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

Abstract

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 $s_{\mathrm{p}}>0$, the onset of the perturbative QCD continuum in Minkowski space \cite{BLR,Rafa,PPR}. To approximate the hadronic spectral function in the region $s>s_{\mathrm{p}}$, 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 ${\Lambda }_{\overline{\mathrm{MS}}}$ and the boundary energy squared $s_{\mathrm{p}}$. We carefully determine the experimental errors on these parameters which come from the errors on the invariant mass squared distribution. For the $\overline{\mathrm{MS}}$ scheme coupling constant, we obtain ${\alpha }_s(m_{\tau }^2)=0.308\pm 0.014_{exp.}$. We show that our numerical analysis is more stable against higher-order corrections than the standard one. The extracted value for the duality point $s_{\mathrm{p}}$ is found surprisingly stable against perturbation theory corrections $s_{\mathrm{d}}=1.71\pm 0.05_{\exp }\pm 0.00_{\mathrm{th}}\mathrm{Ge}{\mathrm{V}}^2$.Additionally, we recalculate the "experimental" Adler function in the infrared region using final ALEPH results. The uncertainty on this function is also determined.

33 pages, 1 postscript figure, 14 Tables. Abstract corrected, two new Tables 1 and 7 added, numbers in Eqs. 43,45,47,52 corrected, numbers in Tables 4,5,6,9,10 corrected, new references [15,16,17,18,21,46,48,50,51] added, presentation of the material improved. see also arXiv: 1112.5958 [hep-ph](2011) a Revised version of the article published in Few-Body Syst(2010) 48:143-169; Erratum: Few Body Syst. 2012 Online First

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