RS-invariant resummations of QCD perturbation theory
We propose a renormalon-inspired resummation of QCD perturbation theory based on approximating the renormalization scheme (RS) invariant effective charge (EC) beta- function coefficients by the portion containing the highest power of b = (11N'-2N(_f))/6, the first beta-function coefficient, for SU(N) QCD with N(_f) quark flavours. This can be accomplished using exact large-N(_f) 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 leading-6 resummation which is RS-dependent. The RS-invariant resummation is used to assess the reliability of fixed-order perturbation theory for the e(^+)e(^-) R-ratio, hadronic tau-decay ratio R(_r), and Deep Inelastic Scattering (DIS) sum rules, by comparing it with the exact NNLO results in the EC RS. For R and R(_r), 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. We show that QCD Minkowski observables such as the R and R(_r) are completely determined by the EC beta-function, p(x), corresponding to the Euclidean QCD vacuum polarization Adler D-function, together with the NLO perturbative coefficient of D. An efficient numerical algorithm is given for evaluating R, Rr from a weighted contour integration of D(se(^10)) around a circle in the complex squared energy .s-plane, with p(x) used to evolve in s around the contour. The difference between the R, R(_r) constructed using the NNLO and leading-b resummed versions of pi(x) provides an estimate of the uncertainty due to the uncalculated higher order corrections. We estimate that at LEP energies ideal data on the R-ratio could determine a(_s)(M(_2)(_Z)) to three-significant figures. For R(_r) we estimate a theoretical uncertainty δa(_s)(m(^2)(_r)) ~ 0.001, corresponding to δa(_s)(m(^2)(_r)) ~ 0.002. This encouragingly small uncertainty is much less than has recently been deduced from comparison with the analogous naive all-orders resummation, which we demonstrate to be extremely RS dependent and hence misleading.