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We consider the standard model with two Higgs doublets with the most generalYukawa coupling terms (``type III''). In the model, the neutral-Higgs-mediatedflavor-changing neutral currents (FCNC's) are allowed, but must be reasonablysuppressed at low energies of probes. It has been known that the existinghierarchies of quark masses render this suppression at low energies rathernatural. On the other hand, the model has been regarded by many as unnaturalbecause of the absence of any symmetry that would ensure persistence of thissuppression as the energy of probes increases. The opinion has been based onthe expectation that the mentioned FCNC's would increase by large factors atincreasing energies. We performed a numerical analysis of the flow of these FCNcoupling parameters as governed by the one-loop renormalization group equations(RGE's), in a simplified case when Yukawa couplings of the first quarkgeneration are neglected. The analysis shows a remarkable persistence of thementioned FCNC suppression and thus indicates that the model is not unnaturalin the RGE sense. Further, we point out two mistakes in the Yukawa RGE's ofMachacek and Vaughn at one-loop level.Comment: LaTeX (REVTeX), 30 pages, 14 eps-figures, slight modifications  performed to adjust to the published versio

Higgs-mediated flavor-changing neutral currents in the general framework with two Higgs doublets: An RGE analysis