A Field Guide to Event-Shape Observables Using Optimal Transport

TL;DR

This work develops a practical framework for event-shape observables defined via the Earth Mover's Distance between collider events and reference geometries, introducing manifold distances that depend on the choice of reference geometry and distance metric. By analyzing isotropic and dipole references and a family of angular distance exponents $\beta$, the authors demonstrate how these choices modulate sensitivity to different QCD structures and hadronization effects, using analytic insights and Pythia-based simulations of hard and soft QCD events. The key contributions include characterizing the dependence on $\beta$, illustrating the gains from multi-differential analyses, and providing actionable prescriptions for tuning observables to the physics of interest, with a practical demonstration that simple combinations like $\bar{\mathcal{I}}/N$ can achieve near-perfect separation of event classes. The findings offer a versatile, geometry-aware toolkit for LHC analyses and future colliders, enabling robust searches for BSM phenomena by tailoring manifold distances to the targeted event topologies, supported by open-source code for computing these observables.

Abstract

We lay out the phenomenological behavior of event-shape observables evaluated by solving optimal transport problems between collider events and reference geometries -- which we name 'manifold distances' -- to provide guidance regarding their use in future studies. This discussion considers several choices related to the metric used to quantify these distances. We explore the differences between the various options, using a combination of analytical studies and simulated minimum-bias and multi-jet events. Making judicious choices when defining the metric and reference geometry can improve sensitivity to interesting signal features and reduce sensitivity to non-perturbative effects in QCD. The goal of this article is to provide a 'field guide' that can inform how choices made when defining a manifold distance can be tailored for the analysis at-hand.

Figures (15)

• Figure 1: A cartoon of the EMD as an event-shape observable. The transverse projection of a collider event $\mathcal{E}$ is drawn such that the weight of the marker corresponds to the particle $p_T$. Some examples of reference geometries to compare against are two particle back-to-back configurations (also called dipole) $\mathcal{E}_2$ from the manifold $\mathcal{M}_2$ and the quasi-uniform $\mathcal{U}_N$. One can imagine the EMD as the minimal work done to spread event $\mathcal{E}$ into either $\mathcal{E}_2$ or $\mathcal{U}_N$.
• Figure 2: A perfectly uniform event (left) compared to the discretization into 256 particles used for computation (right).
• Figure 3: An example of several possible dipole events in $\mathcal{M}_2$. The angle of orientation is a free parameter of the geometry to be minimized for the particular collider event in comparison, comparable to the thrust axis.
• Figure 4: The distance weight as a function of the angular separation for $\beta = 0.5, 1, 2, 4$. We see that distances across small angular separation ($\Delta \phi_{ij} < \pi/2$) are penalized more strongly for low $\beta$, whereas large angular separation ($\Delta \phi_{ij} > \pi/2$) is much more heavily penalized for higher $\beta$.
• Figure 5: Manifold distances as a function of $N$-particle symmetric events (meaning equal energy, evenly distributed particles). The values of event isotropy are shown on the left and exhibit a monotonic behavior, all maximized for the 2-particle ($\mathcal{M}_2$) configuration, decreasing as the $N$-particle event approaches the continuous limit. The values of (unnormalized) dipole distance $\mathcal{D}_{\mathcal{M}_2}$, shown on the right, are not monotonic, and jump around before asymptoting to the infinite particle limit. We show the unnormalized dipole distance values simply for ease of viewing. The dashed lines indicate the value of dipole distance computed for $N = \infty$. To the right of the solid line we only plot the value for $N$ divisible by 2 for ease of analytic computation.