Nucleon Decay in the Minimal Supersymmetric $SU(5)$ Grand Unification

We make a detailed analysis on the nucleon decay in the minimal supersymmetric $SU(5)$ grand unified model. We find that a requirement of the unification of three gauge coupling constants leads to a constraint on a mass $M_{H_C}$ of color-triplet Higgs multiplet as $2 \times 10^{13}~\GeV \leq M_{H_C} \leq 2 \times 10^{17}~\GeV$, taking both weak- and GUT-scale threshold effects into account. Contrary to the results in the previous analyses, the present experimental limits on the nucleon decay turn out to be consistent with the SUSY particles lighter than 1~TeV even without a cancellation between matrix elements contributed from different generations, if one adopts a relatively large value of $M_{H_C}$ ($\ge2\times 10^{16}~\GeV$). We also show that the Yukawa coupling constant of color-triplet Higgs multiplet does not necessarily blow up below the gravitational scale ($2.4\times10^{18}~\GeV$) even with the largest possible value of $M_{H_C}$. We point out that the no-scale model is still viable, though it is strongly constrained.