Estimate of the Three-Loop Perturbative Contribution to Inclusive Semileptonic b to u Decays

Abstract

We utilize asymptotic Pade-approximant methods to estimate the three-loop order MS bar-scheme coefficients within the inclusive semileptonic b to u 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 $\mu \ge m_b$ domain. Given the input values ${\alpha }_s(M_Z)=0.118\pm 0.004$ and $m_b(m_b)=4.17\pm 0.05GeV$, the three-loop expression for the purely-perturbative contribution to the decay rate is minimally sensitive to renormalization scale at $\mu =1.775GeV$, 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 $192{\pi }^3\Gamma (b\to u{\bar{\nu }}_{\ell }{\ell }^{-})/(G_F^2|V_{ub}{|}^2)=2070\mathrm{Ge}{\mathrm{V}}^5\pm 16$, inclusive of theoretical uncertainties from series truncation, the input parameters, and the estimation procedure.

Comments: latex2e, 10 pages, three embedded eps figures. Extended analysis in revised version