A key issue in making precise predictions in QCD is the uncertainty in setting the renormalization scale ${\mu }_r$ and thus determining the correct values of the QCD running coupling ${\alpha }_s\left({\mu }_r\right)$ at each order in the perturbative expansion of a QCD observable. It has often been conventional to simply set the renormalization scale to the typical scale of the process $Q$ and vary it in the range ${\mu }_r\in [Q/2,2Q]$ in order to estimate the theoretical error. This is the practice of Conventional Scale Setting (CSS). The resulting CSS prediction will however depend on the theorist’s choice of renormalization scheme and the resulting pQCD series will diverge factorially. It will also disagree with renormalization scale setting used in QED and electroweak theory thus precluding grand unification. A solution to the renormalization scale-setting problem is offered by the Principle of Maximum Conformality (PMC), which provides a systematic way to eliminate the renormalization scale-and-scheme dependence in perturbative calculations. The PMC method has rigorous theoretical foundations, it satisfies Renormalization Group Invariance (RGI) and preserves all self-consistency conditions derived from the renormalization group. The PMC cancels the renormalon growth, reduces to the Gell-Mann–Low scheme in the $N_c\to 0$ Abelian limit and leads to scale- and scheme-invariant results. The PMC has now been successfully applied to many high-energy processes. In this article we summarize recent developments and results in solving the renormalization scale and scheme ambiguities in perturbative QCD. In particular, we present a recently developed method the PMC${}_{\infty }$ and its applications, comparing the results with CSS. The method preserves the property of renormalizable SU(N)/U(1) gauge theories defined as Intrinsic Conformality (iCF).

This property underlies the scale invariance of physical observables and leads to a remarkably efficient method to solve the conventional renormalization scale ambiguity at every order in pQCD.

This new method reflects the underlying conformal properties displayed by pQCD at NNLO, eliminates the scheme dependence of pQCD predictions and is consistent with the general properties of the PMC. A new method to identify conformal and $\beta$-terms, which can be applied either to numerical or to theoretical calculations is also shown. We present results for the thrust and $C$-parameter distributions in $e^{+}{e}^{-}$ annihilation showing errors and comparison with the CSS. We also show results for a recent innovative comparison between the CSS and the PMC${}_{\infty }$ applied to the thrust distribution investigating both the QCD conformal window and the QED $N_c\to 0$ limit. In order to determine the thrust distribution along the entire renormalization group flow from the highest energies to zero energy, we consider the number of flavors near the upper boundary of the conformal window. In this flavor-number regime the theory develops a perturbative infrared interacting fixed point. These results show that PMC${}_{\infty }$ leads to higher precision and introduces new interesting features in the PMC. In fact, this method preserves with continuity the position of the peak, showing perfect agreement with the experimental data already at NNLO.

We also show a detailed comparison of the PMC${}_{\infty }$ with the other PMC approaches: the multi-scale-setting approach (PMCm) and the single-scale-setting approach (PMCs) by comparing their predictions for three important fully integrated quantities $R_{e^{+}{e}^{-}}$, $R_{\tau }$ and $\Gamma (H\to b\bar{b})$ up to the four-loop accuracy.

在 QCD 中进行精确预测的一个关键问题是设置重正化尺度的不确定性${\mu }_r$从而确定 QCD 运行耦合的正确值${\alpha }_s（{\mu }_r）$QCD 可观测量的微扰展开中的每个阶。通常的做法是简单地将重正化尺度设置为过程的典型尺度$问$并在范围内改变它${\mu }_r\epsilon [问/2,2问]$以便估计理论误差。这就是常规比例设置（CSS）的做法。然而，最终的 CSS 预测将取决于理论家对重整化方案的选择，并且最终的 pQCD 系列将因阶乘而发散。它还不同意 QED 和电弱理论中使用的重正化尺度设置，从而排除大统一。最大共形原理 (PMC) 提供了重正化尺度设置问题的解决方案，它提供了一种系统方法来消除微扰计算中的重正化尺度和方案依赖性。PMC方法具有严格的理论基础，它满足重整化群不变性（RGI）并保留了由重整化群导出的所有自洽条件。PMC 取消了重整增长，简化为 Gell-Mann-Low 方案${氮}_C\to 0$阿贝尔极限并导致尺度不变和方案不变的结果。PMC现已成功应用于许多高能过程。在本文中，我们总结了解决微扰 QCD 中重正化尺度和方案模糊性的最新进展和结果。特别是，我们提出了一种最近开发的方法 PMC${}_{\mathrm{无穷大}}$及其应用，将结果与 CSS 进行比较。该方法保留了可重整化 SU(N)/U(1) 规范理论的属性，定义为固有共形( iCF )。

这一性质是物理可观测量的尺度不变性的基础，并导致一种非常有效的方法来解决 pQCD 中每个阶的传统重整化尺度模糊性。

这种新方法反映了 pQCD 在 NNLO 处显示的潜在共形特性，消除了 pQCD 预测的方案依赖性，并且与 PMC 的一般特性一致。一种识别共形和共形的新方法$\beta$还显示了可应用于数值或理论计算的术语。我们提供推力和$C$-参数分布$e^{+}{e}^{-}$湮灭显示错误并与 CSS 进行比较。我们还展示了 CSS 和 PMC 之间最近的创新比较结果${}_{\mathrm{无穷大}}$应用于研究 QCD 共形窗口和 QED 的推力分布${氮}_C\to 0$限制。为了确定从最高能量到零能量的整个重整化群流的推力分布，我们考虑共形窗口上边界附近的味道数量。在这种风味数体系中，该理论开发了一个微扰红外相互作用固定点。这些结果表明 PMC${}_{\mathrm{无穷大}}$带来更高的精度，并在 PMC 中引入新的有趣功能。事实上，该方法连续地保留了峰的位置，与 NNLO 中已有的实验数据完全一致。

我们还展示了 PMC 的详细比较${}_{\mathrm{无穷大}}$与其他 PMC 方法：多尺度设置方法 (PMCm) 和单尺度设置方法 (PMC)，通过比较它们对三个重要的完全积分量的预测${右}_{e^{+}{e}^{-}}$,${右}_{\tau }$和$\gamma （H\to 乙\overline{乙}）$高达四环精度。