Scale Setting in Top Quark Decays

We compute the $O(\beta_0 \alpha_s^2)$ QCD corrections to the Standard Model decay $t\rightarrow W^+ b$ as well as the non standard decay $t\rightarrow H^+ b$. We then use our results to compute the BLM scale for these decays, and study the dependence of the BLM scale on the mass of the boson in the decay. We find that the BLM procedure gives extremely small scales when top decays into heavy bosons. When we try to improve the BLM scale by reexpressing rates as a functions of the top quark running mass, we find the BLM scale exhibits unnatural dependence on the boson mass, casting doubts on the applicability of the BLM procedure to these processes.