Reconciling the contour-improved and fixed-order approaches for τ hadronic spectral moments. Part II. Renormalon norm and application in αs determinations

In a previous article, we have shown that the discrepancy between the fixed-order (FOPT) and contour-improved (CIPT) perturbative expansions for tau hadronic spectral function moments, which had affected the precision of alpha(s) determinations for many years, may be reconciled by employing a renormalon-free (RF) scheme for the gluon condensate (GC) matrix element. In addition, the perturbative convergence of spectral function moments with a sizeable GC correction can be improved. The RF GC scheme depends on an IR factorization scale R and the normalization N-g of the GC renormalon. In the present work, we use three different methods to determine N-g, yielding a result with an uncertainty of 40%. Following two recent state-of-the-art strong coupling determination analyses at O(alpha(5)(s)), we show that using the renormalon-free GC scheme successfully reconciles the results for alpha(s )(m(tau)(2)) based on CIPT and FOPT. The uncertainties due to variations of R and the uncertainty of N-g only lead to a small or moderate increase of the final uncertainty of alpha(s) (m(tau)(2)), and affect mainly the CIPT expansion method. The FOPT and CIPT results obtained in the RF GC scheme may be consistently averaged. The RF GC scheme thus constitutes a powerful new ingredient for future analyses of tau hadronic spectral function moments.

Automated summary

In this paper, we demonstrate that the discrepancy between the fixed-order and contour-improved perturbative expansions for tau hadronic spectral function moments can be resolved by using a renormalon-free GC scheme. Furthermore, the perturbative convergence can be improved with a considerable GC correction. This new RF GC scheme is a powerful ingredient for future analyses of tau hadronic spectral function moments.