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.