The connection between ghost-free formulations of RG-invariant perturbation theory in the both Euclidean and Minkowskian regions is investigated. Our basic tool is the "double spectral representation", similar to the definition of Adler function, that stems from first principles of local QFT. It relates real functions defined in the Euclidean and Minkowskian regions.On this base we establish a simple relation between -The trick of resummation of the π 2 -terms (known from early 80s) for the invariant QCD coupling and observables in the time-like region and -Invariant Analytic Approach (devised a few years ago) with the "analyticized" coupling α an (Q 2 ) and nonpower perturbative expansion for observables in the space-like domain which are free of unphysical singularities .As a result, we formulate a self-consistent scheme -Analytic Perturbation Theory (APT) -that relates a renorm-invariant, effective coupling functions α an (Q 2 ) andα(s) , as well as non-power perturbation expansions for observables in both space-and time-like domains, that are free of extra singularities and obey better convergence in the infrared region.Then we consider the issue of the heavy quark thresholds and devise a global APT scheme for the data analysis in the whole accessible space-like and time-like domain with various numbers of active quarks.Preliminary estimates indicate that this global scheme produces results a bit different, sometimes even in the five-flavour region, on a few per cent level forᾱ s -from the usual one, thus influencing the total picture of the QCD parameter correlation.

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The renormalization group method allows improving the properties of the QCD perturbative power series in the ultraviolet region. But it ultimately leads to unphysical singularities of observables in the infrared domain. The analytic perturbation theory is the next step in improving the perturbative expansions. Specifically, it involves an additional analyticity requirement based on the causality principle and implemented in the Källen-Lehmann and Jost-Lehmann representations. This approach eliminates spurious singularities of the perturbative power series and enhances the stability of the series with respect to both higher-loop corrections and the choice of the renormalization scheme. This paper is an overview of the basic stages in developing the analytic perturbation theory in QCD, including its recent applications to describing hadronic processes. PreambleThe analytic perturbation theory (APT) method resolves the problem of unphysical (or ghost) singularities of both the invariant charge of quantum chromodynamics (QCD) and the matrix elements of the strong interaction processes. This difficulty (also known as the problem of the Moscow zero or Landau pole) first appeared in quantum electrodynamics (QED) in the mid-1950s. It played a certain dramatic role in the development of quantum field theory (QFT).In the late 1950s, Bogoliubov, Logunov, and Shirkov suggested [1] resolving this problem by merging the renormalization group (RG) method with the Källen-Lehmann representation, which implies the analyticity in the complex variable Q 2 . The APT method in QCD is based on the ideas in [1].The development of the APT over the last decade has revealed a number of new principal features of the analytic approach. Specifically, in addition to resolving the problem of unphysical singularities, the APT leads to the nonpower functional expansion for QCD observables, which has an astonishing stability (compared with the perturbative power series) with respect to both higher-loop corrections and the choice of the renormalization prescription.

We investigate a new "renormalization invariant analytic formulation" of calculations in quantum chromodynamics, where the renormalization group summation is correlated with the analyticity with respect to the square of the transferred momentum Q 2 . The expressions for the invariant charge and matrix elements are then modified such that the unphysical singularities of the ghost pole type do not appear at all, being by construction compensated by additional nonperturbative contributions. Using the new scheme, we show that the results of calculations for a number of physical processes are stable with respect to higher-loop effects and the choice of the renormalization prescription.Having in mind applications of the new formulation to inelastic lepton-nucleon scattering processes, we analyze the corresponding structure functions starting from the general principles of the theory expressed by the Jost-Lehmann-Dyson integral representation. We use a nonstandard scaling variable that leads to modified moments of the structure functions possessing Källén-Lehmann analytic properties with respect to Q 2 . We find the relation between these "modified analytic moments" and the operator product expansion.Take care of the Principles, and the Principles shall take care of you.Scientific achievements of Nikolai Nikolaevich Bogoliubov are characterized by a unique combination of determination in solving concrete scientific problems and a high level of mathematical culture. He could find the shortest path to a physical result using most general principles of the theory.The renormalization-invariant analytic approach to quantum chromodynamics exposed here and its most recent applications are based on the works [1,2,3,4] by Bogoliubov with his closest collaborators. A characteristic feature of these investigations is their strong relation with the fundamental quantum physics principles.