Precision determination of $\alpha_s$ and $m_b$ from QCD sum rules for $b \overline b$.

Abstract

The QCD sum rules for moments of production cross section of $b \overline b$ states in $e^+\,e^-$ annihilation are extremely sensitive to the values of $m_b$ and $\alpha_s$ for moments of large order $n$. This enables one to extract from the existing data on $\Upsilon$ resonances the values of these parameters with a high precision by using a non-relativistic expansion in $1/n$. It is found that the sum rules fit the data with $\alpha_s^{\overline {MS}} (1 \, GeV) = 0.336 \pm 0.011$ and $m_b=4827 \pm 7 MeV$, where the estimate of the errors includes the theoretical uncertainty due to subleading in $1/n$ terms and the experimental uncertainty of the $e^+\,e^-$ annihilation cross section above the $B \overline B$ threshold. The found value of $\alpha_s$, when evolved in two loops up to the $Z$ mass, gives $\alpha_s^{\overline {MS}} (M_Z) = 0.109 \pm 0.001$. The $b$ quark mass $m_b$ corresponds to the `on shell' value appropriate for one-loop perturbative calculations.