Abstract:
We propose a generalization of Grunberg's method of effective charges in which, starting with the effective charge for some dimensionless QCD observable dependent on the single energy scale $Q, R(Q)$, we introduce an infinite set of auxiliary effective charges, each one describing the sub-asymptotic Q-evolution of the immediately preceding effective charge. The corresponding infinite set of coupled integrated effective charge beta-function equations may be truncated. The resulting approximations for $R(Q)$ are the convergents of a continued function. They are manifestly RS-invariant and converge to a limit equal to the Borel sum of the standard asymptotic perturbation series in $\alpha_s({\mu^2})$, with remaining ambiguities due to infra-red renormalons. There are close connections with Pad{\'e} approximation.

Abstract:
We consider the structure of the leading ultra-violet (UV) renormalon singularity associated with the QCD vacuum polarization Adler D-function, in the approximation that only planar Feynman diagrams are retained. This ``planar approximation'' results in some simplifications, in particular three of the four potential contributions from four-fermion operators are shown to be absent. We are able to obtain a fully normalized result for the leading $n\to\infty$ behaviour of the portion of perturbative coefficients proportional to $N_f^{n-r} N^r$, for SU(N) QCD with $N_f$ quark flavours.

Abstract:
We suggest that at any given order of Feynman diagram calculation all renormalization group (RG)-predictable terms should be resummed to all-orders. This ``complete'' RG-improvement (CORGI) serves to separate the perturbation series into infinite subsets of terms which when summed are renormalization scheme (RS)-invariant. Crucially all ultraviolet logarithms involving the dimensionful parameter, Q, on which the observable depends are resummed, thereby building the correct Q-dependence. There are close connections with the effective charge approach of Grunberg.

Abstract:
We show that dimensionful renormalization scheme parameters such as the renormalization or factorization scale can be completely eliminated from perturbative QCD predictions provided that all the ultraviolet logarithms involving the physical energy scale Q are completely resummed.

Abstract:
We perform an all-orders resummation of the QCD Adler D-function for the vector correlator, in which the portion of perturbative coefficients involving the leading power of b, the first beta-function coefficient, is resummed. To avoid a renormalization scale dependence when we match the resummation to the exactly known NLO and NNLO results, we employ the Complete Renormalization Group Improvement (CORGI) approach. These fixed-order and resummed CORGI results are analytically continued by numerically performing a contour integral to obtain corresponding fixed and all-orders ``contour-improved'' results for the e+e- R-ratio ands its tau decay analogue R_{\tau}. The difference between these fixed-order and all-order results is used to estimate the uncertainty in the extraction of {alpha}_{s}({M}_{Z}^{2}} from R_{\tau} measurements, and that in the QED coupling {\alpha}({M}_{Z}^{2}) due to hadronic corrections related to R. Analogous resummations for the scalar correlator are performed, and used to assess the uncertainty in the Higgs decay width to a heavy quark pair. We point out that CORGI fixed-order contour-improved results for R and the Higgs decay width, can be given explicitly in terms of the Lambert-W function and hypergeometric functions, avoiding the need for numerical integration.

Abstract:
For moments of leptoproduction structure functions we show that all dependence on the renormalization and factorization scales disappears, provided that all the ultraviolet logarithms involving the physical energy scale Q are completely resummed. The approach is closely related to Grunberg's method of Effective Charges. A direct and simple method for extracting the universal dimensional transmutation parameter of QCD from experimental data is advocated.

Abstract:
We directly fit the QCD dimensional transmutation parameter, Lambda MS-bar, to experimental data on e+e- jet observables, making use of next-to-leading order (NLO) perturbative calculations. In this procedure there is no need to mention, let alone to arbitrarily vary, the unphysical renormalisation scale mu, and one avoids the spurious and meaningless ``theoretical error'' associated with standard alpha_s determinations. PETRA, SLD, and LEP data are considered in the analysis. An attempt is made to estimate the importance of uncalculated next-NLO and higher order perturbative corrections, and power corrections, by studying the scatter in the values of Lambda MS-bar obtained for different observables.

Abstract:
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 $b$=$\frac{1}{6}(11N$--$2N_{f})$, 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 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 $\tau$-lepton decay ratio $R_{\tau}$, 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 $R_{\tau}$, 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 $\alpha_{s}(M_{Z})$ to three-significant figures, and for the $R_{\tau}$ we estimate a theoretical uncertainty $\delta\alpha_{s}(m_{\tau})\simeq0.008$ corresponding to $\delta\alpha_{s}(M_{Z})\simeq0.001$. This encouragingly small uncertainty is much less than has recently been deduced from comparison with the ambiguous naive resummation.

Abstract:
We show that QCD Minkowski observables such as the $e^{+}e^{-}$ R-ratio and the hadronic tau decay $R_{\tau}$ are completely determined by the effective charge (EC) beta-function, $\rho(x)$, corresponding to the Euclidean QCD vacuum polarization Adler D-function, together with the next-to-leading order (NLO) perturbative coefficient of D. An efficient numerical algorithm is given for evaluating R, $R_{\tau}$ from a weighted contour integration of $D(se^{i\theta})$ around a circle in the complex squared energy s-plane, with $\rho(x)$ used to evolve in s around the contour. The EC beta-function can be truncated at next-to-NLO (NNLO) using the known exact perturbative calculation or the uncalculated N^3 LO and higher terms can be approximated by the portion containing the highest power of b, the first QCD beta-function coefficient. The difference between the R, $R_{\tau}$ constructed using the NNLO and "leading-b" resummed versions of $\rho(x)$ provides an estimate of the uncertainty due to the uncalculated higher order corrections. Simple numerical parametrizations are given to facilitate these fits. For $R_{\tau}$ we estimate an uncertainty $\delta\alpha_{s}(m_{\tau}^{2})\simeq0.01$, corresponding to $\delta\alpha_{s}(M_{Z}^{2})\simeq0.002$. This encouragingly small uncertainty is much less than rather pessimistic estimates by other authors based on analogous all-orders resummations, which we demonstrate to be extremely dependent on the chosen renormalization scheme, and hence misleading.

Abstract:
Exact large-$N_{f}$ results for the QCD Adler $D$-function and Deep Inelastic Scattering sum rules are used to resum to all orders the portion of QCD perturbative coefficients containing the highest power of $b$=$\frac{1}{6}(11N$--$2N_{f})$, for SU($N$) QCD with $N_{f}$ quark flavours. These terms correspond to renormalon singularities in the Borel plane and are expected asymptotically to dominate the coefficients to all orders in the $1/N_{f}$ expansion. Remarkably, we note that this is already apparent in comparisons with the exact next-to-leading order (NLO) and next-to-NLO (NNLO) perturbative coefficients. The ultra-violet ($UV$) and infra-red ($IR$) renormalon singularities in the Borel transform are isolated and the Borel sum (principal value regulated for $IR$) performed. Resummed results are also obtained for the Minkowski quantities related to the $D$-function, the $e^{+}e^{-}$ $R$-ratio and the analogous $\tau$-lepton decay ratio, $R_{\tau}$. The renormalization scheme dependence of these partial resummations is discussed and they are compared with the results from other groups [1--3] and with exact fixed order perturbation theory at NNLO. Prospects for improving the resummation by including more exact details of the Borel transform are considered.