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Renormalons in integrated spectral function moments and $\alpha_s$ extractions
High Energy Physics - Phenomenology · Submitted 27 Feb 2020
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Our results allow for a deeper understanding of the perturbative behaviour of integrated spectral function moments and provide theoretical support for low-$Q^2$ $\alpha_s$ determinations.
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Precise extractions of $\alpha_s$ from $\tau\to {\rm (hadrons)}+\nu_\tau$ and from $e^+e^-\to {\rm (hadrons)}$ below the charm threshold rely on finite energy sum rules (FESRs) where the experimental side is given by integrated spectral function moments. Here we study the renormalons that appear in the Borel transform of polynomial moments in the large-$\beta_0$ limit and in full QCD. In large-$\beta_0$, we establish a direct connection between the renormalons and the perturbative behaviour of moments often employed in the literature. The leading IR singularity is particularly prominent and is behind the fate of moments whose perturbative series are unstable, while those with good perturbative behaviour benefit from partial cancellations of renormalon singularities. The conclusions can be extended to QCD through a convenient scheme transformation to the $C$-scheme together with the use of a modified Borel transform which make the results particularly simple; the leading IR singularity becomes a simple pole, as in large-$\beta_0$. Finally, for the moments that display good perturbative behaviour, we discuss an optimized truncation based on renormalisation scheme (or scale) variation. Our results allow for a deeper understanding of the perturbative behaviour of integrated spectral function moments and provide theoretical support for low-$Q^2$ $\alpha_s$ determinations.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Precise extractions of $\alpha_s$ from $\tau\to {\rm (hadrons)}+\nu_\tau$ and from $e^+e^-\to {\rm (hadrons)}$ below the charm threshold rely on finite energy sum rules (FESRs) where the experimental side is given by integrated spectral function moments. | background 1.00 | 0.05 | 0.03 | asserted 1.00 |
| s1 | Here we study the renormalons that appear in the Borel transform of polynomial moments in the large-$\beta_0$ limit and in full QCD. | contribution 1.00 | 0.90 | 0.04 | asserted 0.88 |
| s2 | In large-$\beta_0$, we establish a direct connection between the renormalons and the perturbative behaviour of moments often employed in the literature. | result 0.70 | 0.91 | 0.11 | asserted 1.00 |
| s3 | The leading IR singularity is particularly prominent and is behind the fate of moments whose perturbative series are unstable, while those with good perturbative behaviour benefit from partial cancellations of renormalon singularities. | result 0.64 | 0.73 | 0.25 | asserted 1.00 |
| s4 | The conclusions can be extended to QCD through a convenient scheme transformation to the $C$-scheme together with the use of a modified Borel transform which make the results particularly simple; the leading IR singularity becomes a simple pole, as in large-$\beta_0$. | method 0.63 | 0.81 | 0.18 | asserted 0.96 |
| s5 | Finally, for the moments that display good perturbative behaviour, we discuss an optimized truncation based on renormalisation scheme (or scale) variation. | method 0.50 | 0.82 | 0.09 | asserted 0.78 |
| s6 | Our results allow for a deeper understanding of the perturbative behaviour of integrated spectral function moments and provide theoretical support for low-$Q^2$ $\alpha_s$ determinations. | interpretation 0.99 | 0.81 | 0.74 | asserted 0.96 |
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The paper
Original abstract
Precise extractions of $\alpha_s$ from $\tau\to {\rm (hadrons)}+\nu_\tau$ and from $e^+e^-\to {\rm (hadrons)}$ below the charm threshold rely on finite energy sum rules (FESRs) where the experimental side is given by integrated spectral function moments. Here we study the renormalons that appear in the Borel transform of polynomial moments in the large-$\beta_0$ limit and in full QCD. In large-$\beta_0$, we establish a direct connection between the renormalons and the perturbative behaviour of moments often employed in the literature. The leading IR singularity is particularly prominent and is behind the fate of moments whose perturbative series are unstable, while those with good perturbative behaviour benefit from partial cancellations of renormalon singularities. The conclusions can be extended to QCD through a convenient scheme transformation to the $C$-scheme together with the use of a modified Borel transform which make the results particularly simple; the leading IR singularity becomes a simple pole, as in large-$\beta_0$. Finally, for the moments that display good perturbative behaviour, we discuss an optimized truncation based on renormalisation scheme (or scale) variation. Our results allow for a deeper understanding of the perturbative behaviour of integrated spectral function moments and provide theoretical support for low-$Q^2$ $\alpha_s$ determinations.
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How the literature uses it
10 arXiv papers cite this in 19 sentences
Papers citing it · 12 sentences without a confident label
Cites it as support 3 papers
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As argued in detail in our previous work, the FO, $FO$ and PV methods of evaluation of the sum rules lead to truncated series[In the PV case, the truncated series refers to the series $δd(e^i ϕ)^[N_t]_D=0$ in Eq. ([ref]) which is free from the leading renormalon singularities and gives the corresponding contour integrals (sum rules) also free from the leading renormalon singularities.] that have the leading renormalon contribution of the Adler function (double UV renormalon pole at $u=+1$) suppressed in them; and that the CI method in the sum rules does not have this property.[These arguments, presented in a very general way in [ref] (especially Appendix A there), are also compatible with somewhat related arguments presented in [ref] .
Using improved Operator Product Expansion in Borel-Laplace Sum Rules with ALEPH $\tau$ decay data, and determination of pQCD coupling · Final results and conclusions · 0.98
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The restriction to relatively low dimensions is in keeping with the expectation that the OPE is an asymptotic expansion, while the absence of a linear term is motivated by the results of Ref. [ref] , which showed evidence for bad perturbative behavior in the case of weights with a linear term, which are strongly sensitive to the gluon-condensate renormalon [ref] .
Quark-hadron duality and the determination of $\alpha_s$ from hadronic $\tau$ decay: facts vs. myths · Logarithms in the condensates and asymptoticity of the OPE · 0.99
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In this context, we recall that the truncated CIPT is neither a perturbation power series nor does it represent a resummation of the evaluated quantity (because it is truncated).[Somewhat related arguments for the preference of the FOPT methods over the CIPT methods, in FESRs of semihadronic $τ$-decays, were presented in Refs. [ref] .]
Determination of perturbative QCD coupling from ALEPH $\tau$ decay data using pinched Borel-Laplace and Finite Energy Sum Rules · Summary of the results and comparison with literature · 0.96
Cites it as background 4 papers
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The analytic properties of the Borel transform $B_δ_w_i(u)$ defined in ([ref]) in the complex $u$ plane have been investigated some time ago in [ref] and more recently in [ref] .
Conformal mapping of the Borel plane: going beyond perturbative QCD · Moments of the spectral function · 0.90
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The non-strange $V+A$ inclusive channel has been thoroughly studied in the literature as a QCD laboratory [ref] .
Semileptonic tau decays beyond the Standard Model · V+A · 1.00
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A number of attempts have been made in the past to estimate or predict the truncation errors and MHOs, see, e.g., refs [ref] .
An analysis of Bayesian estimates for missing higher orders in perturbative calculations · Introduction · 0.98
What it cites · 75 sentences without a confident label
Builds on 1 paper
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The procedure we employ has been suggested for the $τ$ hadronic width in Ref. [ref] but had never been investigated systematically for different integrated moments.
Scheme variations of the QCD coupling and hadronic $\tau$ decays · Introduction · 0.91
Uses the method or tool of 8 papers
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Our results are based on the reconstruction of the higher-order coefficients performed in Ref. [ref] , using Padé approximants.
Higher-order QCD corrections to hadronic $\tau$ decays from Pad\'e approximants · Details on the Borel integrals from Padé approximants · 0.80
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We will show that most of the difficulties in QCD can be circumvented by a convenient scheme transformation, to the so-called $C$ scheme [ref] , together with the use of a modified Borel transform introduced in Ref. [ref] .[The use of modified Borel transforms in combination with the $C$ scheme in similar contexts has been suggested by M.
Large-order perturbation theory for the electromagnetic current-current correlation function · Introduction · 0.94
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We will show that most of the difficulties in QCD can be circumvented by a convenient scheme transformation, to the so-called $C$ scheme [ref] , together with the use of a modified Borel transform introduced in Ref. [ref] .[The use of modified Borel transforms in combination with the $C$ scheme in similar contexts has been suggested by M.
Scheme variations of the QCD coupling and hadronic $\tau$ decays · Introduction · 0.94
Uses data or values from 1 paper
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With these coefficients, given in Tab. 6 of Ref. [ref] , one can obtain the expansion of any moment, in FOPT or CIPT, rather accurately up to order $𝒪(α_s^10)$.
Higher-order QCD corrections to hadronic $\tau$ decays from Pad\'e approximants · Details on the Borel integrals from Padé approximants · 0.98
Compares with 2 papers
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We also compare our results with those of Ref. [ref] .
Perturbative expansion of tau hadronic spectral function moments and alpha_s extractions · Details on the Borel integrals from Padé approximants · 1.00
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The results from the model-independent Padé approximant reconstruction of the series are qualitatively similar to the “Borel model" of Refs. [ref] which attests the robustness of our conclusions.
alpha_s and the tau hadronic width: fixed-order, contour-improved and higher-order perturbation theory · Results in QCD · 0.81
Cites as background 13 papers
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These singularities are the renormalons of perturbation theory [ref] .
Renormalons · Introduction · 0.99
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Several moments of the spectral functions have been used in low-$Q^2$ $α_s$ determinations from hadronic $τ$ data and $$ [ref] .
The strong coupling from hadronic $\tau$ decays: a critical appraisal · Renormalons in spectral function moments · 0.86
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Several moments of the spectral functions have been used in low-$Q^2$ $α_s$ determinations from hadronic $τ$ data and $$ [ref] .
The strong coupling from the revised ALEPH data for hadronic $\tau$ decays · Renormalons in spectral function moments · 0.86
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