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The method of dispersion relations and perturbation theory
http://lt-jds.jinr.ru/record/72292
[Sov.Phys.JETP 10(1960)574]: Both the idea and the results of the present investigation are related to Redmond's recent paper$^1$ on the exclusion of nonphysical poles from the Green's function. In contrast to that work based on the relation between the spectral representations of the Green's function and of the polarization operator we base ourselves on the principle of summing the information obtained from perturbation theory in the integrand of the Källen-Lehmann spectral integral. On summing in this way the contributions from the "principal logarithmic diagrams" we obtain expressions for the photon propagation function in quantum electrodynamics and for the meson propagation function in the symmetric theory which have all the essential properties of Redmond's result: the correct analytic behavior in the complex plane of the momentum variable $p^2$ and a singularity with respect to the variable of the square of the charge $e^2$ at the point $e^2 = 0$. However, in contrast to Redmond's result, which yields correctly only the lowest order of perturbation theory, the expressions obtained by us correspond to terms of arbitrarily high order in the perturbation theory expansions in the region of large $p^2$. By taking into account the lowest order logarithmic terms, it is shown that the region of applicability of the new formulas coincides with the region of applicability of the old formulas containing the logarithmic singularities, since it is restricted by the condition of smallness of the invariant charge. The technique of reducing the expressions so obtained to the renormalization-invariant form is illustrated by the example of the photon Green's function. In conclusion some remarks are made with respect to the possible situation in nonrenormalizable theories. Russian abstract [Zh.Eksp.Teor.Fiz. 37(1959)805]: Эта работа по направлению и по результатам примыкает к недавней работе Редмонда [1] по исключению нефизических полюсов из функций Грина. В отличие от этой работы, основывающейся на связи спектральных представлений для функции Грина и для поляризационного оператора, мы исходим из принципа суммирования информации, полученной из теории возмущений, под знаком спектрального интеграла Челлена—Лемана. Суммируя этим путем вклады от «главных логарифмических диаграмм», мы получаем выражения для функции распространения фотона в квантовой электродинамике и функции распространения мезона в симметричной теории, обладающие всеми существенными свойствами результата Редмонда: правильным аналитическим поведением в комплексной плоскости импульсной переменной $p^2$ и особенностью по переменной квадрата заряда $e^2$ в точке $e^2 = 0$. Однако в отличие от результата Редмонда, правильно передающего лишь низший порядок теории возмущений, полученные выражения соответствуют членам разложений теории возмущений в области больших $p^2$ в любых порядках. Путем учета младших логарифмических членов показано, что область применимости новых формул совпадает с областью применимости старых формул с логарифмическими особенностями, будучи ограничена условием малости инвариантного заряда. На примере функции Грина фотона проиллюстрирована техника приведения полученных выражений к ренормализационно-инвариантному виду. В заключение приведены некоторые соображения относительно возможного положения в неперенормируемых теориях.Bogolyubov, N.N.Mon, 04 Dec 2017 08:32:34 GMThttp://lt-jds.jinr.ru/record/722921960-033rd International Conference on the Renormalization Group
http://lt-jds.jinr.ru/record/68843
Shirkov, Dmitri Vasil'evichThu, 01 Oct 2015 11:06:14 GMThttp://lt-jds.jinr.ru/record/688431998Remarks on simple modified perturbation theory
http://lt-jds.jinr.ru/record/67721
The goal is to devise a pQCD modification that should be regular in the low energy region and could serve practically for the data analysis below 1 GeV up to the infra-red limit. The recently observed “blow-up” of the 4-loop pQCD series for the Bjorken sum rule form-factor around Q ≲ 1 GeV and partial resolving of the issue with the help of the Analytic Perturbation Theory (APT) until Q ∼ 0.6 GeV provided the impetus for this attempt. The “massive pQCD” under construction has two grounds. The first is pQCD with only one parameter added, an effective “glueball mass” m$_{ρ}$ ≲ M$_{glb}$ ≲ 1 GeV, serving as an infrared regulator. Roughly, we introduce it by changing the ultra-violet lnQ$^{2}$ for a massive log, ln(Q$^{2}$ + M$_{glb}^{2}$ ) regular in the low energy region and finite in the infra-red limit. The second stems from the ghost-free APT comprising non-power perturbative expansion that makes it compatible with linear integral transformations.Shirkov, D.V.Wed, 01 Apr 2015 08:07:40 GMThttp://lt-jds.jinr.ru/record/677212015-03-24
REMARKS ON SIMPLE MODIFIED PERTURBATION THEORY
http://lt-jds.jinr.ru/record/66398
D. V. ShirkovTue, 13 Jan 2015 12:12:05 GMThttp://lt-jds.jinr.ru/record/66398Joint Inst. Nucl. Res.2014Large regular QCD coupling at low energy?
http://lt-jds.jinr.ru/record/56371
Shirkov, Dmitry V.Tue, 24 Jan 2012 12:46:19 GMThttp://lt-jds.jinr.ru/record/56371Dubna International Conference on the Renormalization Group
http://lt-jds.jinr.ru/record/42901
Shirkov, Dmitri Vasil'evichFri, 17 Dec 2010 11:52:18 GMThttp://lt-jds.jinr.ru/record/42901urn:ISBN:9789971505738urn:ISBN:9971505738World Scientific19882nd International Conference on the Renormalization Group : Festschrift N N Bogolyubov
http://lt-jds.jinr.ru/record/39656
Shirkov, Dmitri Vasil'evichThu, 16 Dec 2010 08:52:28 GMThttp://lt-jds.jinr.ru/record/39656urn:ISBN:9789810208967urn:ISBN:9810208960World Scientific1992JINR-CERN School of Physics
http://lt-jds.jinr.ru/record/36461
Shirkov, Dmitri Vasil'evichWed, 15 Dec 2010 22:21:18 GMThttp://lt-jds.jinr.ru/record/36461Joint Inst. Nucl. Res.1975ANALYTICAL CALCULATIONS COMPUTER IN PHYSICS AND MATHEMATICS
http://lt-jds.jinr.ru/record/30832
Gerdt, V.P.Fri, 29 Oct 2010 11:45:36 GMThttp://lt-jds.jinr.ru/record/30832Coupling running through the Looking-Glass of dimensional Reduction
http://lt-jds.jinr.ru/record/30743
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.Shirkov, D.V.Fri, 29 Oct 2010 11:16:03 GMThttp://lt-jds.jinr.ru/record/30743Nucleon spin structure and pQCD frontier on the move
http://lt-jds.jinr.ru/record/29952
We discuss the interplay between higher orders of the perturbative QCD expansion and higher twist contributions in the analysis of recent Jefferson Lab (JLab) data on the lowest moments of spin-dependent proton and neutron structure functions $\Gamma_1^{p,n} (Q^2)$ and Bjorken sum rule function $\Gamma_1^{p-n}(Q^2)$ at $0.05<Q^2< 3 {\rm GeV}^2$. We demonstrate that the values of the higher twist coefficients $\mu^{p,n}_{2k} $ extracted from the mentioned data by using the singularity-free analytic perturbation theory provide a better convergence of the higher-twist series than with the standard pQCD. From the high-precision proton data, we extract the value of the singlet axial charge $a_0(1 \GeV^2)=0.33\pm0.05$. We observe a slow $Q^2$-dependence of fitted values of the twist coefficient $\mu_4$ and $a_0$ when going to lower energy scales, which can be explained by the well-known RG evolution of $\mu_4(Q^2)$ and $a_0(Q^2)$. As the main result, a good quantitative description of all the JLab data sets down to $Q \simeq 350 \MeV $ is achieved.Pasechnik, Roman S.Fri, 29 Oct 2010 09:41:01 GMThttp://lt-jds.jinr.ru/record/29952Anatoly Vasilievich Efremov: Known and unknown
http://lt-jds.jinr.ru/record/29870
Shirkov, Dmitry V.Fri, 29 Oct 2010 09:40:32 GMThttp://lt-jds.jinr.ru/record/29870Novel Sets of Coupling Expansion Parameters for low-energy pQCD
http://lt-jds.jinr.ru/record/29764
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.Shirkov, D.V.Fri, 29 Oct 2010 08:43:00 GMThttp://lt-jds.jinr.ru/record/29764Bjorken Sum Rule and pQCD frontier on the move
http://lt-jds.jinr.ru/record/29600
We study the interplay between higher orders of perturbative QCD (pQCD) expansion and higher twist contributions by analyzing recent $0.1 <Q^2 < 3 {\rm GeV}^2$ Jefferson Lab data on the Generalized Bjorken Sum Rule function $\Gamma_1^{p-n} (Q^2)$. The inclusion of the higher-order pQCD corrections is absorbed, with good numerical accuracy, by change of the normalization of the higher-twist terms. {\it Item}, we found that the 'denominator' form of the two-loop effective (running) coupling constant is more suitable at low $Q^2$ than the standard 'PDG' one. At the same time, the inclusion of the third and fourth loops reduces the quality of the fit, which could be a signal of the asymptotic character of the pQCD series at $\alpha_s \sim 0.3-0.4$. Its convergence may be improved by making use of the ghost-free Analytic Perturbation Theory (APT). The values of the higher-twist coefficients extracted using APT approach provide better convergence of the higher-twist series than the standard pQCD. As a main result, we achieved a good quantitative description of the data down to $Q\simeq$ 350 MeV.Pasechnik, Roman S.Fri, 29 Oct 2010 08:41:03 GMThttp://lt-jds.jinr.ru/record/29600Renormalization-group symmetries for solutions of nonlinear boundary value problems
http://lt-jds.jinr.ru/record/29302
Approximately 10 years ago, the method of renormalization-group symmetries entered the field of boundary value problems of classical mathematical physics, stemming from the concepts of functional self-similarity and of the Bogoliubov renormalization group treated as a Lie group of continuous transformations. Overwhelmingly dominating practical quantum field theory calculations, the renormalization-group method formed the basis for the discovery of the asymptotic freedom of strong nuclear interactions and underlies the Grand Unification scenario. This paper describes the logical framework of a new algorithm based on the modern theory of transformation groups and presents the most interesting results of application of the method to differential and/or integral equation problems and to problems that involve linear functionals of solutions. Examples from nonlinear optics, kinetic theory, and plasma dynamics are given, where new analytical solutions obtained with this algorithm have allowed describing the singularity structure for self-focusing of a laser beam in a nonlinear medium, studying generation of harmonics in weakly inhomogeneous plasma, and investigating the energy spectra of accelerated ions in expanding plasma bunches.Kovalev, V.F.Fri, 29 Oct 2010 08:39:34 GMThttp://lt-jds.jinr.ru/record/29302Bound state approach to the QCD coupling at low energy scales
http://lt-jds.jinr.ru/record/29048
We exploit theoretical results on the meson spectrum within the framework of a Bethe-Salpeter (BS) formalism adjusted for QCD, in order to extract an ``experimental'' coupling \alpha_s^{exp}(Q^2) below 1 GeV by comparison with the data. The relativistic BS potential follows from a proper ansatz on the Wilson loop to encode confinement. The experimental points \alpha_s^{exp}(Q^2) exhibit a reasonable agreement with the infrared safe Analytic Perturbation Theory (APT) coupling from 1 GeV down to 200 MeV, while, below this scale, give a hint on the vanishing of \alpha_s(Q^2) as Q \to 0, consistently with a ``massive'' modification of APT. 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 low energy phenomena.Baldicchi, M.Fri, 29 Oct 2010 07:22:44 GMThttp://lt-jds.jinr.ru/record/29048Renorm-group symmetry for functionals of boundary value problem solutions
http://lt-jds.jinr.ru/record/28205
Kovalev, V.F.Fri, 29 Oct 2010 07:19:47 GMThttp://lt-jds.jinr.ru/record/28205Analytic Perturbation Theory Model for QCD and Upsilon Decay
http://lt-jds.jinr.ru/record/28082
An elegant and more precise formula for the 3-loop perturbative QCD coupling is discussed. It improves the common expression (e.g., canonized by PDG) in few GeV region. On its base, we propose simple analytic Model for ghost-free QCD running couplings and their effective powers within the Analytic Perturbation Theory, in both the space-like (Euclidean) and time-like (Minkowskian) regions, very accurate in the range above 1 GeV. Effectiveness of the new Model is illustrated by the example of Upsilon(1S) decay where the standard analysis gives $\alpha_s(M_{\Ups})=0.170\pm 0.004$ value that is inconsistent with the bulk of data. Instead, we obtain $\alpha_s(M_{\Ups})=0.185\pm 0.005$ that corresponds to $\alpha_s(M_Z)=0.120\pm 0.002 $ that is close to the world average.Shirkov, D.V.Fri, 29 Oct 2010 07:19:19 GMThttp://lt-jds.jinr.ru/record/28082Ten years of the Analytic Perturbation Theory in QCD
http://lt-jds.jinr.ru/record/28050
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 Kallen--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.Shirkov, D.V.Fri, 29 Oct 2010 07:19:14 GMThttp://lt-jds.jinr.ru/record/28050QCD effective couplings in Minkowskian and Euclidean domains
http://lt-jds.jinr.ru/record/27367
We argue for essential upgrading of the defining equations (9.5) and (9.6) in Section 9.2 The QCD coupling ... of PDG review and their use for data analysis in the light of recent development of the QCD theory. Our claim is twofold. First, instead of universal expression (9.5) for QCD coupling $\bar{\alpha}_s$, one should use various ghost-free couplings $\alpha_E(Q^2), \alpha_M(s)...$ specific for a given physical representation, Euclidean, Mincowskian etc. Second, instead of power expansion (9.6) for observable, we recommend to use nonpower functional ones over particular functional sets ${{\cal A}_k(Q^2)}$, ${{\mathfrak A}_k(s)}...$ related by suitable integral transformations. We remind that use of this modified prescription results in a better correspondence of reanalyzed low energy data with the high energy ones.Shirkov, D.V.Thu, 28 Oct 2010 18:38:48 GMThttp://lt-jds.jinr.ru/record/27367Analytic perturbation theory for practitioners and upsilon decay
http://lt-jds.jinr.ru/record/27199
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.Shirkov, D.V.Thu, 28 Oct 2010 18:38:20 GMThttp://lt-jds.jinr.ru/record/27199Nonpower expansions for QCD observables at low energies
http://lt-jds.jinr.ru/record/26756
A comprehensive review is presented of the progress made in further developing the ghost-free Analytic Approach to low-energy QCD since QCD-97 meeting. It is now formulated as a logically closed ``Analytic Perturbation Theory algorithm. Its most essential feature is nonpower functional expansions for QCD observables.Shirkov, Dmitry V.Thu, 28 Oct 2010 18:37:01 GMThttp://lt-jds.jinr.ru/record/26756Ghost-free APT analysis of perturbative QCD observables
http://lt-jds.jinr.ru/record/25820
Shirkov, D.V.Thu, 28 Oct 2010 11:41:28 GMThttp://lt-jds.jinr.ru/record/25820On the QCD coupling behavior in the infrared region
http://lt-jds.jinr.ru/record/25489
The summary of nonperturbative results for the QCD invariant coupling bar{alpha}_s obtained by lattice simulations for functional integral and by solution of approximate Dyson--Schwinger equations reveals a puzzling variety of IR behavior of bar{alpha}_s(Q^2) even on a qualitative level. This, in turn, rises a question of correspondence between the results obtained so far by different groups. We analyze this issue in terms of mass-dependent coupling constant transformations and conclude that the question of the IR behavior of effective QCD coupling and of propagators is not a well--defined one and needs to be more specified.Shirkov, D.V.Thu, 28 Oct 2010 11:40:39 GMThttp://lt-jds.jinr.ru/record/25489On the Fourier transformation of renormalization invariant coupling
http://lt-jds.jinr.ru/record/25414
Integral transformations of the QCD invariant (running) coupling and of some related objects are discussed. Special attention is paid to the Fourier transformation, that is to transition from the space-time to the energy--momentum representation. The conclusion is that the condition of possibility of such a transition provides us with one more argument against the real existence of unphysical singularities observed in the perturbative QCD. The second conclusion relates to the way of translation of some singular long--range asymptotic behaviors to the infrared momentum region. Such a transition has to be performed with the due account of the Tauberian theorem. This comment relates to the recent ALPHA collaboration results on the asymptotic behavior of the QCD effective coupling obtained by lattice simulation.Shirkov, D.V.Thu, 28 Oct 2010 11:40:31 GMThttp://lt-jds.jinr.ru/record/25414