Integrable-QFT Casebook, Exact Data, and Limits

Canonical integrable relativistic QFTs share factorized scattering but differ sharply in particle content, internal symmetry, diagonal versus matrix scattering, bound states, operator sectors, ultraviolet behavior, and rigorous status. A method should be selected from those features, not from the word “integrable” alone. Exact data belong to the declared model and conventions; they are neither universal across QFT nor automatically a construction of local observables.

Required background. Classical and quantum conserved charges supplies the quantum-integrability criterion; elasticity and factorization supplies its scattering hypotheses; Yang–Baxter consistency supplies matrix ordering; the exact S-matrix bootstrap supplies analytic and CDD input; bound-state poles and fusion supplies pole classification; Bethe quantization supplies large-volume levels; form-factor bootstrap supplies local-operator matrix elements; and thermodynamic Bethe ansatz supplies finite-size ground-state data. Helpful background. Claim status across methods supplies the distinction among exact calculation, controlled evidence, and rigorous construction.

Comparative casebook

The table is a route selector, not a completeness claim. “Exact data” means closed analytic data within the stated accepted model and hypotheses.

Representative massive integrable relativistic models and the limits of their standard exact-data descriptions

Model	Stable asymptotic content	Scattering structure	Typical exact data	Best first method	Essential boundary
Sinh-Gordon	One neutral massive scalar	Diagonal; pole-free physical strip in the standard coupling range	Scalar S matrix, form factors, TBA	Scalar bootstrap, then TBA or spectral expansion	A bounded regular pole-free subset has a local operator-algebraic construction; arbitrary CDD choices do not inherit it
Sine-Gordon / massive Thirring	Soliton and antisoliton; breathers in the attractive regime	Non-diagonal charged sector with fusion to diagonal neutral sectors	Matrix S matrix, breather spectrum, form factors, finite-size data	Yang–Baxter plus pole and fusion bootstrap	Bosonization parameters, charge sectors, locality phases, and coupling range must be translated explicitly
O(N) sigma model, N greater than 2	Massive O(N) vector multiplet in the standard symmetric phase	Non-diagonal O(N)-invariant scattering	Exact S matrix, Bethe/TBA data, selected form factors	Representation projectors and Yang–Baxter equations	N equals 2 has compact-boson and Berezinskii–Kosterlitz–Thouless physics and is not obtained by unqualified substitution
Discrete-chiral Gross–Neveu	Fermions and model-dependent bound or kink sectors	Generally non-diagonal, symmetry-constrained scattering	Mass spectrum and factorized amplitudes in declared symmetry sectors	Spectrum and channel bootstrap before thermodynamics	Do not transfer spontaneous-breaking language or particle lists to continuous-chiral variants
Continuous-chiral Gross–Neveu variants	Sector and large-N dependent; infrared phase modes require care	Different symmetry and exact-scattering problem from the discrete model	Model-specific Bethe, bosonization, or large-N results	Define the symmetry, finite-N limit, and order of limits first	Coleman's theorem forbids ordinary continuous spontaneous symmetry breaking at finite N in two dimensions
SU(N) principal chiral model	Massive left–right multiplets with a fused rank tower	Non-diagonal matrix scattering with Yangian-type structure	Proposed exact spectrum, S matrix, Bethe/TBA data	Quantum charges, representation theory, and matrix bootstrap	A classical Lax connection alone does not establish quantum charges or the exact spectrum

Sinh-Gordon: the clean scalar benchmark

The standard sinh-Gordon scattering function

$$S_B(\theta) = \frac{\sinh\theta-i\sin(\pi B)} {\sinh\theta+i\sin(\pi B)}, \qquad 0<B<1,$$

has one neutral particle, satisfies scalar unitarity and crossing, and has no physical-strip pole. It is therefore a clean setting for separating:

• exact analytic checks on $S_B$;
• numerical checks on Bethe or TBA equations;
• form-factor truncation errors in correlators; and
• existence of the local theory.

For a class of regular scalar scattering functions that includes the sinh-Gordon example, Lechner’s modular-nuclearity construction yields a bounded rigorous local realization with the qualifications stated on the exact S-matrix page Lechner 2008, Theorems 5.6 and 5.8 and § 6, pp. 848–856. This result does not extend automatically to physical-strip bound-state poles, matrix scattering, or arbitrary CDD deformations.

Sine-Gordon and the Thirring correspondence

The sine-Gordon model supplies topological solitons, antisolitons, and, for attractive coupling, neutral breather bound states. Soliton–antisoliton scattering is non-diagonal, so Yang–Baxter and charge-conjugation conventions matter; breather fusion then produces additional scalar amplitudes. The first breather mass

$$m_1=2M\sin\!\left(\frac{\pi\xi}{2}\right), \qquad \xi=\frac{\beta_{\rm SG}^2}{8\pi-\beta_{\rm SG}^2},$$

is a useful pole-normalization check.

Coleman’s bosonization relation to the massive Thirring model identifies the soliton current with the fermion current and relates couplings in a specific normalization Coleman 1975, §§ IV–V, pp. 2092–2097. It is not permission to equate every local operator without Klein factors, charge sectors, renormalization, and locality phases. The exact factorized scattering and breather bootstrap are developed in Zamolodchikov and Zamolodchikov 1979, §§ 4–5, pp. 267–287.

Choose this family when topological charge, non-diagonal soliton scattering, and bound-state fusion are central. Choose sinh-Gordon instead when one wants a pole-free scalar benchmark without charged asymptotic sectors.

O(N) sigma models: the N greater than 2 boundary

For $N>2$, the two-dimensional O($N$) nonlinear sigma model is asymptotically free and its standard exact-scattering description contains a massive vector multiplet. O($N$) invariance decomposes the two-particle space into singlet, antisymmetric, and symmetric-traceless channels. Yang–Baxter, unitarity, crossing, and minimality relate their scalar amplitudes. The proposed factorized S matrix is given by Zamolodchikov and Zamolodchikov 1978, pp. 525–535.

The qualifier $N>2$ is structural. At $N=2$, the target is a circle, vortices and compact-boson variables control the infrared, and Berezinskii–Kosterlitz–Thouless physics replaces the ordinary asymptotically free massive story. Formulae containing $N-2$ in beta functions or scattering channels cannot be continued to $N=2$ without rebuilding the theory.

Choose the O($N>2$) model when a non-Abelian vector multiplet and a representation-resolved matrix S matrix are the target. It is not a generic template for four-dimensional mass generation.

Gross–Neveu models: symmetry before spectrum

The original O($N$) Gross–Neveu interaction,

$$\mathcal L = \bar\psi_i i\partial\!\!\!/\ \psi_i +\frac{g^2}{2} \left(\bar\psi_i\psi_i\right)^2,$$

has a discrete chiral transformation under which $\bar\psi_i\psi_i$ changes sign. Its large-$N$ saddle, massive phases, and factorized-scattering sectors are developed from a specific finite-$N$ theory Gross and Neveu 1974, §§ II–V, pp. 3237–3253. Exact fermion amplitudes require their own spectrum and representation bootstrap Zamolodchikov and Zamolodchikov 1978, pp. 481–483.

“Continuous-chiral Gross–Neveu,” “chiral Gross–Neveu,” and “Nambu–Jona-Lasinio model in two dimensions” can refer to distinct symmetry and flavor organizations. At finite $N$, ordinary spontaneous breaking of a continuous internal symmetry is obstructed in two dimensions Coleman 1973, pp. 259–264. A large-$N$ saddle that appears to choose an orientation reflects a nonuniform order of limits; it cannot be transferred unchanged to finite $N$. State whether the model has discrete or continuous chiral symmetry, which limit is taken first, and which stable particles actually enter the S matrix.

Choose a Gross–Neveu model only after fixing that symmetry data. The shared four-fermion notation is not enough to identify the exact theory.

Principal chiral model: from Lax data to matrix scattering

For $g(x)\in SU(N)$, the principal chiral model has $SU(N)_L\times SU(N)_R$ symmetry modulo its trivially acting center and a classical spectral-parameter connection. Quantum integrability additionally requires renormalized conserved charges with nontrivial asymptotic action. The proposed exact spectrum has antisymmetric-rank multiplets with sine-law mass ratios

$$\frac{m_r}{m_1} = \frac{\sin(\pi r/N)}{\sin(\pi/N)}, \qquad r=1,\ldots,N-1,$$

and a non-diagonal factorized S matrix Wiegmann 1984, pp. 173–176. Faddeev and Reshetikhin supply a model-specific quantum regularization and integrable lattice route Faddeev and Reshetikhin 1986, §§ 2–5, pp. 231–250.

This model is the right laboratory for left–right non-Abelian representation structure and matrix bootstrap. It is the wrong place to infer quantum integrability merely from the classical monodromy.

Choose the method from the observable

• For stable particle content and mass ratios, begin with physical-strip poles, residue signs, and fusion closure.
• For a matrix amplitude, begin with representation projectors and the Yang–Baxter equation, then impose scalar analytic constraints.
• For large-volume energy levels, use Bethe–Yang only after statistics, twists, and wrapping scales are declared.
• For a local two-point function, solve the operator-specific form-factor axioms and test the spectral tail at the quoted separation.
• For a ground-state scaling function, use TBA with a declared kernel, occupation convention, bulk subtraction, and infrared/ultraviolet checks.
• For a rigorous existence claim, locate an actual construction theorem and verify every hypothesis; algebraic bootstrap consistency is not a substitute.

Where the standard toolkit changes

The massive infinite-line framework must be modified for:

• massless scattering, where left- and right-moving sectors, rapidity origins, and infrared completeness need separate definitions;
• unstable particles, which are resonance poles rather than asymptotic states and cannot label Bethe roots as if stable;
• boundaries and defects, where reflection and transmission matrices obey boundary consistency equations;
• nonintegrable perturbations, which allow production, widths, and diffractive many-body scattering; and
• finite-density or nonequilibrium states, where additional root species, generalized ensembles, and hydrodynamic assumptions enter.

The integrability exact-data chain provides the common logical sequence. The evidence-triangulation graph shows how exact-model calculations can be compared with independent methods without double counting shared input. The exact and rigorous status comparison records the separate status of an exact object, its calculation, and any local construction.

Common pitfalls

Treating a model family name as a definition. Gross–Neveu variants and sigma models with special $N$ can have different symmetry and infrared physics. Write the action, symmetry, dimension, and limit before importing exact data.

Exporting exactness to a nearby theory. A nonintegrable perturbation can turn stable particles into resonances and introduce production. Exact data inside the integrable model remain exact there, not universal nearby.

Calling every closed bootstrap a rigorous QFT. Constructive results exist for bounded classes with explicit hypotheses. Matrix and bound-state solutions require their own existence analysis.

Exercises

1. A proposed calculation needs a local two-point function in an $O(N>2)$ sigma model at large Euclidean distance. Give the minimum sequence of methods and two independent error checks.

Solution

First fix the O($N$) representation channel and accepted exact S matrix. Then solve the form-factor axioms for the named operator, including its normalization, and assemble the spectral expansion. Check the first omitted particle sector and rapidity-cutoff or quadrature stability; an ultraviolet sum rule or independent finite-volume matrix element supplies an additional test. Yang–Baxter consistency of the S matrix does not control the correlator tail.

2. Explain why the large-$N$ continuous-chiral saddle does not prove finite-$N$ spontaneous breaking in two dimensions.

Solution

The $N\to\infty$ limit suppresses fluctuations before the infrared or infinite-volume limit is taken. At finite $N$, long-wavelength fluctuations restore a continuous internal symmetry in the setting of Coleman’s theorem. The limits are nonuniform, so the large-$N$ oriented saddle can organize an expansion without defining a finite-$N$ broken vacuum.

References

• Coleman, Sidney. “There Are No Goldstone Bosons in Two Dimensions.” Communications in Mathematical Physics 31 (1973): 259–264. DOI.
• Coleman, Sidney. “Quantum Sine-Gordon Equation as the Massive Thirring Model.” Physical Review D 11 (1975): 2088–2097. DOI.
• Faddeev, L. D., and N. Yu. Reshetikhin. “Integrability of the Principal Chiral Field Model in (1+1)-Dimension.” Annals of Physics 167 (1986): 227–256. DOI.
• Gross, David J., and André Neveu. “Dynamical Symmetry Breaking in Asymptotically Free Field Theories.” Physical Review D 10 (1974): 3235–3253. DOI.
• Lechner, Gandalf. “Construction of Quantum Field Theories with Factorizing S-Matrices.” Communications in Mathematical Physics 277 (2008): 821–860. DOI. Open PDF.
• Wiegmann, P. “Exact Factorized S-Matrix of the Chiral Field in Two Dimensions.” Physics Letters B 142 (1984): 173–176. DOI.
• Zamolodchikov, Alexander B., and Alexey B. Zamolodchikov. “Exact S-Matrix of Gross-Neveu ‘Elementary’ Fermions.” Physics Letters B 72 (1978): 481–483. DOI.
• Zamolodchikov, Alexander B., and Alexey B. Zamolodchikov. “Factorized S-Matrices in Two Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Models.” Annals of Physics 120 (1979): 253–291. DOI.
• Zamolodchikov, Alexander B., and Alexey B. Zamolodchikov. “Relativistic Factorized S-Matrix in Two Dimensions Having O(N) Isotopic Symmetry.” Nuclear Physics B 133 (1978): 525–535. DOI.