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Meson properties and symmetry emergence based on the deep neural network

Xin Tong, Wei Feng, Weiwei Xu, Chao-Hsi Chang, Guo-Li Wang, Qiang Li

TL;DR

This work tackles the challenge of predicting meson total widths, a difficult problem in non-perturbative QCD, by leveraging a Transformer-based deep neural network (FT-Transformer) trained on a carefully engineered feature set derived from meson quantum numbers and masses. A Gaussian Monte Carlo data augmentation scheme, based on PDG mass uncertainties, expands a small experimental corpus into roughly $2 imes10^5$ samples to bolster robustness. The model achieves near-perfect correlation in log-width space ($R^2 ext{ on log widths }=0.9987$) with sub-percent to a few-percent relative errors across train and test sets, and it generates width spectra for unmeasured states while offering tentative quantum-number assignments for undetermined mesons. Crucially, the network spontaneously exhibits fundamental symmetries—exact charge conjugation symmetry and approximate isospin symmetry—suggesting that data-driven learning can reveal deep physical regularities in hadron structure, and it provides a complementary paradigm to traditional theory and lattice methods for hadron spectroscopy and exotic-state identification.

Abstract

As a key property of hadrons, the total width is quite difficult to obtain in theory due to the extreme complexity of the strong and electroweak interactions. In this work, a deep neural network model with the Transformer architecture is built to precisely predict meson widths in the range of $10^{-14} \sim 625$ MeV based on meson quantum numbers and masses. The relative errors of the predictions are $0.12\%, 2.0\%,$ and $0.54\%$ in the training set, the test set, and all the data, respectively. We present the predicted meson width spectra for the currently discovered states and some theoretically predicted ones. The model is also used as a probe to study the quantum numbers and inner structures for some undetermined states including the exotic states. Notably, this data-driven model is investigated to spontaneously exhibit good charge conjugation symmetry and approximate isospin symmetry consistent with physical principles. The results indicate that the deep neural network can serve as an independent complementary research paradigm to describe and explore the hadron structures and the complicated interactions in particle physics alongside the traditional experimental measurements, theoretical calculations, and lattice simulations.

Meson properties and symmetry emergence based on the deep neural network

TL;DR

This work tackles the challenge of predicting meson total widths, a difficult problem in non-perturbative QCD, by leveraging a Transformer-based deep neural network (FT-Transformer) trained on a carefully engineered feature set derived from meson quantum numbers and masses. A Gaussian Monte Carlo data augmentation scheme, based on PDG mass uncertainties, expands a small experimental corpus into roughly samples to bolster robustness. The model achieves near-perfect correlation in log-width space () with sub-percent to a few-percent relative errors across train and test sets, and it generates width spectra for unmeasured states while offering tentative quantum-number assignments for undetermined mesons. Crucially, the network spontaneously exhibits fundamental symmetries—exact charge conjugation symmetry and approximate isospin symmetry—suggesting that data-driven learning can reveal deep physical regularities in hadron structure, and it provides a complementary paradigm to traditional theory and lattice methods for hadron spectroscopy and exotic-state identification.

Abstract

As a key property of hadrons, the total width is quite difficult to obtain in theory due to the extreme complexity of the strong and electroweak interactions. In this work, a deep neural network model with the Transformer architecture is built to precisely predict meson widths in the range of MeV based on meson quantum numbers and masses. The relative errors of the predictions are and in the training set, the test set, and all the data, respectively. We present the predicted meson width spectra for the currently discovered states and some theoretically predicted ones. The model is also used as a probe to study the quantum numbers and inner structures for some undetermined states including the exotic states. Notably, this data-driven model is investigated to spontaneously exhibit good charge conjugation symmetry and approximate isospin symmetry consistent with physical principles. The results indicate that the deep neural network can serve as an independent complementary research paradigm to describe and explore the hadron structures and the complicated interactions in particle physics alongside the traditional experimental measurements, theoretical calculations, and lattice simulations.

Paper Structure

This paper contains 19 sections, 15 equations, 16 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Meson encoding methods and data enhancement
  3. Features used to denote the mesons
  4. Quark contents encoding
  5. Complete input features
  6. Gaussian Monte Carlo data enhancement
  7. Deep neural network model
  8. Backbone of the model
  9. Loss function to describe the meson widths
  10. Training and validation
  11. Numerical results of the meson width predictions
  12. Overall performance of the model
  13. Meson width predictions and quantum numbers identification
  14. Analysis of quark content encoding
  15. Symmetries reproduced by the neural network
  16. ...and 4 more sections

Figures (16)

  • Figure 1: The mass distribution of $\pi(1300)$ and $\psi(4040)$ after the Gaussian enhancement.
  • Figure 2: The main parts of the FT-Transformer architecture. Both the numerical and categorical features are first embedded to $d$-dimensional vectors, which are then together with the [CLS] token fed into the typical Transformer encoder layer.
  • Figure 3: (a) The Augmented Embedding module to deal with the numerical features; (b) one Transformer layer.
  • Figure 4: The predicted meson widths versus the experimental values in units of MeV under the natural logarithm. The horizontal axis represents the experimental widths reported by PDG, while the vertical axis represents the predicted values. The blue dots denote the results of the training data, while the orange ones denote results of the test sample.
  • Figure 5: The relative error $\epsilon_i$ versus the log widths in units of MeV. The blue (orange) denotes the results of the training (test) data.
  • ...and 11 more figures