Fixing the renormalisation scheme in NNLO perturbative QCD using conformal limit arguments

Abstract

We discuss how the renormalisation scheme ambiguities in QCD can be fixed, when two observables are related, by requiring the coefficients in the perturbative expansion relating the two observables to have their conformal limit values, i.e. to be independent of the $\beta$-function of the renormalised coupling. We show how the next-to-leading order BLM automatic scale fixing method can be extended to next-to-next-to-leading order to fix both the renormalisation scale and $\beta_2$ in a unique way. As an example we apply the method to the relation between Bjorken's sum rule and $R_{e+e-}$ and compare with experimental data as well as other scheme fixing methods.