Three loop estimate of the inclusive semileptonic b→c decay rate

The renormalization-scale (μ) dependence of the two-loop inclusive semileptonic bμcl^-V̄_l decay rate is shown to be significant in the pole mass scheme, and the decay rate is shown to be poorly convergent in the MS̄ scheme. Three-loop contributions to the decay rate are estimated by developing Padé approximant techniques particularly suited to perturbative calculations in the pole mass scheme. An optimized Padé estimate of the three-loop contributions is obtained by comparison of the Padé estimates with the three-loop terms determined by renormalization-group invariance. The resulting three-loop estimate in the pole-mass scheme exhibits minimal sensitivity to the renormalization scale near μ=1.0 GeV, leading to an estimated decay rate of 192π³Γ (b→c l^-v̄_l)/G_F²|V _cb\²)=992±217 GeV² inclusive of theoretical uncertainties and nonperturbative effects.