Summing $O(\beta_0^n \alpha_s^{n+1})$ Corrections to Top Quark Decays

Order $\beta_0^n \alpha_s^{n+1}$ QCD corrections to top quark decays into $W^+$ and $H^+$ bosons are computed to all orders in perturbation theory. Predictions for the radiative corrections to the top quark width are compared with the estimates from BLM scale setting procedures. The results of the summation are shown to greatly improve understanding of higher order corrections in the limit $m_W,~m_H \to m_t$, where the BLM scale setting method is known to fail. Attempts to reduce nonperturbative error by substituting the running mass for the pole mass in the expression for the decay are shown to fail in the limit $m_W,~m_H \to m_t$ because of subtleties in the treatment of phase space.