Principle of maximum conformality and complete renormalization group improvement approaches to optimize QCD observables: Similarity, differences, and preference

Abstract

The principle of maximum conformality (PMC) in perturbative QCD predicts energy scales which are independent of the renormalization scheme and choice of the renormalization scale. In this approach the observable can be divided to the conformal and nonconformal sectors. Nonconformal dependence terms are absorbed into the running coupling constant. By absorbing these terms at any desired order, a unique scale can be obtained at that order, and the final result for the observable is independent of the initial choice of renormalization scale and also the employed scheme. This feature is in accordance with the invariant property of renormalization group. On the other side, we consider another approach which is called the complete renormalization group improvement (CORGI) to optimize the perturbative series of QCD observables. In this approach, using the self-consistency principle, all perturbative terms can be reconstructed such that they are expressed in terms of the quantities which do not depend on the renormalization scale. Each reconstructed term in this approach involves a resummation of infinite terms at a specified order. Therefore the conventional perturbative series of QCD observables appear in terms of scheme-invariant quantities. This approach is then expected to get more reliable results with respect to what we obtain in conventional perturbation theory. Considering these two approaches, we are looking to find the differences and similarities and the probable relation which might exist between them. Since the PMC approach is founded on more strong theoretical bases, the comparison between these two approaches will reveal the priority which naturally should be assigned to the PMC approach. We examine some QCD observables like the R-ratio for ${e}^{+}{e}^{\ensuremath{-}}$ annihilation, ${R}_{\ensuremath{\tau}}$ for $\ensuremath{\tau}$ decay, and Higgs decay width to $b\overline{b}$ and gluon-gluon, and we compare the results for these observable in two approaches. For QCD observables in which there are experimental data, the results we obtain from both CORGI and PMC approaches are in good agrement with the experimental data. In other cases, the results from theses two approaches are near to each other; however, they are a little different from the results of the conventional perturbation theory.