We try to review the main current ideas and points of view on the running coupling constant in QCD. We begin by recalling briefly the classic analysis based on the Renormalization Group Equations with some emphasis on the exact solutions for a given number of loops, in comparison with the usual approximate expressions. We give particular attention to the problem of eliminating the unphysical Landau singularities, and of defining a coupling that remains significant at the infrared scales. We consider various proposal of couplings directly related to the quark-antiquark potential or to other physical quantities (effective charges) and discuss optimization in the choice of the scale parameter and of the renormalization scheme. Our main focus is, however, on dispersive methods, their application, their relation with non-perturbative effects. We try also to summarize the main results obtained by Lattice simulations in particular in various MOM schemes. We conclude briefly recalling the traditional comparison with the experimental data.

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.

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.

A program package, which facilitates computations in the framework of Analytic approach to QCD, is developed and described in details. The package includes the explicit expressions for relevant spectral functions calculated up to the four-loop level and the subroutines for necessary integrals. Nature of problem: Subroutines helping computations within Analytic approach to QCD Solution method: A program package for Maple is provided. It includes the explicit expressions for relevant spectral functions and the subroutines for basic integrals used in the framework of Analytic approach to QCD.