Resummed Distribution Functions: Making Perturbation Theory Positive and Normalized

TL;DR

The paper introduces the Resummed Distribution Function (RDF), a positive, normalized, and finite all-orders completion of fixed-order perturbative predictions for differential cross sections, constructed to match the input to order $M$ while parameterizing all higher-order completions under unitarity. RDFs are defined via an autoregressive PDF ansatz with a tunable function $g$ that splits into a positive $g^*$ and an analytic piece, enabling both analytic and numeric matching to FO results and potential LL-like resummations when informed by higher-order structure. The authors demonstrate RDFs on toy models and QCD observables (jet angularities and thrust), showing that RDFs reproduce FO results, remain well behaved, and provide a natural framework for perturbative uncertainties through nuisance parameters. They apply RDFs to thrust using ALEPH data to extract $\alpha_s$ with perturbative uncertainties, highlighting the method’s potential as a practical tool for theory-error quantification and for cases where traditional resummation is challenging. The work suggests broad applicability to event shapes, multi-observable phase spaces, and potential extensions to include prior information, renormalization-scale dependence, and factorization structures, all while offering open-source code for reproducibility.

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 $α_s$, 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.

Figures (14)

• Figure 1: The RDF unitary completions of the exponential distribution, as given by Eq. (\ref{['eq:exponential_completion']}), for random choices of the higher-order $g_m(t)$ parameters. The FO distributions are shown as thick colored lines, and the true distribution is shown as a black line. Each random choice of $g$ is shown as a thin dotted colored line. Note for $g = 0$, the completion lies exactly on top of the true distribution. To guide the eye, we draw envelopes around the random variations, though these envelopes are not themselves valid distributions. The distributions are shown as a function of $t$ (left) and $x$ (right).
• Figure 2: The same as Fig. \ref{['fig:exponential_completion']}, but for the Rayleigh distribution whose RDF completion is given by Eq. (\ref{['eq:rayleigh_completion']}).
• Figure 3: RDF numeric fits to the exponential toy at $\mathcal{O}(\alpha_s^1)$ (top), $\mathcal{O}(\alpha_s^2)$ (middle), and $\mathcal{O}(\alpha_s^3)$ (bottom), plotted as a function of $t$ (left) and $x$ (right). For several values of $\alpha_s$, the RDF itself is shown as a solid line, and the Taylor expansion of the RDF is shown as a dotted line. For comparison, we show the true exponential distribution as a black dash-dotted line.
• Figure 4: The same as Fig. \ref{['fig:exponential_matching']}, but for the Rayleigh toy rather than the exponential toy.
• Figure 5: RDF analytic matching to the WTA jet angularity at $\mathcal{O}(\alpha_s^1)$, as given by Eqs. (\ref{['eq:rdf_angularity']}) and (\ref{['eq:rdf_angularity_lim']}), plotted as a function of $t$ (left) and the angularity $\lambda^{(\beta = 1)}$ (right). The original fixed-order expressions are shown in light green, and the RDF expressions are in dark green. The soft-collinear (double-log) limits are shown as dashed lines. Random variations of the higher-order terms for the full (i.e. not soft-collinear RDF) of Eq. (\ref{['eq:g_angularity']}) are shown as thin turquoise lines.