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

André H. Hoang, Christoph Regner

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). We demonstrate that the Borel sums obtained from both types of Borel representations in general differ as well and that the apparently conflicting asymptotic behavior of the FOPT and CIPT series, which has been subject to many studies in the past literature, can be understood quantitatively from these results. The discrepancy between the CIPT and FOPT Borel sums, which we call the 'asymptotic separation', can be computed analytically and is related to inverse exponential terms in the strong coupling. The asymptotic separation arises from the singular and non-analytic infrared renormalon structures in the Borel function of the underlying Adler function where the leading dimension four gluon condensate renormalon has the highest weight. The size of the asymptotic difference is in general larger than that of the FOPT Borel sum ambiguity, but it can be modulated in a predictable way by choosing specific spectral function moments. Even though moments can be designed where the asymptotic difference is smaller than the FOPT Borel sum ambiguity, the asymptotic separation can as a matter of principle not be avoided entirely. The asymptotic separation has important implications for the standard operator product expansion approach used for spectral function moment predictions.

Particle Physics, Research Platform Erwin Schrödinger International Institute for Mathematics and Physics
Publication date
Austrian Fields of Science 2012
103012 High energy physics, 103034 Particle physics
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