Resummed Distribution Functions: Making Perturbation Theory Positive and Normalized

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

💡 Research Summary

In the realm of high-energy particle physics, perturbative QCD (pQCD) calculations are indispensable for predicting observable quantities. However, a significant challenge arises from the use of fixed-order perturbative expansions. Because these expansions are truncated at a finite order in the strong coupling constant ($\alpha_s$), they are prone to producing non-physical artifacts. Specifically, these mathematical approximations can lead to negative probability densities, non-normalized distributions, and even divergent (infinite) values in certain kinematic regions—phenomena that are physically impossible and contradict experimental observations.

This paper introduces a groundbreaking framework known as the Resummed Distribution Function (RDF) to bridge the gap between truncated perturbative theory and physical reality. The core objective of the RDF is to provide a mathematical “completion” that preserves the accuracy of the original fixed-order calculation while enforcing essential physical constraints. The RDF is designed to match the original perturbative expression order-by-order, ensuring that no information from the existing calculation is lost. Crucially, the RDF guarantees that the resulting distribution is always positive, normalized to unity, and finite across the entire domain.

One of the most profound aspects of the RDF approach is its ability to parameterize all possible physically valid completions consistent with the fixed-order input. This flexibility allows the framework to incorporate higher-order resummation effects, such as N$^n$LL, within a unified structure. Furthermore, the RDF introduces a more rigorous and direct method for quantifying perturbative uncertainties. By treating the parameters associated with higher-order terms as nuisance parameters, the authors enable a statistical framework where theoretical uncertainties can be directly varied and integrated into the fitting process. This transforms the estimation of theoretical error from a somewhat heuristic approach into a more systematic and statistically sound procedure.

The practical utility of the RDF is demonstrated through its application to the “thrust” observable, an event-shape variable in $e^+e^-$ collisions. By matching the RDF to the $\mathcal{O}(\alpha_s^3)$ perturbative calculation and fitting it to the high-precision ALEPH experimental data, the authors successfully extracted the strong coupling constant $\alpha_s$. The study demonstrates that the RDF not only resolves the mathematical pathologies of fixed-order theory but also provides a robust tool for high-precision parameter extraction with well-defined perturbative uncertainties. This work represents a significant step forward in the quest for precision in quantum chromodynamics.