Functional anatomy of Pythia–Herwig differences with Kolmogorov–Arnold networks

{ "core_finding": "Using an additive Kolmogorov-Arnold network (KAN) to decompose classifier-derived log density ratios, the study shows that Pythia-Herwig differences are driven primarily by jet multiplicity at the shower level, shift toward jet mass and shape after hadronization, and develop a mixed shape-multiplicity structure in the full generator configuration.", "caveats": "The staged comparison cannot isolate a single microscopic mechanism because the underlying shower states at Stage A already differ between Pythia and Herwig, and the jet-mass factors in the KAN-based framework are constrained by poor statistical support in the tails.", "markdown": "## Research Question and Approach\n\nHigh-energy event generators like Pythia and Herwig model the complex evolution of collision events using different theoretical prescriptions, particularly for parton showering and hadronization. While global classifiers can quantify the disagreement between these generators, they typically act as black boxes that fail to identify which specific physical observables drive the discrepancy. This paper investigates the functional anatomy of Pythia-Herwig differences by asking how generator discrepancies evolve through successive stages of collision simulation—specifically from shower-only (Stage A) to hadronized (Stage B) and full-generator configurations (Stage C)—while keeping the underlying hard dijet scattering fixed.\n\nThe author addresses this by employing additive Kolmogorov-Arnold networks (KANs). Unlike standard multilayer perceptrons, KANs replace scalar edge weights with trainable one-dimensional functions represented by splines. In a deliberately additive KAN formulation, the learned log density ratio factorizes into explicit, exportable response functions for each input observable. Using a common eight-observable jet representation covering constituent multiplicity, mass, momentum fraction, and girth, the framework allows for exhaustive subset recomposition and Shapley allocation across all generator stages, as well as downstream transport of individual shower-level functional components.\n\n## Stage-Dependent Anatomy of Generator Differences\n\nAnalyzing the eight-observable jet representation across the three generator stages reveals a clear evolution in the physical sources of disagreement between Pythia and Herwig. At the shower-only stage, the Pythia-Herwig difference is dominated overwhelmingly by constituent multiplicity, reflecting the distinct ways the two generators handle perturbative radiation and kinematic ordering. Once hadronization is introduced at Stage B, the primary driver of the difference shifts significantly toward jet mass and shape observables. Finally, in the full-generator configuration that incorporates multiparton interactions and underlying-event modeling, the disagreement develops a mixed structure driven by both shape and multiplicity.\n\n## Functional Transport and Limitations\n\nTo test whether structural differences identified early in the event generation chain remain meaningful later, the author transports individual shower-level functional responses downstream to hadronized and full-generator events. The analysis demonstrates that shower-level multiplicity information retains its reweighting power when applied to later stages. However, the corresponding shower-level shape responses do not necessarily retain their efficacy, even though shape observables regain importance in later stages. Furthermore, attempts to transport jet-mass factors are heavily constrained by poor statistical support in the tails, highlighting important limitations regarding where learned density ratios can be safely applied.\n\n## Key Terms and Definitions\n- Kolmogorov-Arnold Network (KAN) — A neural network architecture that replaces fixed activation functions and scalar weights with trainable one-dimensional functions, enabling an additive decomposition of learned log density ratios.

• Additive KAN Representation — A network structure where the total logit is expressed as a sum of individual, observable-dependent response functions, allowing for exact factorization and isolation of generator discrepancies.
• Stage-Dependent Anatomy — The staged evaluation of how generator-model differences reorganize across different levels of simulation, from parton showering to hadronization and full event generation.
• Eight-Observable Representation — A common, low-dimensional coordinate basis consisting of constituent multiplicities, invariant masses, momentum sharing parameters, and girths for leading and subleading jets.
• Shapley Allocation — A game-theoretic method used to quantify the relative contribution of each individual observable and its subsets to the overall generator separation.
• Common Hard-Event Cohort — A strictly controlled set of identical matrix-element hard scatterings that are successfully reconstructed and defined across all generator configurations and stages." }