Three loop estimate of the inclusive semileptonicb→cdecay rate

Abstract

The renormalization-scale $(\ensuremath{\mu})$ dependence of the two-loop inclusive semileptonic $\stackrel{\ensuremath{\rightarrow}}{b}{\mathrm{cl}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{l}$ decay rate is shown to be significant in the pole mass scheme, and the decay rate is shown to be poorly convergent in the $\overline{\mathrm{MS}}$ scheme. Three-loop contributions to the decay rate are estimated by developing Pad\'e approximant techniques particularly suited to perturbative calculations in the pole mass scheme. An optimized Pad\'e estimate of the three-loop contributions is obtained by comparison of the Pad\'e 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 $\ensuremath{\mu}=1.0 \mathrm{GeV},$ leading to an estimated decay rate of $192{\ensuremath{\pi}}^{3}\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{b}{\mathrm{cl}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{l}{)/(G}_{F}^{2}|{V}_{\mathrm{cb}}{|}^{2})=992\ifmmode\pm\else\textpm\fi{}217 {\mathrm{GeV}}^{5}$ inclusive of theoretical uncertainties and nonperturbative effects.