Reconciling the contour-improved and fixed-order approaches for τ hadronic spectral moments. Part I. Renormalon-free gluon condensate scheme

Abstract

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.

Reconciling the Contour-Improved and Fixed-Order Approaches for $τ$ Hadronic Spectral Moments II: Renormalon Norm and Application in $α_s$ Determinations

“…w , was extensively discussed in Part I of this work [5]. A review of these results is given in Sec.…”

“…where the weight function w(s) can be any analytic function; in practice one most often chooses polynomials. To obtain the physical decay rate, one must set s 0 = m 2 τ and use the weight function determined by the kinematics of the hadronic τ decay [2,5]. Smaller values of the upper limit in the integration, s 0 < m 2 τ , are also frequently used in the literature as a means to further constrain the theoretical description [12,[21][22][23], but it is important to keep s 0 within the perturbative regime.…”

“…The perturbative coefficients are known to O(α 4 s ) (5-loop) [16] As is customary in state-of-the-art analyses of hadronic τ decay spectral function moments, we in addition employ an estimate for the 6-loop coefficient c5,1 [5],…”

“…In a recent paper [5], to which we will refer often as 'Part I', we laid the foundations of a method to eliminate the CIPT-FOPT discrepancy. The method was motiviated from the work of Hoang and Regner [6,7] where it was shown that the discrepancy can be of infra-red (IR) origin and caused by contributions in the CIPT expansion terms that are incompatible with the standard form commonly adopted for the higher dimensional corrections in the operator product expansion (OPE).…”

“…We note that, the RF GC scheme is formulated using the C-scheme for the strong coupling [18] for the concrete value C = 0, see Sec. 2.2 of Part I [5]. This scheme, which we just refer to as the C-scheme, has the special feature that the QCD β-function depends only on the universal one-and two-loop coefficients β 0 and β 1 and can be written down in closed form.…”