Optimal Renormalization Scales and Commensurate Scale Relations

Commensurate scale relations relate observables to observables and thus are independent of theoretical conventions, such as the choice of intermediate renormalization scheme. The physical quantities are related at commensurate scales which satisfy a transitivity rule which ensures that predictions are independent of the choice of an intermediate renormalization scheme. QCD can thus be tested in a new and precise way by checking that the observables track both in their relative normalization and in their commensurate scale dependence. For example, the radiative corrections to the Bjorken sum rule at a given momentum transfer $Q$ can be predicted from measurements of the e+e- annihilation cross section at a corresponding commensurate energy scale $\sqrt s \propto Q,$ thus generalizing Crewther's relation to non-conformal QCD. The coefficients that appear in this perturbative expansion take the form of a simple geometric series and thus have no renormalon divergent behavior. We also discuss scale-fixed relations between the threshold corrections to the heavy quark production cross section in e+e- annihilation and the heavy quark coupling $\alpha_V$ which is measurable in lattice gauge theory.