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TBA equations for excited states in the O(3) and O(4) nonlinear $σ$-model

J. Balog, A. Hegedus

TL;DR

The paper develops excited-state Thermodynamic Bethe Ansatz equations for 1-particle states in the sausage- and SS-models and their $O(3)$ and $O(4)$ nonlinear sigma-model limits, enabling the exact finite-volume mass gap to be computed in conjunction with ground-state TBA and Lüscher’s formula. By leveraging the infinite-volume Y-system structure and an efficient cutoff-infinite approach, the authors seed and iteratively solve the TBA equations to obtain high-precision finite-volume energies across volume ranges, and they validate the results against three-loop perturbation theory and Monte Carlo data. The key contributions are the proposal of explicit excited-state TBA/Y-system forms for these models, the demonstration of their numerical solvability, and the demonstration that the sigma-model mass gap agrees with independent nonperturbative and perturbative benchmarks. The work provides a consistent framework for finite-volume spectroscopy in integrable QFTs, with potential applicability to other deformations and sigma-model limits.

Abstract

TBA integral equations are proposed for 1-particle states in the sausage- and SS-models and their $σ$-model limits. Combined with the ground state TBA equations the exact mass gap is computed in the O(3) and O(4) nonlinear $σ$-model and the results are compared to 3-loop perturbation theory and Monte Carlo data.

TBA equations for excited states in the O(3) and O(4) nonlinear $σ$-model

TL;DR

The paper develops excited-state Thermodynamic Bethe Ansatz equations for 1-particle states in the sausage- and SS-models and their and nonlinear sigma-model limits, enabling the exact finite-volume mass gap to be computed in conjunction with ground-state TBA and Lüscher’s formula. By leveraging the infinite-volume Y-system structure and an efficient cutoff-infinite approach, the authors seed and iteratively solve the TBA equations to obtain high-precision finite-volume energies across volume ranges, and they validate the results against three-loop perturbation theory and Monte Carlo data. The key contributions are the proposal of explicit excited-state TBA/Y-system forms for these models, the demonstration of their numerical solvability, and the demonstration that the sigma-model mass gap agrees with independent nonperturbative and perturbative benchmarks. The work provides a consistent framework for finite-volume spectroscopy in integrable QFTs, with potential applicability to other deformations and sigma-model limits.

Abstract

TBA integral equations are proposed for 1-particle states in the sausage- and SS-models and their -model limits. Combined with the ground state TBA equations the exact mass gap is computed in the O(3) and O(4) nonlinear -model and the results are compared to 3-loop perturbation theory and Monte Carlo data.

Paper Structure

This paper contains 11 sections, 133 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. S-matrix data
  3. Ground state TBA equations and Y-systems
  4. Excited states in the SG model
  5. Lüscher's formula
  6. The $l \rightarrow \infty$ solution of the one particle Y-systems
  7. One-particle TBA equations of the SS-model
  8. One-particle TBA equations of the sausage-model
  9. Numerical results
  10. Monte Carlo and perturbative results
  11. Summary and conclusion

Figures (8)

  • Figure 1: Finite volume mass gap of the $O(3)$ model. Comparison of numerical solution of the TBA integral equations (dots) to three-loop perturbation theory (solid line). The two-loop perturbative curve (dashed line) is also shown.
  • Figure 2: Finite volume mass gap of the $O(4)$ model. Comparison of numerical solution of the TBA integral equations (dots) to three-loop perturbation theory (solid line). The two-loop perturbative result is too close to the three-loop result and is not shown here.
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  • ...and 3 more figures