Systematic Scale-Setting to All Orders: The Principle of Maximum Conformality and Commensurate Scale Relations

TL;DR

The paper develops a systematic, automation-friendly method (PMC) to set renormalization scales in perturbative QCD to all orders by absorbing nonconformal $\beta$-dependent terms into the running coupling, yielding a conformal series free of scheme and scale ambiguities. By employing a generalized δ-Renormalization scheme, it exposes the pattern and degeneracy of $\beta$-terms, enabling precise scale setting at each perturbative order. The authors demonstrate the approach on four key observables up to ${\cal O}(\alpha_s^4)$—$e^+e^- \to$ hadrons, $\tau$ decays, Bjorken/GLS sum rules, and the static quark potential—showing conformal coefficients and generalized Crewther relations, and establishing commensurate scale relations between observables. The framework is consistent with QED in the Abelian limit and supports automation for multi-scale, multi-process QCD calculations, offering scheme- and scale-fixed predictions with reduced theoretical error.

Abstract

We present in detail a new systematic method which can be used to automatically eliminate the renormalization scheme and scale ambiguities in perturbative QCD predictions at all orders. We show that all of the nonconformal β-dependent terms in a QCD perturbative series can be readily identified by generalizing the conventional renormalization schemes based on dimensional regularization. We then demonstrate that the nonconformal series of pQCD at any order can be resummed systematically into the scale of the QCD coupling in a unique and unambiguous way due to a special degeneracy of the β-terms in the series. The resummation follows from the principal of maximum conformality (PMC) and assigns a unique scale for the running coupling at each perturbative order. The final result is independent of the initial choices of renormalization scheme and scale, in accordance with the principles of the renormalization group, and thus eliminates an unnecessary source of systematic error in physical predictions. We exhibit several examples known to order α_s^4; i.e. i) the electron-positron annihilation into hadrons, ii) the tau-lepton decay to hadrons, iii) the Bjorken and Gross-Llewellyn Smith (GLS) sum rules, and iv) the static quark potential. We show that the final series of the first three cases are all given in terms of the anomalous dimension of the gluon field, in accordance with conformality, and with all non-conformal properties encoded in the running coupling. The final expressions for the Bjorken and GLS sum rules directly lead to the generalized Crewther relations, exposing another relevant feature of conformality. The static quark potential shows that PMC scale setting in the Abelian limit is to all orders consistent with QED scale setting. Finally, we demonstrate that the method applies to any renormalization scheme and [...]