Metric Space of Collider Events

When are two collider events similar? Despite the simplicity and generality of this question, there is no established notion of the distance between two events. To address this question, we develop a metric for the space of collider events based on the earth mover’s distance: the “work” required to rearrange the radiation pattern of one event into another. We expose interesting connections between this metric and the structure of infrared- and collinear-safe observables, providing a novel technique to quantify event modifications due to hadronization, pileup, and detector effects. We showcase how this metrization unlocks powerful new tools for analyzing and visualizing collider data without relying upon a choice of observables. More broadly, this framework paves the way for data-driven collider phenomenology without specialized observables or machine learning models.

Figure 1

The optimal movement to rearrange one top jet (red) into another (blue). Particles are shown as points in the rapidity-azimuth plane with areas proportional to their transverse momenta. Darker lines indicate more transverse momentum movement. The energy mover’s distance in Eq. (1) is the total “work” required to perform this rearrangement.

Figure 2

Two-dimensional histogram of the EMD between 30 000 QCD jets before and after hadronization versus the corresponding $\beta =1$ angularity modification. The red region is excluded based on the bound in Eq. (3), shown as a dashed red line. The bound is clearly satisfied and is nearly saturated for $\mathrm{EMD}\lesssim 10\mathrm{GeV}$.

Figure 3

ROC curves showing the boosted $W$ classification performance of a $k=32$ nearest-neighbor EMD classifier, which requires no choice of observables or parametrized machine learning architectures. The EMD classifier is competitive with machine learning techniques known to be good multiprong classifiers, such as PFNs, EFNs, and EFPs.

Figure 4

The correlation dimension of top, $W$, and QCD jets as a function of the energy scale $Q$ using hadrons (solid), partons (dashed), and hard decay products (dotted). Generally, QCD jets are the lowest dimensional and top jets are the highest dimensional. By comparing partons and hadrons, one sees that hadronization affects the structure of the space at scales below about 30 GeV. Similarly, the hard decay structure of top and $W$ jets governs their dimension at high scales. Below about 10 GeV, the data become very high dimensional and sparse, making dimension estimation difficult.

Figure 5

The embedding of $W$ jets into a two-dimensional space with t-SNE. The gray contours represent the density of embedded jets. Examples of $W$ jets are shown throughout the space. The color of each jet corresponds to its angularity ${\lambda }^{(\beta =1/2)}$ fractile to quantify the energy sharing of the two prongs. An annulus emerges with jets in the lower (upper) region of the manifold having a more energetic lower (upper) subjet. More complex topologies with the largest angularity values populate the center of the manifold.

Figure 6

The jet mass distribution for QCD jets, with $k=3$ medoids shown above each bin. This visualization highlights that simple one-prong topologies dominate low jet masses and complex two-prong topologies exist at high jet masses.