# Novel All-Orders Single-Scale Approach to QCD Renormalization Scale-Setting

## TL;DR

This paper introduces a single-scale approach to the Principle of Maximal Conformality (PMC) in QCD, simplifying scale-setting while maintaining scheme independence and improving convergence, with applications to key high-energy processes.

## Contribution

The paper presents a novel single-scale PMC method that simplifies the multi-scale approach, maintaining scheme independence and improving perturbative series convergence.

## Key findings

- Single-scale PMC matches multi-scale predictions for key processes.
- Eliminates renormalization scale ambiguity, enhancing Standard Model tests.
- Improves perturbative QCD convergence by removing divergent renormalon terms.

## Abstract

The Principle of Maximal Conformality (PMC) provides a rigorous method for eliminating renormalization scheme-and-scale ambiguities for perturbative QCD predictions. The PMC uses the renormalization group equation to fix the $\beta$-pattern of each order in an arbitrary pQCD approximant, and it then determines the optimal renormalization scale by absorbing all $\{\beta_{i}\}$ terms into the running coupling at each order. The resulting coefficients of the pQCD series match the scheme-independent conformal series with $\beta=0$. As in QED, different renormalization scales appear at each order; we call this the multi-scale approach. In this paper, we present a novel single-scale approach for the PMC, in which a single effective scale is constructed to eliminate all non-conformal $\beta$-terms up to a given order simultaneously. The PMC single-scale approach inherits the main features of the multi-scale approach; for example, its predictions are scheme-independent, and the pQCD convergence is greatly improved due to the elimination of divergent renormalon terms. As an application of the single-scale approach, we investigate the $e^+e^-$ annihilation cross-section ratio $R_{e^+e^-}$ and the Higgs decay-width $\Gamma(H \to b \bar{b})$, including four-loop QCD contributions. The resulting predictions are nearly identical to the multi-scale predictions for both the total and differential contributions. Thus in many cases, the PMC single-scale approach PMC-s, which requires a simpler analysis, could be adopted as a reliable substitution for the PMC multi-scale approach for setting the renormalization scale for high-energy processes, particularly when one does not need detailed information at each order. The elimination of the renormalization scale uncertainty increases the precision of tests of the Standard Model at the LHC.