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.
Highlights
Within the framework of the perturbative quantum chromodynamics theory, a physical observable (ρ) can be expanded up to nth order in the strong coupling constant αs asXn ρn 1⁄4 CRi ðμÞðaRs ÞpþiðμÞ; ð1Þ i1⁄40 where R stands for the chosen renormalization scheme, p is the power of the strong coupling constant associated with the tree-level term, aRs ðμÞ 1⁄4 αRs ðμÞ/π
To show how the renormalization scheme dependence changes when more loop terms have been included, we discuss the sensitivity of perturbative QCD (pQCD) prediction on the scheme parameters by using the scheme-dependent fβm≥2g-terms
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
Summary
Within the framework of the perturbative quantum chromodynamics (pQCD) theory, a physical observable (ρ) can be expanded up to nth order in the strong coupling constant αs as. When n → ∞, the infinite series ρn→∞ corresponds to the exact value of the observable and is independent to the choices of renormalization scheme and scale This is the standard renormalization group (RG) invariance. The fixed-order prediction obtained by using the above mentioned guessed scale depends heavily on the renormalization scheme which is itself arbitrary. The fβRm≥2g-terms in the pQCD approximant of a physical observable can be inversely adopted to get the correct running behavior of the coupling constant. For the PMC scale-setting approach, if one can tick out which fβRi g-terms pertain to which perturbative order, one can achieve the correct running behavior by using the RG equation and set the correct scale for the strong coupling at this particular order. Refs. [7,23] extended the RG equation to the extended RG equations to incorporate both the scalerunning and scheme-running behaviors of the coupling, especially via this way the strong coupling at different scales and schemes can be reliably related via a continuous way, since along the evolution trajectory described by the extended RG equations, no dissimilar scales or schemes are involved
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