Scale setting using the extended renormalization group and the principle of maximum conformality: The QCD coupling constant at four loops

Abstract

A key problem in making precise perturbative QCD predictions is to set the proper renormalization scale of the running coupling. The extended renormalization group equations, which express the invariance of the physical observables under both the renormalization scale- and scheme-parameter transformations, provide a convenient way for estimating the scale- and scheme-dependence of the physical process. In this paper, we present a solution for the scale equation of the extended renormalization group equations at the four-loop level. Using the principle of maximum conformality (PMC)/Brodsky-Lepage-Mackenzie (BLM) scale-setting method, all nonconformal ${{\ensuremath{\beta}}_{i}}$ terms in the perturbative expansion series can be summed into the running coupling, and the resulting scale-fixed predictions are independent of the renormalization scheme. The PMC/BLM scales can be fixed order-by-order. As a useful reference, we present a systematic and scheme-independent procedure for setting PMC/BLM scales up to next-to-next-to-leading order. An explicit application for determining the scale setting of ${R}_{{e}^{+}{e}^{\ensuremath{-}}}(Q)$ up to four loops is presented. By using the world average ${\ensuremath{\alpha}}_{s}^{\overline{\mathrm{MS}}}({M}_{Z})=0.1184\ifmmode\pm\else\textpm\fi{}0.0007$, we obtain the asymptotic scale for the 't Hooft scheme associated with the $\overline{\mathrm{MS}}$ scheme, ${\ensuremath{\Lambda}}_{\overline{\mathrm{MS}}}^{\ensuremath{'}tH}={245}_{\ensuremath{-}10}^{+9}\text{ }\text{ }\mathrm{MeV}$, and the asymptotic scale for the conventional $\overline{\mathrm{MS}}$ scheme, ${\ensuremath{\Lambda}}_{\overline{\mathrm{MS}}}={213}_{\ensuremath{-}8}^{+19}\text{ }\text{ }\mathrm{MeV}$.