Measuring $α_s(Q^2)$ in $τ$ Decays

Abstract

The decay rate of the $τ$ lepton into hadrons of invariant mass smaller than $Q\ggΛ_{\rm QCD}$ can be calculated in QCD using the OPE. Using experimental data on the hadronic mass distribution, the running coupling constant $α_s(Q^2)$ is extracted in the range $0.85~\mbox{GeV}<Q<m_τ$, where its value changes by about a factor~2. At $Q=m_τ$, the result is $α_s(m_τ^2)=0.33\pm 0.03$, corresponding to $α_s(m_Z^2)=0.119\pm 0.004$. The running of the coupling constant is in excellent agreement with the QCD prediction based on the three-loop $β$-function.