Disentangling running coupling and conformal effects in QCD

TL;DR

The paper investigates how an Abelian-like skeleton expansion in QCD can disentangle running-coupling effects from conformal coefficients, potentially mitigating renormalon-related divergences. It develops BLM scale-setting within a skeleton-scheme framework and shows that, independently of the skeleton details, the resulting conformal coefficients match those defined in the Banks-Zaks conformal limit, offering a renormalon-free template for QCD predictions. The authors also explore the effective-charge (ECH) approach as an alternative resummation tool and analyze explicit examples, including the Banks-Zaks expansion and Crewther-type relations, highlighting the often small and simple nature of conformal coefficients. They discuss the limitations arising from the lack of a complete diagrammatic skeleton construction in QCD and the implications for the convergence and universality of conformal relations. Overall, the work provides a structured way to connect perturbative QCD, conformal (renormalon-free) expansions, and phenomenological relations, with potential improvements in the reliability of high-order predictions and power-correction analyses.

Abstract

We investigate the relation between a postulated skeleton expansion and the conformal limit of QCD. We begin by developing some consequences of an Abelian-like skeleton expansion, which allows one to disentangle running-coupling effects from the remaining skeleton coefficients. The latter are by construction renormalon-free, and hence hopefully better behaved. We consider a simple ansatz for the expansion, where an observable is written as a sum of integrals over the running-coupling. We show that in this framework one can set a unique Brodsky-Lepage-Mackenzie (BLM) scale-setting procedure as an approximation to the running-coupling integrals, where the BLM coefficients coincide with the skeleton ones. Alternatively, the running-coupling integrals can be approximated using the effective charge method. We discuss the limitations in disentangling running coupling effects in the absence of a diagrammatic construction of the skeleton expansion. Independently of the assumed skeleton structure we show that BLM coefficients coincide with the conformal coefficients defined in the small $β_0$ (Banks-Zaks) limit where a perturbative infrared fixed-point is present. This interpretation of the BLM coefficients should explain their previously observed simplicity and smallness. Numerical examples are critically discussed.