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

TL;DR

Using QCD sum rules for $b\overline{b}$ production in $e^+e^-$ annihilation, the study employs a non-relativistic $1/n$ expansion and Coulomb resummation to connect high-order moments to near-threshold dynamics. Incorporating running coupling effects and a short-distance radiative correction, the analysis fits to $\Upsilon$ resonance data and yields $\alpha_s^{\overline{MS}}(1\,\mathrm{GeV})=0.336\pm0.011$ and $m_b=4827\pm7\,\mathrm{MeV}$, with $\alpha_s^{\overline{MS}}(M_Z)=0.109\pm0.001$ after two-loop evolution. The results interpret $m_b$ as the on-shell mass appropriate for one-loop perturbation theory and demonstrate the robustness of a low-energy, short-distance QCD approach complementary to LEP and $\tau$-decay determinations. The method emphasizes near-threshold physics, suppresses continuum uncertainties, and suggests that improved high-energy cross-section data could further sharpen the precision of $\alpha_s$ and $m_b$ extracted from heavy-quark sum rules.

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 $α_s$ for moments of large order $n$. This enables one to extract from the existing data on $Υ$ 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 $α_s^{\overline {MS}} (1 \, GeV) = 0.336 \pm 0.011$ 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 $α_s$, when evolved in two loops up to the $Z$ mass, gives $α_s^{\overline {MS}} (M_Z) = 0.109 \pm 0.001$. The $b$ quark mass $m_b$ corresponds to the `on shell' value appropriate for one-loop perturbative calculations.