Strong coupling constant to four loops in the analytic approach to QCD

The QCD analytic running coupling αan which has no nonphysical singularities for all Q2 > 0 is considered for the initial perturbation theory approximations up to four loop order. The finiteness of the analytic coupling at zero is shown to be a consequence of the asymptotic freedom property of the initial theory. The nonperturbative contributions to the analytic coupling are extracted explicitly. For all Q > Λ they are represented in the form of an expansion in inverse powers of Euclidean momentum squared. The effective method for a precise calculation of the analytic running coupling is developed on the basis of the stated expansion. The energy scale evolution of the analytic running coupling for the oneto fourloop cases is studied and the higher loop stability and low dependence on the quark threshold matching conditions in comparison with the perturbative running coupling were found. Normalizing the analytic running coupling at the scale of the rest mass of the Z boson with the world average value of the strong coupling constant, αan(M 2 Z) = 0.1181 ± 0.002, one obtains as a result of the energy scale evolution of the analytic running coupling αan(M 2 τ ) = 0.2943 +0.0111 −0.0106 that is notably lower than the estimations of the coupling strength available at the scale of the mass of the τ lepton. PACS number(s): 12.38.Aw, 12.38.Lg ∗Electronic address: [email protected]