Decoupling or nondecoupling: is that the $R_b$ question?

The top quark is well known for the nondecoupling effects it implies in $\rho$ and $R_b$. The recent experimental $R_b$ data exhibits a disagreement with the SM prediction at more than the $3 \sigma$ level. It is tempting to explore whether this might be due to nondecoupling New Physics effects, opposite to those of the top. We investigate this issue in the context of models with an extra family of right or left handed, singlet or doublet quarks. It is shown that, contrary to what one might naively expect, the nondecoupling properties of a mirror $t^\prime$ do not have an impact on $R_b$, due to a conspiracy of the mixing angles, imposed by the requirement that there be no $b$ - $b^\prime$ mixing. Our analysis agrees with an analysis performed independently, which includes this model as a particular case.