Extending the Predictive Power of Perturbative QCD Using the Principle of Maximum Conformality and Bayesian Analysis

TL;DR

This paper enhances perturbative QCD predictions by combining the Principle of Maximum Conformality to eliminate scale ambiguities with Bayesian analysis to reliably estimate unknown higher-order contributions, improving predictive accuracy.

Contribution

It introduces a novel Bayesian approach integrated with PMC to estimate unknown higher-order terms in pQCD, ensuring more reliable and convergent series.

Findings

PMC removes renormalization scheme-and-scale ambiguities.

Bayesian analysis provides reliable estimates of unknown higher-order contributions.

Application to $R_{e^+e^-}$ and $R_ au$ shows improved predictive power.

Abstract

In addition to the evaluation of high-order loop contributions, the precision and predictive power of perturbative QCD (pQCD) predictions depends on two important issues: (1) how to achieve a reliable, convergent fixed-order series, and (2) how to reliably estimate the contributions of unknown higher-order terms. The recursive use of renormalization group equation, together with the Principle of Maximum Conformality (PMC), eliminates the renormalization scheme-and-scale ambiguities of the conventional pQCD series. The result is a conformal, scale-invariant series of finite order which also satisfies all of the principles of the renormalization group. In this paper we propose a novel Bayesian-based approach to estimate the size of the unknown higher order contributions based on an optimized analysis of probability distributions. We show that by using the PMC conformal series, in combination with the Bayesian analysis, one can consistently achieve high degree of reliability estimates for the unknown high order terms. Thus the predictive power of pQCD can be greatly improved. We illustrate this procedure for two pQCD observables: $R_{e^+e^-}$ and $R_\tau$, which are each known up to four loops in pQCD. Numerical analyses confirm that by using the scale-independent and more convergent PMC conformal series, one can achieve reliable Bayesian probability estimates for the unknown higher-order contributions.