⚛️ phenomenology

Collider event distances from the phase space sphere

This paper introduces a polynomial-time distance-based classification method that leverages the global topology of Lorentz-invariant phase space and reconstructed masses to effectively distinguish between complex collider event processes, such as four-top production versus $t\bar t+\text{jets}$ .

Original authors: Nathaniel Craig, Brian Henning, Andrew J. Larkoski, Iñigo Valenzuela Lombera

Published 2026-10-20

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CC BY 4.0

Original authors: Nathaniel Craig, Brian Henning, Andrew J. Larkoski, Iñigo Valenzuela Lombera

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

In the high-energy world of particle physics, scientists smash protons together at nearly the speed of light to break them apart and reveal the fundamental building blocks of nature. When these collisions happen, they produce a spray of new particles that fly out in every direction. To understand what happened, researchers must reconstruct the entire event from these flying fragments, measuring their energy and direction. This collection of measurements exists in a vast, abstract mathematical space known as phase space. Traditionally, physicists have focused on the volume of this space to calculate how often certain events occur, treating the geometry of the space itself as a secondary detail. However, the shape and connectivity of this space hold a deeper story about how particles move and interact. If two different physical processes produce similar sprays of particles, they might still leave distinct footprints in the geometry of this space, offering a new way to tell them apart.

A team of researchers has developed a new method to distinguish between different types of particle collisions by treating the entire event as a single geometric shape. Instead of looking for specific patterns or counting particles one by one, they mapped every collision onto a sphere, a shape that naturally describes the possible arrangements of momentum and mass. In this approach, each particle in a collision is assigned a position on the sphere based on its energy and direction. By measuring the distance between the shapes formed by two different events, the researchers can determine how similar or different the underlying physical processes are. This technique allows them to compare complex collisions that produce dozens of particles, a task that becomes incredibly difficult when trying to match individual particles between two events because there are too many possible ways to pair them up.

To solve this matching problem, the team used a mathematical algorithm that efficiently finds the best way to pair particles between two events without having to check every single possibility. They then tested this method on three different scenarios using computer simulations. The first scenario compared the production of top quarks, heavy particles that decay into jets of other particles, against a background of ordinary particle sprays. The second, more difficult test looked for the production of four top quarks at once, a rare event that is hard to distinguish from a background of two top quarks plus extra radiation. The third scenario simulated the creation of microscopic black holes, a hypothetical phenomenon predicted by theories of extra dimensions, and tried to separate these signals from standard particle sprays.

The results showed that the new geometric method was highly effective at distinguishing between signal and background events. In the test involving four top quarks, the method maintained its ability to tell the difference even as the number of detected particles increased, whereas older methods lost their effectiveness. The most successful version of the new approach treated all particles in the event with equal importance, rather than focusing only on the most energetic ones. This "democratic" view allowed the classifier to pick up on subtle structural differences in the way energy was shared among the particles. For the black hole simulation, the method achieved a high level of accuracy, correctly identifying the signal in the vast majority of cases.

The researchers found that the ability to distinguish between different types of collisions improved as they looked at the events with finer detail, resolving more individual particles. However, this improvement did not continue forever; for some processes, the method reached a point of diminishing returns where adding more particles did not significantly help. The study demonstrated that by preserving the masses of the reconstructed particles and using the global shape of the event space, physicists can create powerful tools for finding new physics. This approach does not require knowing the specific details of how a particle decays or what it is called; it relies solely on the four-dimensional coordinates of the particles. This makes the method flexible and applicable to a wide range of searches, from confirming known processes to hunting for entirely new phenomena that might be hiding in the data.

Technical Summary: Collider Event Distances from the Phase Space Sphere

Problem Statement
The paper addresses the challenge of distinguishing collider events produced by different physical processes using the global topology of Lorentz-invariant phase space. While traditional approaches focus on the phase space volume measure to calculate cross-sections, this work posits that the geometric structure of phase space itself contains discriminative information. Previous applications of phase-space distances (e.g., Ref. [7]) utilized massless approximations and lightcone coordinates, which discard reconstructed object masses and lose accuracy at coarse resolutions. Furthermore, existing distance-based methods often struggle with the computational complexity of matching particles between events with high multiplicity, where explicit minimization over all permutations is prohibitive. The authors aim to develop a classification framework that retains reconstructed masses, handles variable multiplicities efficiently, and leverages the spherical topology of fixed-mass phase space extended to variable-mass configurations.

Methodology
The authors construct a geometric framework for event comparison through the following steps:

Phase Space Topology and Representation:
- Fixed-Mass Shell: They establish that $N$ -particle fixed-mass phase space $\Phi_N(M; m)$ has the topology of a $(3N-4)$ -dimensional sphere ( $S^{3N-4}$ ). A radial map decomposes momenta into a magnitude and a unit vector, defining an angular distance.
- Variable-Mass Extension: To accommodate reconstructed jets with fluctuating invariant masses, they enlarge the space by treating masses as coordinates. This forms a "collider dome" $C^+_N(M)$ , topologically a closed ball ( $B^{4N-4}$ ), where the signed shell is a sphere $S^{4N-4}$ .
- Convex Representation: They introduce a non-linear transformation mapping the collider dome to a convex "spectrahedron" $K_N(M)$ in energy-momentum space, allowing for straight-line Euclidean distances.
Event Representations:
The paper defines four distinct representations for comparing events, all retaining normalized four-vectors but weighting kinematic differences differently:
- Radial Spherical: Angular distance between normalized mass-momentum vectors.
- Convex Energy-Momentum: Euclidean distance between energy-momentum vectors in the event rest frame.
- Democratic Moving Frame: A spherical representation where each object's momentum and mass are scaled by $1/\sqrt{E_i}$ before normalization, treating all objects symmetrically.
- Pivoted Moving Frame: Similar to the democratic frame but uses the highest- $p_T$ object as a fixed reference (pivot) to align events.
Scalable Assignment and Alignment:
To compare events without preferred particle labeling, the method must minimize distances over particle permutations and transverse rotations/reflections ( $O(2)$ ). Since exhaustive minimization is $O(N!)$ , the authors employ a polynomial-time sequential approximation:
- Hungarian Algorithm: Finds the optimal one-to-one particle matching in the input frame.
- Transverse Alignment: Optimizes the $O(2)$ rotation/reflection for the fixed assignment.
- This sequential procedure approximates the joint minimum over assignment and alignment.
Classification Pipeline:
Events are preprocessed by boosting to the center-of-momentum frame and normalizing by invariant mass ( $M=1$ ). Pairwise distances are computed using the representations above, converted to Gaussian kernel similarities, and fed into a Support Vector Machine (SVM) for binary classification.

Key Contributions

Mass-Aware Phase Space Geometry: The work extends phase-space distance methods to include reconstructed masses, moving beyond the massless approximations of prior classifiers.
Topological Foundation: It rigorously establishes the spherical and convex topologies of the variable-mass event space, defining metrics on these manifolds.
Efficient High-Multiplicity Matching: The integration of the Hungarian algorithm with transverse alignment provides a scalable polynomial-time approximation for event comparison at high multiplicities ( $N \sim 30-45$ ).
Democratic Representation: The introduction of the "democratic" moving-frame representation, which scales features by energy, is shown to outperform other representations in high-multiplicity regimes by giving greater relative influence to subleading kinematic structures.

Results
The authors evaluate the method on three benchmarks at particle level:

$t\bar{t}$ vs. QCD Dijets: The updated massless prescription (using Hungarian assignment) significantly improves upon legacy rank-ordering, retaining discrimination at high multiplicity ( $N=30$ ). Among massive representations, the democratic moving frame achieves the highest Area Under the Curve (AUC), reaching $0.9510 \pm 0.0009$ at $N=30$ .
Fully Hadronic Four-Top vs. $t\bar{t}$ + Jets: This challenging benchmark (signal vs. background with real tops) shows that moving-frame representations maintain performance as resolution increases ( $N$ up to 45), whereas radial and convex representations degrade. The democratic frame reaches a plateau near $N=15$ with an AUC of $0.8579 \pm 0.0013$ .
Conditioned BlackMax vs. QCD Multijets: In a semi-classical black hole search scenario with suppressed global discriminants, the democratic and pivoted moving frames again lead, achieving broad plateaus near $N=20$ with AUCs around $0.91$.

Across all benchmarks, the democratic moving-frame representation consistently yields the highest discrimination power at the largest tested multiplicities. The convex energy-momentum and massless branches tend to saturate or decline at high $N$ .

Significance and Claims
The paper claims that treating collider events as points on a geometric manifold allows for effective classification without relying on process-specific feature engineering or assumed decay chains. The primary significance lies in demonstrating that:

Retaining reconstructed masses and utilizing the global topology of phase space improves classification performance over massless approximations.
The "democratic" weighting of kinematic features is particularly effective for high-multiplicity events, suggesting that subleading radiation carries significant discriminative information.
The proposed scalable assignment procedure makes these geometric comparisons computationally feasible for modern collider analyses involving large numbers of final-state objects.

The authors note that while the sequential assignment is an approximation, it performs well empirically. They conclude that the framework provides a direct route from pseudojet four-vectors to event classification, with the useful multiplicity range depending on the specific signal and background processes. Future work is suggested to explore the physical origins of the discrimination (e.g., energy sharing vs. angular structure) and to refine the joint minimization of assignment and alignment.

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