  In this work, we revisit the one-loop renormalization group equations (RGEs)
among non-degenerate seesaw scales, i.e., threshold effects in the canonical
seesaw mechanism, which have been obtained for more than two decades. We find
some contributions from the Weinberg operator to its Wilson coefficient, the
neutrino Yukawa coupling matrix, and the Higgs quartic coupling absent in the
previous calculations. Based on the updated one-loop RGEs, we derive the RGE of
the effective neutrino mass matrix's determinant without any approximation.
Then, for the first time, we provide a strict proof that the one-loop RG
running effects among non-degenerate seesaw scales can not generate a non-zero
mass for the initial massless neutrino in the minimal type-I seesaw mechanism
or in the canonical one with a rank-degenerate neutrino Yukawa coupling matrix.
One has to include two- or higher-loop corrections to achieve a non-zero mass
for the massless neutrino.
