Setting the Renormalization Scale in QCD: The Principle of Maximum Conformality

Abstract

A key problem in making precise perturbative QCD predictions is the uncertainty in determining the renormalization scale $μ$ of the running coupling $α_s(μ^2).$ The purpose of the running coupling in any gauge theory is to sum all terms involving the $β$ function; in fact, when the renormalization scale is set properly, all non-conformal $β\ne 0$ terms in a perturbative expansion arising from renormalization are summed into the running coupling. The remaining terms in the perturbative series are then identical to that of a conformal theory; i.e., the corresponding theory with $β=0$. The resulting scale-fixed predictions using the "principle of maximum conformality" (PMC) are independent of the choice of renormalization scheme -- a key requirement of renormalization group invariance. The results avoid renormalon resummation and agree with QED scale-setting in the Abelian limit. The PMC is also the theoretical principle underlying the BLM procedure, commensurate scale relations between observables, and the scale-setting method used in lattice gauge theory. The number of active flavors $n_f$ in the QCD $β$ function is also correctly determined. We discuss several methods for determining the PMC scale for QCD processes. We show that a single global PMC scale, valid at leading order, can be derived from basic properties of the perturbative QCD cross section. The elimination of the renormalization scale ambiguity and the scheme dependence using the PMC will not only increase the precision of QCD tests, but it will also increase the sensitivity of collider experiments to new physics beyond the Standard Model.