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We discuss the new model expressionᾱ an (Q 2 ) recently obtained for the QCD running coupling with a regular ghost-free behavior in the "low Q 2 " region. Being deduced from the standard "asymptotic-freedom" expression by imposing the Q 2 -analyticity -without any adjustable parameters -it obeys nice features: (i) The universal limiting valueᾱ an (0) = 4π/β 0 ≃ 1.4 expressed only via group symmetry factors and independent of experimental estimates on the running couplingᾱ s (Q 2 ) (of QCD scale parameter Λ). This value turns out to be stable with respect to higher order corrections; (ii) Stability of IR behavior with respect to higher-loop effects; (iii) Coherence between the experimentalᾱ an (M 2 τ ) value and integral information on IRᾱ s (Q 2 ) behavior as extracted from jet physics 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.

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The problem of investigation of ultraviolet asymptotics in quantum field theory has met recently with specific difficulties related to the asymptotic character of power expansions of quantum perturbation theory. This article is a review on the present status of this problem. First, we discuss the saddle‐point method for the path integral, by which many important results have been obtained in the last years. Then a sketch of results is given concerning the asymptotic series in problems of quantum field theory and quantum mechanics. Next we consider the problem of “summation” of such series, which arises in attempting to reach the region of not small values of the coupling constant g.