Measuring $\alpha_{s}$(Q$^{2}$) in $\tau$ decays

The decay rate of the \tau lepton into hadrons of invariant mass smaller than Q\gg\Lambda_{\rm QCD} can be calculated in QCD using the OPE. Using experimental data on the hadronic mass distribution, the running coupling constant \alpha_s(Q^2) is extracted in the range 0.85~\mbox{GeV}<Q<m_\tau, where its value changes by about a factor~2. At Q=m_\tau, the result is \alpha_s(m_\tau^2)=0.33\pm 0.03, 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.