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

We utilize asymptotic Padé-approximant methods to estimate the three-loop order MS-scheme coefficients within the inclusive b→uν̄ℓℓ− 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 Padé-approximant estimate of the three-loop term over the entire μ≥mb domain. Given the input values αs(MZ)=0.118±0.004 and mb(mb)=4.17±0.05 GeV, the three-loop expression for the purely-perturbative contribution to the b→uν̄ℓℓ− decay rate is minimally sensitive to renormalization scale at μ=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 192π3Γ(b→uν̄ℓℓ−)/GF2|Vub|2=2070 GeV5±16%, inclusive of theoretical uncertainties from series truncation, the input parameters, and the estimation procedure.