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Lorentz-Equivariant Geometric Algebra Transformers for High-Energy Physics
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Proposes L-GATr, a Lorentz-equivariant geometric-algebra Transformer for high-energy physics, applied to amplitude surrogates, top tagging, and a Riemannian flow-matching generative model of LHC events.
Problem
Existing architectures for high-energy physics do not systematically respect Lorentz symmetry and scale poorly, while the field needs precise, data-efficient models for regression, classification, and generation of particle data.
Approach
Represent particles as multivectors in the geometric algebra Cl(1,3) and build a Transformer whose linear maps, attention, normalization, and nonlinearities are Lorentz-equivariant; generative modelling uses a continuous normalizing flow with Riemannian flow matching in physically motivated coordinates that respects phase-space boundaries.
Key results
- Top tagging: L-GATr reaches AUC 0.9870 and accuracy 0.9423, on par with the best baselines (PELICAN 0.9870 AUC), with 1/eps_B = 540 at eps_S=0.5.
- Generative modelling: L-GATr CFM achieves lower NLL than Transformer, MLP, E(3)-GATr flows and JetGPT across all t tbar+n jets processes; classifier two-sample ROC AUC 0.51-0.56 vs 0.58-0.85 for Transformer/MLP.
- Riemannian flow matching lowers NLL from -30.11 (Euclidean) to -32.65 on the ttbar+0j test set.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use L-GATr instead of the strongest reported baseline | Top tagging · Top tagging | accuracy: 0.9426 ± 0.0002 vs 0.9423 ± 0.0002 | No clear effect |
| Use L-GATr instead of PELICAN | Top tagging · Top tagging | AUC: 0.9870 ± 0.0001 vs 0.9870 ± 0.0001 | No clear effect |
| Use L-GATr instead of the strongest reported baseline | Top tagging · Top tagging | 1/epsilon_B (epsilon_S=0.3): 2240 ± 70 vs 2250 ± 75 | No clear effect |
| Use Riemannian flow matching probability paths instead of Euclidean paths (ablation) | L-GATr CFM · ttbar + 0j test set | NLL: -30.11 ± 0.98 -> -32.65 ± 0.01 | Helped |
| Use L-GATr as the score network instead of Transformer or MLP score networks | Conditional flow matching generative models · ttbar + jets processes | log likelihood | Helped |
| Use L-GATr flow instead of Transformer or MLP flows | Generative models · ttbar + jets processes | classifier two-sample test ROC AUC: 0.51 and 0.56; 0.58 and 0.85 | Helped |
| Use L-GATr instead of DSI | Amplitude surrogate · QFT amplitudes, low particle multiplicity | surrogate error | Hurt |
| Use L-GATr instead of the baselines | Amplitude surrogate · QFT amplitudes, high particle multiplicity | surrogate error | Helped |
Surprising findings
- For a small number of particles the domain-specific DSI baseline is slightly better than L-GATr in amplitude surrogates, while L-GATr wins at high multiplicity.
Limitations
- Computational overhead compared to efficient vanilla Transformer implementations due to more complex linear maps.
- Not investigated as a foundation model or for multi-task pretraining.
- Generative models still fail to capture the top mass peak at percent-level precision.
Experimental setup
Datasets: q qbar -> Z+ng MadGraph amplitudes top tagging reference dataset (Kasieczka et al.) t tbar + n jets MadGraph/Pythia/Delphes simulation
Benchmarks: top tagging reference dataset q qbar -> Z+ng amplitude surrogate t tbar + n jets generative benchmark
Metrics: accuracy AUC inverse background rejection at fixed signal efficiency surrogate error negative log likelihood classifier two-sample test ROC AUC
Baselines: CGENN Transformer Deep Sets with invariants (DSI) Geometric Algebra Perceptron MLP LorentzNet PELICAN ParticleNet ParT JetGPT E(3)-GATr
Builds on: GATr Riemannian flow matching conditional flow matching Clifford group equivariant neural networks Deep Sets
Closest papers by meaning
- Lorentz Local Canonicalization: How to make any Network Lorentz-Equivariant (NeurIPS 2025 · similarity 0.89)
- Lorentz Group Equivariant Neural Network for Particle Physics (ICML 2020 · similarity 0.89)
- Geometric Algebra Transformer (NeurIPS 2023 · similarity 0.88)
- GLGENN: A Novel Parameter-Light Equivariant Neural Networks Architecture Based on Clifford Geometric Algebras (ICML 2025 · similarity 0.83)
- Large Language-Geometry Model: When LLM meets Equivariance (ICML 2025 · similarity 0.83)
- Equivariance via Minimal Frame Averaging for More Symmetries and Efficiency (ICML 2024 · similarity 0.83)
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