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

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