All-Orders Renormalon Resummations for some QCD Observables

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.

Comments: 25 pages, LaTeX. 4 figures in separate uuencoded PS file. Revisions - changes to some expressions in sections 3 & 4