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 q{sup 2}=0 (low-energy), q{sup 2}=4m{sup 2} (threshold, where m is the quark mass), and q{sup 2}{yields}-{infinity} (high-energy) in a systematic way. We exemplify the method for the perturbative vector correlator at O({alpha}{sub s}{sup 2}) and O({alpha}{sub s}{sup 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 operator-product expansion, and a fixed-sign component given by the logarithms of the threshold expansion in the external momentum q{sup 2}.