Renormalization scheme dependence of high-Order perturbative QCD predictions

TL;DR

This paper investigates how the dependence of high-order perturbative QCD predictions on renormalization schemes and scales evolves with more loop terms, highlighting the importance of proper scale-setting methods like the principle of minimum sensitivity.

Contribution

It demonstrates that including higher-loop terms does not significantly reduce scheme dependence under conventional scale setting, emphasizing the need for optimal scale-setting approaches.

Findings

Scheme dependence remains significant with more loops under conventional scale setting.

Proper scale-setting methods can effectively reduce scheme dependence.

The principle of minimum sensitivity offers a practical solution for optimal scheme and scale choice.

Abstract

Conventionally, one adopts typical momentum flow of a physical observable as the renormalization scale for its perturbative QCD (pQCD) approximant. This simple treatment leads to renormalization scheme-and-scale ambiguities due to the renormalization scheme and scale dependence of the strong coupling and the perturbative coefficients do not exactly cancel at any fixed order. It is believed that those ambiguities will be softened by including more higher-order terms. In the paper, to show how the renormalization scheme dependence changes when more loop terms have been included, we discuss the sensitivity of pQCD prediction on the scheme parameters by using the scheme-dependent $\{\beta_{m \geq 2}\}$-terms. We adopt two four-loop examples, $e^+ e^- \to {\rm hadrons}$ and $\tau$ decays into hadrons, for detailed analysis. Our results show that under the conventional scale setting, by including more-and-more loop terms, the scheme dependence of the pQCD prediction cannot be reduced as efficiently as that of the scale dependence. Thus a proper scale-setting approach should be important to reduce the scheme dependence. We observe that the principle of minimum sensitivity could be such a scale-setting approach, which provides a practical way to achieve optimal scheme and scale by requiring the pQCD approximate be independent to the "unphysical" theoretical conventions.