Analytic Perturbation Theory for Practitioners and Upsilon Decay

Within the ghost-free Analytic Perturbation Theory (APT), devised in the last decade for low energy QCD, simple approximations are proposed for 3-loop analytic couplings and their effective powers, in both the space-like (Euclidean) and time-like (Minkowskian) regions, accurate enough in the large range (1--100 GeV) of current physical interest.\par Effectiveness of the new Model is illustrated by the example of $\Upsilon(1\mathrm{S})$ decay where the standard analysis gives $\alpha_s(M_{\Upsilon})=0.170\pm 0.004$ value that is inconsistent with the bulk of data for $\alpha_s$. Instead, we obtain $\alpha_s^{Mod}(M_{\Upsilon})=0.185\pm 0.005$ that corresponds to $\alpha_s^{Mod}(M_Z)=0.120\pm 0.002$ that is close to the world average.\par The issue of scale uncertainty for $\Upsilon$ decay is also discussed.Comment: 12 pages, 0 figures. Model slightly modified to increase its accuracy. Numerical results upgraded, references added. The issue of scale uncertainty is discusse

Ten years of the Analytic Perturbation Theory in QCD

The renormalization group method enables one to improve the properties of the QCD perturbative power series in the ultraviolet region. However, it ultimately leads to the unphysical singularities of observables in the infrared domain. The Analytic Perturbation Theory constitutes the next step of the improvement of perturbative expansions. Specifically, it involves additional analyticity requirement which is based on the causality principle and implemented in the K\"allen--Lehmann and Jost--Lehmann representations. Eventually, this approach eliminates spurious singularities of the perturbative power series and enhances the stability of the latter with respect to both higher loop corrections and the choice of the renormalization scheme. The paper contains an overview of the basic stages of the development of the Analytic Perturbation Theory in QCD, including its recent applications to the description of hadronic processes.Comment: 26 pages, 9 figures, to be published in Theor. Math. Phys. (2007

Novel Sets of Coupling Expansion Parameters for low-energy pQCD

In quantum theory, physical amplitudes are usually presented in the form of Feynman perturbation series in powers of coupling constant \al . However, it is known that these amplitudes are not regular functions at $\alpha=0 .$ For QCD, we propose new sets of expansion parameters {\bf w}_k(\as) that reflect singularity at \as=0 and should be used instead of powers \as^k. Their explicit form is motivated by the so called Analytic Perturbation Theory. These parameters reveal saturation in a strong coupling case at the level \as^{eff}(\as\gg1)={\bf w}_1(\as\gg 1) \sim 0.5 . They can be used for quanitative analysis of divers low-energy amplitudes. We argue that this new picture with non-power sets of perturbation expansion parameters, as well as the saturation feature, is of a rather general nature.Comment: 8 pages, 1 figure, submitted to Part. Nucl. Phys. Let

Renorm-group, Causality and Non-power Perturbation Expansion in QFT

The structure of the QFT expansion is studied in the framework of a new "Invariant analytic" version of the perturbative QCD. Here, an invariant (running) coupling $a(Q^2/\Lambda^2)=\beta_1\alpha_s(Q^2)/4\pi$ is transformed into a "$Q^2$--analytized" invariant coupling $a_{\rm an}(Q^2/\Lambda^2) \equiv {\cal A}(x)$ which, by constuction, is free of ghost singularities due to incorporating some nonperturbative structures. Meanwhile, the "analytized" perturbation expansion for an observable $F$, in contrast with the usual case, may contain specific functions ${\cal A}_n(x)= [a^n(x)]_{\rm an}$, the "n-th power of $a(x)$ analytized as a whole", instead of $({\cal A}(x))^n$. In other words, the pertubation series for $F(x)$, due to analyticity imperative, may change its form turning into an {\it asymptotic expansion \`a la Erd\'elyi over a nonpower set} $\{{\cal A}_n(x)\}$. We analyse sets of functions $\{{\cal A}_n(x)\}$ and discuss properties of non-power expansion arising with their relations to feeble loop and scheme dependence of observables. The issue of ambiguity of the invariant analytization procedure and of possible inconsistency of some of its versions with the RG structure is also discussed.Comment: 12 pages, LaTeX To appear in Teor. Mat. Fizika 119 (1999) No.

The 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 "plateau-like" 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 $\tau$ lepton decay. The effects due to the pion mass play an essential role here as well, affecting the value of the QCD scale parameter $\Lambda$ 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.Comment: 13 pages, 7 figures, revtex

Infrared enhanced analytic coupling and chiral symmetry breaking in QCD

We study the impact on chiral symmetry breaking of a recently developed model for the QCD analytic invariant charge. This charge contains no adjustable parameters, other than the QCD mass scale $\Lambda$, and embodies asymptotic freedom and infrared enhancement into a single expression. Its incorporation into the standard form of the quark gap equation gives rise to solutions for the dynamically generated mass that display a singular confining behaviour at the origin. Using the Pagels-Stokar method we relate the obtained solutions to the pion decay constant $f_{\pi}$, and estimate the scale parameter $\Lambda$, in the presence of four active quarks, to be about 880 MeV.Comment: 14 pages, 3 figures; to appear in J. Phys.

Nonperturbative Contributions in an Analytic Running Coupling of QCD

In the framework of analytic approach to QCD the nonperturbative contributions in running coupling of strong interaction up to 4-loop order are obtained in an explicit form. For all $Q>\Lambda$ they are shown to be represented in the form of an expansion in inverse powers of Euclidean momentum squared. The expansion coefficients are calculated for different numbers of active quark flavors $n_f$ and for different number of loops taken into account. On basis of the stated expansion the effective method for precise calculation of the analytic running coupling can be developed.Comment: 9 pages, LaTeX, 1 table, 1 eps figur