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The renormalization scale-setting problem in QCD

机译：QCD中的重新规范化尺度设定问题

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A key problem in making precise perturbative QCD predictions is to set the proper renormalization scale of the running coupling. The conventional scale-setting procedure assigns an arbitrary range and an arbitrary systematic error to fixed-order pQCD predictions. In fact, this ad hoc procedure gives results which depend on the choice of the renormalization scheme, and it is in conflict with the standard scale-setting procedure used in QED. Predictions for physical results should be independent of the choice of the scheme or other theoretical conventions. We review current ideas and points of view on how to deal with the renormalization scale ambiguity and show how to obtain renormalization scheme- and scale-independent estimates. We begin by introducing the renormalization group (RG) equation and an extended version, which expresses the invariance of physical observables under both the renormalization scheme and scale-parameter transformations. The RG equation provides a convenient way for estimating the scheme- and scale-dependence of a physical process. We then discuss self-consistency requirements of the RG equations, such as reflexivity, symmetry, and transitivity, which must be satisfied by a scale-setting method. Four typical scale setting methods suggested in the literature, i.e., the Fastest Apparent Convergence (FAC) criterion, the Principle of Minimum Sensitivity (PMS), the Brodsky-Lepage-Mackenzie method (BLM), and the Principle of Maximum Conformality (PMC), are introduced. Basic properties and their applications are discussed. We pay particular attention to the PMC, which satisfies all of the requirements of RG invariance. Using the PMC, all non-conformal terms associated with the β-function in the perturbative series are summed into the running coupling, and one obtains a unique, scale-fixed, scheme-independent prediction at any finite order. The PMC provides the principle underlying the BLM method, since it gives the general rule for extending BLM up to any perturbative order; in fact, they are equivalent to each other through the PMC-BLM correspondence principle. Thus, all the features previously observed in the BLM literature are also adaptable to the PMC. The PMC scales and the resulting finite-order PMC predictions are to high accuracy independent of the choice of the initial renormalization scale, and thus consistent with RG invariance. The PMC is also consistent with the renormalization scale-setting procedure for QED in the zero-color limit. The use of the PMC thus eliminates a serious systematic scale error in perturbative QCD predictions, greatly improving the precision of empirical tests of the Standard Model and their sensitivity to new physics.
机译：进行精确的扰动QCD预测时的关键问题是设置运行耦合的适当的重归一化尺度。常规的比例设置过程将任意范围和任意系统误差分配给固定顺序的pQCD预测。实际上，该临时过程所提供的结果取决于重归一化方案的选择，并且与QED中使用的标准比例设置过程相冲突。物理结果的预测应独立于方案或其他理论惯例的选择。我们回顾了有关如何处理重归一化尺度歧义的当前观点和观点，并展示了如何获得与重归一化方案和尺度无关的估计。我们首先介绍重归一化组（RG）方程和一个扩展版本，它表示在重归一化方案和比例参数转换下物理可观测值的不变性。 RG方程提供了一种方便的方法来估算物理过程的方案依赖性和规模依赖性。然后，我们讨论了RG方程的自洽性要求，例如自反性，对称性和可传递性，这必须通过比例设置方法来满足。文献中提出了四种典型的标度设置方法，即最快表观收敛（FAC）准则，最小灵敏度原理（PMS），Brodsky-Lepage-Mackenzie方法（BLM）和最大适形原理（PMC） ，介绍。讨论了基本属性及其应用。我们特别注意PMC，它满足RG不变性的所有要求。使用PMC，将与扰动序列中的β函数相关的所有非保形项相加到运行耦合中，并且可以以任意有限阶数获得唯一的，与比例无关的，与方案无关的预测。 PMC提供了BLM方法的基本原理，因为它给出了将BLM扩展到任意扰动阶数的一般规则。实际上，通过PMC-BLM对应原理，它们彼此等效。因此，以前在BLM文献中观察到的所有功能也都适用于PMC。 PMC尺度和所得的有限阶PMC预测具有较高的准确性，而与初始重整化尺度的选择无关，因此与RG不变性一致。 PMC还与零色范围内QED的重归一化比例设置过程一致。因此，PMC的使用消除了扰动QCD预测中的严重系统尺度误差，从而大大提高了标准模型的经验测试的精度以及它们对新物理的敏感性。

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