Relating physical observables in QCD using the extended renormalization group method

We discuss the St\"uckelberg-Peterman extended renormalization group equations in perturbative QCD, which express the invariance of physical observables under renormalization-scale and scheme-parameter transformations. We introduce a universal coupling function that covers all possible choices of scale and scheme. Any perturbative series in QCD is shown to be equivalent to a particular point in this function. This function can be computed from a set of first-order differential equations involving the extended β functions. We propose the use of these evolution equations instead of a perturbative series for numerical evaluation of physical observables. This formalism is free of scale-scheme ambiguity and allows a reliable error analysis of higher-order corrections. It also provides a precise definition for Λ_-MS as the pole in the associated 't Hooft scheme. A concrete application to R(e⁺e^-→hadrons) is presented.