## Renormalization Group Invariance and Optimal QCD Renormalization Scale-Setting

Preprint number: CP3-Origins-2014-14 DNRF90 and DIAS-2014-14
Authors: Stanley J. Brodsky (SLAC, Stanford University), Hai-Bing Fu (Dep. of Physics, Chongqing University), Hong-Hao Ma (Dep. of Physics, Chongqing University), Yang Ma (Dep. of Physics, Chongqing University), Matin Mojaza (CP3-Origins & DIAS), Sheng-Quan Wang (Dep. of Physics, Chongqing University), and Xing-Gang Wu (Dep. of Physics, Chongqing University)
A valid prediction from quantum field theory for a physical observable should be independent of the choice of renormalization scheme — this is the primary requirement of renormalization group invariance (RGI). Satisfying scheme invariance is a challenging problem for perturbative QCD (pQCD), since truncated perturbation series do not automatically satisfy the requirements of the renormalization group. Two distinct approaches for satisfying the RGI principle have been suggested in the literature. One is the “Principle of Maximum Conformality” (PMC) in which the terms associated with the $\beta$-function are absorbed into the scale of the running coupling at each perturbative order; its predictions are scheme and scale independent at every finite order. The other approach is the “Principle of Minimum Sensitivity” (PMS), which is based on local RGI; the PMS approach determines the optimal renormalization scale by requiring the slope of the approximant of an observable to vanish. In this paper, we present a detailed comparison of the PMC and PMS procedures by analyzing two physical observables $$R_{e+e-}$$ and $$\Gamma(H\to b\bar{b})$$ up to four-loop order in pQCD. At the four-loop level, the PMC and PMS predictions for both observables agree within small errors with those of conventional scale setting, and each prediction shows small scale dependences. However, the convergence of the pQCD series at high orders, behaves quite differently: The PMC displays the best pQCD convergence since it eliminates the divergent renormalon terms; in contrast, the convergence of the PMS prediction is questionable, often even worse than the conventional prediction based on an arbitrary guess for the renormalization scale. PMC predictions also have the property that any residual dependence on the choice of initial scale is highly suppressed even for low-order predictions. Thus the PMC, based on the standard RGI, has a rigorous foundation; it eliminates a unnecessary systematic error for high precision pQCD predictions and can be widely applicable to many high-energy hadronic processes, including multi-scale problems.