Conformal mappings in perturbative QCD

TL;DR

This paper explores the use of conformal mappings in perturbative QCD to improve series convergence and potentially offer an alternative to the operator product expansion, with applications to the Adler function and lattice QCD.

Contribution

It introduces a conformal mapping method for Borel transforms in perturbative QCD, analyzing its properties and potential to replace standard OPE expansions.

Findings

Modified expansions show better convergence properties.

Conformal mappings can incorporate nonperturbative features.

Application to Adler function and lattice QCD demonstrates practical utility.

Abstract

We discuss the method of conformal mappings applied to perturbative QCD. The approach is based on the Borel-Laplace integral regulated with the principal value prescription and the expansion of the Borel transform in powers of the variable which performs the conformal mapping of the cut Borel plane onto the unit disk. We write down the expression of the conformal mapping for the most general location of the singularities of the Borel transform and review the properties of the corresponding expansions of the correlators. Unlike the standard perturbative expansions, which are divergent, the modified expansions have a tamed behaviour at large orders and may even converge under some conditions. On the other hand, the expansion functions exhibit nonperturbative features similar to those of the expanded function. Using these properties, it was suggested recently that the expansions based on the conformal mapping of the Borel plane may provide an alternative to the standard OPE. We briefly review the arguments in favour of this conjecture and discuss the application of the method to the Adler function for massless quarks and the static quark self-energy calculated in lattice QCD.