Resummation of threshold, low- and high-energy expansions for heavy-quark correlators

With the help of the Mellin-Barnes transform, we show how to simultaneously resum the expansion of a heavy-quark correlator around q2=0 (low-energy), q2=4m2 (threshold, where m is the quark mass), and q2→- (high-energy) in a systematic way. We exemplify the method for the perturbative vector correlator at O(αs2) and O(αs3). We show that the coefficients, Ω(n), of the Taylor expansion of the vacuum polarization function in terms of the conformal variable ω 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 operator-product expansion, and a fixed-sign component given by the logarithms of the threshold expansion in the external momentum q2. © 2010 The American Physical Society.