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A Unified Geometric Framework for Understanding Collider Event Manifolds

This paper introduces the multi-reference relative representation (M3R) framework to unify and systematically compare distinct collider event metrics, revealing how they resolve complementary geometric structures and decision boundaries within a cohesive coordinate system.

Original authors: Tianji Cai, Hancheng Li

Published 2026-10-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tianji Cai, Hancheng Li

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

High-energy particle colliders are massive machines that smash protons together at nearly the speed of light, creating a chaotic spray of new particles in the process. To physicists, these collisions are not just random explosions; they are the fingerprints of the fundamental laws of nature. When a heavy particle like a Higgs boson decays, or when a common quark radiates energy, the resulting spray of debris follows specific patterns. The challenge for scientists is to tell these patterns apart. They need a way to measure how similar or different two collision events are, much like a geologist might measure the distance between two rock formations to understand their history. In recent years, researchers have developed mathematical tools to define this "distance" between events, turning the chaotic spray of particles into a structured shape, or a manifold, that can be studied. However, there is a problem: scientists have invented several different ways to measure this distance, each based on different physical ideas. One method might focus on the total energy flowing through space, while another looks at the angles between particles. Because these methods are built on different principles, they often produce different maps of the same data, making it difficult to know which map tells the true story or how to combine them.

A team of researchers has introduced a new framework called Multi-Reference Relative Representation, or M3R, to solve this confusion. Instead of trying to force all these different maps into a single, rigid coordinate system, they proposed a simpler approach: describe every collision event by how far it is from a small, fixed set of reference events. Imagine trying to describe the location of a house in a vast, unfamiliar city. Instead of giving complex latitude and longitude coordinates, you could simply say how far the house is from the library, the train station, and the park. If you know the distances to these three landmarks, you have a unique and reliable description of where the house is, regardless of the city's original map. The researchers applied this same logic to particle collisions. They selected a handful of "landmark" collisions from their data and measured the distance from every other collision to these landmarks using three different mathematical rules. This allowed them to translate the complex, high-dimensional data of particle physics into a common, manageable language where different measurement methods could be compared side-by-side.

Using this new language, the team analyzed millions of simulated particle collisions, including those from standard quark and gluon interactions as well as those from the decay of heavy particles like the Higgs boson and the top quark. They discovered that the shape of the data depends heavily on which measurement rule is used. Some rules, which focus on the broad, overall flow of energy, create simple, smooth shapes that are easy to describe with just a few landmarks. Other rules, which look at the fine details of individual particles, create much more intricate and complex shapes that require many more landmarks to describe accurately. Surprisingly, the researchers found that the specific type of particle collision mattered less than the choice of measurement rule. Whether the data came from a Higgs boson or a standard quark, the complexity of the shape was determined primarily by the mathematical tool used to measure it.

The study also revealed a fascinating twist regarding how well these shapes help scientists distinguish between different types of particles. While the most detailed measurement rule created the most complex and difficult-to-reconstruct shapes, it was also the most effective at separating different physical processes. In other words, the rule that made the data look the most complicated was actually the best at telling a Higgs boson apart from a background quark. Conversely, the simpler measurement rules, which produced smoother shapes, were harder to use for precise separation. This suggests that the extra complexity captured by the detailed rules contains vital clues that the simpler rules miss. By combining the different measurement approaches into a single, unified description, the researchers found they could improve their ability to identify rare events. The different rules provided complementary information, filling in the gaps left by the others. For instance, adding information about the specific types of particles produced, such as whether they were charged or neutral, further sharpened the distinction between similar-looking events.

The work demonstrates that there is no single, perfect way to view the geometry of particle collisions. Instead, the universe offers multiple valid perspectives, each revealing different layers of structure. The new framework provides a way to hold these perspectives together, allowing physicists to see both the broad strokes and the fine details simultaneously. This approach does not just offer a better way to sort data; it suggests that the geometry of particle collisions is a rich, multi-faceted object that can be understood by combining different notions of similarity. By treating these different mathematical views as parts of a whole, scientists can build a more complete picture of the fundamental forces at work, turning the chaotic spray of a collider into a clear, readable map of the subatomic world.

Technical Summary: A Unified Geometric Framework for Understanding Collider Event Manifolds

Problem Statement
Collider event spaces possess rich geometric structures determined by the choice of distance metric used to define event similarity. While various physically motivated metrics exist—such as optimal transport (Wasserstein) distances and relativistic phase space metrics—each induces a distinct geometry on the same ensemble of events. A conceptual gap remains in systematically comparing these heterogeneous metric spaces to understand how they resolve underlying physics, how they organize events within a single process, and how they separate different physical processes. Furthermore, the computational cost of analyzing these geometries is prohibitive, as resolving the geometry of $N$ events typically requires $O(N^2)$ pairwise metric evaluations. There is a need for a unified framework that allows for the comparison of different event metrics on equal footing while providing a scalable alternative to exhaustive pairwise comparisons.

Methodology
The authors introduce the Multi-Reference Relative Representation (M3R) framework. Instead of defining a global coordinate system for the event space, M3R characterizes each event $x$ by its vector of distances to a finite set of reference events $R = \{r_1, \dots, r_{N_{ref}}\}$ .
$\Phi_R^{(\alpha)}(x) = (d_\alpha(x, r_1), d_\alpha(x, r_2), \dots, d_\alpha(x, r_{N_{ref}})) \in \mathbb{R}^{N_{ref}}$
where $d_\alpha$ is a specific event metric. This maps events from potentially infinite-dimensional or complex metric spaces into a common, finite-dimensional Euclidean space.

The framework is applied to three specific kinematic metrics:

Energy-Flow 2-Wasserstein ( $W_2$ ): Measures the cost of transporting transverse momentum on the rapidity-azimuth plane.
Spectral 2-Wasserstein ( $SW_2$ ): Measures the cost of transporting distributions of pairwise angular separations (spectral functions).
Relativistic Phase Space (PS): Measures distances based on the intrinsic geometry of the $N$ -body phase space manifold.

Additionally, non-kinematic information (particle identification and charge) is incorporated via simple scalar distances and combined within the M3R framework.

The analysis utilizes the JetClass dataset, focusing on ten classes of jets (QCD, $W$ , $Z$ , $H$ , and top quark decays). The study employs two primary operational measures:

Manifold Complexity: Quantified by the Jaccard Area Under the Curve (JAUC) and Modified Average Decision Ordering (MADO), which measure how well the M3R representation reconstructs the original metric geometry (neighborhood preservation and distance ordering) as a function of the number of references ( $N_{ref}$ ).
Decision Boundary Geometry: Analyzed using Support Vector Machines (SVMs) with Radial Basis Function (RBF) kernels. Performance is measured by the Area Under the Curve (AUC), and boundary complexity is quantified by the fraction of training events retained as Support Vectors (SVs).

Key Contributions and Results

Metric-Dependent Manifold Complexity:
The geometric complexity of an individual event manifold is governed primarily by the choice of metric rather than the underlying physical process. A hierarchy of reconstruction complexity is established: PS < $SW_2$ < $W_2$ .
- PS is the most compressible, requiring few references to achieve high reconstruction fidelity because it integrates out fine-grained radiation.
- $W_2$ is the most complex to reconstruct, as it resolves the finest details of the energy flow.
- $SW_2$ occupies an intermediate position.
  Crucially, the convergence complexity is low for all metrics; most geometric information is recovered with only $O(10)$ reference events, significantly fewer than the $10^4$ events sampled per process.
Manifold Alignment and Process Dependence:
The alignment of principal directions between manifolds induced by different metrics depends strongly on the physical process.
- For QCD jets (dominated by soft/collinear radiation), $W_2$ and $SW_2$ exhibit strong alignment in their leading principal directions, consistent with their known perturbative correspondence.
- For multi-pronged top jets, this alignment breaks down; $W_2$ resolves distinct triangular configurations that $SW_2$ maps to degenerate spectral functions, leading to weak alignment.
- Phase Space (PS) shows substantial alignment with both Wasserstein metrics in leading directions across processes, suggesting a shared coarse kinematic structure.
Reversed Hierarchy in Decision Boundaries:
A striking reversal is observed when analyzing the geometry relevant for distinguishing different processes (inter-manifold geometry). While $W_2$ induces the most complex individual manifolds, it often yields the simplest decision boundaries (lowest fraction of support vectors) and best preserves classification performance with few references.
- In contrast, PS, while easy to reconstruct individually, often produces more complex decision boundaries (higher SV fractions) and lower classification performance for difficult tasks (e.g., Higgs vs. Higgs discrimination).
- This suggests that the fine-grained structure resolved by $W_2$ provides cleaner separation between processes, whereas the compressed PS description may collapse discriminating directions.
Unified Representation and Complementarity:
Combining M3R representations from multiple metrics (PS, $SW_2$ , $W_2$ ) and non-kinematic PID information into a unified feature vector improves classification performance, particularly for challenging tasks.
- The information is complementary: $SW_2$ and $W_2$ provide significant additional discriminative power beyond PS, whereas adding PS to $SW_2$ / $W_2$ yields diminishing returns.
- PID information provides the largest gains for distinguishing processes with similar kinematics but different particle content (e.g., $W$ vs. $Z$ , or $H \to c\bar{c}$ vs. $H \to gg$ ).

Significance and Claims
The paper claims to provide a general language for comparing and composing different notions of event similarity, moving beyond the analysis of a single prescribed metric. By introducing M3R, the authors offer a scalable method to analyze collider event geometries that avoids the $O(N^2)$ bottleneck of pairwise distance calculations while retaining physical interpretability.

The work demonstrates that collider event space is not a fixed geometric structure but a compositional object shaped jointly by the underlying physics and the specific metric used to resolve it. The authors argue that intra-manifold complexity (how hard it is to describe a single process) and inter-manifold complexity (how easy it is to separate processes) are distinct properties that do not necessarily correlate. This framework opens a path toward a systematic geometric description of collider event space, allowing researchers to probe which structures are resolved by specific metrics and how these structures contribute to distinguishing physical processes. The authors note that future work could connect these operational definitions to intrinsic mathematical invariants (e.g., intrinsic dimension, metric entropy) and extend the framework to Beyond the Standard Model (BSM) signatures and learned representations from machine learning models.

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