In 1+11+1 dimensions continuous symmetries cannot be spontaneously broken, yet the SU(NN) Thirring model (equivalently the chiral Gross–Neveu model) admits a large-NN saddle with dynamically massive fermions and a massless phase mode θ\theta. Witten (1978) showed that θ\theta is not a Goldstone boson and argued that, in leading approximation, it decouples from the fermion SS-matrix because the chiral current satisfies the free wave equation. The 1978–79 exact-SS-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 mr=m1sin⁡(rπ/N)/sin⁡(π/N)m_r = m_1\sin(r\pi/N)/\sin(\pi/N), and the infrared-singular θ\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(NN) Thirring model so that a massless θ\theta survives yet fails to decouple from scattering, and what the spectrum, correlator exponents, and SS-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+11+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-NN (or 1/N1/N) expansion of the SU(NN) Thirring model, the two-dimensional theory of NN massless fermion flavors ψk\psi_k with Lagrangian

L=∫d2x [i ψˉk∂̸ ψk+gN((ψˉkψk)2−(ψˉkγ5ψk)2)],\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 ψˉ=ψ†γ0\bar\psi = \psi^\dagger\gamma^0, γ5\gamma_5 is the chirality matrix (which distinguishes left- from right-moving fermion components in 1+11+1 dimensions), and repeated flavor indices are summed. The interaction is equivalently the current–current form of the chiral Gross–Neveu (Nambu–Jona-Lasinio2_2) model, so the two names denote the same theory; it is invariant under a continuous U(1) chiral symmetry ψk→eiαγ5ψk\psi_k \to e^{i\alpha\gamma_5}\psi_k.

The 1/N1/N expansion is a semiclassical expansion in powers of 1/N1/N around a saddle point. At the saddle a chiral condensate ⟨ψˉψ⟩≠0\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 σ+iπ=ρ eiθ\sigma + i\pi = \rho\,e^{i\theta} exposes a massless phase excitation θ\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|x|^{-1/N}, the effective theory of the phase is a free massless scalar with action N4π∫d2x (∇θ)2\frac{N}{4\pi}\int d^2x\,(\nabla\theta)^2, and the 1/N1/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 β2=4π+1/a2\beta^2 = 4\pi + 1/a^2. In both settings the massless mode θ\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 ∇2Aμ=0\nabla^2 A_\mu = 0) shows that the θ\theta quantum is created by the conserved current and decouples from the fermion SS-matrix in leading approximation.

A factorized SS-matrix in 1+11+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 SS-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(NN) model, Berg–Weisz (1978), Köberle–Kurak–Swieca (1979), and Abdalla–Berg–Weisz (1979) proposed the exact SS-matrix and confronted the infrared (IR) singularities that the would-be Goldstone mode causes in the naive 1/N1/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=12(1−1/N)s = \frac{1}{2}(1 - 1/N) forming N−1N-1 multiplets of masses mr=m1sin⁡(rπ/N)/sin⁡(π/N)m_r = m_1\sin(r\pi/N)/\sin(\pi/N), and the IR-singular π\pi (i.e. θ\theta) sector decouples from the rest of the model. A separate later framework — massless factorized SS-matrices describing renormalization-group flows between critical points — establishes that massless particles in 1+11+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+11+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(NN) Thirring (chiral Gross–Neveu) model — in which a massless mode θ\theta of the would-be Goldstone type survives but does not decouple from the SS-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|x|^{-1/N} in the parent model) are changed or destabilized by the coupling of θ\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+11+1 dimensions. Alternatively, prove an obstruction: that under precisely stated hypotheses (e.g. unbroken continuous chiral symmetry, locality, Poincaré invariance in 1+11+1 dimensions) a massless would-be Goldstone mode necessarily decouples from the SS-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

  1. 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
  2. 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
  3. 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
  4. 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
  5. 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
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  7. 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 SS-matrix decoupling question.)
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