Calculations of binding energies and masses of heavy quarkonia using renormalon cancellation

Abstract

We use various methods of Borel integration to calculate the binding ground energies and masses of $b\overline{b}$ and $t\overline{t}$ quarkonia. The methods take into account the leading infrared renormalon structure of the $(\mathrm{hard}+)$soft part of the binding energies $E(s),$ and of the corresponding quark pole masses ${m}_{q},$ where the contributions of these singularities in ${M(s)=2m}_{q}+E(s)$ cancel. Beforehand, we carry out the separation of the binding energy into its $(\mathrm{hard}+)$soft and ultrasoft parts. The resummation formalisms are applied to expansions of ${m}_{q}$ and $E(s)$ in terms of quantities which do not involve renormalon ambiguity, such as $\overline{\mathrm{MS}}$ mass ${m}_{q}$ and ${\ensuremath{\alpha}}_{s}(\ensuremath{\mu}).$ The renormalization scales $\ensuremath{\mu}$ are different in calculations of ${m}_{q},$ $E(s)$ and $E(\mathrm{us}).$ The mass ${m}_{b}$ is extracted, and the binding energies ${E}_{t\overline{t}}$ and the peak (resonance) energies ${E}_{\mathrm{res}.}$ for $t\overline{t}$ production are obtained.