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Using standard mathematical methods for asymptotic series and thelarge-$\beta_0$ approximation, we define a Minimum Distance between theFixed-Order perturbative series and the Contour-Improved perturbative series inthe strong coupling $\alpha_s$ for finite-energy sum rules as applied tohadronic $\tau$ decays. This distance is similar, but not identical, to theAsymptotic Separation of Hoang and Regner, which is defined in terms of thedifference of the two series after Borel resummation. Our results confirm anonzero nonperturbative result in $\alpha_s$ for this Minimum Distance as ameasure of the intrinsic difference between the two series, as well as aconflict with the Operator Product Expansion for Contour-Improved PerturbationTheory.

On the difference between Fixed-Order and Contour-Improved Perturbation Theory