Strong Coupling Constant from $\tau$ Decay within a Dispersive Approach to Perturbative QCD

We present a new dispersive framework for the extraction of the strong coupling constant $\alpha_s$ from $\tau$-lepton decays. A new feature of our procedure is the use of the quark-hadron duality on the limited region $s_{\rm d}<s<m_{\tau}^{2}$. The duality point $s_{\rm d}$ and the $\bar{\rm MS}$ strong coupling constant $\alpha_{s}(m_{\tau}^{2})$ are self-consistently extracted from the $\tau$ data for the non-strange vector spectral function. We use 2005 ALEPH and 1998 OPAL experimental data on the vector spectral function. We compare the new framework with the contour improved perturbation theory up to order $\alpha_s^{5}$. The new procedure yields systematically lower values for $\alpha_s$. From the 2005 ALEPH data, we obtain $\alpha_{s}(m_{\tau}^{2})=0.308\pm 0.014_{\rm exp}\pm 0.005_{\rm th}$ which corresponds to $\alpha_{s}(M_{\rm_{z}}^{2})=0.1170\pm 0.0018_{\rm exp}\pm 0.0007_{\rm th}\pm 0.0005_{\rm ev}$. The extracted value for the duality point $s_{\rm d}$ is found surprisingly stable against perturbation theory corrections $s_{\rm d}= 1.71\pm 0.05_{\rm exp}\pm 0.00_{\rm th}\,\, {\rm GeV^{2}}$. From the 1998 OPAL data, we obtain $\alpha_{s}(m_{\tau}^{2})=0.290\pm 0.023_{\rm exp}$ and $s_{\rm d}=1.68\pm 0.10_{\rm exp}\,\, {\rm GeV^{2}}$.