Estimate of the three-loop perturbative contribution to inclusive semileptonic b→u decays

We utilize asymptotic Pade-approximant methods to estimate the three-loop order MS-scheme coefficients within the inclusive b --> u<(nu)over bar>(i)l(-) decay rate for four and five active quark flavours. The estimates we obtain for the three renormalization-group-accessible coefficients within the three-loop contribution are all found to be within 5.1% of their true values, using a least-squares procedure in conjunction with an asymptotic Pade-approximant estimate of the three-loop term over the entire mu greater than or equal to m(b) domain. Given the input values alpha(s)(M-Z) = 0.118 +/- 0.004 and m(b)(m(b)) = 4.17 +/- 0.05 GeV, the three-loop expression for the purely-perturbative contribution to the b --> u<(nu)over bar>(i)l(-) decay rate is minimally sensitive to renormalization scale at mu = 1.775 GeV, at which scale the three-loop contribution is estimated to be only 1.4% of the leading tree-order contribution. We estimate the full perturbative decay rate to be 192 pi(3)Gamma(b --> u<(nu)over bar>(i)l(-))(..)/(G(F)(2)\V-ub\(2)) = 2070 GeV5 +/- 16%, inclusive of theoretical uncertainties from series truncation, the input parameters, and the estimation procedure. (C) 2000 Elsevier Science B.V. All rights reserved.