0 Reducing the Theoretical Uncertainty in Extracting | V ub | from the Inclusive B → X u l − ν l Decay Rate

Utilizing asymptotic Padé-approximant methods, we estimate the three-loop order MS coefficients of α s [log(μ/m b (μ)] terms [k = {0, 1, 2, 3}] within the b → ulνl decay rate. Except for the coefficient of the k = 0 term, all other coefficients may also be obtained via renormalization-group (RG) methods. The relative errors in asymptotic Padé-approximant estimates of the k = {1, 2, 3} terms are found to be 5.1% or less, thereby providing an estimate of the theoretical uncertainty in the asymptotic Padé-approximant estimate of the RG-inaccessible k = 0 term. By a judicious choice in the renormalization scale parameter, we are able to extract |Vub| from the inclusive decay rate to within ±7% theoretical uncertainty. INTRODUCTION The inclusive semileptonic B → Xul νl decay rate is sensitive only to a single mass parameter, the b-quark mass mb, since the u-quark, charged lepton, and associated antineutrino are all very light compared to mb. This property makes the calculated decay rate particularly attractive for the extraction of the CKM mixing parameter |Vub|. QCD corrections to the rate have recently been calculated to two-loop order within an MS scheme [1]. If the renormalization scale μ is chosen to be mb(mb) ∼= 4.2GeV [2], i.e., if μ = mb(μ), then the two loop rate is given by Γ(B → Xul νl) = Km 5 b(mb) [1 + 0.30 + 0.14] , K ≡ G 2 F |Vub| /192π. (1) Determinations of |Vub| based upon comparison of experimental results to this expression are necessarily subject to theoretical uncertainties associated with the premature truncation of the slowly convergent perturbative series in (1). In the present work, we demonstrate how asymptotic Padé-approximant methods in conjunction with renormalization-group (RG-) methods can dramatically reduce such theoretical uncertainties . . . 1) by providing an estimate of three loop contributions to the decay rate that can be tested against RG-accessible three-loop coefficients in the rate, and 2) by demonstrating that the physically-motivated choice of scale parameter μ which minimizes scale sensitivity [3] is quite close to that which minimizes the size of the four-loop term, and consequently, the truncation error for an asymptotic series. DETERMINATION OF RG-ACCESSIBLE THREE-LOOP COEFFICIENTS The series portion S[x, L] of the perturbative MS expression for the inclusive semileptonic rate,