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RS-invariant all-orders renormalon resummations for some QCD observables
High Energy Physics - Phenomenology · Submitted 20 Jun 1996
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.85 | 0.91 | 0.10 | asserted 0.92 |
| s1 | This can be accomplished using exact large-$N_{f}$ all-orders results. | method 0.99 | 0.40 | 0.03 | asserted 0.61 |
| s2 | 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. | contribution 0.47 | 0.87 | 0.13 | asserted 1.00 |
| s3 | 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. | background 0.49 | 0.16 | 0.08 | asserted 1.00 |
| s4 | 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. | method 0.54 | 0.91 | 0.09 | asserted 0.96 |
| s5 | 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. | result 0.82 | 0.88 | 0.25 | hedged possibility 0.90 |
| s6 | For DIS sum rules, which have a leading infra-red renormalon singularity, the performance is rather poor. | result 0.98 | 0.88 | 0.12 | asserted 0.99 |
| s7 | 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 $\delta\alpha_{s}(M_{Z})\simeq0.001$. | result 0.84 | 0.89 | 0.25 | hedged possibility 0.98 |
| s8 | This encouragingly small uncertainty is much less than has recently been deduced from comparison with the ambiguous naive resummation. | interpretation 0.78 | 0.67 | 0.16 | asserted 0.98 |
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Original 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.
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