Commensurate Scale Relations and the Abelian Correspondence Principle

Commensurate scale relations are perturbative QCD predictions which relate observable to observable at fixed relative scales, independent of the choice of intermediate renormalization scheme or other theoretical conventions. A prominent example is the "generalized Crewther relation" which connects the Bjorken and Gross-Llewellyn Smith deep inelastic scattering sum rules to measurements of the $e^+ e^-$ annihilation cross section. Commensurate scale relations also provide an extension of the standard minimal subtraction scheme which is analytic in the quark masses, has non-ambiguous scale-setting properties, and inherits the physical properties of the effective charge $\alpha_V(Q^2)$ defined from the heavy quark potential. I also discuss a property of perturbation theory, the "Abelian correspondence principle", which provides an analytic constraint on non-Abelian gauge theory for $N_C \to 0.$