Behavior of the effective QCD couplingατ(s)at low scales

Abstract

The hadronic decays of the $\ensuremath{\tau}$ lepton can be used to determine the effective charge ${\ensuremath{\alpha}}_{\ensuremath{\tau}}{(m}_{{\ensuremath{\tau}}^{\ensuremath{'}}}^{2})$ for a hypothetical $\ensuremath{\tau}$ lepton with a mass in the range $0l{m}_{{\ensuremath{\tau}}^{\ensuremath{'}}}l{m}_{\ensuremath{\tau}}.$ This definition provides a fundamental definition of the QCD coupling at low mass scales. We study the behavior of ${\ensuremath{\alpha}}_{\ensuremath{\tau}}$ at low mass scales directly from first principles and without any renormalization-scheme dependence by looking at the experimental data from the OPAL Collaboration. The results are consistent with the freezing of the physical coupling at mass scales ${s=m}_{{\ensuremath{\tau}}^{\ensuremath{'}}}^{2}$ of order $1{\mathrm{GeV}}^{2}$ with a magnitude ${\ensuremath{\alpha}}_{\ensuremath{\tau}}\ensuremath{\sim}0.9\ifmmode\pm\else\textpm\fi{}0.1.$