Using Gauge Coupling Unification and Proton Decay to Test Minimal Supersymmetric SU(5)

We derive a one-loop expression, including all thresholds, for the mass of the proton decay mediating color triplets, $M_{D^c}$, in minimal supersymmetric SU(5). The result for $M_{D^c}$ does not depend on other heavy thresholds or extra representations with SU(5) invariant masses which might be added to the minimal model. We numerically correct our result to two-loop accuracy. Choosing inputs to maximize $M_{D^c}$ and $\tau_P$, within experimental limits on the inputs and a $1~TeV$ naturalness bound, we derive a strict bound $\alpha_3>0.117$. We discuss how this bound will change as experimental limits improve. Measurements of $\alpha_3$ from deep inelastic scattering and the charmonium spectrum are below the bound $\alpha_3>0.117$ by more than $3\sigma$. We briefly review several ideas of how to resolve the discrepancy between these low values of $\alpha_3$ and the determinations of $\alpha_3$ from LEP event shapes.