We propose a simple and easy-to-implement scheme for a renormalon-free gluon condensate (GC) matrix element, which is analogous to implementations of short-distance heavy-quark mass renormalization schemes existing in the literature already for a long time. Because the scheme is based on a perturbative subtraction at the level of the matrix element, with a freely adaptable infrared factorization scale, it can be implemented with little effort for any observable where the GC is relevant. The scheme depends on the renormalon norm of the GC which has to be supplemented independently. We apply the scheme to the fixed-order (FOPT) and contour-improved (CIPT) perturbative expansions of τ hadronic spectral function moments. These expansions exhibit a long-standing discrepancy for moments used in high-precision determinations of the strong coupling in the commonly used GC scheme that is not renormalon-free. We show that the scheme is capable of resolving the FOPT-CIPT discrepancy problem. At the same time, the perturbative behaviour of the moments that previously showed bad convergence properties and for which the non-perturbative corrections from the GC are sizeable, is substantially improved. The new GC scheme may provide a powerful theoretical tool for future phenomenological applications.

In a previous article, we have shown that the discrepancy between the fixed-order (FOPT) and contour-improved (CIPT) perturbative expansions for τ hadronic spectral function moments, which had affected the precision of αs 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 Ng of the GC renormalon. In the present work, we use three different methods to determine Ng, yielding a result with an uncertainty of 40%. Following two recent state-of-the-art strong coupling determination analyses at $$ \mathcal{O} $$ O ($$ {\alpha}_s^5 $$ α s 5 ), we show that using the renormalon-free GC scheme successfully reconciles the results for αs($$ {m}_{\tau}^2 $$ m τ 2 ) based on CIPT and FOPT. The uncertainties due to variations of R and the uncertainty of Ng only lead to a small or moderate increase of the final uncertainty of αs($$ {m}_{\tau}^2 $$ m τ 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 τ hadronic spectral function moments.

In a recent work it was suggested that the discrepancy observed in the perturbation series behavior of the τ hadronic decay rate determined in the FOPT and CIPT approaches can be explained from a different infrared sensitivity inherent to both methods, assuming that the major source of the discrepancy is the asymptotic behavior of the series related to the gluon condensate renormalon. This implies that the predictions of both methods may be reconciled in infrared subtracted perturbation theory. In this talk we explore this implication concretely in the large-β 0 approximation, where the perturbation series is known to all orders, using a renormalon-free scheme for the gluon condensate.

In a previous article, we have shown that the discrepancy between the fixedorder (FOPT) and contour-improved (CIPT) perturbative expansions for τ hadronic spectral function moments, which had affected the precision of α 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(α 5 s ), we show that using the renormalon-free GC scheme successfully reconciles the results for α s (m 2 τ ) 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 α s (m 2 τ ), 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 τ hadronic spectral function moments.

Recently it has been clarified by Hoang and Regner that the longstanding discrepancy between the CIPT and FOPT expansion approaches in αs determinations from the τ hadronic spectral function moments has been caused by an inconsistency of CIPT with the standard OPE approach. This inconsistency arises in the presence of IR renormalons in the underlying Adler function and is numerically dominated by the dimension-4 gluon condensate renormalon. In this talk we report on an approach to reconcile the CIPT based on a perturbative definition of a renormalon-free and scale-invariant gluon condensate scheme, called RF GC scheme. The scheme implies perturbative subtractions which eliminate the CIPT inconsistency for all practical applications of the τ hadronic spectral function moments. The scheme depends on the gluon condensate renormalon norm Ng as an independent input and on an IR subtraction scale R. We discuss three different approaches to determine Ng which yield consistent results and we apply the RF GC scheme in two full-fledged phenomenological αs determinations based on the truncated OPE and the duality violation model approach. In the RF GC scheme the long-standing CIPT-FOPT discrepancy problem is gone and the CIPT and FOPT αs determinations can be consistently combined.