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Enhanced $Γ(p\to K^0μ^+)/Γ(p\to K^+\barν_μ)$ as a Signature of Minimal Renormalizable SUSY $SO(10)$ GUT

The ratio of the partial widths of some dimension-5 proton decay modes can be predicted without detailed knowledge of SUSY particle masses, and thus allows us to experimentally test various SUSY GUT models without discovering SUSY particles. In this paper, we study the ratio of the partial widths of the $p\to K^0μ^+$ and $p\to K^+\barν_μ$ decays in the minimal renormalizable SUSY $SO(10)$ GUT, under only a plausible assumption that the 1st and 2nd generation left-handed squarks are mass-degenerate. In the model, we expect that the Wilson coefficients of dimension-5 operators responsible for these modes are on the same order and that the ratio of $p\to K^0μ^+$ and $p\to K^+\barν_μ$ partial widths is $O(0.1)$. Hence, we may be able to detect both $p\to K^0μ^+$ and $p\to K^+\barν_μ$ decays at Hyper-Kamiokande, thereby gaining a hint for the minimal renormalizable SUSY $SO(10)$ GUT. Moreover, since this partial width ratio is quite suppressed in the minimal $SU(5)$ GUT, it allows us to distinguish the minimal renormalizable SUSY $SO(10)$ GUT from the minimal $SU(5)$ GUT. In the main body of the paper, we perform a fitting of the quark and lepton masses and flavor mixings with the Yukawa couplings of the minimal renormalizable $SO(10)$ GUT, and derive a concrete prediction for the partial width ratio based on the fitting results. We find that the partial width ratio generally varies in the range 0.05-0.6, confirming the above expectation.