Renormalization-Mass Scale Dependence in QCD Contributions to Semileptonic $b \to u$ Decay

QCD contributions to the $b \to u \ell^- \bar{\nu}_\ell$ decay rate, which are known to two-loop order in the $\bar{MS}$ scheme, exhibit sufficient dependence on the renormalization mass $\mu$ to compromise phenomenological predictions for inclusive semileptonic $B \to X_u$ processes. Such scale dependence is ameliorated by the renormalization-group (RG) extraction and summation of all leading and RG-accessible subleading logarithms occurring subsequent to two-loop order in the perturbative series. This optimal RG-improvement of the known portion of the perturbative series virtually eliminates $\mu$-dependence as a source of theoretical uncertainty in the predicted semileptonic $B \to X_u$ inclusive rate.