QCD coupling up to third order in standard and analytic perturbation theories

Abstract

We analyze two sets of specific functions, that/which form the basis of the nonpower asymptotic expansions both in the timelike and spacelike regions for single scale dependent QCD observables in the Shirkov--Solovtsov's Analytic Perturbation Theory (APT) free of unphysical singularities. These functions are explicitly derived up to the third order in the closed form in terms of the Lambert-W function. As an input we used the exact two loop and the three loop (corresponding to Pad\'e transformed beta-function) RG solutions for common invariant coupling \alpha_s. The elegant recurrence formulas, helpful for numerical analysis, are obtained for the both sets of the APT functions. Then we construct the global versions of APT functions using the continuity conditions (at the quark thresholds) on the \alpha_s in the \bar{MS} scheme and give numerical results. For first three of these functions \mathfrak{A}_n(s) and {\cal A}_n(Q^2); n=1,2,3 in the large interval of the momentum transfer and energy (1 GeV <Q,{\sqrt s}< 170 GeV), numerical tables are presented. From these we observe that, for the timelike arguments, the differences between functions \mathfrak{A}_n(s) and the corresponding powers of the standard iteratively approximated coupling \alpha_s^n(s) are not negligible even for moderate energies in the five--flavor region.