Bquark low-scale running mass fromΥsum rules

Abstract

The b quark low-scale running mass ${m}_{\mathrm{kin}}$ is determined from an analysis of the $\ensuremath{\Upsilon}$ sum rules in the next-to-next-to-leading order (NNLO). It is demonstrated that using this mass one can significantly improve the convergence of the perturbation series for the spectral density moments. We obtain ${m}_{\mathrm{kin}}(1 \mathrm{GeV})=4.56\ifmmode\pm\else\textpm\fi{}0.06 \mathrm{GeV}$. Using this result we derive the value of the modified minimal subtraction scheme mass $\overline{m}: \overline{m}(\overline{m})=4.2\ifmmode\pm\else\textpm\fi{}0.1 \mathrm{GeV}$. Contrary to the low-scale running mass, the pole mass of the b quark cannot be reliably determined from the sum rules. As a by-product of our study we find the NNLO analytical expression for the cross section ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}Q\overline{Q}$ of the quark-antiquark pair production in the threshold region, as well as the energy levels and the wave functions at the origin for the ${}^{1}{S}_{3}$ bound states of $Q\overline{Q}$.