Closed-form scattering of an arbitrary number of bound states and quantum integrability of the $O(2)_L\times O(2)_R$ chiral Gross–Neveu (Zakharov–Mikhailov) model

The $O(2)_L\times O(2)_R$ chiral Gross–Neveu model in 1+1 dimensions — dual to the one-flavor 'perfect' Gross–Neveu (pGN) model, i.e. the quantized Zakharov–Mikhailov (ZM) spinor model — was solved at large $N$ by Thies (Phys. Rev. D 107, 076024 (2023)): its two vacuum sectors (chiral-symmetry-breaking and Cooper-paired), kinks as domain walls between them, DHN-type baryons, breathers, mesons and phase diagram are known, but only three-kink and baryon–kink scattering were worked out. The problem, explicitly left open there, is to derive closed-form analytical expressions for the scattering of an arbitrary number of bound states — the analogue of the complete multisoliton formulae of the one-flavor Gross–Neveu and Nambu–Jona-Lasinio models — and thereby to definitively confirm the quantum integrability of the model through the elastic, factorized character of all multi-particle collisions. The problem remains open.

Background

The Gross–Neveu (GN) model (Gross and Neveu, 1974) is a $(1+1)$-dimensional field theory of $N$ identical ('colored') Dirac fermions with a four-fermion interaction; it is asymptotically free and generates a fermion mass dynamically. In the large-$N$ limit the theory is governed by self-consistent mean fields: the Hartree–Fock (HF) approximation becomes exact as $N\to\infty$, with quantum corrections organized as a systematic $1/N$ expansion. Dashen, Hasslacher and Neveu (1975) used this structure to construct 'baryons' — semiclassical bound states in which valence fermions fill bound levels of a self-consistent potential — and the same machinery yields kinks (domain walls of the chiral condensate) and breathers (oscillating bound states) as exact time-dependent HF (TDHF) solutions.

A key technical tool is the theory of 'transparent' (reflectionless) one-dimensional Dirac Hamiltonians: self-consistent HF mean fields supporting finitely many bound states are reflectionless potentials, and multisoliton configurations describing the collision of many such objects can be built in closed form. For the one-flavor GN model the general $N$-baryon problem was solved exactly by Fitzner and Thies (2011), with analytical and numerical evidence for elastic, factorized scattering of the composite states (Fitzner and Thies, 2012); the analogous full time-dependent HF solutions of the chiral (Nambu–Jona-Lasinio-type, NJL$_2$) model — whose semiclassical bound-state spectrum goes back to Shei (1976) — were constructed by Dunne and Thies (2013, 2014). In these models, closed analytical expressions for the scattering of any number of bound states exist, which is regarded as the hallmark of quantum integrability at the level of the large-$N$ solution.

Thies (2014) introduced a further integrable variant, the 'perfect' Gross–Neveu (pGN) model: a one-flavor four-fermion theory whose interaction preserves the full Pauli–Gürsey symmetry — the particle–antiparticle mixing symmetry of free massless fermions — so that particle–hole condensation (chiral symmetry breaking, CSB) and particle–particle condensation (Cooper pairing) can coexist and compete. Written in Majorana fermion components, its Lagrangian coincides with one of the classical spinor models proved integrable by Zakharov and Mikhailov (1980) (the ZM model), suggesting that the corresponding quantum theory is integrable as well.

Thies (2023) settled the large-$N$ solvability of this model. He constructed a two-flavor GN model with $O(2)_L\times O(2)_R$ chiral symmetry — where $O(2)$ is a continuous group with two connected components (rotations and reflections), matching the two possible condensation channels — proved its duality with the one-flavor pGN model (the $O(2)$ GN model with $N$ colors is dual to the pGN model with $2N$ colors), and solved it semiclassically at large $N$: two vacuum sectors (CSB and Cooper paired), kinks as domain walls between them, DHN-type baryons (bound states of two twisted kinks, interpretable as Josephson junctions between normal and superconducting phases), breathers, the meson spectrum, and the phase diagram including chiral-spiral and inhomogeneous (LOFF-type, after Larkin–Ovchinnikov–Fulde–Ferrell) phases; the twisted-kink building blocks in the multiflavor chiral GN model were developed earlier by Thies (2022). Scattering, however, was only worked out up to three-kink and baryon–kink collisions, with expressions 'too lengthy to be written down'; the general formulae were left open.

Problem Statement

Derive closed-form analytical expressions for the scattering of an arbitrary number of bound states in the $O(2)_L\times O(2)_R$ chiral Gross–Neveu model — equivalently the one-flavor 'perfect' Gross–Neveu (pGN) model, i.e. the quantized Zakharov–Mikhailov spinor model — at leading order in the $1/N$ expansion, in direct analogy with the complete multisoliton formulae available for the one-flavor Gross–Neveu and NJL$_2$ models. Concretely: construct the general self-consistent time-dependent Hartree–Fock solutions describing arbitrary collisions among the model's bound states — kinks (domain walls between the chiral-symmetry-breaking and Cooper-paired vacua), DHN-type baryons (twisted-kink bound states) and breathers. The single overarching objective is thereby to definitively confirm the quantum integrability of this model at this order, i.e. to demonstrate from these solutions that all many-particle collisions of its bound states are elastic and factorized into freely moving constituents with additive time delays and phase shifts, as in the one-flavor Gross–Neveu and NJL$_2$ models. A valid answer may alternatively refute the premise by exhibiting, within the same leading-order self-consistent large-$N$ framework, a concrete inelastic or non-factorizing many-body collision of the model's bound states.

References

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