Improving parton shower predictions via precision moments of energy flow polynomials

Abstract

In this paper, we study various conceptual and practical aspects of using maximum-entropy reweighting to upgrade parton-shower event samples based on higher-accuracy theoretical constraints. Our approach produces strictly positive per-event weights that improve parton-shower predictions while preserving full event-level exclusivity, allowing any observable to be computed on the reweighted sample without rebinning or regeneration. On the conceptual side, we explain how theoretical principles can help determine which constraints to use and which kinds of priors lead to efficient reweighting. On the practical side, we perform a proof-of-concept study with hemisphere observables in $e^+e^-\!\to$ hadrons, and show that even when the parton-shower prior is purposefully degraded by removing the non-singular parts of the QCD splitting functions, a small set of precision calculations can nevertheless restore the desired physical behavior. We use energy flow polynomials (EFPs) as a systematic basis to organize infrared- and collinear-safe constraints, and study how information transfers from constrained observables to unconstrained ones. We find rapid information saturation, where constraints from a compact set of EFP moments achieve broad improvements across observable space, including for standard hemisphere observables never used in training. Physics-motivated basis reductions guided by collinear power counting achieve comparable performance to complete bases, and mixed moments combining polynomial and logarithmic terms outperform pure alternatives. These results suggest a systematic approach to improving parton-shower event generators, where theoretical constraints of highest accuracy can be translated into full phase-space predictions of experimental relevance.

Figures (21)

• Figure 1: Schematic of information-theoretic reweighting. A parton-shower prior $q_{\rm PS}(\Phi)$ is updated to a posterior $p^\star(\Phi)$ by a maximum-entropy projection that enforces precision constraints $\langle m_i \rangle = c_i$ via strictly positive per-event weights. A minimal prior $q_{\mathrm{uni.}}(\Phi)$ sets $|\mathcal{M}_N|^2 = \text{const.}$ for each particle multiplicity $N$, while more realistic priors incorporate additional physical effects. The resulting $p^\star(\Phi)$ is closer to the truth distribution $p_{\rm QCD}(\Phi)$.
• Figure 2: Graphical representation of various EFPs. Each vertex contributes an energy factor of $z_i$ and each edge contributes an angular factor of $\theta_{ij}$ when $\beta = 1$. Multiple edges between the same pair of vertices indicate higher powers of the corresponding angle. The degree $d$ of an EFP counts the total number of edges (with multiplicity). The value of an EFP involves a sum over all possible assignments of particles to nodes.
• Figure 4: Comparison of EFP basis choices for degrees $d=1$ to $d=4$. The prime basis includes all connected multigraphs. The 1-collinear basis retains graphs that are linearly independent when all emissions are collinear-soft relative to a single hard parton. The strongly-ordered (SO) basis further reduces to graphs independent under hierarchical ordering of emissions in energy and angle. The bases are nested: strongly-ordered $\subset$ 1-collinear $\subset$ prime. In general, to have a complete linear basis, one has to also include composite EFPs built from these prime elements.
• Figure 5: Comparison of marginal distributions with $d_{\rm max}=5$ reweighting with mixed moments for (a) $\mathrm{EFP}_{1,1}$ ($\chi=2$, trained), (b) $\mathrm{EFP}_{3,3}$ ($\chi=3$, trained), and (c) $\mathrm{EFP}_{6,29}$ ($\chi=4$, transfer). Black points show the target with statistical uncertainties, dashed lines show the unweighted priors, and solid lines show the reweighted distribution. Colors indicate the $\alpha_s$ variations: central (green), $+20\%$ (red), $-20\%$ (blue). Lower panels show the ratio to target with gray bands indicating the target statistical uncertainty. The complete graph $K_4$ (i.e. $\mathrm{EFP}_{6,29}$) is outside the training set yet it is substantially corrected, demonstrating transfer across chromatic complexity.
• Figure 6: The 10 homeomorphically irreducible trees with $n=10$ vertices, constituting the complete degree-9 EFP set used for validation in Fig. \ref{['fig:saturation_prime_d5']}. Their enumeration is a classic problem in combinatorial graph theory Harary1973Gessel2023.