The Equivalence Of Two Renormalization Scale Setting Methods For The Principle Of Maximum Conformality

The dynamics for the strong interactions is known as Quantum Chromodynamics(QCD). The perturbative QCD(pQCD) is studied by using the fact that the strong coupling constant is small in high energy or short distance interactions, thus allowing perturbation theory techniques to be applied. The renormalization is needed when high order perturbative calculation is taken into account. According to the renormalization group invariance, the predictions for physical results should be independent of the choice of the renormalization scheme and scale. However, at any finite order, the conventional scale-setting procedure assigns an arbitrary range and an arbitrary systematic error, leading to renormalization-scheme and renormalization-scale ambiguities. These ambiguities always lead to errors between the experimental analysis and pQCD predictions, what’s worse, the ambiguities even give wrong predictions. How to set a proper renormalization scale of the running coupling and achieve a precise prediction becomes a key problem when applying the pQCD theory.The Principle of Maximum Conformality(PMC) eliminates QCD renormalization scale-setting uncertainties using fundamental renormalization group methods. PMC satisfy all of the principles of the renormalization group, such as existence and uniqueness of the renormalization scale, reflexivity, symmetry and transitivity. The resulting scale-fixed pQCD predictions after PMC are independent of the choice of renormalization scheme and show rapid convergence. The coefficients of the scale-fixed couplings are identical to the corresponding ‘conformal’ series with zero i?-functions.In this work, we study two scale setting methods for PMC, i.e. PMC-I and PMC-II. PMC-I is based on the PMC-BLM correspondence; the other, more recent, PMC-II uses the R?-scheme, a systematic generalization of the minimal subtraction renormalization scheme. We show that PMC-I and PMC-II scale-setting methods are in practice equivalent to each other, although two PMC differ slightly in effective scales, both methods can lead to the same resummed(‘conformal’) series up to all orders. We illustrate this equivalence for the four-loop calculations of the annihilation ratio e eR?-and the Higgs partial width ?(H ?bb) and verify that the small scale differences between the two approaches are reduced as additional renormalization group i?-terms in the pQCD expansion are taken into account. We also show that special degeneracy relations, which underly the equivalence of the two PMC approaches and the resulting conformal features of the pQCD series, are in fact general properties of non-Abelian gauge theory.