Functional-analysis based tool for testing quark-hadron duality

TL;DR

This paper introduces a mathematical tool based on minimal distance in the complex plane to study quark-hadron duality violations in QCD, providing a new way to analyze the accuracy of the operator product expansion.

Contribution

It presents a novel functional-analysis-based method using Hankel matrices to quantify duality violations in QCD correlators, linking experimental data with theoretical expansions.

Findings

The minimal distance can be computed exactly from spectral data.

The method offers an alternative to traditional chi-squared analyses.

Illustrated on a toy model for QCD polarization function.

Abstract

Quark-hadron duality is a key concept in QCD, allowing for the description of physical hadronic observables in terms of quark-gluon degrees of freedom. The modern theoretical framework for its implementation is Wilson's operator product expansion (OPE), supplemented by analytic extrapolation from large Euclidean momenta, where the OPE is defined, to the Minkowski axis, where observable quantities are defined. Recently, the importance of additional terms in the expansion of QCD correlators near the Minkowski axis, responsible for quark-hadron duality violations (DVs), was emphasized. In this paper we introduce a mathematical tool that might be useful for the study of DVs in QCD. It is based on finding the minimal distance, measured in the $L^\infty$ norm along a contour in the complex momentum plane, between a class of admissible functions containing the physical amplitude and the asymptotic expansion predicted by the OPE. This minimal distance is given by the norm of a Hankel matrix that can be calculated exactly, using as input the experimental spectral function on a finite interval of the timelike axis. We also comment on the relation between the new functional tool and the more commonly used $\chi^2$-based analysis. The approach is illustrated on a toy model for the QCD polarization function recently proposed in the literature.