Constructing $1+1$-dimensional chiral fermion models in which the massless would-be Goldstone mode $\theta$ fails to decouple from the $S$-matrix

In $1+1$ dimensions continuous symmetries cannot be spontaneously broken, yet the SU( $N$ ) Thirring model (equivalently the chiral Gross–Neveu model) admits a large- $N$ saddle with dynamically massive fermions and a massless phase mode $\theta$ . Witten (1978) showed that $\theta$ is not a Goldstone boson and argued that, in leading approximation, it decouples from the fermion $S$ -matrix because the chiral current satisfies the free wave equation. The 1978–79 exact- $S$ -matrix program and the 2010 form-factor construction for the unmodified model confirmed this decoupling nonperturbatively: the on-shell theory contains only massive anyonic multiplets of masses $m_r = m_1\sin(r\pi/N)/\sin(\pi/N)$ , and the infrared-singular $\theta$ sector decouples together with a compensating negative-norm field. The question left open — whether one can add extra fields or interactions to Witten's exactly soluble Yukawa-type example or to the SU( $N$ ) Thirring model so that a massless $\theta$ survives yet fails to decouple from scattering, and what the spectrum, correlator exponents, and $S$ -matrix of such models would be — has never been taken up in the literature. Resolving it requires either an explicit construction or a general obstruction argument, and would clarify whether the on-shell invisibility of would-be Goldstone modes in two dimensions is a theorem or an artifact of the exactly solved cases.

Background

In $1+1$ space-time dimensions the Mermin–Wagner–Coleman theorem forbids spontaneous breaking of continuous symmetries: fluctuations of the would-be Goldstone mode destroy long-range order. Witten (Nucl. Phys. B 145 (1978) 110) studied how this coexists with the large- $N$ (or $1/N$ ) expansion of the SU( $N$ ) Thirring model, the two-dimensional theory of $N$ massless fermion flavors $\psi_k$ with Lagrangian

\mathcal{L} = \int d^2x\,\Big[ i\,\bar\psi_k \not\partial\,\psi_k + \frac{g}{N}\big( (\bar\psi_k\psi_k)^2 - (\bar\psi_k\gamma_5\psi_k)^2 \big) \Big],

where $\bar\psi = \psi^\dagger\gamma^0$ , $\gamma_5$ is the chirality matrix (which distinguishes left- from right-moving fermion components in $1+1$ dimensions), and repeated flavor indices are summed. The interaction is equivalently the current–current form of the chiral Gross–Neveu (Nambu–Jona-Lasinio $_2$ ) model, so the two names denote the same theory; it is invariant under a continuous U(1) chiral symmetry $\psi_k \to e^{i\alpha\gamma_5}\psi_k$ .

The $1/N$ expansion is a semiclassical expansion in powers of $1/N$ around a saddle point. At the saddle a chiral condensate $\langle\bar\psi\psi\rangle \neq 0$ appears to break chiral symmetry, the fermions acquire a dynamical mass, and writing the auxiliary fields in polar form $\sigma + i\pi = \rho\,e^{i\theta}$ exposes a massless phase excitation $\theta$ . Witten argued, by reasoning analogous to the Berezinskii–Kosterlitz–Thouless analysis of the two-dimensional XY model (in which vortex proliferation destroys true long-range order while correlations still decay as powers of distance), that the exact theory instead exhibits algebraic (quasi-long-range) order: the chiral-symmetry-breaking correlator decays as $|x|^{-1/N}$ , the effective theory of the phase is a free massless scalar with action $\frac{N}{4\pi}\int d^2x\,(\nabla\theta)^2$ , and the $1/N$ expansion remains a good qualitative guide (massive physical fermions plus one massless mode) even though no symmetry is broken.

The paper also analyzed an exactly soluble example: a Yukawa-type model of massless fermions coupled to scalar and pseudoscalar fields, solvable by bosonization (the exact rewriting of two-dimensional fermion fields as vertex operators of a scalar boson) followed by a linear redefinition of the bosonic fields into a free massless scalar and a sine-Gordon field (an exactly solvable scalar field with a periodic cosine self-interaction) with coupling $\beta^2 = 4\pi + 1/a^2$ . In both settings the massless mode $\theta$ is 'not a Goldstone boson' — its quanta do not supply the low-energy-theorem singularities that broken symmetry would require in the Ward identities — and a field-theoretic argument (the chiral current satisfies the free wave equation $\nabla^2 A_\mu = 0$ ) shows that the $\theta$ quantum is created by the conserved current and decouples from the fermion $S$ -matrix in leading approximation.

A factorized $S$ -matrix in $1+1$ dimensions is a two-particle scattering matrix that decomposes into elastic two-body amplitudes depending only on the difference of rapidities (parameters labeling on-shell momenta); integrable models admit such exact $S$ -matrices, constrained by unitarity, crossing symmetry, and the Yang–Baxter equations (the consistency condition that makes three-particle scattering factorize into two-particle amplitudes independently of the scattering order). For the chiral SU( $N$ ) model, Berg–Weisz (1978), Köberle–Kurak–Swieca (1979), and Abdalla–Berg–Weisz (1979) proposed the exact $S$ -matrix and confronted the infrared (IR) singularities that the would-be Goldstone mode causes in the naive $1/N$ expansion. Two off-shell mechanisms competed: 'infrared clouds' (Swieca and collaborators, introducing compensating negative-norm bosonic fields) and the elaboration of Witten's own proposal; they agree on shell. The form-factor construction of Babujian–Foerster–Karowski (2010) (form factors are matrix elements of local operators between the vacuum and multi-particle states) selects the Swieca-type solution: the physical particles are anyons (particles with fractional exchange statistics) of spin $s = \frac{1}{2}(1 - 1/N)$ forming $N-1$ multiplets of masses $m_r = m_1\sin(r\pi/N)/\sin(\pi/N)$ , and the IR-singular $\pi$ (i.e. $\theta$ ) sector decouples from the rest of the model. A separate later framework — massless factorized $S$ -matrices describing renormalization-group flows between critical points — establishes that massless particles in $1+1$ dimensions can scatter nontrivially (between left- and right-movers), but no such construction has been realized for a would-be Goldstone mode of a chiral fermion model.

Problem Statement

Construct, or rule out, $1+1$ -dimensional quantum field theories of Witten's chiral-fermion type — obtained by adding extra fields or interactions to the exactly soluble Yukawa-type example of Witten's section 2, or to the SU( $N$ ) Thirring (chiral Gross–Neveu) model — in which a massless mode $\theta$ of the would-be Goldstone type survives but does not decouple from the $S$ -matrix. For at least one explicit such model, the following are to be determined: (i) the particle spectrum, in particular whether the massless mode appears as an asymptotic scattering state and how the massive multiplet structure is modified; (ii) the exact or controlled asymptotic behavior of chiral-symmetry-breaking correlators — whether the algebraic decay exponents (e.g. $|x|^{-1/N}$ in the parent model) are changed or destabilized by the coupling of $\theta$ to other degrees of freedom; and (iii) the scattering amplitudes involving the massless mode, including the nontrivial left–right-mover and massless–massive amplitudes, within a well-defined framework for massless scattering in $1+1$ dimensions. Alternatively, prove an obstruction: that under precisely stated hypotheses (e.g. unbroken continuous chiral symmetry, locality, Poincaré invariance in $1+1$ dimensions) a massless would-be Goldstone mode necessarily decouples from the $S$ -matrix, which would elevate Witten's leading-order argument to a theorem and explain the outcome of the 1978–2010 analyses of the unmodified model.

References

E. Witten, "Chiral symmetry, the 1/N expansion and the SU(N) Thirring model", Nuclear Physics B 145 (1978) 110-118. doi:10.1016/0550-3213(78)90416-9. https://doi.org/10.1016/0550-3213(78)90416-9
B. Berg, P. Weisz, "Exact S-matrix of the chiral invariant SU(N) Thirring model", Nuclear Physics B 146 (1978) 205-214. doi:10.1016/0550-3213(78)90438-8. https://doi.org/10.1016/0550-3213(78)90438-8
E. Abdalla, B. Berg, P. Weisz, "More about the S-matrix of the chiral SU(N) Thirring model", Nuclear Physics B 157 (1979) 387-391. doi:10.1016/0550-3213(79)90110-x. https://doi.org/10.1016/0550-3213(79)90110-x
R. Köberle, V. Kurak, J. A. Swieca, "Scattering theory and 1/N expansion in the chiral Gross-Neveu model", Physical Review D 20 (1979) 897-902. doi:10.1103/PhysRevD.20.897. https://doi.org/10.1103/PhysRevD.20.897
H. M. Babujian, A. Foerster, M. Karowski, "Exact form factors of the SU(N) Gross–Neveu model and 1/N expansion", Nuclear Physics B 825 (2010) 396-425. arXiv:0907.0662, doi:10.1016/j.nuclphysb.2009.09.023. https://arxiv.org/abs/0907.0662, https://doi.org/10.1016/j.nuclphysb.2009.09.023
B. Schroer, "Jorge A. Swieca's contributions to quantum field theory in the 60s and 70s and their relevance in present research", The European Physical Journal H 35 (2010) 53-88. arXiv:0712.0371, doi:10.1140/epjh/e2010-00004-1. https://arxiv.org/abs/0712.0371, https://doi.org/10.1140/epjh/e2010-00004-1
M. Faber, A. N. Ivanov, "Goldstone bosons in the massless Thirring model. Witten's criterion", unpublished preprint (2003), arXiv:hep-th/0305174. https://arxiv.org/abs/hep-th/0305174 (Unrefereed; a dissenting reading of Witten's Goldstone criterion for the unmodified massless Thirring model, which does not bear on the modified-model construction or the $S$ -matrix decoupling question.)
S. Frolov, D. Polvara, A. Sfondrini, "Exchange relations and crossing", Journal of Physics A: Mathematical and Theoretical 58 (2025) 415402. arXiv:2506.04096, doi:10.1088/1751-8121/ae0edb. https://arxiv.org/abs/2506.04096, https://doi.org/10.1088/1751-8121/ae0edb

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Constructing $1+1$ -dimensional chiral fermion models in which the massless would-be Goldstone mode $\theta$ fails to decouple from the $S$ -matrix

Background

Problem Statement

References