Higgs-mediated flavor-changing neutral currents in the general framework   with two Higgs doublets -- an RGE analysis
  We consider the standard model with two Higgs doublets with the most general Yukawa coupling terms (``type III''). In the model, the neutral-Higgs-mediated flavor-changing neutral currents (FCNC's) are allowed, but must be reasonably suppressed at low energies of probes. It has been known that the existing hierarchies of quark masses render this suppression at low energies rather natural. On the other hand, the model has been regarded by many as unnatural because of the absence of any symmetry that would ensure persistence of this suppression as the energy of probes increases. The opinion has been based on the expectation that the mentioned FCNC's would increase by large factors at increasing energies. We performed a numerical analysis of the flow of these FCN coupling parameters as governed by the one-loop renormalization group equations (RGE's), in a simplified case when Yukawa couplings of the first quark generation are neglected. The analysis shows a remarkable persistence of the mentioned FCNC suppression and thus indicates that the model is not unnatural in the RGE sense. Further, we point out two mistakes in the Yukawa RGE's of Machacek and Vaughn at one-loop level. 
G. Cvetic, C.S. Kim and S.S. Hwang