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Violation of Parity and Flavor Symmetries in a Nambu-Jona-Lasinio Model

Yukimi Goto, Tohru Koma

TL;DR

The paper proves, for a lattice two-flavor NJL model with continuous chiral symmetry and no Wilson term, that the ground state can spontaneously break both parity and flavor. It builds a rigorous framework based on fermion reflection positivity, Gaussian domination, and infrared bounds to show simultaneous long-range order in two noncommuting order parameters, $O_\Lambda^{(1)}$ and $O_{5,\Lambda}^{(2)}$, in the infinite-volume limit at low temperature and sufficiently strong coupling (small $|\kappa|/g$). The results hinge on a careful plane-reflection analysis, decomposition of the Hamiltonian, and a variational construction that yields nonzero magnetizations for both order parameters. The work connects to Aoki-phase-type flavor-parity structures in lattice gauge theories while highlighting caveats about the absence of a proper chiral anomaly and potential ground-state mixtures.

Abstract

We study a lattice Nambu-Jona-Lasinio model with certain continuous chiral and two-flavor symmetries. For the Hamiltonian of the model, we construct a ground state which breaks the parity and flavor symmetries. In our argument, the chiral symmetry plays a crucial role for proving the violation of the parity and flavor symmetries, although the model does not contain the so-called Wilson term.

Violation of Parity and Flavor Symmetries in a Nambu-Jona-Lasinio Model

TL;DR

The paper proves, for a lattice two-flavor NJL model with continuous chiral symmetry and no Wilson term, that the ground state can spontaneously break both parity and flavor. It builds a rigorous framework based on fermion reflection positivity, Gaussian domination, and infrared bounds to show simultaneous long-range order in two noncommuting order parameters, and , in the infinite-volume limit at low temperature and sufficiently strong coupling (small ). The results hinge on a careful plane-reflection analysis, decomposition of the Hamiltonian, and a variational construction that yields nonzero magnetizations for both order parameters. The work connects to Aoki-phase-type flavor-parity structures in lattice gauge theories while highlighting caveats about the absence of a proper chiral anomaly and potential ground-state mixtures.

Abstract

We study a lattice Nambu-Jona-Lasinio model with certain continuous chiral and two-flavor symmetries. For the Hamiltonian of the model, we construct a ground state which breaks the parity and flavor symmetries. In our argument, the chiral symmetry plays a crucial role for proving the violation of the parity and flavor symmetries, although the model does not contain the so-called Wilson term.
Paper Structure (17 sections, 1 theorem, 246 equations)

This paper contains 17 sections, 1 theorem, 246 equations.

Table of Contents

  1. Introduction
  2. Hamiltonian of the Nambu--Jona-Lasinio model
  3. Continuous chiral symmetry of the Hamiltonian
  4. Spontaneous breaking of both flavor and parity symmetries
  5. Reflection positivity: Preliminary
  6. Reflection with respect to the plane $x^{(1)}=1/2$
  7. Two unitary transformations
  8. Kinetic Hamiltonian
  9. Interaction Hamiltonian $\tilde{H}_{\rm int,R}^{(\Lambda)}(h)$
  10. Interaction Hamiltonian $\tilde{H}_{\rm int,I}^{(\Lambda)}$
  11. Reflection with respect to the plane $x^{(2)}=1/2$
  12. Gaussian domination and infrared bound
  13. Long-range order
  14. Chiral rotational symmetry
  15. A lower bound for $\mathcal{E}^{(\Lambda)}$
  16. ...and 2 more sections

Key Result

Theorem 4.1

There exists a positive number $\alpha_0$ small enough such that for all the hopping parameter $\kappa$ and the coupling constant $g$ of the interaction which satisfy $\vert\kappa\vert/g \le \alpha_0$.

Theorems & Definitions (2)

  • Theorem 4.1
  • Remark 4.2