1111.6175 (Stanley J. Brodsky et al.)

Scale Setting Using the Extended Renormalization Group and the Principle
of Maximum Conformality: the QCD Coupling Constant at Four Loops    [PDF]

Stanley J. Brodsky, Xing-Gang Wu

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 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 non-conformal
$\{\beta_i\}$ 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. Different schemes lead to different effective
PMC/BLM scales, but the final results are scheme independent. Conversely, from
the requirement of scheme independence, one not only can obtain
scheme-independent commensurate scale relations among different observables,
but also determine the scale displacements among the PMC/BLM scales which are
derived under different schemes. In principle, the PMC/BLM scales can be fixed
order-by-order, and as a useful reference, we present a systematic and
scheme-independent procedure for setting PMC/BLM scales up to NNLO. An explicit
application for determining the scale setting of $R_{e^{+}e^-}(Q)$ up to four
loops is presented. By using the world average $\alpha^{\bar{MS}}_s(M_Z)
=0.1184 \pm 0.0007$, we obtain the asymptotic scale for the 't Hooft associated
with the $\bar{MS}$ scheme, $\Lambda^{'tH}_{\bar{MS}}= 245^{+9}_{-10}$ MeV, and
the asymptotic scale for the conventional $\bar{MS}$ scheme,
$\Lambda_{\bar{MS}}= 213^{+19}_{-8}$ MeV.

View original: http://arxiv.org/abs/1111.6175