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

Coupling running through the Looking-Glass of dimensional Reduction

The dimensional reduction, in a form of transition from four to two dimensions, was used in the 90s in a context of HE Regge scattering. Recently, it got a new impetus in quantum gravity where it opens the way to renormalizability and finite short-distance behavior. We consider a QFT model $g\,\varphi^4\,$ with running coupling defined in both the two domains of different dimensionality; the \gbar(Q^2)\, evolutions being duly conjugated at the reduction scale $\,Q\sim M.$ Beyond this scale, in the deep UV 2-dim region, the running coupling does not increase any more. Instead, it {\it slightly decreases} and tends to a finite value \gbar_2(\infty) \,< \, \gbar_2(M^2)\, from above. As a result, the global evolution picture looks quite peculiar and can propose a base for the modified scenario of gauge couplings behavior with UV fixed points provided by dimensional reduction instead of leptoquarks.Comment: 8 pages, 4 figures,Version to match the one which (besides the Appendix) will appear in "Particles and Nuclei (PEPAN), Letters", v.7, No 6(162) 2010 pp 625-631. Slightly edited, one more reference and related numerical estimate adde

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

QCD coupling below 1 GeV from quarkonium spectrum

In this paper we extend the work synthetically presented in Ref.[1] and give theoretical details and complete tables of numerical results. We exploit calculations within a Bethe-Salpeter (BS) formalism adjusted for QCD, in order to extract an ``experimental'' strong coupling \alpha_s^{exp}(Q^2) below 1 GeV by comparison with the meson spectrum. The BS potential follows from a proper ansatz on the Wilson loop to encode confinement and is the sum of a one-gluon-exchange and a confinement terms. Besides, the common perturbative strong coupling is replaced by the ghost-free expression \alpha_E(Q^2) according to the prescription of Analytic Perturbation Theory (APT). The agreement of \alpha_s^{exp}(Q^2) with the APT coupling \alpha_E(Q^2) turns out to be reasonable from 1 GeV down to the 200 MeV scale, thus confirming quantitatively the validity of the APT prescription. Below this scale, the experimental points could give a hint on the vanishing of \alpha_s(Q^2) as Q approaches zero. This infrared behaviour would be consistent with some lattice results and a ``massive'' generalization of the APT approach. As a main result, we claim that the combined BS-APT theoretical scheme provides us with a rather satisfactory correlated understanding of very high and rather low energy phenomena from few hundreds MeV to few hundreds GeV.Comment: Preliminary revision. Typos corrected, comments and references adde

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.

Solutions of the Klein-Gordon equation on manifolds with variable geometry including dimensional reduction

We develop the recent proposal to use dimensional reduction from the four-dimensional space-time D=(1+3) to the variant with a smaller number of space dimensions D=(1+d), d < 3, at sufficiently small distances to construct a renormalizable quantum field theory. We study the Klein-Gordon equation on a few toy examples ("educational toys") of a space-time with variable special geometry, including a transition to a dimensional reduction. The examples considered contain a combination of two regions with a simple geometry (two-dimensional cylindrical surfaces with different radii) connected by a transition region. The new technique of transforming the study of solutions of the Klein-Gordon problem on a space with variable geometry into solution of a one-dimensional stationary Schr\"odinger-type equation with potential generated by this variation is useful. We draw the following conclusions: (1) The signal related to the degree of freedom specific to the higher-dimensional part does not penetrate into the smaller-dimensional part because of an inertial force inevitably arising in the transition region (this is the centrifugal force in our models). (2) The specific spectrum of scalar excitations resembles the spectrum of the real particles; it reflects the geometry of the transition region and represents its "fingerprints". (3) The parity violation due to the asymmetric character of the construction of our models could be related to violation of the CP symmetry.Comment: laTeX file, 9 pages, 8 figures. Significant corrections in the title, abstract, text. Corrected formulas and figures. Added new references, amendments in English. Acceptred for publication in Theoretical and Mathematical Physics. To appear in vol. 167, may 201

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.

Exact renormalization group approach in scalar and fermionic theories

The Polchinski version of the exact renormalization group equation is discussed and its applications in scalar and fermionic theories are reviewed. Relation between this approach and the standard renormalization group is studied, in particular the relation between the derivative expansion and the perturbation theory expansion is worked out in some detail.Comment: 15 pages, 2 Postscript figures, Latex, uses sprocl.sty which is included; contribution to the Proceedings of the Meeting "Renormalization Group - 96" (August 26 - 31, 1996, Dubna, Russia); misprints are corrected, some minor changes are made and one reference is added in the revised versio

Applying generalized Pad\'e approximants in analytic QCD models

A method of resummation of truncated perturbation series, related to diagonal Pad\'e approximants but giving results independent of the renormalization scale, was developed more than ten years ago by us with a view of applying it in perturbative QCD. We now apply this method in analytic QCD models, i.e., models where the running coupling has no unphysical singularities, and we show that the method has attractive features such as a rapid convergence. The method can be regarded as a generalization of the scale-setting methods of Stevenson, Grunberg, and Brodsky-Lepage-Mackenzie. The method involves the fixing of various scales and weight coefficients via an auxiliary construction of diagonal Pad\'e approximant. In low-energy QCD observables, some of these scales become sometimes low at high order, which prevents the method from being effective in perturbative QCD where the coupling has unphysical singularities at low spacelike momenta. There are no such problems in analytic QCD.Comment: 14 pages; extended presentation of the analytic QCD models in Sec.IV; two references added ([37,38]); version to appear in Phys.Rev.

Analogs of noninteger powers in general analytic QCD

In contrast to the coupling parameter in the usual perturbative QCD (pQCD), the coupling parameter in the analytic QCD models has cuts only on the negative semiaxis of the Q^2-plane (where q^2 = -Q^2 is the momentum squared), thus reflecting correctly the analytic structure of the spacelike observables. The Minimal Analytic model (MA, named also APT) of Shirkov and Solovtsov removes the nonphysical cut (at positive Q^2) of the usual pQCD coupling and keeps the pQCD cut discontinuity of the coupling at negative Q^2 unchanged. In order to evaluate in MA the physical QCD quantities whose perturbation expansion involves noninteger powers of the pQCD coupling, a specific method of construction of MA analogs of noninteger pQCD powers was developed by Bakulev, Mikhailov and Stefanis (BMS). We present a construction, applicable now in any analytic QCD model, of analytic analogs of noninteger pQCD powers; this method generalizes the BMS approach obtained in the framework of MA. We need to know only the discontinuity function of the analytic coupling (the analog of the pQCD coupling) along its cut in order to obtain the analytic analogs of the noninteger powers of the pQCD coupling, as well as their timelike (Minkowskian) counterparts. As an illustration, we apply the method to the evaluation of the width for the Higgs decay into b+(bar b) pair.Comment: 29 pages, 5 figures; sections II and III extended, appendix B is ne