Three Loop Estimate of the Inclusive Semileptonic $b\to c$ Decay Rate

Abstract

The renormalization-scale ($\mu$) dependence of the two-loop inclusive semileptonic $b\to c{\ell }^{-}{\bar{\nu }}_{\ell }$ decay rate is shown to be significant in the pole mass scheme, and the decay rate is shown to be poorly convergent in the MS-bar scheme. Three-loop contributions to the decay rate are estimated by developing Pade approximant techniques particularly suited to perturbative calculations in the pole mass scheme. An optimized Pade estimate of the three-loop contributions is obtained by comparison of the Pade 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 $\mu =1.0GeV$, leading to an estimated decay rate of $192{\pi }^3\Gamma (b\to c{\ell }^{-}{\bar{\nu }}_{\ell })/(G_F^2|V_{cb}{|}^2)=992\pm 217Ge{V}^5$ inclusive of theoretical uncertainties and non-perturbative effects.

Comments: latex2e using amsmath. 17 pages with 5 eps figures embedded in mansucript. Corrections made to eqns (47), (57) and Reference [1]