sin\theta_{\s t})^2
\nn\\
&& \times \left[
\frac{\sqrt2}6(\tan\theta_W\,N_{11}+3N_{12})\right]^2
m_{\s t_1}\left(1-\frac{m_{\s \chi_1^0}^2}{m_{\s t_1}^2}\right)^2
\end{eqnarray}
%
where the prefactor is $\sim 6 \times 10^{-7}$ 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_{\s c_L}^2-m_{\s
t_L}^2}
\sim 1
\end{equation}
%
is basically independent of the initial conditions due to the $m_0$
dependence both in the numerator as in the denominator and
\begin{equation}
\delta_A=\frac{m_t(A_b+\frac12A_t)}{m_{\s c_L}^2-m_{\s t_L}^2}
\end{equation}
%
 
Note however, that the one-step approximation includes only the third
generation Yukawa couplings and neglects the running of the soft
breaking terms~\cite{HikKob,baer,porod,Boehm:1999tr}.  Such an
approximation is rather poor for our purposes, since we will be
interested in comparing with other \rp violating decay modes (see the
next section). In order to have an accurate calculation of the
respective branching ratios we need to go beyond the one-step
approximation.  We therefore use an improved calculation for the FCNC
process $\s t \to c\, \s\chi_1^0$ in which the running of the Yukawa
couplings and soft breaking terms is taken into account.  First we
note that we have checked that indeed the effect of the Yukawas from
the two first generations is negligible. However the same is not true
for the running of the soft breaking terms.  As can be seen from
Figure~\ref{exact-os} the range of variation that we obtain from the
numerical solution is
%
\begin{equation}
\Gamma(\s t_1\to c\,\s\chi_1^0)\sim(10^{-16}\hbox{ -- }10^{-6})\hbox{GeV}
\end{equation}
%
depending on the assumed value of $m_{1/2}$ and $\tan\beta$. In this
figure we have compared the decay width obtained from
Eq.~(\ref{tause}) with the approximate formulae in Eq.~(\ref{gtn2cos})
for two fixed values of $m_{1/2}$, $\tan\beta$ and taking $A_0=0$. The
approximate formulae only reproduce well the numerical result for the
academic case of no SUSY breaking gaugino mass $m_{1/2}=0$. For the
more realistic case $m_{1/2}>100\,$GeV, the exact solution is usually
one decade smaller than the approximate one. In the one-step
approximation $\Gamma(\s t_1\to c\,\s\chi_1^0)$ can be arbitrarily small
if the two terms $\delta_{m_0^2} \cos\theta_{\s t}$ and
$\delta_A\sin\theta_{\s t}$ in Eq.~(\ref{gtn2coshb}) cancel. This
behaviour can be illustrated in Figure~\ref{exact-os} by the dashed
line labelled 358, which corresponds to $m_0 = 358$ GeV.  One sees
clearly that while the approximate solution goes to zero, while the
numerical one reaches a minimum value around $10^{-11}$ GeV.  The
wrong behaviour of the approximate solution indicates that the
$\delta_A$ depends strongly on the scale.  For example, the RGE for
$A_b$ is very sensitive on $m_{1/2}$ and $\tan\beta$ and in the
one-step approximation there is no explicit dependence on $m_{1/2}$,
which is crucial. Both solutions increase with $\tan\beta$, as
expected by the bottom Yukawa dependence explicit in
Eq.~(\ref{gtn2coshb}) and remain practically constant for large
$m_{1/2}$ values.
%
\begin{figure}
\centerline{\protect\hbox{
\psfig{file=gtn2c-gtn2cos.eps,height=8cm}}}
\caption{\small 
Comparison between the exact numerical calculation (ordinate) and the
one--step approximation (abscissa) for the $\s t_1\to c\,\s\chi_1^0$
decay width for various values of $\tan\beta$ and $m_{1/2}$ with
$A_0=0$ and $m_0$ varying in the indicated range. The solid left
diagonal line would signify the equality between the estimates, while
the right diagonal line would indicate one order of magnitude
difference. Results of both estimates indicated in lower right
legend. More details in text.}
\label{exact-os}
\end{figure}

\section{Two-Body Decays of the Lightest Stop: the \rp violating  case}

With this at hand we can easily compute the \rp violating stop decay
width $\tilde t_1\to b\,\tau$, where $\tau=F_3^+$ from 
eq.~(\ref{sqtocha}):
%
\begin{eqnarray}
\Gamma(\s t_1 \to b\,\tau)&=&
\frac{g^2\lambda^{1/2}(m_{\s t_1}^2,m_{b}^2,m_{\tau}^2)}{
16\pi m_{\s t_1}^3}\{-4U_{32}^*\hat h_bc_{\theta_{\s t}}(V_{32}^*\hat
h_ts_{\theta_{\s t}}-V_{31}^*c_{\theta_{\s t}})m_{b}m_{\tau}\nn\\
&&+[(V_{32}^*
\hat h_ts_{\theta_{\s t}}-V_{31}^*c_{\theta_{\s t}})^2+U_{32}^{*2}
\hat h_b^2c^2_{\theta_{\s t}}](m_{\s t_1}^2-m_{b}^2-m_{\tau}^2)\}
\label{taue}
\end{eqnarray}
%
which coincides with the result in ref.~\cite{porod}. 
In~\cite{akeroyd} it was shown that, except for $U_{32}$ which
determines the SU(2)-conserving mixing of the Higgsino with the
left-handed $\tau$, all $V_{3i}$ and $U_{3i}$ are proportional to
$v_3^{II}$ and therefore to the tau neutrino mass.  Neglecting these
terms we have from Eq.~(\ref{taue})
%
\begin{equation}
\Gamma(\s t_1\to b\,\tau)\approx
\frac{g^2\lambda^{1/2}(m_{\s t_1}^2,m_{b}^2,m_{\tau}^2)}{
16\pi m_{\s t_1}^3}\sin^2\xi'
\hat h_b^2c^2_{\theta_{\s t}}(m_{\s t_1}^2-m_{b}^2-m_{\tau}^2)
\label{tausa}
\end{equation}
%
noting that, to a good approximation,
%
\begin{equation}
|U_{32}|\approx\left|\frac{\epsilon_3}{\mu'}\right|=|\sin\xi'|
\label{u32}
\end{equation}

The dependence on the factor $\sin^2\xi' h_b^2$ may be more easily
seen in basis II, where $\epsilon^{II}_3=0$.  In this case $v_3^{II}$
is proportional to the tau neutrino mass and therefore in this basis
all the elements $U_{3i}$ and $V_{3i}$ are small~\cite{javi}.
Neglecting these terms, $\Gamma(\s t_1\to b\,\tau)$ may be written
directly from the interaction term $\s t_Lb_R\tau_L$. It is induced by
the trilinear term in the $\epsilon^{II}_3=0$--basis given in
Eq.~(\ref{supxip}) as
\begin{equation}
\lambda_3^{II}=\left(\epsilon_3/\mu'\right)h_b=h_b\sin\xi'
\end{equation}
%
which reproduces Eq.~(\ref{tausa}). In our numerical calculation to be
described in the next section we have used for $\Gamma(\s t_1\to
b\,\tau)$ the full expression given in Eq.~(\ref{tause}) of the
appendix.

In the next section we will determine the conditions under which the
\rp violating decay width $\Gamma(\s t_1 \to b\,\tau)$ can be dominant
over the \rp conserving ones, $\Gamma(\s t_1\to c\,\s\chi_1^0)$ and
$\Gamma(\s t_1\to b\,\s\chi_1^+)$.

\subsection{Region I}

Using the one-step approximation for $\Gamma(\s t_1\to c\,\s\chi_1^0)$
one finds
%
\begin{equation}
\Gamma(\s t_1\to c\,\s\chi_1^0)\sim 10^{-6}h_b^4m_{\s t_1}
\end{equation}
%
Using the Eq.~(\ref{tausa}) and neglecting charm, tau and bottom
masses we get
%
\begin{equation}
\frac{\Gamma(\s t_1\to c\,\s\chi_1^0)}{\Gamma(\s t_1\to b\,\tau)}\sim
10^{-5}\frac{h_b^2}{\sin^2\xi'}
\end{equation}
%
Therefore $\Gamma(\s t_1\to c\,\s\chi_1^0)$ will start to compete with
$\Gamma(\s t_1\to b\,\tau)$ from $\sin\xi'\lsim 5\times 10^{-3}$
($10^{-4}$) for $\tan\beta$ large (small). In fig \ref{reg1-2} we
compare $BR({\tilde t}_1\to c\,{\tilde{\chi}_1^0})$ with $BR({\tilde
t}_1\to b\,\tau)$ within the restricted region of the $m_{{\tilde
t}_1}$--$m_{{\tilde\chi}_1^0}$ plane where only those two decay modes
are open. We consider different \mnt values (these correspond to
relatively small values of the \rp parameters $|\epsilon_3|, |v_3|
\lsim 1$ GeV. We vary the MSSM parameters randomly obeying the condition
$m_{\tilde t_1}<m_{\s\chi^{\pm}_1}+m_b$ and depict the corresponding
region in dark or light grey (yellow or green, respectively, in the
coloured version).  The upper--left triangular region corresponds to
$m_{\tilde t_1}<m_{\s\chi^0_1}+m_c$ so that $BR(\tilde t_1 \to b\,\tau)=100 \: \%$. The lower--right triangular region corresponds to
$m_{\tilde t_1}>m_{\s\chi^0_1}+m_b$.  One notices from fig
\ref{reg1-2} that in the central region the dominant stop decay mode
is $\tilde t_1 \to b\,\tau$ with branching ratio $BR(\tilde t_1 \to b\,
\tau)>0.9$.  The dotted lines in the light grey region indicate maximum
\nt mass values obtained in the scan. Therefore if the lightest stop
only decays into the two modes considered here, the processes $\s
t_1\to b\,\tau$, will be important for the light tau neutrino mass range
indicated by the simplest oscillation interpretation of the
Super-Kamiokande atmospheric neutrino data. This may also be seen in
Fig.~\ref{bstbtatm} where the r\^ole played by $\tan\beta$ is
manifest. In this figure we have shown $BR(\s t_1\to b\,\tau)$ as
function of the lighter stop mass for tau neutrino mass in the sub--eV
range. We have obtained this kind of tau neutrino masses numerically
allowing only one decade of cancellation between the two terms that
contribute to $\sin\xi$ in \eq{sinxi}. The degree of suppression for
$\epsilon_3/\mu$ obtained numerically agrees very well with the
expectations from the approximate formula for the minimal tau neutrino
mass in Eq.~(\ref{mNeutrinods}).
%
\begin{figure}
\centerline{\protect\hbox{
\psfig{file=x1-t1-1.eps,height=7.0cm,width=7.5cm}}}
\caption{\small   
Regions where the ${\tilde t}_1\to b\,\tau$ decay branching ratio
exceeds 90\% in the $m_{{\tilde t}_1}$--$m_{{\tilde\chi}_1^0}$ plane
for different \mnt values.  The MSSM parameters are randomly varied as
indicated in the text under the restriction $m_{\tilde
t_1}<m_{\s\chi^{\pm}_1}+m_b$.  The upper--left triangular region
corresponds to $m_{\tilde t_1}<m_{\s\chi^0_1}+m_c$ so that only the
$\tilde t_1 \to b\,\tau$ decay channel is open. The lower--right
triangular region corresponds to $m_{\tilde
t_1}>m_{\s\chi^+_1}+m_b$. }
\label{reg1-2}
\end{figure}
%
\begin{figure}
\centerline{\protect\hbox{
\psfig{file=bstbtau-mst.eps,height=7.0cm,width=7.5cm}}}
\caption{\small  $BR(\s t_1\to b\,\tau)$ as function of
the lighter stop mass for tau neutrino mass in the sub--eV range and
two different values of $\tan\beta$ and $\epsilon_3/\mu$.  This
prediction is natural in the sense that we have allowed only up to one
order of magnitude of cancellation between the two terms that
contribute to $\sin\xi$.  }
\label{bstbtatm}
\end{figure}
%


\subsection{Region II}

In region II the R--Parity conserving decay mode $\tilde
t_1\rightarrow b\tilde\chi^+_1$ is open and competes with the
R--Parity violating mode $\tilde t_1\rightarrow b\,\tau$. Replacing the
subindex 3 by 1 on the diagonalization matrices $U$ and $V$ in
eq.~(\ref{taue}) we get the corresponding expression for $\Gamma(\s
t_1\to b\,\s\chi_1^+)$. In order to get an approximate expression for
the ratio of the two main decay rates in this region, consider that in
MSUGRA the lightest chargino is usually gaugino-like implying that
$V_{11}^2\sim 1$. In addition, the lightest stop is usually right,
hence $\sin^2\theta_{\s t}\gsim\cos^2\theta_{\s t}$. This way we find
%
\begin{equation}
\frac{\Gamma(\tau)}{\Gamma(\s\chi_1^+)}\equiv
\frac{\Gamma(\s t_1\to b\,\tau)}{\Gamma(\s t_1\to b\,\s\chi_1^+)}\approx
\frac{\sin^2\xi'\hat h_b^2\cos^2{\theta_{\s t}}}
{\left[(V_{11}^{*}\cos{\theta_{\s t}}-
V_{12}^{*}\hat h_t\sin{\theta_{\s t}})^2+U_{12}^{*2}\hat h_b^2
\cos^2{\theta_{\s t}}\right]}\,K
\label{Gtc}
\end{equation}
%
where $K$ is a kinematical factor that tends to increase the ratio, and
$\hat h_{t,b}\equiv h_{t,b}/g$. The presence of the bottom quark Yukawa 
coupling indicates that large values of $\tan\beta$ are necessary to have 
large R--Parity violating branching ratios. In fact, we have numerically 
checked with the exact expressions that in Region II (RII) 
${\Gamma(\tau)}/{\Gamma(\s\chi_1^+)}\gsim1$ only for large $\tan\beta$ as 
we will see in the next figures. Note that if the stops have a small 
mixing ($\cos{\theta_{\s t}}\approx 0$), then 
${\Gamma(\tau)}/{\Gamma(\s\chi_1^+)}\ll 1$ in RII. 

In Fig.~\ref{reg2} we plot in the $m_{\chi^{\pm}_1}-m_{\tilde t_1}$ plane 
the regions where $BR(\tilde t_1\rightarrow b\,\tau)$ dominates over
$BR(\tilde t_1\rightarrow b\widetilde\chi^+_1)$. In the upper--left region
the decay mode $\tilde t_1\rightarrow b\widetilde\chi^+_1$ is not allowed
and corresponds to Region I. Below and to the right of this forbidden zone, 
and above and to the left of the inclined lines, lies the region where 
${\Gamma(\tau)}/{\Gamma(\s\chi_1^+)}>1$ of RII. Three lines have been drawn
corresponding to $|\epsilon_3|<80$ GeV (dashed), $|\epsilon_3|<60$ GeV 
(dotted), and $|\epsilon_3|<40$ GeV (dot--dashed). The proximity to the
forbidden zone indicates that the the RPV decay dominates for large 
values of the RPV parameters only close to the threshold where there is a
high kinematical suppression through the factor $K$.
%
\begin{figure}
\centerline{\protect\hbox{
\psfig{file=x1-t1-z2.eps,height=8cm}}}
\caption{\small 
Contours of $BR({\tilde t}_1\to b\,\tau)>BR({\tilde t}_1\to b\s\chi_1^+)$  
in the $m_{{\tilde t}_1}$--$m_{{\s\chi}_1^+}$ plane for $|v_3|<10$ GeV.
Three different maximum values for $|\epsilon_3|$ are considered: 
$|\epsilon_3|<40$ GeV (dot-dash), $|\epsilon_3|<60$ GeV (dots), and
$|\epsilon_3|<80$ GeV (dashes). The region where 
$m_{\tilde t_1}<m_b+m_{\widetilde\chi^+_1}$ corresponds to the previously 
studied Region I.}  
\label{reg2}
\end{figure}
%

An even more simpler expression for the ratio of decay rates in 
eq.~(\ref{Gtc}) is obtained if we take $V_{11}\approx1$ and assume no 
kinematical suppression in eq.~(\ref{Gtc}) through the factor $K$:
%
\begin{equation}
\frac{\Gamma(\tau)}{\Gamma(\s\chi_1^+)}\sim
\sin^2\xi'\hat h_b^2\,.
\label{Gta}
\end{equation}
%
Note that the presence of the parameter $\sin\xi'=\epsilon_3/\mu'$ 
indicates that the R--Parity violating decay mode is not proportional to 
the neutrino mass, and rather proportional to the BRpV parameter 
$\epsilon_3^2$.

In Fig.~\ref{Gtaumnu} we plot $\Gamma(\tau)/\Gamma(\s\chi_1^+)$ in RII as a 
function of the tau neutrino mass. Both decay rates have been calculated 
numerically from the exact formulas. In this figure we have imposed 
$m_{H_1}^2=m_{L_3}^2$ at the GUT scale, but we have not imposed
universality between $B_3$ and $B$. If we accept cancellation within one
decade between the $\Delta B$ and $\Delta m^2$ terms in the neutrino mass
formula of eq.~(\ref{sinxi}), then the allowed region is at the right and
down the corresponding dashed inclined line. If larger cancellation are
accepted, the left boundary of the allowed region moves to the left as
indicated in the figure. We have set the right boundary of the allowed
region at the collider experimental limit of the tau neutrino mass, and 
have chosen fixed values of $\epsilon_3/\mu=1$, 0.1, and 0.01. The allowed
region for $\epsilon_3/\mu=1$ is above the dashed line. In the case of
$\epsilon_3/\mu=0.1$ (0.01) the allowed region lies between the solid
(dotted) lines. The effect of $\tan\beta$ is to increase the ratio 
$\Gamma(\tau)/\Gamma(\s\chi_1^+)$: the minimum value of the ratio is
obtained for $\tan\beta\approx 2$ and the maximum for $\tan\beta\approx60$.
The extreme values of $\tan\beta$ are dictated by perturbativity. 

%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centerline{\protect\hbox{
\psfig{file=gtau-gchar-nu.eps,height=7.0cm,width=7.5cm}}}
\caption{\small
Zones for $\Gamma({\tilde t}_1\to b\,\tau)/\Gamma({\tilde t}_1\to b\, 
{\tilde{\chi}_1^+})$ as a function of the tau neutrino mass for different 
amounts of cancellation between the two terms that contribute to the 
neutrino mass. We impose the universality condition $m_{H_1}^2=m_{L_3}^2$ 
at the unification scale, but $B$ and $B_3$ are not universal. We take 
$|\epsilon_3/\mu|=1$ (inside the dashed lines), $|\epsilon_3/\mu|=0.1$ 
(solid lines), and $|\epsilon_3/\mu|=0.01$ (dotted lines).}
\label{Gtaumnu}
\end{figure}

A number of statistically less significant points appear outside the 
drawn regions in Fig.~\ref{Gtaumnu} and are not included. They correspond
to points with $m_{\tilde t_1}-m_b-m_{{\tilde \chi}_1^\pm}<10\,$GeV
which appear above the diagonal line, and points with 
$\cos\theta_{\tilde t}<0.1$ which appear below the horizontal line
corresponding to the lowest values of $\tan\beta$. In the last case, our 
approximation in eq.~(\ref{Gta}) does not work any more. On the other hand, 
If $\cos\theta_{\s t}>0.1$, then eq.~(\ref{Gta}) predicts very well the 
behavior of $\Gamma(\tau)/\Gamma(\s\chi_1^+)$. For example considering 
$\epsilon_3/\mu=1$, which is equivalent to $\sin\xi'=1/\sqrt2$, we expect
from eq.~(\ref{Gta}) a maximum value of order 1 for large $\tan\beta$ 
($h_b\approx1$) and a minimum value of order $10^{-3}$ for small 
$\tan\beta$ ($h_b^2\approx 10^{-3}$), and this is confirmed by 
Fig.~\ref{Gtaumnu}. In addition, if we accept only cancellation within a 
decade between the two terms that contribute to the tau neutrino mass, 
then our approximate formula in eq.~(\ref{mNeutrinods}) which predict the 
minimum tau neutrino mass, works very well.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centerline{\protect\hbox{
\psfig{file=gtau-gchar-nu-u.eps,height=7.0cm,width=7.5cm}}}
\caption{\small
Zones for $\Gamma({\tilde t}_1\to b\,\tau)/\Gamma({\tilde t}_1\to b\, 
{\tilde{\chi}_1^+})$ as a function of the tau neutrino mass, but now the 
universality condition $B=B_3$ at the unification scale condition is 
imposed at 0.1\% as indicated. Its effect is to alter the maximum 
attainable tau neutrino mass. The dot-dash line corresponds to the case
where $\Delta B=0$ at the weak scale.}
\label{nueuni}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In Fig.~\ref{nueuni} we plot the ratio
$\Gamma(\tau)/\Gamma(\s\chi_1^+)$ in RII as a function of the
tau--neutrino mass, but this time imposing universality $B=B_3$ at the
GUT scale within 0.1\%. Cancellation between the $\Delta m^2$ and
$\Delta B$ terms in the neutrino mass are accepted only within 1
decade.  As a reference we have drawn the line corresponding to
$\Delta B=0$ and $\Delta m^2=\Delta m^2_{min}$ ($\Delta m^2$ is
negative and its magnitude is bounded from below by $\Delta
m^2_{min}$) at the weak scale, which gives an idea of the value of the
neutrino mass when there is no cancellation between the $\Delta B$ and
$\Delta m^2$ terms. The region of high neutrino mass and small
R--Parity violating branching ratio is disallowed, but this region is
not theoretically interesting anyway. High values of the R--parity
violating branching ratio for large values of $\epsilon_3$ are highly
restricted for large $\tan\beta$. This can be understood as
follows. In the case of $\epsilon_3/\mu=1$ and $\tan\beta=60$
acceptable neutrino masses are obtained only if $\sin\xi\sim 1$. On
the other hand, in this regime from eq.~(\ref{sinxi}) we find that the
$\Delta B$ term is large because of the high value of $\tan\beta$, and
that the $\Delta m^2$ term is large because $m_{H_1}^2$ becomes
negative and $\Delta m^2=m_{H_1}^2-m_{L_3}^2$ grows in magnitude. In
this way, acceptable neutrino masses are achieved only with
cancellation within more than one decade. We think that
Fig.~\ref{nueuni} is very conservative considering that in MSSM--SUGRA
with unification of top-bottom-tau Yukawa couplings, the large value
of $\tan\beta$ implies that a cancellation of four decades among vev's
is needed.

%
\begin{figure}
\centerline{\protect\hbox{
\psfig{file=b2u-bu-ao.eps,height=8cm}}}
\caption{\small 
Universality condition $B=B_3$ at the unification scale as a function
of $A_0$. As $\tan\beta$ increase, the allowed values of $A_0$ are more 
constrained in order to get universality at the GUT scale.}
\label{b2ubu}
\end{figure}

\subsection{Effects of non--Universality}

We now study the effect of possible non-universality of soft-breaking
SUSY parameters on our previous results. SUGRA spectra are typically
found for given values of $m_{1/2}$, $m_0$, $A_0$, $\tan\beta$ and
Sgn($\mu$). In our case we have in addition $\sin\xi'$ (or
equivalently, $\epsilon_3$). The value of $v_3$ is determined by the
previous parameters, and a relation between $A_0$ and the ratio
$B_3/B$ at the GUT scale emerges, which indicates the degree of
universality. This relation can be seen in Fig.~\ref{b2ubu} for
$\epsilon_3/\mu=1$ and the values $\tan\beta=3$, 40, and 60. The
relation becomes more and more restrictive as $\tan\beta$ is
increased, starting from $-1000<A_0<1000$ GeV allowed for
$\tan\beta=3$, until practically only a single value of $A_0$ being
compatible with unification if $\tan\beta=60$.

%
\begin{figure}
\centerline{\protect\hbox{
\psfig{file=mh1sq-ml3sq.eps,height=8.0cm}}}
\caption{\small Comparison between the ratio $m_{H_1}^2/m_{L_3}^2$ at the
weak and the unification scales for $\tan\beta=3$. The universality at
the unification scale, $m_{H_1}^2/m_{L_3}^2=1$, implies a minimum
values for this rate at the weak scale.}
\label{Dm2uni}
\end{figure}
%
In Fig.~\ref{Dm2uni} we illustrate the effect of relaxing the universality
condition $m_{H_1}^2=m_{L_3}^2$ at the GUT scale. For $\tan\beta=3$ we
plot the ratio $m_{H_1}^2/m_{L_3}^2$ at the weak scale as a function of 
the same ratio at the unification scale $M_{GUT}$. The shaded region is
allowed implying an upper bound on the ratio at the weak scale for a given
value of the ratio at the GUT scale. In particular, if we consider the
universality $(m_{H_1}^2/m_{L_3}^2)_{GUT}=1$ case, we find an upper bound
for $(m_{H_1}^2/m_{L_3}^2)_{weak}$, implying that $\Delta m^2$
is negative and its magnitude has a minimum value $|\Delta m^2|_{min}$
which appears in Fig.~\ref{nueuni}. From this latest figure we see that 
for $\tan\beta=3$ and $|\epsilon_3/\mu|=1$ the minimum value of the 
neutrino mass is 2$\times 10^{-3}$ MeV, and if no cancellation is allowed 
($\Delta B=0$ and $\Delta m^2=\Delta m^2_{min}$), the neutrino mass is 
$4\times 10^{-2}$ MeV. The one--step approximation for $\Delta m^2$ give 
us a good estimation of this latest value. Smaller neutrino masses are 
obtained either with smaller values of $\epsilon_3$ or accepting a larger 
cancellation. We also see from Fig.~\ref{Dm2uni} that a relaxation of 
universality of 0.5\% or more is enough to make 
$(m_{H_1}^2/m_{L_3}^2)_{weak}=0$ available, meaning that smaller neutrino 
masses are attainable without having to rely on a cancellation between the 
$\Delta m^2$ and $\Delta B$ terms or small values of $\epsilon_3$.

%
\begin{figure}
\centerline{\protect\hbox{
\psfig{file=mh1sqml3sq-tgb.eps,height=8.0cm}}}
\caption{\small
Misalignment in $\Delta m^2$ at the weak scale as a function of 
$\tan\beta$. It is always different from zero, negative and increase
with $\tan\beta$.}
\label{Dm2unitgb}
\end{figure}
%
However as we increase $\tan\beta$ the upper bound on $(\Delta m^2)_{weak}$ 
decreases, and thus, the required non universality between $m_{H_1}$ and 
$m_{L_3}$ at unification scale grows drastically. In Fig.~\ref{Dm2unitgb}
we show the ratio $m_{H_1}^2/m_{L_3}^2$ at the weak scale as a function
of $\tan\beta$. We appreciate clearly the growing of $|\Delta m^2|_{min}$
with $\tan\beta$. We remind the reader that this kind of non--universality 
in the soft terms is not uncommon in string models \cite{BIM}, or GUT
models based on $SU(5)$ \cite{Pomarol} or $SO(10)$ \cite{so10} for example.

%
\begin{figure}
\centerline{\protect\hbox{
\psfig{file=mnu-sinxi.eps,height=8cm}}}
\caption{\small 
Minimum value of the tau neutrino mass as a function of $\sin\xi$ for 
different values of $m_{H_1}/m_{L_3}$ at the GUT scale and two values of
$\tan\beta$. The ratio $\epsilon_3/\mu$ is fixed to the shown value, 
leading to a nearly constant value for $\Gamma({\tilde t}_1\to b\,
\tau)/\Gamma({\tilde t}_1\to b\,{\tilde{\chi}_1^+})$. Only a cancellation 
within one decade between the two term contributing to the tau neutrino 
mass is allowed.}
\label{mnu-sinxi}
\end{figure}
%
The effect of non--universality it is also explored in Fig.~\ref{mnu-sinxi} 
where it is shown the relation between the neutrino mass and the 
parameter $\sin\xi$ for $\epsilon_3/\mu=1$. Two different bands are shown:
one for $\tan\beta=3$ and $\Gamma(\tau)/\Gamma(\s\chi_1^+)=2\times10^{-3}$,
and a second one for $\tan\beta=46$ and 
$\Gamma(\tau)/\Gamma(\s\chi_1^+)=0.4\pm0.2$. Inside the bands it is 
indicated the degree of universality at the GUT scale that is necessary.
For example, in order to have neutrino masses of the order of eV for 
$\tan\beta=3$, $m_{H_1}^2$ needs to be at least 0.2\% larger than 
$m_{L_3}^2$. Similarly, for $\tan\beta=46$ we need a $m_{H_1}^2$ twice as
large as $m_{L_3}^2$ at the GUT scale in order to have neutrino masses of
1 eV. We stress the fact that for Fig.~\ref{mnu-sinxi} we have accepted 
cancellation within 1 decade only, and we consider this a very conservative 
choice for reasons explained before. 

\section{Conclusions}

We have studied the decays of the lightest stop quark in SUGRA models
with and without R-parity. We have improved the calculation for the
decay $\s t_1 \to c\,\s\chi^0$ by numerically solving the
renormalization group equations (RGE's) of the MSSM including all
generation mixing in the RGE's for Yukawa couplings as well as soft
SUSY breaking parameters. The width decay is in general one order of
magnitude smaller than the one obtained in the usual one--step
approximation. This result will therefore enlarge the regions of the
parameter space where the four--body decays of lightest stop dominate
over the decay into a charm quark and the lightest neutralino. It will
also affect the present experimental lower bound on the $\s t_1$ mass
even in the R-parity conserving case~\cite{Boehm:1999tr}. In the RPV
case, the lightest stop can be the LSP decaying with 100\% rate into
bottom and tau. We have shown that the decay mode $\s t_1 \to b\,
\tau$ dominate over $\s t_1 \to c\,\s\chi^0$ even for neutrino masses
in the range suggested by the simplest oscillation interpretation of
the Super-Kamiokande atmospheric neutrino data. This result has a
strong impact on the searches for the stop at
LEP~\cite{Abbiendi:1999yz} and TEVATRON~\cite{Holck:1999tp} where it
is assumed that the decay mode $\s t_1 \to c\,\s\chi^0$ is the main
decay channel. In addition to the signal of two jets and two taus
present when the two produced stops decays through the R-parity
violating channel, one expects a plethora of exotic high--multiplicity
fermion events since in RPV the neutralino will decay inside the
detector even for neutrino masses in the range suggested by the
$\nu_\mu$ to $\nu_\tau$ oscillation interpretation of the atmospheric
neutrino anomaly~\cite{Romao:1999up}. We have also compared the decay
$\s t_1 \to b\,\tau$ with the R-parity and flavor conserving mode $\s
t_1 \to b\,\s\chi^+$ and shown that the rate of the former can be
comparable or even bigger than the second one if the tau neutrino mass
is in the MeV range and $\tan\beta$ is large. However it is possible
to have a sizeable branching of $\s t_1 \to b\,\tau$ in the case of
suppressed tree level neutrino mass if larger cancelations between the
two terms that contributes to $\sin\xi$ are accepted, or in some
regions of the parameter space of non-univesal SUGRA models with
$(m^2_{H_1}/m^2_{L_3})_{GUT}\neq1$. A detailed analysis of the
detectability prospects of the related signatures at present and
future accelerators lies outside of the scope of the present paper and
it will be shown elsewhere \cite{willfernando}


\section*{Acknowledgments}

We thank J. Ferrandis, O. J. P. Eboli and W. Porod for useful
discussions.  This work was supported by DGICYT under grants PB95-1077
and by the TMR network grant ERBFMRXCT960090 of the European
Union. M.A.D. was supported by a DGICYT postdoctoral grant and by the
U.S. Department of Energy under contract number
DE-FG02-97ER41022. D.R. was supported by Colciencias fellowship.

\appendix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Please comment the following three lines if you like to use the usual
%format for appendix. Also change \sectionapp by \section
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\def\sectionapp#1{\setcounter{equation}{0}
\section{#1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\sectionapp{Feynman Rules}
\label{appendixb}
In this Appendix we derive the Feynman rules $F_j^0q_i\tilde q_k$
(involving a neutralino/tau-neutrino, a quark, and a squark) and
$F_j^{\pm}q_i\tilde q'_k$ (involving a chargino/tau, a quark, and a
squark of different electric charge) in the case of three generations
and RPV in the third generation. This is a generalization of the
Feynman rules contained in \cite{GH}, which are done for the R--Parity
conserving MSSM and for one generation of quarks and squarks.

Following \cite{BBMR} we work in a quark interaction basis where 
$d_{L,R}=d_{L,R}^0$, $u_{L}=K u_{L}^0$, and $u_{R}=u_{R}^0$ (we 
denote $q$ and $q^0$ the mass and current eigenstates respectively), 
as opposed to ref.~\cite{Rosiek} where a more general basis is used. 
In addition, we implement the notation $\tilde q_{L,R}\equiv \tilde 
q^0_{L,R}$ for the interaction basis.

The starting point is the following piece of the Lagrangian
%
\begin{eqnarray}
{\cal L}_{u \tilde u F^0}&=& 
-g{\bar u}_i^0
\Bigg\{
\sqrt2
\left[
\sin\theta_W e_U {N'}_{J1}+\frac 1{\cos\theta_W}
(\half-e_U\sin^2\theta_W){N'}_{J2}
\right]
\tilde u_{iL}\nonumber\\
&&+
\frac{m^{0U}_{ij}}{\sqrt2m_W\sin\beta\sin\theta}{N'}_{J4}\tilde u_{jR}
\Bigg\}
P_R F_J^0\nonumber\\
&&+g{\bar u}^0_i
\Bigg\{
\sqrt2
\left[
\sin\theta_W e_U {N'}_{J1}^*+\frac 1{\cos\theta_W}(-e_U\sin^2\theta_W)
{N'}_{J2}^*
\right]
\tilde u_{iR}\nonumber\\
&&-
\frac{m^{0U\dagger}_{ij}}{\sqrt2m_W\sin\beta\sin\theta}{N'}_{J4}^*\tilde 
u_{jL}\Bigg\}P_L F_J^0+\textrm{h.c.}
\label{nusu}
\end{eqnarray}
%
written in the quark interaction basis. The $5\times5$ matrix $N'$ 
diagonalizes the neutralino/neutrino mass matrix in the 
$(\tilde\gamma,\tilde Z,\tilde H_1^0,\tilde H_2^0,\nu_{\tau})$ basis 
as defined in \cite{v3cha}, with the index $J=1...5$. The $3\times3$ 
up--type quark mass matrix $m^{0U}$ is not diagonal, with the indexes 
$i,j=1,2,3$. 

In order to write the above Lagrangian with mass eigenstates we use
the basic relations mentioned before, in particular, $u_{iL}^0=
(K^\dagger)^{ij}u_{jL}$, which implies that ${\bar u}_{iL}^0={\bar
u}_{jL}K^{ji}$. We need the following relations:
%
\begin{eqnarray}
{\bar u}_{iL}^0\tilde u_{iL}&=&
{\bar u}_{iL}\left(\Gamma_{UL}^{*}{(K^\dagger)}^*\right)^{ki}\tilde u_{k}
\nonumber\\
{\bar u}_{iL}^0m^{0U}_{ij}\tilde u_{jR}
&=&{\bar u}_{iL}(\Gamma_{UR}^{*}m^U)^{ki}\tilde u_{k}
\nonumber\\
{\bar u}_{iR}\tilde u_{iR}&=&{\bar u}_{iR}\Gamma_{UR}^{*ki}\tilde u_{k}
\\
{\bar u}_{iR}m^{0U}_{ij}\tilde u_{jL}&=& 
{\bar u}_{iR}(\Gamma_{UL}^{*}K^*{m}^U)^{ki}\tilde u_{k}
\nonumber
\end{eqnarray}
%
where $i,j=1,2,3$ label the quark flavours, $k=1...6$ labels the squarks, 
and $m^U\equiv{\mathrm{diag}}\,\{m_u,m_c,m_t\}$ is the diagonal up--type
quark mass matrix. In this way, the Lagrangian in Eq.~(\ref{nusu}) can
be written as
%
\begin{equation}
{\cal L}_{u \tilde u F^0}= 
-g{\bar u}_i[(\sqrt2G_{0UL}^{*jki}+H_{0UR}^{*jki})P_R-
(\sqrt{2}G_{0UR}^{*jki}-H_{0UL}^{*jki})P_L]F_j^0\tilde u_k+\textrm{h.c.}
\label{luuN}
\end{equation}
%
where the different couplings are
% 
\begin{eqnarray}
G_{0UL}^{jki}&=&\left[
\sin\theta_W e_U {N'}^*_{j1}+\frac 1{\cos\theta_W}
(\half-e_U\sin^2\theta_W){N'}^*_{j2}
\right]\left(\Gamma_{UL}K^\dagger\right)^{ki}\nonumber\\
G_{0UR}^{jki}&=&\left[
\sin\theta_W e_U {N'}_{j1}+\frac 1{\cos\theta_W}(-e_U\sin^2\theta_W)
{N'}_{j2}
\right]\Gamma_{UR}^{ki}\label{gh}\\
H_{0UL}^{jki}&=&{N'}_{j4}(\Gamma_{UL}K^{\dagger}{\hat h}_U)^{ki}
\nonumber\\
H_{0UR}^{jki}&=&{N'}^*_{j4}(\Gamma_{UR}\hat h_U)^{ki}\nonumber
\end{eqnarray}
%
and $\hat h_U\equiv$diag$\,(m_u,m_c,m_t)/(\sqrt2m_W\sin\beta\sin\theta)$.
Graphically, the $F_j^0u_i\tilde u_k$ Feynman rules are given by

%%%%%%Neutralino--(u)quark--(u)squark%%%%%%%%%%%%%%%
\begin{picture}(120,60)(0,25) % y_2 for equation position
\DashArrowLine(60,25)(110,0){6}
\Vertex(60,25){3}
\ArrowLine(10,25)(60,25)
\ArrowLine(110,60)(60,25)
\Text(35,35)[]{$F_j^0$}
\Text(85,55)[]{$u_i$}
\Text(85,0)[]{${\tilde u}_k$}
\end{picture}
$
-ig[(\sqrt2G_{0UL}^{jki}+H_{0UR}^{jki})P_L
-(\sqrt{2}G_{0UR}^{jki}-H_{0UL}^{jki})P_R]
$

\begin{picture}(120,70)(0,25) % y_2 for equation position
\DashArrowLine(110,0)(60,25){6}
\Vertex(60,25){3}
\ArrowLine(60,25)(10,25)
\ArrowLine(60,25)(110,60)
\Text(35,35)[]{$F_j^0$}
\Text(85,55)[]{$u_i$}
\Text(85,0)[]{${\tilde u}_k$}
\end{picture}
$
-ig[(\sqrt2G_{0UL}^{*jki}+H_{0UR}^{*jki})P_R
-(\sqrt{2}G_{0UR}^{*jki}-H_{0UL}^{*jki})P_L]
$
\vspace{30pt} \hfill \\
%
The analogous Feynman rules in the MSSM are obtained by replacing
$F_i^0 \to \tilde\chi^0_i$, by interpreting the matrix $N$
as the usual $4\times4$ neutralino mixing matrix, and by setting
$\theta=\pi/2$ in the formula for the Yukawa couplings.

Similarly, replacing all $u(\tilde u)$ by $d(\tilde d)$ in 
Eq.~(\ref{nusu}) and
starting from Eq.(5.3) of \cite{GH} we can obtain the complete Feynman
rules for the neutralino/tau-neutrino and chargino/tau with quarks and 
squarks. The results, that complements the obtained in \cite{BBMR}, are


Neutralino--(d)quark--(d)squark
%%%%%%%%%%Neutralino--(d)quark--(d)squark%%%%%%%%%

\begin{picture}(120,60)(0,25) % y_2 for equation position
\DashArrowLine(60,25)(110,0){6}
\Vertex(60,25){3}
\ArrowLine(10,25)(60,25)
\ArrowLine(110,60)(60,25)
\Text(35,35)[]{$F_j^0$}
\Text(85,55)[]{$d_i$}
\Text(85,0)[]{${\tilde d}_k$}
\end{picture}
$
-ig[(\sqrt2G_{0DL}^{jki}+H_{0DR}^{jki})P_L
-(\sqrt{2}G_{0DR}^{jki}-H_{0DL}^{jki})P_R]
$

\begin{picture}(120,70)(0,25) % y_2 for equation position
\DashArrowLine(110,0)(60,25){6}
\Vertex(60,25){3}
\ArrowLine(60,25)(10,25)
\ArrowLine(60,25)(110,60)
\Text(35,35)[]{$F_j^0$}
\Text(85,55)[]{$d_i$}
\Text(85,0)[]{${\tilde d}_k$}
\end{picture}
$
-ig[(\sqrt2G_{0DL}^{*jki}+H_{0DR}^{*jki})P_R
-(\sqrt{2}G_{0DR}^{*jki}-H_{0DL}^{*jki})P_L]
$
\vspace{30pt} \hfill \\

\noindent
The mixing matrices $G_{0D}$ and $H_{0D}$ are defined as 
\begin{eqnarray}
G_{0DL}^{jki}&=&\left[
\sin\theta_W e_D {N'}_{j1}^*+\frac 1{\cos\theta_W}
(T_{3D}-e_D\sin^2\theta_W){N'}^*_{j2}
\right]\Gamma_{DL}^{ki}\nonumber\\
G_{0DR}^{jki}&=&\left[
\sin\theta_W e_D {N'}_{j1}+\frac 1{\cos\theta_W}(-e_D\sin^2\theta_W)
{N'}_{j2}
\right]\Gamma_{DR}^{ki}\nonumber\\
H_{0DL}^{jki}&=&{N'}_{j3}(\Gamma_{DL}{\hat h}_D)^{ki}\nonumber\\
H_{0DR}^{*jki}&=&{N'}_{j3}^*(\Gamma_{DR}\hat h_D)^{ki}
\end{eqnarray}

%\newpage

Chargino/tau--(d)quark--(u)squark

%%%%%% Chargino--(d)quark--(u)squark%%%%%%%%%%%%%%

\begin{picture}(120,60)(0,25) % y_2 for equation position
\DashArrowLine(60,25)(110,0){6}
\Vertex(60,25){3}
\ArrowLine(10,25)(60,25)
\ArrowLine(110,60)(60,25)
\Text(35,35)[]{$F_j^+$}
\Text(85,55)[]{$d_i$}
\Text(85,0)[]{${\tilde u}_k$}
\end{picture}
$
-ig(-C^{-1})[(G_{UL}^{jki}-H_{UR}^{jki})P_L-H_{UL}^{jki}P_R]
$

\vglue0.5truecm

\begin{picture}(120,70)(0,25) % y_2 for equation position
\DashArrowLine(110,0)(60,25){6}
\Vertex(60,25){3}
\ArrowLine(60,25)(10,25)
\ArrowLine(60,25)(110,60)
\Text(35,35)[]{$F_j^+$}
\Text(85,55)[]{$d_i$}
\Text(85,0)[]{${\tilde u}_k$}
\end{picture}
$
-ig[(G_{UL}^{*jki}-H_{UR}^{*jki})P_R-H_{UL}^{*jki}P_L]C
$
\vspace{30pt} \hfill \\

\noindent
where $C$ is the charge conjugation matrix (in spinor space) and the
mixing matrices $G_U$ and $H_U$ are defined as
\begin{eqnarray}
G_{UL}^{jki}&\equiv& V_{j1}^*\Gamma_{UL}^{ki},\qquad
H_{UL}^{jki}\equiv U_{j2}^*(\Gamma_{UL}\hat h_D)^{ki},\nonumber\\
H_{UR}^{jki}&\equiv&V_{j2}^*(\Gamma_{UR}\hat h_UK)^{ki},
\end{eqnarray}
Chargino/tau--(u)quark--(d)squark
%%%%%%%%%% chargino--(u)quark--(d)squark%%%%%%%%%

\begin{picture}(120,60)(0,25) % y_2 for equation position
\DashArrowLine(60,25)(110,0){6}
\Vertex(60,25){3}
\ArrowLine(10,25)(60,25)
\ArrowLine(110,60)(60,25)
\Text(35,35)[]{$F_j^+$}
\Text(85,55)[]{$u_i$}
\Text(85,0)[]{${\tilde d}_k$}
\end{picture}
$
-ig(-C^{-1})[(G_{DL}^{jki}-H_{DR}^{jki})P_L-H_{DL}^{jki}P_R]
$

\begin{picture}(120,70)(0,25) % y_2 for equation position
\DashArrowLine(110,0)(60,25){6}
\Vertex(60,25){3}
\ArrowLine(60,25)(10,25)
\ArrowLine(60,25)(110,60)
\Text(35,35)[]{$F_j^+$}
\Text(85,55)[]{$u_i$}
\Text(85,0)[]{${\tilde d}_k$}
\end{picture}
$
-ig[(G_{DL}^{*jki}-H_{DR}^{*jki})P_R-H_{DL}^{*jki}P_L]C
$
\vspace{30pt} \hfill \\

\noindent
where the mixing matrices $G_D$ and $H_D$ are defined as
\begin{eqnarray}
G_{DL}^{jki}&\equiv& U_{j1}^*(\Gamma_{DL}K^\dagger)^{ki},\qquad
H_{DL}^{jki}\equiv V_{j2}^*(\Gamma_{DL}K^\dagger\hat h_U)^{ki},\nonumber\\
H_{DR}^{jki}&\equiv&U_{j2}^*(\Gamma_{DR}\hat h_DK^\dagger)^{ki},
\end{eqnarray}
%
In order to derive the decays widths we write, for example 
Eq.~(\ref{luuN}) as
\begin{equation}
{\cal L}_{u \tilde u F^0}= 
g{\bar u}_i^0(f^{*jki}_UP_R
+h^{*jki}_UP_L)F_j^0\tilde u_k+\textrm{h.c}
\end{equation}
The result is
%
\begin{eqnarray}
\Gamma(\tilde q_k\to q_i+F_j^0)&=&
\frac{g^2\lambda^{1/2}(m_{\tilde q_k}^2,m_{q_i}^2,m_{F_j^0}^2)}{
16\pi m_{\tilde q_k}^3}\bigg[-4h^{jki}_Qf^{jki}_Q
m_{q_i}m_{F_j^0}
\nonumber\\
&&+\bigg((h^{jki}_Q)^2+(f^{jki}_Q)^2\bigg)\bigg(
m_{\tilde q_k}^2-m_{q_i}^2-m_{F_j^0}^2\bigg)\bigg]
\label{tause}\\
\Gamma(\tilde q_k\to q'_i+F_j^\pm)&=&
\frac{g^2\lambda^{1/2}(m_{\tilde q_k}^2,m_{q'_i}^2,m_{F_j^\pm}^2)}{
16\pi m_{\tilde q_k}^3}\bigg[-4l^{jki}_{Q}H^{jki}_{QL}
m_{q'_i}m_{F_j^\pm}
\nonumber\\
&&+\bigg((l^{jki}_{Q})^2+(H^{jki}_{QL})^2\bigg)\bigg(
m_{\tilde q_k}^2-m_{q'_i}^2-m_{F_j^\pm}^2\bigg)\bigg]
\label{sqtocha}
\end{eqnarray}
%
where $Q=U,D$ refers to $\tilde q$ and
%
\begin{eqnarray}
f^{jki}_Q&=&-(\sqrt2G_{0QL}^{jki}+H_{0QR}^{jki})\\
h^{jki}_Q&=&\sqrt{2}G_{0QR}^{jki}-H_{0QL}^{jki}\\
l^{jki}_Q&=&H_{QR}^{jki}-G_{QL}^{jki}
\end{eqnarray}
%
with the $G$ and $H$ couplings defined earlier in this appendix.

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\end{document}
