Extending the Predictive Power of Perturbative QCD

TL;DR

This paper enhances the predictive accuracy of perturbative QCD by combining the Principle of Maximum Conformality with Padé Approximation to better estimate higher-order terms, ensuring scheme independence and consistency with fundamental relations.

Contribution

It introduces a novel approach combining PMC and PAA to improve higher-order predictions in pQCD, maintaining scheme invariance and aligning with fundamental QCD relations.

Findings

PMC eliminates scheme and scale ambiguities.

PMC + PAA provides reliable higher-order estimates.

Predictions agree with fundamental QCD relations.

Abstract

The predictive power of perturbative QCD (pQCD) depends on two important issues: (1) how to eliminate the renormalization scheme-and-scale ambiguities at fixed order, and (2) how to reliably estimate the contributions of unknown higher-order terms using information from the known pQCD series. The Principle of Maximum Conformality (PMC) satisfies all of the principles of the renormalization group and eliminates the scheme-and-scale ambiguities by the recursive use of the renormalization group equation to determine the scale of the QCD running coupling $\alpha_s$ at each order. Moreover, the resulting PMC predictions are independent of the choice of the renormalization scheme, satisfying the key principle of renormalization group invariance. In this letter, we show that by using the conformal series derived using the PMC single-scale procedure, in combination with the Pad\'e Approximation Approach (PAA), one can achieve quantitatively useful estimates for the unknown higher-order terms from the known perturbative series. We illustrate this procedure for three hadronic observables $R_{e^+e^-}$, $R_{\tau}$, and $\Gamma(H \to b \bar{b})$ which are each known to 4 loops in pQCD. We show that if the PMC prediction for the conformal series for an observable (of leading order $\alpha_s^p$) has been determined at order $\alpha^n_s$, then the $[N/M]=[0/n-p]$ Pad\'e series provides quantitatively useful predictions for the higher-order terms. We also show that the PMC + PAA predictions agree at all orders with the fundamental, scheme-independent Generalized Crewther relations which connect observables, such as deep inelastic neutrino-nucleon scattering, to hadronic $e^+e^-$ annihilation. Thus, by using the combination of the PMC series and the Pad\'e method, the predictive power of pQCD theory can be greatly improved.