Padé - Improved Estimate of Perturbative Contributions to Inclusive Semileptonic

Padé-approximant methods are used to estimate the three-loop perturbative contributions to the inclusive semileptonic b → u decay rate. These improved estimates of the decay rate reduce the theoretical uncertainty in the extraction of the CKM matrix element |V ub | from the measured inclusive semileptonic branching ratio. In this paper we briefly review the development of Padé approximation techniques to QCD quantities satisfying a renormalization group equation, 1 and the application of these techniques to the estimate of three-loop contributions to the inclusive semileptonic b → u decay rate. 2 The QCD perturbative contributions to the inclusive semileptonic decay rate Γ (b → X u ℓ − ¯ ν ℓ) are known to two-loop order. 3 The theoretical prediction of the decay rate is an interesting phenomenological quantity since it depends on the CKM matrix element |V ub |. Moreover, the two-loop calculation is mainly sensitive to m b since the b mass is much larger than final state particle masses (m u , m ℓ), and m c only enters the partial rate b → uℓ¯ ν ℓ c¯ c or in virtual corrections. The M S scheme obviates the poor convergence of the perturbative series in on-shell schemes, leading to the following two-loop result for the decay rate for five active flavours: