Generating particle physics Lagrangians with transformers

TL;DR

This work treats particle-physics Lagrangians as symbolic expressions and trains a transformer to map a list of fields and quantum numbers to a gauge-invariant Lagrangian under the SM group $SU(3)_C \times SU(2)_L \times U(1)_Y$. Using a dataset generation pipeline that combines AutoEFT with custom code, a ~ $3.57\times 10^8$-parameter BART model learns to produce kinetic, mass, and interaction terms that respect both gauge and Lorentz invariance, with an emphasis on trilinear couplings. The model achieves high accuracy (>90%) on in-distribution test sets and demonstrates meaningful out-of-distribution generalization up to higher field counts, aided by embedding analyses that reveal learned symmetry structures and a conjugation axis. The authors publicly release the model and data, and discuss next steps toward a broader foundation-model for theoretical physics that can incorporate more complex symmetry groups, flavor structures, and experimental constraints. $S$-level outcomes suggest that symbolic-structure learning is feasible for symbolic physics tasks and can guide future ML-assisted theory development.

Abstract

In physics, Lagrangians provide a systematic way to describe laws governing physical systems. In the context of particle physics, they encode the interactions and behavior of the fundamental building blocks of our universe. By treating Lagrangians as complex, rule-based constructs similar to linguistic expressions, we trained a transformer model -- proven to be effective in natural language tasks -- to predict the Lagrangian corresponding to a given list of particles. We report on the transformer's performance in constructing Lagrangians respecting the Standard Model $\mathrm{SU}(3)\times \mathrm{SU}(2)\times \mathrm{U}(1)$ gauge symmetries. The resulting model is shown to achieve high accuracies (over 90\%) with Lagrangians up to six matter fields, with the capacity to generalize beyond the training distribution, albeit within architectural constraints. We show through an analysis of input embeddings that the model has internalized concepts such as group representations and conjugation operations as it learned to generate Lagrangians. We make the model and training datasets available to the community. An interactive demonstration can be found at: \url{https://huggingface.co/spaces/JoseEliel/generate-lagrangians}.

Figures (11)

• Figure 1: Data generation and model task. The left side shows the data generation pipeline, where fields and their quantum numbers are used to write a Lagrangian. The right side shows the model task, where the model takes input tokens and generates output tokens.
• Figure 2: Training Data Distribution of both sampled (left) and uniform (right) datasets.
• Figure 3: Distribution of the Lagrangian scores (Eq. \ref{['eq:lag_score']}) for the sampled model (left) and the uniform model (right).
• Figure 4: Cumulative distribution of the fraction of wrong terms in predicted Lagrangian with different number of fields. On the left (solid lines) is the sampled model, and on the right (dashed lines) is the uniform model.
• Figure 5: Cumulative distribution of the fraction of wrong terms in predicted Lagrangian with and without trilinear interactions. On the left (solid lines) is the sampled model, and on the right (dashed lines) is the uniform model.