Two-body decays of the lightest stop in minimal 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 t~1→cχ+1, the lightest stop decays mainly into third generation fermions, t~1→bτ instead of the R-parity conserving mode t~1→cχ01, even for tiny tau--neutrino mass values. Moreover we show that, even above the threshold for decays into charginos, the decay t~1→bτ 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 t~1→bτ as well as the cascades originated by the conventional t~1→cχ01 decay followed by the R-parity violating neutralino decays can provide new signatures for stop production at LEP and the Tevatron.