Authors Referees APS Members |
December 2007, to be published in Physical Review D1
QCD coupling below 1 GeV from the quarkonium spectrum
In this paper we extend the work synthetically presented in Ref.~\cite{Baldicchi:2007ic} and give theoretical details and complete set of numerical results. We exploit calculations within a Bethe-Salpeter (BS) formalism adjusted for QCD, in order to extract an ``experimental'' strong coupling $\alpha_{\rm s}^{\rm exp}(Q^2)$ below 1~GeV by comparison with the meson spectrum. The BS potential follows from a proper ansatz on the Wilson loop to encode confinement and is the sum of a one-gluon-exchange and a confinement terms. Besides, the common perturbative strong running coupling is replaced by the ghost-free expression $\alpha_{\rm E}(Q^2)$ according to the prescription of Analytic Perturbation Theory~(APT).\\ The agreement of $\alpha_{\rm s}^{\rm exp}(Q^2)$ with the APT coupling $\alpha_{\rm E}(Q^2)$ turns out to be reasonably good from 1~GeV down to 200~MeV, thus confirming quantitatively the validity of the APT prescription in this range. Below this scale, the experimental points could give a hint on the vanishing of $\alpha_{\rm s}(Q^2)$ as $Q^2$ approaches zero. This infrared behaviour would be consistent with some lattice results as well as with the ``massive'' modification of the APT approach.\\ As a main result, we claim that the combined BS-APT theoretical scheme provides quite satisfactory correlated understanding of very high and rather low energy phenomena from few hundreds MeV to few hundreds~GeV. © 2008 The American Physical Society.
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