RS-invariant all-orders renormalon resummations for some QCD 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 $\delta {\alpha }_s(m_{\tau })\simeq 0.008$ corresponding to $\delta {\alpha }_s(M_Z)\simeq 0.001$. This encouragingly small uncertainty is much less than has recently been deduced from comparison with the ambiguous naive resummation.

25 pages, uses LaTeX, 12 Postscript figures, epsfig.sty 'elsart.sty' and 'elsart12.sty' are available via anonymous-ftp at ftp://ftp.tex.ac.uk/tex-archive/macros/latex/contrib/supported/elsevier

References in corpus (2)

• Dispersive Approach to Power-Behaved Contributions in QCD Hard Processes
• Improved Evaluation of the Next-Next-To-Leading Order QCD Corrections to the e+e- Annihilation Into Hadrons

Cited by in corpus (20)

• Renormalons
• The Physics of Hadronic Tau Decays
• Masses, decays and mixings of gluonia in QCD
• The Determination of alpha_s from Tau Decays Revisited
• Renormalization Group Invariance and Optimal QCD Renormalization Scale-Setting
• On the Behavior of the Effective QCD Coupling alpha_tau(s) at Low Scales
• On the role of power expansions in quantum field theory
• All-orders infrared freezing of observables in perturbative QCD
• Precision physics with inclusive QCD processes
• Perturbative relations between $e^{+}{e}^{-}$ annihilation and $\tau$ decay observables including resummation effects
• Renormalon-inspired resummations for vector and scalar correlators- estimating the uncertainty in {alpha}_{s}({m}_τ^{2}) and and α({M}_{Z}^{2})
• Resummation analysis of the $\tau$ decay width using the four-loop $\beta$-function
• ALEPH Tau Spectral Functions and QCD
• Large-order Behaviour of the QCD Adler D-function in Planar Approximation
• The uncertainty in ${\alpha }_s(M_Z^2)$ determined from hadronic tau decay measurements
• QCD Tests from Tau Decays
• Renormalization scheme dependence and the problem of determination of alpha_s and the condensates from the semileptonic tau decays
• Resummation of Large Logarithms within the Method of Effective Charges
• The Banks-Zaks expansion in perturbative QCD: an update
• A new determination of higher-order QCD corrections to hadronic $\tau$ decays