Two-loop renormalization factors of dimension-six proton decay operators in the supersymmetric standard models

The renormalization factors of the dimension-six effective operators for proton decay are evaluated at two-loop level in the supersymmetric grand unified theories. For this purpose, we use the previous results in which the quantum corrections to the effective Kähler potential are evaluated at two-loop level. Numerical values for the factors are presented in the case of the minimal supersymmetric SU(5) grand unified model. We also derive a simple formula for the one-loop renormalization factors for any higher-dimensional operators in the Kähler potential, assuming that they are induced by the gauge interactions