Vacuum Stability with Tachyonic Boundary Higgs Masses in No-Scale Supersymmetry or Gaugino Mediation

No-scale supersymmetry or gaugino mediation augmented with large negative Higgs soft masses at the input scale provides a simple solution to the supersymmetric flavor problem while giving rise to a neutralino LSP. However, to obtain a neutralino LSP it is often necessary to have tachyonic input Higgs soft masses that can give rise to charge-and-color-breaking (CCB) minima and unbounded-from-below (UFB) directions in the low energy theory. We investigate the vacuum structure in these theories to determine when such problematic features are present. When the standard electroweak vacuum is only metastable, we compute its lifetime under vacuum tunneling. We find that vacuum metastability leads to severe restrictions on the parameter space for larger $\tan\beta \sim 30$, while for smaller $\tan\beta\sim 10$, only minor restrictions are found. Along the way, we derive an exact bounce solution for tunneling through an inverted parabolic potential.