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

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 ωadmit, 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.

arXiv comment27 pages, 8 figures. We fix typos in Eqs. (18), (27), (55) and (56). Results unchanged

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• Life of Π
• Precise ${\alpha }_s$ determination from charmonium sum rules
• Hadronic vacuum polarization using gradient flow
• Precise determination of ${\alpha }_s$ from relativistic quarkonium sum rules
• Low- and High-Energy Expansion of Heavy-Quark Correlators at Next-To-Next-To-Leading Order
• Large-Nc QCD, Harmonic Sums and the Riemann Zeros
• Hadronic Vacuum Polarization and the MUonE proposal
• Small-momentum expansion of heavy-quark correlators in the large-${\beta }_0$ limit and ${\alpha }_s$ extractions
• Charm and Bottom Masses from Sum Rules with a Convergence Test
• On the Time Momentum Representation of Hadronic Vacuum Polarization and ${\mathbf{g}}_{\mathbf{\mu }}-2$