Mathematical aspects of the asymptotic expansion in contour improved perturbation theory for hadronic tau decays

In a previous work by one of the authors, it was demonstrated that the discrepancy between the fixed order and contour-improved (CIPT) perturbative expansions for & tau;-lepton decay hadronic spectral function moments, which had been affecting the precision of & alpha;s determinations for many years, is related to the CIPT expansion being inconsistent with the standard formulation of the operator product expansion. Even though the problem can be alleviated phenomenologically for the most part by employing a renormalonfree scheme for the gluon-condensate matrix element, the principal inconsistency of CIPT remains. The CIPT expansion is special because it is not a power expansion, but represents an asymptotic expansion in a sequence of functions of the strong coupling. In this article we provide a closer look at the mathematical aspects of the asymptotic sequence of the functions the CIPT method is based on, and we expose the origin of the CIPT inconsistency as well as the reasons for its apparent good convergence at low orders. Our results are of general interest, and may in particular provide a useful tool to check for the consistency of expansion methods that are similar to CIPT.

Automated summary

In this study, the authors investigate the inconsistency between the fixed order and contour-improved (CIPT) perturbative expansions in measuring the hadronic spectral function moments of τ-lepton decay. This inconsistency has affected the precision of αs determinations for a long time and is found to be related to the CIPT expansion being inconsistent with the standard formulation of the operator product expansion. The authors provide a mathematical analysis of the asymptotic sequence of functions used in the CIPT method and explain the origin of the CIPT inconsistency as well as the reasons for its apparent good convergence at low orders. The results have general interest and can serve as a useful tool to check the consistency of similar expansion methods to CIPT.