Test of the running of $\alpha_{s}$ in $\tau$ decays

The decay rate of the \tau lepton into hadrons of invariant mass smaller than \sqrt{s_0}\gg\Lambda_{\rm QCD} can be calculated in QCD assuming global quark--hadron duality. It is shown that this assumption holds with high accuracy for s_0>0.7~GeV^2. From measurements of the hadronic mass distribution, the running coupling constant \alpha_s(s_0) is extracted in the range 0.7~GeV^2<s_0<m_\tau^ 2, where its value changes by a factor~2. At s_0=m_\tau^2, the result is \alpha_s(m_\tau^2)=0.329\pm 0.030, corresponding to \alpha_s(m_Z^2)=0.119\pm 0.004. The running of the coupling constant is in excellent agreement with the QCD prediction based on the three-loop \beta-function.