Infrared-safe energy weighting does not guarantee small nonperturbative effects

Infrared and collinear (IRC) safety has long been used a proxy for robustness when developing new jet substructure observables. This guiding philosophy has been carried into the deep learning era, where IRC-safe neural networks have been used for many jet studies. For graph-based neural networks, the most straightforward way to achieve IRC safety is to weight particle inputs by their energies. However, energy-weighting by itself does not guarantee that perturbative calculations of machine-learned observables will enjoy small nonperturbative corrections. In this paper, we demonstrate the sensitivity of IRC-safe networks to nonperturbative effects, by training an energy flow network (EFN) to maximize its sensitivity to hadronization. We then show how to construct Lipschitz energy flow networks (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>L</a:mi></a:mrow></a:math>-EFNs), which are both IRC safe and relatively insensitive to nonperturbative corrections. We demonstrate the performance of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>L</c:mi></c:math>-EFNs on generated samples of quark and gluon jets, and showcase fascinating differences between the learned latent representations of EFNs and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>L</e:mi></e:math>-EFNs.