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<p>Fixed-order perturbative calculations for differential cross sections can suffer from non-physical artifacts: they can be non-positive, non-normalizable, and non-finite, none of which occur in experimental measurements. We propose a framework, the <i>Resummed Distribution Function</i> (RDF), that, given a perturbative calculation for an observable to some finite order in <i>α</i><sub><i>s</i></sub>, will “resum” the expression in a way that is guaranteed to match the original expression order-by-order and be positive, normalized, and finite. Moreover, our ansatz parameterizes <i>all</i> possible finite, positive, and normalized completions consistent with the original fixed-order expression, which can include N<sup><i>n</i></sup>LL resummed expressions. The RDF also enables a more direct notion of perturbative uncertainties, as we can directly vary higher-order parameters and treat them as nuisance parameters. We demonstrate the power of the RDF ansatz by matching to thrust to <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">O</mi> <mfenced close=")" open="("> <msubsup> <mi>α</mi> <mi>s</mi> <mn>3</mn> </msubsup> </mfenced> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{O}\left({\alpha}_s^3\right) \)</EquationSource> </InlineEquation> and extracting <i>α</i><sub><i>s</i></sub> with perturbative uncertainties by fitting the RDF to ALEPH data.</p>

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Resummed distribution functions: making perturbation theory positive and normalized

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  • 发表日期：

    2026年6月23日

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  • Rikab Gambhir,
  • Radha Mastandrea

摘要

Fixed-order perturbative calculations for differential cross sections can suffer from non-physical artifacts: they can be non-positive, non-normalizable, and non-finite, none of which occur in experimental measurements. We propose a framework, the Resummed Distribution Function (RDF), that, given a perturbative calculation for an observable to some finite order in αs, will “resum” the expression in a way that is guaranteed to match the original expression order-by-order and be positive, normalized, and finite. Moreover, our ansatz parameterizes all possible finite, positive, and normalized completions consistent with the original fixed-order expression, which can include NnLL resummed expressions. The RDF also enables a more direct notion of perturbative uncertainties, as we can directly vary higher-order parameters and treat them as nuisance parameters. We demonstrate the power of the RDF ansatz by matching to thrust to O α s 3 \( \mathcal{O}\left({\alpha}_s^3\right) \) and extracting αs with perturbative uncertainties by fitting the RDF to ALEPH data.