Conformal Symmetry as a Template: Commensurate Scale Relations and Physics Renormalization Schemes

Commensurate scale relations are perturbative QCD predictions which relate observable to observable at fixed relative scale, such as 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. We show how conformal symmetry provides a template for such QCD predictions, providing relations between observables which are present even in theories which are not scale invariant. All non-conformal effects are absorbed by fixing the ratio of the respective momentum transfer and energy scales. In the case of fixed-point theories, commensurate scale relations relate both the ratio of couplings and the ratio of scales as the fixed point is approached. In the case of the $\alpha_V$ scheme defined from heavy quark interactions, virtual corrections due to fermion pairs are analytically incorporated into the Gell-Mann Low function, thus avoiding the problem of explicitly computing and resumming quark mass corrections related to the running of the coupling. Applications to the decay width of the Z boson, the BFKL pomeron, and virtual photon scattering are discussed.