Resummed Distribution Functions: Making Perturbation Theory Positive and Normalized

TL;DR

The paper introduces the Resummed Distribution Function (RDF), a framework that ensures perturbative calculations of differential cross sections are positive, normalized, and finite, improving their physical reliability and interpretability.

Contribution

It proposes a novel RDF framework that resums fixed-order calculations to guarantee physical properties and includes a method to estimate perturbative uncertainties.

Findings

Successfully matched RDF to thrust at $ ext{O}( ext{α}_s^3)$

Extracted $ ext{α}_s$ with perturbative uncertainties from ALEPH data

Demonstrated RDF's ability to produce physically consistent distributions

Abstract

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 $\alpha_s$, 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 N$^n$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 $\mathcal{O}(\alpha_s^3)$ and extracting $\alpha_s$ with perturbative uncertainties by fitting the RDF to ALEPH data.