Analytic Running Coupling Research Articles

On the analytically-improved running coupling in QCD

In this paper, we examine, in the ’t Hooft renormalization scheme, the analytic running coupling [Formula: see text] in QCD, using the two-loop [Formula: see text]-function with positive expansion parameters [Formula: see text] and [Formula: see text]. An exact integral representation is derived for this causal coupling, which is fully expressed in terms of the imaginary part of the Lambert function [Formula: see text]. This integral form manifestly accounts for the universal value of the infrared limit [Formula: see text].

Synthetic Running Coupling of QCD

Based on a study of the analytic running coupling obtained from the standard perturbation theory results up to four-loop order, the QCD ``synthetic'' running coupling \alpha_{syn} is built. In so doing the perturbative time-like discontinuity is preserved and nonperturbative contributions not only remove the nonphysical singularities of the perturbation theory in the infrared region but also decrease rapidly in the ultraviolet region. In the framework of the approach, on the one hand, the running coupling is enhanced at zero and, on the other hand, the dynamical gluon mass m_g arises. Fixing the parameter which characterize the infrared enhancement corresponding to the string tension \sigma and normalization, say, at M_\tau completely define the synthetic running coupling. In this case the dynamical gluon mass appears to be fixed and the higher loop stabilization property of m_g is observed. For \sigma = (0.42 GeV)^2 and \alpha_{syn}(M^2_\tau) = 0.33 \pm 0.01 it is obtained that m_g = 530 \pm 80 MeV.

The QCD analytic running coupling and chiral symmetry breaking

We study the dependence on the pion mass of the QCD effective charge by employing the dispersion relations for the Adler D function. This new massive analytic running coupling is compared to the effective coupling saturated by the dynamically generated gluon mass. A qualitative picture of the possible impact of the former coupling on the chiral symmetry breaking is presented.

THE QCD ANALYTIC EFFECTIVE CHARGE AND ITS DEPENDENCE ON THE PION MASS

A new model for the QCD analytic running coupling, which incorporates the effects due to the π meson mass, is proposed. The properties of this invariant charge in spacelike and timelike regions are examined. Its main distinctive features are a finite infrared limiting value, which depends on the pion mass, and the "plateau–like" behavior in the deep infrared domain of the timelike region.

Massive analytic invariant charge in QCD

The low-energy behavior of a recently proposed model for the massive analytic running coupling of QCD is studied. This running coupling has no unphysical singularities, and in the absence of masses displays infrared enhancement. The inclusion of the effects due to the mass of the lightest hadron is accomplished by employing the dispersion relation for the Adler $D$ function. The presence of the nonvanishing pion mass tames the aforementioned enhancement, giving rise to a finite value for the running coupling at the origin. In addition, the effective charge acquires a plateaulike behavior in the low-energy region of the timelike domain. This plateau is found to be in agreement with a number of phenomenological models for the strong running coupling. The developed invariant charge is applied in the processing of experimental data on the inclusive $\ensuremath{\tau}$ lepton decay. The effects due to the pion mass play an essential role here as well, affecting the value of the QCD scale parameter $\ensuremath{\Lambda}$ extracted from these data. Finally, the massive analytic running coupling is compared with the effective coupling arising from the study of Schwinger-Dyson equations, whose infrared finiteness is due to a dynamically generated gluon mass. A qualitative picture of the possible impact of the former coupling on the chiral symmetry breaking is presented.

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

The QCD analytic running coupling alpha_{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>Lambda 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 four-loop 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, alpha_{an}(M_Z^2)=0.1181^{+0.002}_{-0.002}, one obtains as a result of the energy 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 the coupling strength available at the scale of the mass of the tau lepton.

Effective method for calculating the analytic QCD coupling constant

The analytic running coupling constant αan for strong interactions is considered for approximations of standard perturbation theory up to the three-loop level. Nonperturbative contributions are singled out explicitly in αan. They are represented in the form of an expansion in a series in inverse powers of the Euclidean momentum squared. It is shown that two-and three-loop corrections lead to a partial compensation of the nonperturbative one-loop contribution of order 1/q2, which is leading in the ultraviolet region. An efficient method for calculating the analytic running coupling constant for all q>Λ is developed on the basis of the above expansion. A comparative analysis of perturbative and nonperturbative contributions is performed in the infrared region, where the latter play the most important role. A simultaneous consideration of the momentum dependence of αan and its perturbative component for one-to three-loop cases leads to the conclusion that the analytic running coupling constant is stable with respect to higher corrections and that it depends only slightly on conditions imposed in matching solutions that involve different numbers nf of active-quark flavors.

NEW ANALYTIC RUNNING COUPLING IN QCD: HIGHER LOOP LEVELS

The properties of the new analytic running coupling are investigated at the higher loop levels. The expression for this invariant charge, independent of the normalization point, is obtained by invoking the asymptotic freedom condition. It is shown that at any loop level the relevant β-function has the universal behaviors at small and large values of the invariant charge. Due to this feature the new analytic running coupling possesses the universal asymptotics in both the ultraviolet and infrared regions irrespective of the loop level. The consistency of the model considered with the general definition of the QCD invariant charge is shown.

New analytic running coupling in spacelike and timelike regions

The new model for the QCD analytic running coupling, proposed recently, is extended to the timelike region. This running coupling naturally arises under unification of the analytic approach to QCD and the renormalization group (RG) formalism. A new method for determining the coefficients of the ``analytized'' RG equation is elaborated. It enables one to take into account the higher loop contributions to the new analytic running coupling (NARC) in a consistent way. The expression for the new analytic running coupling, independent of the normalization point, is obtained by invoking the asymptotic freedom condition. It is shown that the difference between the values of the NARC in respective spacelike and timelike regions is rather valuable for intermediate energies. This is essential for the correct extracting of the running coupling from experimental data. The new analytic running coupling is applied to the description of the inclusive $\ensuremath{\tau}$ lepton decay. The consistent estimation of the parameter ${\ensuremath{\Lambda}}_{\mathrm{QCD}}$ is obtained here.

INVESTIGATION OF A NEW ANALYTIC RUNNING COUPLING IN QCD

The mathematical properties of the new analytic running coupling (NARC) in QCD are investigated. This running coupling naturally arises under "analytization" of the renormalization group equation. One of the crucial points in our consideration is the relation established between the NARC and its inverse function. The latter is expressed in terms of the so-called Lambert W function. This relation enables one to present explicitly the NARC in the renorminvariant form and to derive the corresponding β function. The asymptotic behavior of this β function is examined. The consistent estimation of the parameter Λ QCD is given.

Quark-antiquark potential in the analytic approach to QCD

The quark-antiquark potential is constructed by making use of a new analytic running coupling in QCD. This running coupling arises under ``analytization'' of the renormalization group equation. The rising behavior of the quark-antiquark potential at large distances, which provides the quark confinement, is shown explicitly. At small distances, the standard behavior of this potential originating in the QCD asymptotic freedom is revealed. The higher loop corrections and the scheme dependence of the approach are briefly discussed.