Testing the Concept of Quark-Hadron Duality with the ALEPH $\tau$ Decay Data

We propose a modified procedure for extracting the numerical value for the strong coupling constant $\alpha_s$ from the $\tau$ lepton hadronic decay rate into non-strange particles in the vector channel. We employ the concept of the quark-hadron duality specifically, introducing a boundary energy squared $s_{\rm p}>0$, the onset of the perturbative QCD continuum in Minkowski space \cite{BLR,Rafa,PPR}. To approximate the hadronic spectral function in the region $s>s_{\rm p}$, we use Analytic Perturbation Theory (APT) up to the fifth order. A new feature of our procedure is that it enables us to extract from the data simultaneously the QCD scale parameter $\Lambda_{\bar{\rm MS}}$ and the boundary energy squared $s_{\rm p}$. We carefully determine the experimental errors on these parameters which come from the errors on the invariant mass squared distribution. For the $\bar{\rm MS}$ scheme coupling constant, we obtain $\alpha_s(m^{2}_{\tau})=0.308\pm 0.014_{\rm exp.}$. We show that our numerical analysis is more stable against higher-order corrections than the standard one. The extracted value for the duality point $s_{\rm p}$ is found surprisingly stable against perturbation theory corrections $s_{\rm d}= 1.71\pm 0.05_{\rm exp}\pm 0.00_{\rm th}\,\, {\rm GeV^{2}}$.Additionally, we recalculate the "experimental" Adler function in the infrared region using final ALEPH results. The uncertainty on this function is also determined.