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

Using standard mathematical methods for asymptotic series and the large-$\beta_0$ approximation, we define a Minimum Distance between the Fixed-Order perturbative series and the Contour-Improved perturbative series in the strong coupling $\alpha_s$ for finite-energy sum rules as applied to hadronic $\tau$ decays. This distance is similar, but not identical, to the Asymptotic Separation of Hoang and Regner, which is defined in terms of the difference of the two series after Borel resummation. Our results confirm a nonzero nonperturbative result in $\alpha_s$ for this Minimum Distance as a measure of the intrinsic difference between the two series, as well as a conflict with the Operator Product Expansion for Contour-Improved Perturbation Theory.