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Fiz. }{\bf #1} (19#2) #3} \def\zetf#1#2#3{ {Z. Eksp. Teor. Fiz. }{\bf #1} (19#2) #3} \def\zetfpr#1#2#3{ {Z. Eksp. Teor. Fiz. Pisma. Red. }{\bf #1} (19#2) #3} \def\jetp#1#2#3{ {JETP }{\bf #1} (19#2) #3} \def\mpl#1#2#3{ {Mod. Phys. Lett. }{\bf #1} (19#2) #3} \def\ufn#1#2#3{ {Usp. Fiz. Naut. }{\bf #1} (19#2) #3} \def\sp#1#2#3{ {Sov. Phys.-Usp.}{\bf #1} (19#2) #3} \def\ppnp#1#2#3{ {Prog. Part. Nucl. Phys. }{\bf #1} (19#2) #3} \def\cnpp#1#2#3{ {Comm. Nucl. Part. Phys. }{\bf #1} (19#2) #3} \def\ijmp#1#2#3{ {Int. J. Mod. Phys. }{\bf #1} (19#2) #3} \def\tp{these proceedings} \def\pc{private communication} \def\opc{\hbox{{\sl op. cit.} }} \def\ip{in preparation} %******************************************** \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 $m_{{\tilde\chi}_1^+}+m_bm_{{\tilde\chi}_1^0}+m_t$ } \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 $m_{\tilde t_1}m_{\s\chi^0_1}+m_b$. 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 $m_{\tilde t_1}m_{\s\chi^+_1}+m_b$. } \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 $m_{\tilde t_1}0.1$, 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