RS-Invariant All-Orders Renormalon Resummations for some QCD Observables

We propose a renormalon-inspired resummation of QCD perturbation theory based on approximating the renormalization scheme (RS) invariant effective charge beta-function coefficients by the portion containing the highest power of b=16(11N– 2Nf ), for SU(N) QCD with Nf quark flavours. This can be accomplished using exact large-Nf all-orders results. The resulting resummation is RS-invariant and the exact next-to-leading order (NLO) and next-to-NLO (NNLO) coefficients in any RS are included. This improves on a previously employed naive resummation of the leading-b piece of the perturbative coefficients which is RS-dependent, making its comparison with fixed-order perturbative results ambiguous. The RS-invariant resummation is used to assess the reliability of fixed-order perturbation theory for the e+e− R-ratio, the analogous τ -lepton decay ratio Rτ , and Deep Inelastic Scattering (DIS) sum rules, by comparing it with the exact NNLO results in the effective charge RS. For the R-ratio and Rτ , where large-order perturbative behaviour is dominated by a leading ultra-violet renormalon singularity, the comparison indicates fixed-order perturbation theory to be very reliable. For DIS sum rules, which have a leading infra-red renormalon singularity, the performance is rather poor. In this way we estimate that at LEP/SLD energies ideal data on the R-ratio could determine αs(MZ) to three-significant figures, and for the Rτ we estimate a theoretical uncertainty δαs(mτ ) ≃ 0.008 corresponding to δαs(MZ) ≃ 0.001. This encouragingly small uncertainty is much less than has recently been deduced from comparison with the ambiguous naive resummation.