Renormalization Group Invariance And Optimal QCD Renormalization Scale-setting

According to the requirement of renormalization group invariance?RGI?, a valid prediction for a physical observable should be dependent on neither the choice of the renormalization scheme or that of the renormalization scale. However, for truncated finite order calculations in perturbative quantum chromodynamics?pQCD?, it is a challenging problem to satisfy the scheme and scale invariance, i.e., pQCD predictions that are truncated at fixed order have an unphysical dependence on the renormalization procedure.It is conventional to assume the renormalization scale in pQCD calculations to be equal to a typical momentum transfer of the process and vary it over an arbitrary range for error estimation. This ad hoc procedure based on a guessed scale leads to an arbitrary systematic error for the fixed-order pQCD predictions and it also leads to incorrect predictions when applied to quantum electrodynamics?QED? processes. In principle the error can be suppressed by including more-and-more loop corrections, but it is quite difficult to complete higher order p QCD calculations. Moreover, there exist the divergentn n!?ⁿ?_s^s renormalon terms, for which the pQCD series might not converge well at high orders. The elimination of such ambiguities is quite important for obtaining precise tests of the standard model?SM? at colliders such as LHC as well as for increasing the sensitivity of experimental measurements to new physics beyond SM.It is essential to have an objective way to solve the renormalization scale ambiguity problem. In order to solve the renormalization scheme and scale ambiguities, one should answer the question of how to set optimal scale systematically for any physical processes up to any orders from some basic principle of QCD theory. In this paper, we begin by introducing the renormalization group equation?RGE? and its extended version, which describe the invariance of physical observables under both the renormalization scheme and renormalization scale transformations. Using the extended RGE, we have a detailed discussion on the RGI itself as well as the renormalization scale setting problem. After a brief overview on previous ideas to set the renormalization scale, we pay particular attention to two scale setting strategies that are based on RGI, i.e., the Principle of Maximum Conformality?PMC? which respects the standard RGI, and the Principle of Minimum Sensitivity?PMS? which is based on local RGI but breaks the standard RGI. The PMC suggests that the non-conformal b terms associated with the renormalization group equation should be absorbed from the perturbation coefficients into the scale of running coupling constant at each perturbation order, which provides the underlying principle of the Brodsky-Lepage-Mackenzie?BLM? method and unambiguously extends it up to any order. The PMS determines an optimal choice of scheme and scale parameters for a pQCD prediction by requiring its slope over the scheme and scale changes to vanish. We provide the algorithms for both the PMC and the PMS scale setting and have a detailed discussion on their different properties. The comparison study is based on the analysis of R_e⁺e^-,R_? and ??H?b???? up to the four loop level. Our results show that both the PMC and the PMS can suppress the divergence caused by the renormalon terms and provide more accurate predictions with high enough order pQCD corrections taken into account. However, they behave differently: the series convergence of the PMS results is questionable while the PMC provides more convergent series. Moreover, the PMS fails to achieve a proper error estimation for its own next-to-leading order predictions, which indicates that the PMS is only a practical way to set the renormalization scale but the PMC is a more fundamental scale setting approach.