Representation of the RG-Invariant Quantities in Perturbative QCD through Powers of the Conformal Anomaly

Abstract

In this work we consider the possibility of representing the perturbative series for renormalization group invariant quantities in QCD in the form of their decomposition in powers of the conformal anomaly \({{\beta ({{\alpha }_{s}})} \mathord{\left/ {\vphantom {{\beta ({{\alpha }_{s}})} {{{\alpha }_{s}}}}} \right. \kern-0em} {{{\alpha }_{s}}}}\) in the \(\overline {{\text{MS}}} \)-scheme. We remind that such expansion is possible for the Adler function of the process of \({{e}^{ + }}{{e}^{ - }}\) annihilation into hadrons and the coefficient function of the Bjorken polarized sum rule for the deep-inelastic electron-nucleon scattering, which are both related by the CBK relation. In addition, we study the discussed decomposition for the static quark-antiquark Coulomb-like potential, its relation with the quantity defined by the cusp anomalous dimension and the coefficient function of the Bjorken unpolarized sum rule of neutrino-nucleon scattering. In conclusion we also present the formal results of applying this approach to the non-renormalization invariant ratio between the pole and \(\overline {{\text{MS}}} \)-scheme running mass of heavy quark in QCD and compare them with those already known in the literature. The arguments in favor of the validity of the considered representation in powers of \({{\beta ({{\alpha }_{s}})} \mathord{\left/ {\vphantom {{\beta ({{\alpha }_{s}})} {{{\alpha }_{s}}}}} \right. \kern-0em} {{{\alpha }_{s}}}}\) for all mentioned renorm-invariant perturbative quantities are discussed.