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We show that the Borel representations of tau hadronic spectral functionmoments based on contour-improved perturbation theory (CIPT) in general differfrom those obtained within fixed-order perturbation theory (FOPT) in thepresence of IR renormalons in the underlying Adler function. The Borel sumsobtained from both types of Borel representations in general differ as well,and the apparently conflicting behavior of the FOPT and CIPT spectral functionmoment series at intermediate orders, which has been subject to many studies inthe past literature, can be understood quantitatively using concrete Borelfunction models. The difference between the CIPT and FOPT Borel sums, which wecall the "asymptotic separation", can be computed analytically for any Borelfunction model and is proportional to inverse exponential terms in the strongcoupling. Even though moments can be designed where the asymptotic separationis strongly suppressed, it is as a matter of principle unavoidable. If theBorel function of the Euclidean Adler function has a sizeable gluon condensaterenormalon cut, the asymptotic separation can explain the observed disparity ofthe CIPT and FOPT spectral function moments at the 5-loop level. The existenceof the asymptotic separation implies that the power corrections in the operatorproduct expansion for the spectral function moments in the CIPT expansionapproach do not have the commonly assumed analytic standard form.

Borel Representation of $\tau$ Hadronic Spectral Function Moments in Contour-Improved Perturbation Theory