Constraints on higher-order perturbative corrections in b $rightarrow$ u semileptonic decays from residual renormalization-scale dependence

The constraint of a progressive decrease in residual renormalization scale dependence with increasing loop order is developed as a method for obtaining bounds on unknown higher-order perturbative corrections to renormalization-group-invariant quantities. This technique is applied to the inclusive semileptonic process b --> u (v) over bar (l)l(-) (explicitly known to two-loop order) in order to obtain bounds on the three- and four-loop perturbative contributions that are not accessible via the renormalization group. Combining this technique with the principle of minimal sensitivity, we obtain an estimate for the perturbative contributions to Gamma(b --> u (v) over bar (l)l(-)) that incorporates theoretical uncertainty from as-yet-undetermined higher-order QCD corrections