We exploit theoretical results on the meson spectrum within the framework of a Bethe-Salpeter (BS) formalism adjusted for QCD, in order to extract an experimental coupling alpha(s)exp (Q2) below 1 GeV by comparison with the data. Our results for alpha(s)exp (Q2) exhibit a good agreement with the infrared safe analytic perturbation theory (APT) coupling from 1 GeV down to 200 MeV. As a main result, we claim that the combined BS-APT theoretical scheme provides us with a rather satisfactory correlated understanding of very high- and low-energy phenomena.

We compute the cb spectrum from a first principles Salpeter equation obtained in a previous paper. For comparison we report also the heavy-light quarkonium spectrum and the hyperfine separations previously presented only in a graphical form. Notice that all results are parameter free.

The perturbative expression of the running strong coupling constant α s (Q 2 ) has an unphysical singularity for Q 2 = Λ 2 QCD . Various modification have been proposed for the infrared region. The effect of some of such proposals on the quark-antiquark spectrum is tested on a Bethe-Salpeter (second order) formalism which was successfully applied in previous papers to an overall evaluation of the spectrum in the light-light, light-heavy and heavy-heavy sectors (the only serious discrepancy with data being for the light pseudoscalar meson masses). In this paper only the cc, bb and qq (q = u or d) cases are considered and fine structure is neglected. It is found that in the bb and cc cases the results are little sensitive to the specific choice. In the light-light case the Dokshitzer et al. prescription is again essentially equivalent to the truncation prescription used in the previous calculation and it is consistent with the same a priori fixing of the quark light masses on the typical current values m u = m d = 10 MeV (only the pion mass resulting completely out of scale of about 500 MeV). With the Shirkov-Solovtsov prescription, on the contrary, a reasonable agreement with the data is obtained only at the price of using a phenomenological momentum dependent effective mass for the quark. The use of such an effective mass should amount to a correction of the free quark propagator. It is remarkable that this has also the effect of bringing the pion mass in the correct range.

In this paper we extend the work synthetically presented by M. Baldicchi, A. V. Nesterenko, G. M. Prosperi, D. V. Shirkov, and C. Simolo [Phys. Rev. Lett. 99, 242001 (2007)] and give theoretical details and a 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 exp s 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 term and a confinement term. Besides, the common perturbative strong running coupling is replaced by the ghost-free expression E Q 2 according to the prescription of analytic perturbation theory (APT). The agreement of exp s Q 2 with the APT coupling 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 s Q 2 as Q 2 approaches zero. This infrared behavior 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 a few hundred MeV to a few hundred GeV.

Abstract. In the framework of the Bethe-Salpeter formalism used in previous papers to evaluate the quarkonium spectrum, here we reverse the point of view to extract an "experimental" running coupling α exp s (Q 2 ) in the infrared (IR) region from the data. The values so obtained agree within the erros with the Shirkov-Solovtsov analytic coupling for 200 MeV < Q < 1.2 GeV, thus giving a very satisfactory unifying description of high and low energy phenomena. Below 1 GeV however α exp s (Q 2 ) seems to vanish as Q → 0 . The paper is based on a work in progress in collaboration with D. V. Shirkov.