# The QCD Renormalization Group Equation and the Elimination of Fixed-Order Scheme-and-Scale Ambiguities Using the Principle of Maximum Conformality

## TL;DR

This paper discusses how the Principle of Maximum Conformality (PMC) can eliminate renormalization scheme and scale ambiguities in perturbative QCD predictions, leading to more precise and scheme-independent results.

## Contribution

It introduces the application of PMC to fix renormalization scales in QCD, reducing scheme and scale ambiguities and improving the reliability of perturbative series.

## Key findings

- PMC achieves scheme and scale independence at all orders.
- Conformal series with PMC provides more reliable estimates of uncalculated higher-order terms.
- Comparison shows conformal series with PMC outperforms conventional scale-dependent approaches.

## Abstract

The conventional approach to fixed-order perturbative QCD predictions is based on an arbitrary choice of the renormalization scale, together with an arbitrary range. This {\it ad hoc} assignment of the renormalization scale causes the coefficients of the QCD running coupling at each perturbative order to be strongly dependent on the choice of both the renormalization scale and the renormalization scheme. However, such ambiguities are not necessary, since as a basic requirement of renormalization group invariance (RGI), any physical observable must be independent of the choice of both the renormalization scheme and the renormalization scale. In fact, if one uses the {\it Principle of Maximum Conformality} (PMC) to fix the renormalization scale, the coefficients of the pQCD series match the series of conformal theory, and they are thus scheme independent. It has been found that the elimination of the scale and scheme ambiguities at all orders relies heavily on how precisely we know the analytic form of the QCD running coupling $\alpha_s$. In this review, we summarize the known properties of the QCD running coupling and its recent progresses, especially for its behavior within the asymptotic region. We also summarize the current progress on the PMC and some of its typical applications, showing to what degree the conventional renormalization scheme-and-scale ambiguities can be eliminated after applying the PMC. We also compare the PA approach for the conventional scale-dependent pQCD series and the PMC scale-independent conformal series. We observe that by using the conformal series, the PA approach can provide a more reliable estimate of the magnitude of the uncalculated terms. And if the conformal series for an observable has been calculated up to $n_{\rm th}$-order level, then the $[N/M]=[0/n-1]$-type PA series provides an important estimate for the higher-order terms.