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

Abstract

The strong corrections to the R-ratio of electronpositron annihilation into hadrons are studied at the higherloop levels. Specifically, the derivation of a general form of the commonly employed approximate expression for the Rratio (which constitutes its truncated re-expansion at high energies) is delineated, the appearance of the pertinent π 2 -terms is expounded, and their basic features are examined. It is demonstrated that the validity range of such approximation is strictly limited to √ s/Λ > exp(π/2) 4.81 and that it converges rather slowly when the energy scale approaches this value. The spectral function required for the proper calculation of the R-ratio is explicitly derived and its properties at the higher-loop levels are studied. The developed method of calculation of the spectral function enables one to obtain the explicit expression for the latter at an arbitrary loop level. By making use of the derived spectral function the proper expression for the R-ratio is calculated up to the five-loop level and its properties are examined. It is shown that the loop convergence of the proper expression for the R-ratio is better than that of its commonly employed approximation. The impact of the omitted higher-order π 2 -terms on the latter is also discussed.

Renormalization scheme and gauge (in)dependence of the generalized Crewther relation: what are the real grounds of the β-factorization property?

“…The Adler function and the R-ratio are related through the following Källen-Lehmann dispersion representation 4) and begin to differ in the Minkowski region from three-loop level due to the effects of the analytical continuation, studied in refs. [4][5][6]: The second term C NS Bjp,Born in the l.h.s. of eq.…”

Explicit form of the R-ratio of electron–positron annihilation into hadrons

“…The function V m n (s) (41) constitutes the generalization of the function v m n (s) specified in Refs. [25,26] and the details of its derivation are quite similar to those given therein. It is worthwhile to note that in Eqs.…”

“…Moreover, the approximation R (ℓ) appr (s) (24) becomes quite inaccurate when s approaches the lower bound of its validity range √ s/Λ > exp(π/2) ≃ 4.81 and its loop convergence is worse than that of the expression (19), see Refs. [20][21][22][23][24][25][26] and references therein for a detailed discussion of these issues. It is worthwhile to mention also that the explicit expression for the perturbative spectral function ρ (ℓ) (σ) (20) has recently been derived at an arbitrary loop level in Refs.…”

Calculation of the pion-photon transition form factor using dispersion relations and renormalization-group summation

“…To avoid this problem, analytic versions of the power-series expansion in α s (better say, non-power expansions) have been proposed by various authors, e.g., [31][32][33][34][35][36][37][38] (see [39] for a review and further references while more recent developments are discussed, for instance, in [40,41]). Such schemes make use of dispersion relations in the spacelike and the timelike regions in order to implement causality while preserving the RG properties-see [42] for a broad review of such methods.…”