Systematic scale-setting to all orders: The principle of maximum conformality and commensurate scale relations

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 $\beta$-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 $\beta$ 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 ${\alpha }_s^4$; 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 photon field in $SU(N)$, in accordance with conformality, and with all nonconformal 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 can be used to derive commensurate scale relations between measurable effective charges, which provide nontrivial tests of QCD to high precision. This work extends Brodsky-Lepage-Mackenzie (BLM) scale-setting to any perturbative order, with no ambiguities in identifying $\beta$ terms in pQCD, demonstrating that BLM scale-setting follows from a principle of maximum conformality.