Ten years of the analytic perturbation theory in QCD

Abstract

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.

Electron–positron annihilation into hadrons at the higher-loop levels

“…(11) hereinafter, which makes Eq. (10) identical to that of both the foregoing DPT [31][32][33][34] and the so-called analytic perturbation theory 2 (APT) [54][55][56][57]. It has to be noted that, in general, the perturbative spectral function at small values of its argument may be altered by the terms of an intrinsically nonperturbative nature.…”

Search for short strings ine+e−-annihilation

“…We plan to perform an analogous analysis in the framework of the analytic perturbation theory [32][33][34].…”

Modified contour-improved perturbation theory

“…Alternatively, evaluation of low energy QCD observables can be performed in the context of analytic QCD [14] (anQCD; for reviews and further references see [15][16][17]). Such approaches have a number of advantages.…”