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A one-parameter integrable deformation of the Dirac--sinh-Gordon system

Laith H. Haddad

Abstract

We establish the integrability of a one-parameter family of coupled Dirac--scalar field theories in $(1+1)$ dimensions that interpolates between the known Dirac--sinh-Gordon and Dirac--sine-Gordon systems. The deformation is controlled by a phase parameter that modifies the Yukawa coupling and simultaneously rescales the scalar backreaction. For all values of the parameter, we construct an explicit zero-curvature representation based on an $sl(2,\mathbb{C})$-valued Lax pair and show that the deformation preserves integrability. We further prove that the family is physically non-trivial, in the sense that distinct parameter values are not related by admissible field redefinitions. In addition, we derive the continuity relation for the fermion bilinear, show that the spatial bilinear constraint follows from the zero-curvature equations, and construct the first conserved densities of the hierarchy. At the two endpoints, the family reduces to the standard integrable Dirac--sinh-Gordon model and, after analytic continuation, to the Dirac--sine-Gordon system which is dual to the massive Thirring model.

A one-parameter integrable deformation of the Dirac--sinh-Gordon system

Abstract

We establish the integrability of a one-parameter family of coupled Dirac--scalar field theories in dimensions that interpolates between the known Dirac--sinh-Gordon and Dirac--sine-Gordon systems. The deformation is controlled by a phase parameter that modifies the Yukawa coupling and simultaneously rescales the scalar backreaction. For all values of the parameter, we construct an explicit zero-curvature representation based on an -valued Lax pair and show that the deformation preserves integrability. We further prove that the family is physically non-trivial, in the sense that distinct parameter values are not related by admissible field redefinitions. In addition, we derive the continuity relation for the fermion bilinear, show that the spatial bilinear constraint follows from the zero-curvature equations, and construct the first conserved densities of the hierarchy. At the two endpoints, the family reduces to the standard integrable Dirac--sinh-Gordon model and, after analytic continuation, to the Dirac--sine-Gordon system which is dual to the massive Thirring model.
Paper Structure (27 sections, 7 theorems, 63 equations)

This paper contains 27 sections, 7 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Setup and endpoint systems
  3. Conventions
  4. Component form of the Dirac equation
  5. The $\theta_{0}$-deformed Lax pair
  6. Zero-curvature computation
  7. Decomposition by grade
  8. Grade $+1$: the $\mathsf{E}_{+}$ component
  9. Grade $-1$: the $\mathsf{E}_{-}$ component
  10. Grade $0$: the $\mathsf{H}$ component
  11. Gauge analysis and physical non-triviality
  12. The phase-generating gauge transformation
  13. Why the physical system is non-trivial
  14. The fermion bilinear constraint
  15. Anomalous continuity equation
  16. ...and 12 more sections

Key Result

Theorem 4.1

The zero-curvature condition eq:ZC for the Lax pair eq:Ap_def--eq:Am_def is equivalent to the system eq:theta_scalar--eq:theta_Dirac together with the constraint $\partial_x(\bar{\psi}\psi) = 0$. The phase $\theta_{0}$ cancels from all three grade components of the zero-curvature condition. In parti

Theorems & Definitions (15)

  • Definition 3.1: $\theta_{0}$-deformed Lax pair
  • Theorem 4.1: Zero-curvature condition
  • Lemma 5.1
  • proof
  • Corollary 5.2
  • Theorem 5.3: Physical non-triviality
  • Remark 5.4
  • Proposition 6.1: Anomalous continuity equation
  • proof
  • Proposition 6.2: Constraint from zero-curvature
  • ...and 5 more