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The QCD analytic running coupling alpha_{an} which has no nonphysicalsingularities for all Q^2>0 is considered for the initial perturbation theoryapproximations up to four loop order. The finiteness of the analytic couplingat zero is shown to be a consequence of the asymptotic freedom property of theinitial theory. The nonperturbative contributions to the analytic coupling areextracted explicitly. For all Q>Lambda they are represented in the form of anexpansion in inverse powers of Euclidean momentum squared. The effective methodfor a precise calculation of the analytic running coupling is developed on thebasis of the stated expansion. The energy scale evolution of the analyticrunning coupling for the one- to four-loop cases is studied and the higher loopstability and low dependence on the quark threshold matching conditions incomparison with the perturbative running coupling were found. Normalizing theanalytic running coupling at the scale of the rest mass of the Z boson with theworld average value of the strong coupling constant,alpha_{an}(M_Z^2)=0.1181^{+0.002}_{-0.002}, one obtains as a result of theenergy scale evolution of the analytic running coupling alpha_{an}(M_tau^2)=0.2943^{+0.0111}_{-0.0106} that is notably lower than the estimations of thecoupling strength available at the scale of the mass of the tau lepton.Comment: 30 pages, LATEX, 4 tables, 8 figure

Strong Coupling Constant to Four Loops in the Analytic Approach to QCD