New Constraints on Superpartner Masses

We consider the Minimal Supersymmetric Standard Model (MSSM) without imposing relations on the superpartner masses that arise in grand unified theories. Given an arbitrary pattern of superpartner masses (consistent with experimental constraints), it may happen that the scalar potential is actually unstable, even though all scalar masses-squared are positive at the weak scale $M_W$. This is most likely to happen if the running mass-squared in a ``flat'' direction in field space becomes negative at some scale $Q_0$ which is well below the GUT scale. In this case, either this pattern of masses is ruled out, or there must be new physics (beyond the MSSM) at or below the scale $Q_0$.