Yukawa Unification and Unstable Minima of the Supersymmetric Scalar Potential

Motivated by the possibilities of $b-\tau$ or $t-b-\tau$ Yukawa unification in the supersymmetric Grand Unified Theories, we consider the dangerous directions of the supersymmetric potential for large values of $\tan\beta (\geq 30)$, in two versions of the minimal supergravity model with and without common soft breaking scalar masses at the GUT scale, where the potential may become unbounded from below. We find that for the common trilinear coupling $A_0 \leq 0$ the requirement of $b-\tau$ unification in conjunction with the stability condition on the potential yields highly restrictive sparticle spectra with upper, and in many cases, lower bounds stronger than the available experimental lower bounds, on the soft SUSY breaking common scalar mass and the common gaugino mass. Over a significant region of the parameter space, the model becomes even more restrictive if the common sfermion soft mass is different from the soft mass for the Higgs sector. We also find that the bulk of this restricted parameter space can be probed at the LHC. Im models with $t-b-\tau$ Yukawa unification, $A_0 \leq 0$ is ruled out from potential constraints.