Two- and Three-Dimensional Constructive Model Atlas

Low-dimensional QFT contains genuine interacting constructions, but “constructed” does not name one uniform endpoint. Depending on the model, the theorem may deliver an infinite-volume Euclidean measure, a Schwinger hierarchy, OS reconstruction, a mass gap, an isolated particle, scattering states, or only a specific scaling field. A useful atlas keeps those columns separate and never transfers a result from two or three dimensions to four.

Required background. The constructive model-by-dimension comparison fixes the status columns, while existence, uniqueness, and equivalence fixes their logical strength. Helpful background. The Schwinger model supplies the gauge-theory laboratory, and factorized scattering supplies a distinct route to rigorous two-dimensional models.

A model-by-model atlas

The following comparison uses an evidence cutoff of 2026-08-10. “Continuum” always means the limit specified in the cited theorem, not an unqualified continuum QFT.

Model	Constructed object	Separately established properties	Boundary of the result
Massive stable $P(\phi)_2$	Infinite-volume interacting Euclidean measure and Schwinger functions after volume removal	OS/Wightman axioms; in controlled regimes clustering, mass gap, particle structure, and scattering results	Does not imply four-dimensional scalar existence or universal asymptotic completeness
$\phi^4_3$	Ultraviolet- and volume-cutoff removal with tuned mass and vacuum counterterms	Wightman axioms and a mass gap at weak coupling; non-Gaussian correlations	Dimension-specific superrenormalizable estimates do not extend to $d=4$
Massive Gross–Neveu$_2$	Cutoff-independent tempered Schwinger hierarchy at small renormalized coupling	Euclidean covariance, nonzero truncated four-point function, and stretched-exponential clustering in the 2024 construction	The cited theorem does not assert arbitrary coupling or every Hilbert-space scattering property
Schwinger model	Exact two-dimensional gauge model and gauge-invariant observable solution	Massive bosonic excitation and charge screening in the exact solution	Screening is not four-dimensional color confinement
$SU(2)$ Yang–Mills–Higgs scaling	Weak-coupling, large-Higgs-length lattice field converges after projection	Massive Gaussian Proca random one-form in any $d\ge2$ under the theorem’s joint scaling	Higgs matter remains; the limit is Gaussian; pure Yang–Mills and a non-Gaussian limit remain open

For $P(\phi)_2$, Glimm, Jaffe, and Spencer prove Wightman axioms and particle structure for the constructed model 1974, Theorems 1–4, pp. 585–632. Feldman and Osterwalder establish Wightman axioms and a mass gap for weakly coupled $\phi^4_3$ 1976, §§2–9, pp. 80–135. The different dimensions entail different counterterms and scale estimates; their placement in adjacent rows is comparative, not an interpolation theorem.

The modern Gross–Neveu construction is especially instructive. For $N\ge2$ and sufficiently small renormalized coupling, Duch proves convergence of every smeared Schwinger function as both cutoffs are removed, Euclidean invariance, a nonzero truncated four-point function, and stretched-exponential cluster decay Duch 2024, Theorem 1.1. The nonzero fourth cumulant proves non-Gaussianity. The theorem is stronger than fixed-order perturbation theory, yet its exact field class, small-coupling range, and Euclidean outputs must remain visible.

First application: five rows with identical questions

At the low-dimensional confinement and screening laboratory, ask the same questions of every row:

1. What regulated measure, Grassmann functional, Hamiltonian, or lattice field is defined?
2. Which ultraviolet and volume limits are taken, in what order and topology?
3. Are all local correlations controlled, or only a projected field?
4. Is reflection positivity proved and is a Lorentzian Hilbert theory reconstructed?
5. Which gap, particles, charges, and scattering channels are actually established?

For the Schwinger model, the exact observable analysis identifies a massive neutral boson and screening of electric charge Lowenstein and Swieca 1971, §§2–4, pp. 172–184. This is a powerful counterexample to identifying “mass gap” with “confinement”: the spectrum is gapped while external charges are screened.

For Yang–Mills–Higgs, the scaling is exceptionally narrow. With lattice spacing $\varepsilon\to0$, the gauge coupling tends rapidly to zero and the Higgs length tends to infinity while their product is tied to $\varepsilon$. After unitary gauge fixing and stereographic projection, the gauge field converges as a random distributional one-form to a massive Gaussian field. Chatterjee proves the $U(1)$ and $SU(2)$ versions in 2026, Theorems 3.1–3.2, pp. 10–14 and explicitly leaves non-Gaussian scaling open. This is neither a construction of pure $SU(2)$ Yang–Mills nor a solution of the four-dimensional mass-gap problem.

Implication checks

Each stronger column requires a new theorem. A probability measure need not be reflection positive. OS reconstruction need not yield an isolated one-particle shell. Haag–Ruelle scattering-state existence need not yield asymptotic completeness. Conversely, an exact factorized S-matrix proposal does not by itself construct local algebras whose scattering operator it is.

An independent check uses connected correlations. A claimed interacting scalar or fermionic limit should exhibit a nonzero connected correlation of order greater than two or another theorem-level nontriviality criterion. The Gaussian Yang–Mills–Higgs limit correctly fails that check: this is a feature of its proved scaling, not a contradiction.

Failure test: evidence placed in the construction column

Insert a Monte Carlo spectrum or a formal asymptotic expansion as a constructed continuum theory. Neither supplies a tight family of probability laws, a complete limiting hierarchy, or a Hilbert-space reconstruction. The strongest surviving statement is numerical or perturbative evidence for named observables at named cutoffs. A second failure erases the subscript from $\phi^4_3$; the change of dimension alters power counting and invalidates the cited bounds.

Exercises

Why does a massive Gaussian scaling limit not establish an interacting Yang–Mills continuum theory?

Solution

A Gaussian law has vanishing connected correlations above order two. The cited limit also contains Higgs matter and uses a joint weak-coupling scaling. It establishes the projected Proca field in that regime, not non-Abelian self-interaction, pure-gauge continuum existence, or the Clay spectral theorem.

References

• Chatterjee, Sourav. “A Scaling Limit of $SU(2)$ Lattice Yang–Mills–Higgs Theory.” Probability and Mathematical Physics 7 (2026): 339–381. DOI; Open PDF.
• Duch, Paweł. “Construction of the Gross–Neveu Model Using the Polchinski Flow Equation.” arXiv:2403.18562 (2024). arXiv.
• Feldman, Joel, and Konrad Osterwalder. “The Wightman Axioms and the Mass Gap for Weakly Coupled $\phi^4_3$ Quantum Field Theories.” Annals of Physics 97 (1976): 80–135. DOI.
• Glimm, James, Arthur Jaffe, and Thomas Spencer. “The Wightman Axioms and Particle Structure in the $P(\phi)_2$ Quantum Field Model.” Annals of Mathematics 100 (1974): 585–632. DOI.
• Lowenstein, Joel H., and John A. Swieca. “Quantum Electrodynamics in Two Dimensions.” Annals of Physics 68 (1971): 172–195. DOI.