⚛️ phenomenology

Lorentz-Equivariant Geometric Algebra Transformers for Top-Quark Event Reconstruction

This paper introduces Lorentz-equivariant geometric algebra transformers (LGaTr), which leverage a common Lorentz-covariant representation of physical objects to outperform baseline networks in top-quark event reconstruction tasks such as $t\bar{t}/tWb$ separation and FCNC detection.

Original authors: E. Abasov, L. Dudko, A. Markina, M. Perfilov, P. Volkov, G. Vorotnikov, A. Zaborenko

Published 2026-10-22

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CC BY 4.0

Original authors: E. Abasov, L. Dudko, A. Markina, M. Perfilov, P. Volkov, G. Vorotnikov, A. Zaborenko

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 heart of high-energy physics, scientists study the smallest building blocks of the universe by smashing particles together at incredible speeds. When these collisions occur, they create a fleeting shower of new particles that fly out in every direction, much like debris from a shattered vase. To understand what happened, researchers must reconstruct the original event from this chaotic spray. A central challenge in this work is the top quark, the heaviest known elementary particle, which decays almost instantly into other particles. Because it vanishes so quickly, physicists cannot see it directly; they can only infer its existence and properties by carefully piecing together the paths and energies of the particles it left behind. This reconstruction relies heavily on the laws of motion and symmetry that govern how these particles behave, particularly how their descriptions change when viewed from different angles or speeds.

A team of researchers at Moscow State University has developed a new way to teach computers how to perform this reconstruction. They created a specialized type of artificial intelligence designed to understand the unique geometry of these particle collisions. Instead of feeding the computer a list of numbers representing positions and speeds, which can be confusing when the viewpoint changes, the researchers gave the computer a language that naturally respects the rules of physics. This new system, called a Lorentz-equivariant geometric algebra transformer, treats the particles' energy and momentum as a single, unified object that transforms correctly no matter how the observer is moving. By using this approach, the computer learns the relationships between particles more intuitively, much like a person who understands the shape of a shadow without needing to calculate the light source's position every time.

The researchers tested this new system on three different challenges involving top quarks. In the first test, the computer had to distinguish between two very similar types of particle collisions: one where a single top quark is produced alongside a W boson, and another where a pair of top quarks is created. This is a difficult task because the final debris looks nearly identical in both cases. The new system successfully separated these two scenarios with a high degree of accuracy, outperforming older methods that relied on manually designed features. It achieved this while using significantly fewer adjustable settings than the traditional models, suggesting that its design is more efficient at capturing the essential physics.

In a second test, the team asked the computer to identify a rare and exotic process called a flavor-changing neutral current, where a top quark interacts with other particles in a way that the standard laws of physics usually forbid. Here, the new system performed slightly better than standard transformer networks, which are a common type of artificial intelligence used in many fields. However, it did not quite match the performance of a simpler model that used hand-crafted features specifically tailored for this problem. This result suggests that while the new geometric approach is powerful, there are still specific details in the data that human-designed tools can currently capture more effectively.

The most ambitious test involved a universal reconstruction task where the computer had to take a messy collection of detector signals and figure out exactly which particles were created, how they decayed, and what their original energies were. This required the system to predict multiple things at once, such as the identity of particles and the specific paths they took. In this complex scenario, the new system matched the performance of the best existing models for identifying and assigning particles to their parent decays. It significantly outperformed older, simpler models in these tasks. However, when it came to predicting the precise energy and direction of the particles, the new system still lagged slightly behind the most advanced existing architectures. The researchers noted that this gap might be due to a lack of specific types of collision data in their training set, particularly those involving certain types of particle decays that are rare in their current samples.

The study concludes that this new geometric language offers a promising path forward for analyzing particle collisions. It provides a common framework that allows computers to see the underlying structure of events without getting lost in the details of coordinate systems. While the system is not yet perfect at every single task, it demonstrates that building artificial intelligence with the laws of physics built into its very structure can lead to more efficient and accurate tools for discovery. The researchers plan to refine their system by feeding it more diverse data and adjusting how it handles specific types of measurements, aiming to close the remaining gaps in its performance. This work represents a step toward a future where computers can help physicists decode the universe's most complex events with greater clarity and speed.

Technical Summary: Lorentz-Equivariant Geometric Algebra Transformers for Top-Quark Event Reconstruction

Problem Statement
The reconstruction and interpretation of top quark events in high-energy physics rely heavily on the relationships between four-momenta, particle identities, and decay chains. Conventional machine learning approaches typically infer these relations from coordinate components and manually constructed observables. This paper addresses the need for representations that encode the transformation properties of event objects directly, specifically under Lorentz transformations. The authors investigate whether Geometric Algebra (GA), which provides a unified language for four-momenta and their covariant combinations, can serve as the foundation for more efficient and generalizable neural network architectures.

Methodology
The study introduces the Lorentz-Equivariant Geometric Algebra Transformer (LGaTr). The core methodology relies on the Clifford algebra $Cl(1, 3)$ of Minkowski spacetime with signature $(+, -, -, -)$ .

Representation: Event objects are encoded as multivectors where the grade-one component represents the four-momentum ( $p_\mu \gamma^\mu$ ) and grade-zero components encode discrete labels (flavor, charge, b-tag).
Equivariance: The network utilizes the "sandwich" action of rotors ( $X \mapsto R X \tilde{R}$ ) to ensure that multivector transformations remain consistent with proper, orthochronous Lorentz transformations without requiring explicit coordinate matrix representations.
Architecture: The LGaTr layers are constructed from equivariant bilinear maps derived from geometric products followed by grade projections. The model employs geometric-product layers, per-grade equivariant linear layers, and Lorentz-invariant attention heads.

The methodology is evaluated across three distinct physics tasks:

$tW$ vs. $t\bar{t}$ Separation: Distinguishing single-resonant $tW$ contributions from double-resonant $t\bar{t}$ contributions in the $pp \to tWb$ final state at the parton level.
FCNC Single Top Classification: Identifying flavor-changing neutral-current (FCNC) single top quark production against Standard Model backgrounds (including $t$ -, $tW$ -, and $s$ -channel production, $t\bar{t}$ , $W$ +jets, and QCD multijets) using detector-level objects.
Universal Event Reconstruction: A multiprocess, multi-task setting mapping detector-level inputs (missing transverse momentum, leptons, jets) to parton-level targets. This involves simultaneous regression of four-momenta, classification of particle identities, assignment of decay chains, and reconstruction of high-level observables (HLR).

Key Contributions

Unified Framework: The paper demonstrates that geometric algebra allows kinematic vectors, decay planes, and scalar observables to be represented within a single Lorentz-covariant framework, simplifying complex calculations such as top quark spin correlations.
Architecture Design: The authors provide a specific implementation of a Transformer-based architecture that operates directly on multivector inputs, maintaining Lorentz equivariance throughout the network.
Comprehensive Benchmarking: The study evaluates LGaTr against two baselines:
- A fully connected Multilayer Perceptron (MLP) trained on hand-crafted high-level features (e.g., invariant masses, transverse momenta).
- Standard Transformer architectures (e.g., DETR) that process detector tokens without explicit Lorentz-equivariant constraints.

Results

$tW$ vs. $t\bar{t}$ Separation: LGaTr achieved an Area Under the Curve (AUC) of 0.9653, outperforming the reference MLP (0.9594). Notably, LGaTr achieved this with significantly fewer trainable parameters ( $1.51 \times 10^5$ vs. $5.4 \times 10^5$ ).
FCNC Classification: In this task, the baseline MLP (0.8368) outperformed LGaTr (0.8325) and a standard Transformer (0.8261). The authors attribute this to the MLP's use of engineered features which contain more event information than the raw four-momenta provided to the LGaTr. However, LGaTr did show a slight improvement over the standard Transformer, suggesting the benefit of the equivariant architecture on similar inputs.
Universal Event Reconstruction:
- Assignment: LGaTr significantly outperformed the MLP baseline in decay assignment tasks, which the authors identify as the most difficult subtask.
- Regression and HLR: LGaTr performed on par with DETR-like architectures but fell short of DETR in the reconstruction of high-level variables and four-momenta regression.
- Limitations: The authors note specific areas where performance was suboptimal, including $W \to j$ assignment (due to class imbalance favoring leptonic decays) and the reconstruction of the azimuthal angle $\phi$ , which exhibited poor correlation and required specific architectural adjustments.

Significance and Claims
The paper claims that geometric algebra provides a "common language" for collider event kinematics that naturally encodes Lorentz covariance. The primary significance lies in demonstrating that Lorentz-equivariant networks can achieve performance comparable to or better than standard baselines while utilizing fewer parameters, particularly in tasks where the underlying physics is dominated by Lorentz-invariant relationships (such as $tW$ separation).

The authors modestly conclude that while GA-based networks are highly effective for parton-level separation and assignment tasks, they currently require further refinement (such as better handling of hadronic decays and angle-periodic readouts) to match the performance of DETR-like architectures in high-level regression tasks. The work establishes a foundation for using geometric algebra to unify the representation of physical objects and their transformations in deep learning models for high-energy physics.

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