Bottom Quark Mass from $\Upsilon$ Sum Rules to ${\cal O}(\alpha_s^3)$

Abstract

We use the ${\cal O}(\alpha_s^3)$ approximation of the heavy-quark vacuum polarization function in the threshold region to determine the bottom quark mass from nonrelativistic $\Upsilon$ sum rules. We find very good stability and convergence of the perturbative series for the bottom quark mass in $\rm\overline{MS}$ renormalization scheme. Our final result is $\overline{m}_b(\overline{m}_b)=4.169\pm 0.008_{th}\pm 0.002_{\alpha_s}\pm 0.002_{exp}$.