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

摘要

We propose a modified procedure for extracting the numerical value for thestrong coupling constant $\alpha_s$ from the $\tau$ lepton hadronic decay rateinto non-strange particles in the vector channel. We employ the concept of thequark-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 theregion $s>s_{\rm p}$, we use Analytic Perturbation Theory (APT) up to the fifthorder. A new feature of our procedure is that it enables us to extract from thedata simultaneously the QCD scale parameter $\Lambda_{\bar{\rm MS}}$ and theboundary energy squared $s_{\rm p}$. We carefully determine the experimentalerrors on these parameters which come from the errors on the invariant masssquared distribution. For the $\bar{\rm MS}$ scheme coupling constant, weobtain $\alpha_s(m^{2}_{\tau})=0.308\pm 0.014_{\rm exp.}$. We show that ournumerical analysis is more stable against higher-order corrections than thestandard one. The extracted value for the duality point $s_{\rm p}$ is foundsurprisingly stable against perturbation theory corrections $s_{\rm d}= 1.71\pm0.05_{\rm exp}\pm 0.00_{\rm th}\,\, {\rm GeV^{2}}$.Additionally, we recalculatethe "experimental" Adler function in the infrared region using final ALEPHresults. The uncertainty on this function is also determined.