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

Abstract

The QCD analytic running coupling α an which has no nonphysical singularities for all Q 2 > 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 one-to 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, α an (M 2 Z ) = 0.1181 ± 0.002, one obtains as a result of the energy scale evolution of the analytic running coupling α an (M 2 τ ) = 0.2943−0.0106 that is notably lower than the estimations of the coupling strength available at the scale of the mass of the τ lepton.

On the running coupling constant in QCD

“…Here γ E is the Euler constant, B 2 = β 0 β 2 /β (105)), one can extract the three-loop scaling constant Λ (n f =5) ≃ 210 MeV [87], which lies within the errors of the pure perturbative estimate (see sec. 2.3), being the non-perturbative tail negligible around the normalization point.…”

“…with the standard three or four-loop asymptotic solution (31), one has to face with the leading singularity in z = 1 of the form (39), beside the IR log-oflog generated cut; terms accounting for these divergences acquire the form of cumbersome finite limits integral as in (113). Nonetheless, the effects of non-perturbative contributions have been widely investigated up to four-loop [86,87], both in IR (where they play the most prominent role) and UV region, by using asymptotic solution (31) as a perturbative input. While confirming at once IR stability due to the universal freezing value of the analytized coupling, its UV tail has been reduced [87] in the form of power type corrections analogous to (114).…”

“…Nonetheless, the effects of non-perturbative contributions have been widely investigated up to four-loop [86,87], both in IR (where they play the most prominent role) and UV region, by using asymptotic solution (31) as a perturbative input. While confirming at once IR stability due to the universal freezing value of the analytized coupling, its UV tail has been reduced [87] in the form of power type corrections analogous to (114). Within this approximation the n f -dependent coefficients c n are all negative up to four-loop, and their absolute values monotonously increase with n. In the large Q 2 limit, however, there is no need to sum a high number of terms, and truncation of e.g.…”

“…Obviously the main discrepancies emerge in the low energy region, where non-perturbative contributions slow down the rise of the curve. By using continuous matching up to three-loop, at the MS quark masses m b = 4.3 GeV and m c = 1.3 GeV, for the three-loop analytized case one finds roughly [87] …”

Renormalization of local quark-bilinear operators forNf=3flavors of stout link nonperturbative clover fermions

“…The expressions for the coupling g S (µ) in the MS and in the RI ′ -MOM schemes coincide to three loops and read [17]:…”