Explicit expressions for Euclidean and Minkowskian QCD observables in analytic perturbation theory

Abstract

Technical aspects of the Shirkov-Solovtsov's analytic perturbation theory (APT) are considered. We construct explicitly two sets of specific functions, ${\mathfrak{A}_n(s)}$ and ${{\cal A}_n(Q^2)}$ that determine the nonpower as ymptotic expansions for Minkowskian and Euclidean QCD observables in APT. The results, up to third order, are written in terms of the Lambert W-functions. As an input we used the exact two loop and the three loop (corresponding to Pad\'e transformed be ta-function) RG solutions for common invariant coupling $\alpha_s$. In addition, the exact three-loop coupling is expanded in powers of the exact two-loop solution. The excellent accuracy is achieved with few terms of this series. We derive order by order elegant systems of equations for both sets of the functions. Then we construct the global versions of the APT functions with quark thresholds in the $\bar{MS}$ scheme and give numerical results.