Conditions for Suppressing Dimension-five Proton Decay in Renormalizable SUSY $SO(10)$ GUT

The SUSY $SO(10)$ GUT is in severe tension with the experimental bounds on proton partial lifetimes because proton decay mediated by colored Higgsinos (dimension-five proton decay) is too rapid. In this paper, we pursue the possibility that a texture of the Yukawa coupling matrices in a renormalizable SUSY $SO(10)$ GUT model suppresses dimension-five proton decay. We focus on a general renormalizable SUSY $SO(10)$ GUT model which contains ${\bf 10}+{\bf 126}+{\bf \overline{126}}+{\bf 120}$ representation fields and where the Yukawa coupling matrices of the ${\bf 16}$ matter fields with the ${\bf 10}$, ${\bf \overline{126}}$, ${\bf 120}$ fields, $Y_{10},Y_{126},Y_{120}$, provide the quark and lepton Yukawa couplings and Majorana mass of the singlet neutrinos. We find that if components in certain flavor bases, $(Y_{10})_{u_R d_R}$, $(Y_{126})_{u_R d_R}$, $(Y_{10})_{u_R s_R}$, $(Y_{126})_{u_R s_R}$, $(Y_{10})_{u_L d_L}$, $(Y_{126})_{u_L d_L}$, $(Y_{10})_{u_L s_L}$, $(Y_{126})_{u_L s_L}$, $(Y_{10})_{u_L u_L}$, $(Y_{126})_{u_L u_L}$, are all on the order of the up quark Yukawa coupling, dimension-five proton decay can be suppressed while the Yukawa coupling matrices still reproduce the realistic quark and lepton masses and flavor mixings. We numerically obtain specific Yukawa coupling matrices satisfying the above conditions, calculate proton partial lifetimes from them and evaluate how dimension-five proton decay is suppressed when these conditions are met.