Resummation of Threshold, Low- and High-Energy Expansions for Heavy-Quark Correlators

TL;DR

This paper introduces a systematic Mellin-Barnes transform-based method to resum low-energy, threshold, and high-energy expansions of heavy-quark correlators simultaneously, improving understanding of their asymptotic behavior.

Contribution

It presents a novel approach to resum multiple energy expansions of heavy-quark correlators using Mellin-Barnes transforms, with explicit large-n expansion calculations.

Findings

Derived exact large-n expansion for Taylor coefficients Omega(n)

Identified sign-alternating and fixed-sign components in the expansion

Applied method to vector correlator at O(alpha_s^2) and O(alpha_s^3)

Abstract

With the help of the Mellin-Barnes transform, we show how to simultaneously resum the expansion of a heavy-quark correlator around q^2=0 (low-energy), q^2= 4 m^2 (threshold, where m is the quark mass) and q^2=-\infty (high-energy) in a systematic way. We exemplify the method for the perturbative vector correlator at O(alpha_s^2) and O(alpha_s^3). We show that the coefficients, Omega(n), of the Taylor expansion of the vacuum polarization function in terms of the conformal variable \omega admit, for large n, an expansion in powers of 1/n (up to logarithms of n) that we can calculate exactly. This large-n expansion has a sign-alternating component given by the logarithms of the OPE, and a fixed-sign component given by the logarithms of the threshold expansion in the external momentum q^2.