Measurement of the strong coupling constant $\alpha_{\rm s}$ and the vector and axial-vector spectral functions in hadronic tau decays

Abstract

The spectral functions of the vector current and the axial-vector current have been measured in hadronic $\tau$ decays using the OPAL detector at LEP. Within the framework of the Operator Product Expansion a simultaneous determination of the strong coupling constant $\alpha_{\rm s}$ , the non-perturbative operators of dimension 6 and 8 and of the gluon condensate has been performed. Different perturbative descriptions have been compared to the data. The Contour Improved Fixed Order Perturbation Theory gives $\alpha_{\rm s}(m_\tau^2) = 0.348 \pm 0.009_{\rm exp} \pm 0.019_{\rm theo}$ at the $\tau$ -mass scale and $\alpha_{\rm s}(m^2_{\rm Z}) = 0.1219 \pm 0.0010_{\rm exp} \pm 0.0017_{\rm theo}$ at the ${\rm Z}^0$ -mass scale. The values obtained for $\alpha_{\rm s}(m^2_{\rm Z})$ using Fixed Order Perturbation Theory or Renormalon Chain Resummation are 2.3% and 4.1% smaller, respectively. The ‘running’ of the strong coupling between $s_0 \simeq 1.3 {\rm GeV}^2$ and $s_0 = m_\tau^2$ has been tested from direct fits to the integrated differential hadronic decay rate $R_\tau(s_0)$ . A test of the saturation of QCD sum rules at the $\tau$ -mass scale has been performed.