$\{\beta\}$-expansion in QCD, its conformal symmetry limit: theory + applications

Abstract

The basis of the $\{\beta\}$-expansion for the perturbative series evaluated in the $\overline{MS}$ scheme for the renormalization group invariant quantities is summarized.Comparison with a similar representation,used within the BLM-motivated Principle of Maximal Conformality,is discussed.We stress that the original $\{\beta\}$-expansion contains a completed list of terms rather than its PMC analog. The arguments in favour of the complete $\{\beta\}$-expansion are presented. They are based on the relations which follow from the power $\beta$-function generalization of the Crewther relation for the nonsinglet $\overline{MS}$ contributions to the Adler $D^{NS}$-functionand to the Bjorken sum rule $C^{Bjp}_{NS}$ of the polarized lepton-nucleon scattering. The terms of the complete $\{\beta\}$-expansionat the $O(\alpha_s^3)$ level for $D^{NS}$ and $C^{Bjp}_{NS}$ are presented. These perturbative results are expressed in the PMC-type form. The problem of applications of these expressions for phenomenological applications is summarized.