Rational approximations in Analytic QCD

TL;DR

This paper introduces rational approximations for the modified Minimal Analytic (mMA) coupling in Analytic QCD, demonstrating their convergence properties and potential for improved data fitting.

Contribution

It establishes the convergence of Pade approximants for the mMA coupling and develops rational approximations for higher powers, enhancing analytic QCD modeling.

Findings

Pade approximants converge to the mMA coupling in the complex plane

Rational approximations effectively model higher powers of the coupling

Proposed improvements may better reproduce experimental data

Abstract

We consider the ``modified Minimal Analytic'' (mMA) coupling that involves an infrared cut to the standard MA coupling. The mMA coupling is a Stieltjes function and, as a consequence, the paradiagonal Pade approximants converge to the coupling in the entire $Q^2$-plane except on the time-like semiaxis below the cut. The equivalence between the narrow width approximation of the discontinuity function of the coupling, on the one hand, and this Pade (rational) approximation of the coupling, on the other hand, is shown. We approximate the analytic analogs of the higher powers of mMA coupling by rational functions in such a way that the singularity region is respected by the approximants.Several comparisons, for real and complex arguments $Q^2$, between the exact and approximate expressions are made and the speed of convergence is discussed. Motivated by the success of these approximants, an improvement of the mMA coupling is suggested, and possible uses in the reproduction of experimental data are discussed.