Detailed comparison of renormalization scale-setting procedures based on the principle of maximum conformality

The Principle of Maximum Conformality (PMC), which generalizes the conventional Gell-Mann-Low method for scale-setting in perturbative QED to non-Abelian QCD, provides a rigorous method for achieving unambiguous scheme-independent, fixed-order predictions for physical observables consistent with the principles of the renormalization group. In addition to the original multi-scale-setting approach (PMCm), two variations of the PMC have been proposed to deal with ambiguities associated with the uncalculated higher order terms in the pQCD series, i.e. the single-scale-setting approach (PMCs) and the procedures based on "intrinsic conformality" (PMC infinity). In this paper, we will give a detailed comparison of these PMC approaches by comparing their predictions for three important quantities Re+e-, Rr, and P(H -> b b over bar ) up to four-loop pQCD corrections. The PMCs approach determines an overall effective running coupling alpha s(Q) by the recursive use of the renormalization group equation, whose argument Q represents the actual mo