Tauberian theorems for the Borel summability and analytic continuation of the asymptotics of QCD sum rules

Using QCD sum rules as inputs, the conditions for and proof of the Borel summability and analytic continuation of QCD asymptotics expansions for current propagators are given. The corollary is that duality averages can be ''coarse grain'', i.e. performed over the mass-squared interval s-bar → ∞, corresponding to short distances (t → 0), or equivalently ''fine grain'', i.e. performed over the intervals s-bar → 0, corresponding to long distances (t → ∞). The former is the usual formulation of duality, the latter, the paradoxical novelty proposed by Shifman, Vainshtein and Zakharov, which pushes duality to the limit of being applicable at a point. The two limits are related so that QCD is, surprisingly, relevant to both. The relationship is a consequence of the covariance of dilatation convolutions, which define duality averages, with respect to the conformal inversion. Underlying both Borel summability and the short-distance operator product expansion is SO(2, 1) symmetry