Renormalization-scale-invariant continuation of truncated QCD (QED) series -- an analysis beyond large-beta_0 approximation

An approximation algorithm is proposed to transform truncated QCD (or QED) series for observables. The approximation is a modification of the Baker-Gammel approximants, and is independent of the renormalization scale (RScl) $\mu$ -- the coupling parameter $\alpha(\mu)$ in the series and in the resulting approximants can evolve according to the perturbative renormalization group equation (RGE) to any chosen loop order. The proposed algorithm is a natural generalization of the recently proposed method of diagonal Pad\'e approximants, the latter making the result RScl-invariant in large-$\beta_0$ approximation for ${\alpha}(\mu)$. The algorithm described below can extract large amount of information from a calculated available truncated perturbative series for an observable, by implicitly resumming large classes of diagrams.