Behavior of the effective QCD coupling ${\alpha }_{\tau }\mathrm{}\left(s\right)\mathrm{}$ at low scales

The hadronic decays of the $\tau$ lepton can be used to determine the effective charge ${\alpha }_{\tau }{(m}_{{\tau }^{\prime }}^2)$ for a hypothetical $\tau$ lepton with a mass in the range $0<\mathrm{}{m}_{{\tau }^{\prime }}<m_{\tau }.$ This definition provides a fundamental definition of the QCD coupling at low mass scales. We study the behavior of ${\alpha }_{\tau }$ 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 ${s=m}_{{\tau }^{\prime }}^2$ of order $1{\mathrm{GeV}}^2$ with a magnitude ${\alpha }_{\tau }\sim 0.9\pm \mathrm{0.1.}$