Two-Body Decays of the Lightest Stop in Supergravity with and without R-Parity

We study the decays of the lightest top squark 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 $\tilde t_1\to c\chi^+_1$, the lightest stop decays mainly into third generation fermions, $\tilde t_1\to b\tau$ instead of the R-parity conserving mode $\tilde t_1\to c\chi^0_1$, even for tiny tau--neutrino mass values. Moreover we show that, even above the threshold for decays into charginos, the decay $\tilde t_1\to b\tau$ may be dominant. We study the role played by the universality of the boundary conditions on the soft supersymmetry breaking terms. This new decay mode $\tilde t_1\to b\tau$ as well as the cascades originated by the conventional $\tilde t_1\to c\chi^0_1$ decay followed by the R-parity violating neutralino decays can provide new signatures for stop production at LEP and the Tevatron.