⚛️ phenomenology

Top Spin and CP from Equivariant Transformers

This paper demonstrates that the Lorentz-equivariant transformer L-GATr-slim significantly enhances spin analyzing power for hadronic top quarks and, when augmented with a pseudoscalar channel, effectively learns and interprets optimal observables for measuring CP properties in processes like WBF-Higgs and top pair production.

Original author: Henning Bahl, Marco Menen, Antoine Petitjean, Tilman Plehn

Published 2026-11-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original author: Henning Bahl, Marco Menen, Antoine Petitjean, Tilman Plehn

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

At the heart of the Large Hadron Collider, the world's most powerful particle accelerator, lies a quest to understand the fundamental rules that govern our universe. Among the many particles created in these high-energy collisions, the top quark holds a special place. It is the heaviest known elementary particle, a fleeting speck of matter that exists for only a fraction of a second before it decays into other particles. Because it dies so quickly, it never has time to form the complex structures that other particles do; instead, it passes its internal state directly to its debris. This makes the top quark a unique messenger, carrying information about its own spin and its behavior under the laws of physics that can be read by scientists. One of the most intriguing mysteries in physics is whether the universe treats matter and antimatter exactly the same way, or if there is a subtle difference known as CP violation. Finding this difference is crucial because it might explain why the universe is made of matter at all, rather than having been annihilated by antimatter in its earliest moments.

A team of researchers has developed a new, highly sophisticated method to read these messages from the top quark, using a type of artificial intelligence designed to respect the deep symmetries of space and time. Their work focuses on two main challenges: first, figuring out exactly how the top quark was spinning when it died, and second, detecting the faint fingerprints of CP violation in the way pairs of top quarks are produced. In the past, identifying the specific particles that carry the top quark's spin information was like trying to find a specific needle in a haystack of similar-looking needles. The researchers used a machine learning tool called a Lorentz-equivariant transformer, which is built to understand the geometry of the universe in a way that standard computer programs cannot. By teaching this system to recognize the subtle patterns in the debris of top quark collisions, they were able to significantly improve the accuracy of their measurements.

The researchers first tackled the problem of "hadronic" top quark decays, where the top quark breaks apart into a spray of other particles, including a down-type quark. Identifying this specific down-type quark is notoriously difficult because it looks very much like other particles in the spray. Previous methods could guess correctly about 68 percent of the time, but the new system, trained on millions of simulated collision events, achieved a much higher level of precision. By analyzing the energy and direction of the particles with extreme care, the system learned to distinguish the down-type quark with an accuracy that translates to a spin-analyzing power of 0.721. This means the new method extracts far more information about the top quark's spin than any previous technique, especially when focusing on the most confident predictions. The system was tested on data that included the effects of real-world detectors, proving it works not just in theory but in the messy reality of a particle collider.

Having mastered the art of reading the spin, the team then turned their attention to the more elusive search for CP violation. They modified their artificial intelligence to look for a specific kind of asymmetry that would signal a difference between matter and antimatter. In the language of physics, they taught the system to separate "scalar" information, which stays the same under certain transformations, from "pseudoscalar" information, which flips sign. This distinction is vital because the signature of CP violation lives in that flipping sign. The researchers tested this approach on two different scenarios. First, they looked at the production of Higgs bosons, a process where the theoretical rules for CP violation are well understood, to see if their system could learn the correct answer. It did, successfully separating events that violated CP from those that did not.

Finally, they applied this refined tool to the production of top quark pairs, a process that is much more complex and occurs at the detector level. Here, they trained the system to distinguish between standard particle interactions and those influenced by a hypothetical new force that would cause CP violation. The artificial intelligence learned to construct an optimal observable, essentially a mathematical recipe for spotting the difference, directly from the data without needing a human to write the formula beforehand. When they compared the sign of the CP-violating effect, the system correctly identified the direction of the violation. The results showed that the new method could detect these subtle effects with greater sensitivity than traditional techniques, particularly when the violation was strong. While the study relied on simulations of detector data rather than live experimental results, the findings demonstrate that these symmetry-aware artificial intelligence tools can unlock new levels of precision in the search for physics beyond our current understanding.

The work represents a significant step forward in how physicists analyze the data from the Large Hadron Collider. By building artificial intelligence that respects the fundamental laws of physics, the researchers have created a tool that is not only more accurate but also more interpretable. They showed that the system could learn to separate different types of physical information on its own, effectively discovering the best way to look for new physics. This approach suggests that the future of particle physics may lie in partnerships between human intuition and machines that understand the geometry of the universe, allowing scientists to see patterns that were previously hidden in the noise of billions of collisions.

Technical Summary: Top Spin and CP from Equivariant Transformers

Problem Statement
The precision program at the Large Hadron Collider (LHC) relies heavily on the top quark as a probe for fundamental properties, particularly spin correlations and CP violation. While spin correlations in purely leptonic top decays are well-measured, extending these measurements to hadronic decays remains challenging due to the difficulty in identifying the down-type quark ( $d$ ) among the $W$ -boson decay products. Current methods, ranging from simple kinematic assignments to calibrated probabilities and Graph Neural Networks (GNNs), fail to achieve optimal spin analyzing power. Furthermore, constructing optimal observables for CP violation in top pair production is difficult at the detector level. Traditional CP-odd observables rely on the Levi-Civita tensor and require explicit construction of phase-space correlations, which are often suboptimal or hard to define explicitly in the presence of detector effects.

Methodology
The authors employ L-GATr-slim, a Lorentz-equivariant transformer architecture based on geometric algebra (Clifford algebra $Cl_{1,3}(\mathbb{R})$ ). The methodology is divided into two main applications:

Hadronic Top Polarization:
- The authors utilize L-GATr-slim to classify the two light quark jets from a hadronic $W$ decay to identify the down-type quark.
- The network inputs include 4-momenta (vectors), particle IDs, charges, and kinematic features encoded as scalars.
- To handle the symmetry requirements, the architecture is modified to include a pseudoscalar channel alongside the standard scalar and vector channels. This allows the network to distinguish between CP-even and CP-odd information.
- The output is a calibrated probability $p(d=q_{hard}|x)$ , which is used to construct an optimal "down-probe vector" $\vec{k}_{opt}$ . This vector replaces the simple assignment of the down quark, significantly improving the reconstruction of the helicity angle.
CP Violation in Top Pairs and WBF Higgs:
- The authors extend the L-GATr-slim architecture to separate scalar (CP-even) and pseudoscalar (CP-odd) outputs.
- Theoretical Basis: The network is trained to distinguish between CP-conjugate processes (e.g., $f_+$ vs. $f_-$ ). Theoretical arguments suggest that the optimal classifier output corresponds to the log-likelihood ratio, where the pseudoscalar component $p(x)$ converges to the optimal CP-odd observable, while the scalar component $s(x)$ captures CP-even kinematic differences.
- Applications:
  - WBF Higgs Production: Tested at the parton level to distinguish Standard Model (SM) events from those with a dimension-6 SMEFT operator ( $c_{H\tilde{W}}$ ) inducing CP violation.
  - Top Pair Production: Tested at the detector level (including full simulation via Delphes) to distinguish SM events from those with a complex Wilson coefficient $Im(c_{tG})$ in the chromomagnetic dipole operator. The task involves both identifying the presence of CP violation and determining the sign of the interference term.

Key Contributions

Improved Spin Analyzing Power: The paper demonstrates that L-GATr-slim significantly outperforms previous methods (simple assignment, W-helicity probes, and GNNs) in tagging the down-type quark in hadronic top decays.
Architecture for CP Separation: The authors introduce a specific modification to the L-GATr-slim architecture that explicitly separates scalar and pseudoscalar multivector elements. This allows the network to learn and interpret optimal CP-odd observables directly from data without manual construction.
Optimal CP Observables: The study shows that the learned pseudoscalar output acts as an optimal CP-odd observable (equivalent to the log-likelihood ratio), capable of distinguishing CP-conjugate states and determining the sign of CP-violating Wilson coefficients.

Results

Down-Quark Tagging:
- The classifier achieves an Area Under the Curve (AUC) of $0.785 \pm 0.001$ for distinguishing the down quark flavor.
- The resulting spin analyzing power is $\beta_{L-GATr} = 0.721 \pm 0.001$ at the detector level. This is a significant improvement over the GNN benchmark ( $\beta_{GNN} \approx 0.68$ ) and the W-helicity probe ( $\beta \approx 0.64$ ).
- By selecting the top 20% of events with the highest network confidence, the spin analyzing power reaches $0.935 \pm 0.002$ .
CP Violation in WBF Higgs:
- When distinguishing SM from $c_{H\tilde{W}} = +1$ , the network achieves an AUC of $0.607$. The scalar output captures the CP-even interference, while the pseudoscalar output isolates the CP-odd interference.
- When distinguishing $c_{H\tilde{W}} = +1$ from $c_{H\tilde{W}} = -1$ (purely CP-odd task), the network achieves an AUC of $0.691$, relying almost exclusively on the pseudoscalar output.
CP Violation in Top Pairs:
- For distinguishing SM from $Im(c_{tG}) = 2.6$ , the network achieves an AUC of $0.565$ (true down quark) and $0.556$ (tagged down quark). The scalar output dominates this classification due to kinematic changes, while the pseudoscalar output correctly identifies the CP-odd interference.
- For determining the sign of $Im(c_{tG})$ (distinguishing $+2.6$ from $-2.6$), the network achieves an AUC of $0.551$. The pseudoscalar output provides a clear separation, acting as the learned optimal observable.
- The asymmetry $A_{OL-GATr}$ for $Im(c_{tG}) = 2.6$ is found to be $0.0733$ (lepton-lepton) and $0.0518$ (lepton-hadron). The authors find that for $Im(c_{tG}) = 2.6$ , the learned observables from L-GATr-slim substantially improve over their $O_1$ counterparts in the lepton-hadron channel, while they agree within statistical uncertainties for $Im(c_{tG}) = 0.1$ .

Significance and Claims
The paper claims that Lorentz-equivariant transformers based on geometric algebra provide a powerful framework for BSM physics searches at the LHC. By incorporating fundamental symmetries (Lorentz and CP) directly into the network architecture, the authors demonstrate that:

Spin analyzing power in hadronic top decays can be significantly boosted, addressing a long-standing bottleneck in top polarization measurements.
Optimal CP-odd observables can be learned directly from data, bypassing the need for explicit, often suboptimal, analytical constructions.
The separation of scalar and pseudoscalar channels allows for the interpretation of network outputs as distinct physical quantities (CP-even vs. CP-odd), facilitating the extraction of CP-violating parameters even in the presence of detector effects.

The authors conclude that this approach offers access to relevant information for BSM physics through output channels with well-defined Lorentz representations, validating the use of equivariant transformers for precision LHC physics.

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