Interacting Constructive QFT in Four Dimensions

Which interacting four-dimensional continuum QFTs can be constructed rigorously? For the families examined here, the evidence separates a proved Gaussian scalar scaling limit, partial constructive gauge results, and a recent massive but Gaussian Higgs limit. None supplies the interacting pure Yang–Mills theory on all of spacetime required by the standard existence-and-mass-gap problem. That problem remains open according to the Clay Mathematics Institute, checked 7 September 2026. This is a comparison of specified constructions, not a no-go theorem for every four-dimensional theory.

Evidence cutoff. 7 September 2026. The finite search and its limitations are described below.

Required background. Four-dimensional scalar QFT existence and triviality supplies the precise scalar problem. Constructive existence by model, dimension, and observable distinguishes a measure bound, Schwinger functions, and a reconstructed Lorentzian theory.

Helpful background. Regulators, cutoffs, and continuum limits supplies the distinction between a controlled ordered limit and an interchange of limits. Rigorous construction status and open problems distinguishes numerical and perturbative evidence from existence. Ultraviolet and infrared fixed points explains why asymptotic freedom or a candidate fixed point does not itself construct a continuum measure.

What counts as a four-dimensional construction

A Euclidean route begins with regulated Schwinger functions $S_n^{(a,L)}$, with lattice spacing $a$ and physical box size $L$. One possible construction is

$$S_n=\lim_{L\to\infty}\lim_{a\to0}S_n^{(a,L)},$$

with convergence in a specified distributional or smeared-correlation topology. A different construction may first take infinite volume, or prove a joint limit along a declared trajectory. Existence in one controlled order does not require that the orders commute. Independence of the trajectory and uniqueness are additional claims requiring their own estimates.

The limiting functions must satisfy a complete named reconstruction axiom set, including the appropriate regularity, covariance and positivity conditions. Reflection positivity alone is insufficient. Interaction must also survive in the intended theory, rather than occurring only in its regulated action.

Coordinate	Required evidence
Dimension and spacetime	Four Euclidean dimensions with reconstruction to 3+1 Lorentzian dimensions, or a direct Lorentzian construction
Cutoffs	Ultraviolet regulator removed and infinite-volume/infrared limit controlled
Axioms	All hypotheses of the selected Osterwalder–Schrader reconstruction theorem, or a complete stated Wightman/Haag–Kastler alternative
Gauge structure	Gauge-invariant observables, positivity, and control of gauge fixing/Gribov or lattice-to-continuum issues
Interaction	A criterion distinguishing the intended theory from a free theory; for a specified scalar scaling field, a surviving higher connected cumulant excludes Gaussianity of that field
Dynamics	Locality, covariance and the spectrum condition; a mass gap or scattering sector must be proved when claimed
Non-example	A finite lattice, a formal power series, a UV bound with a fixed IR cutoff, or a continuum correlator inferred only numerically

The Yang–Mills problem further requires the intended short-distance gauge-invariant observable content and a positive mass gap. Its official formulation quantifies over compact simple gauge groups; existence for one fixed group would be significant partial progress. An isolated one-particle shell, confinement and asymptotic completeness are stronger questions, not interchangeable definitions of a vacuum mass gap Jaffe and Witten, official problem statement, §§ 4–5, p. 6, PDF.

Scalar Gaussianity and its precise reach

For one real scalar component with nearest-neighbor ferromagnetic coupling and a stable quartic single-site weight, divide the smeared block field by the square root of the block-sum variance. First take the box-to-block ratio $R/\ell\to\infty$, then the block scale $\ell\to\infty$, allowing bare parameters to vary and subsequences to be taken. Every limiting field in this class with the stated two-point decay at large separation is a generalized Gaussian process. The associated critical and near-critical Ising result is also proved Aizenman and Duminil-Copin 2021, arXiv v4, Eqs. (1.10), (1.14), Definition 1.1 and Theorem 1.2, pp. 3–5, PDF.

The 2024 correction repairs exponential-moment bounds and absolute-value factors without changing the conclusions Aizenman and Duminil-Copin 2024, p. 479. Gaussianity does not determine every limiting covariance or the rate of convergence. Nor does it extend automatically to arbitrary multicomponent, non-ferromagnetic, nonpolynomial or gauge-coupled scalar models. The canonical scalar comparison explains how logarithmic corrections can coexist with Gaussian scaling.

Gauge constructions with different endpoints

Small-field renormalization. In the cited 1987 paper, Bałaban constructs localized effective actions, beta functions and a recursive coupling renormalization within the small-field approximation. This particular result must not be relabeled as a complete all-field ultraviolet construction Bałaban 1987, Abstract. The broader program contains further results; their full proofs are outside this comparison.

A fixed infrared cutoff. Magnen, Rivasseau and Sénéor report a construction of four-dimensional pure $SU(2)$ Schwinger functions in the trivial topological sector with the ultraviolet cutoff removed, a fixed infrared cutoff, regularized axial gauge, and nonperturbative Slavnov identities Magnen, Rivasseau and Sénéor 1993, Abstract. The infrared restriction prevents this result from completing the theory on all of $\mathbb R^4$. Here the source supports the reported result and its scope; the full construction proof is not reproduced.

A massive continuum Higgs limit. Chatterjee constructs a different limit of lattice $SU(2)$ Yang–Mills coupled to a fixed-length fundamental Higgs field. In unitary gauge, a stereographically projected and rescaled gauge field converges to three independent Gaussian Euclidean Proca fields. Writing $g$ for the gauge coupling and $\alpha$ for the source’s Higgs-coupling parameter, the theorem for $d\ge2$ takes infinite-volume lattice subsequential limits first, then lattice spacing $\epsilon\to0$ with $\alpha g=c\epsilon$, $c>0$, and $g=O(\epsilon^{50d})$; the limiting Proca parameter is $c^2/2$. Thus $d=4$ is included, but neither pure Yang–Mills nor a surviving non-Abelian interaction is established. The result does not assert prelimit lattice exponential decay Chatterjee 2026, arXiv v4, Theorem 3.2 and following paragraph, pp. 12–13, PDF. Slower coupling decay and general Higgs potentials remain separate questions Chatterjee 2026, § 3.5, p. 16, PDF.

Comparing the claims

Family and limit	Positive result	What it does not establish
Nearest-neighbor one-component ferromagnetic scalar scaling	Gaussianity in the specified limiting class	An interacting limit of that scaling field, or a universal theorem about every scalar model
Bałaban’s cited four-dimensional small-field analysis	Localized effective actions and coupling renormalization	Control of the complete field domain or all infrared limits
Infrared-regulated pure $SU(2)$	Reported ultraviolet-cutoff removal and Slavnov identities	The infinite-volume theory with a uniform positive gap
Chatterjee’s $SU(2)$ Higgs scaling	A massive Gaussian continuum limit in a specified weak-coupling regime	An interacting pure-gauge theory
Self-dual scalar theory on Moyal space	An exact solution of the planar-sector integral equation	The ordinary local scalar theory on commutative spacetime

The last row is a serious example of changing the problem’s hypotheses. Grosse, Hock and Wulkenhaar solve the Fredholm equation controlling the planar sector of a self-dual matrix model on four-dimensional Moyal space. Its matrix/deformation limit and observable sector are explicit; the result is not a counterexample within the nearest-neighbor scalar theorem Grosse, Hock and Wulkenhaar 2019, arXiv v1, § 1, pp. 1–3, PDF. This comparison does not classify all noncommutative, supersymmetric or conformal constructions.

Method limits and common overclaims

Uniform boundedness of a regulated partition function is not enough: correlation functions can converge to a free theory. Borel summability of perturbation theory around a cutoff model is not enough unless the summed objects survive all required limits and satisfy positivity. A lattice continuum extrapolation with shrinking statistical and discretization errors is evidence for a limit, but not a proof of tightness and uniqueness over all scales.

Likewise, a perturbative beta function describes running within its regime; its sign alone does not construct a probability measure or a Hilbert space. Agreement among several calculations using the same expansion, regulator assumptions or imported theorem is not independent proof of those shared inputs.

A composite-field trap. For a free centered Gaussian scalar with covariance $C_{ij}=\langle\phi(x_i)\phi(x_j)\rangle$, the normal-ordered composite $O=:\!\phi^2\!:$ has, at distinct points,

$$\langle O(x_1)O(x_2)O(x_3)\rangle_c =8C_{12}C_{23}C_{31}.$$

Each connected Wick pairing has one contraction between each pair of points, and there are eight such pairings. The underlying theory remains free. Thus a higher connected correlation disproves Gaussianity of the selected field, but arbitrary nonlinear composites require a stronger interaction test, such as an appropriate nontrivial scattering amplitude when that sector exists. Merely changing the interpolating observable cannot establish interacting dynamics.

Concrete resolution criteria

A positive construction should provide:

1. a regulator family and uniform estimates that permit ultraviolet removal and infinite volume in a declared order or joint limit;
2. limiting gauge-invariant correlation functions or local algebras satisfying a complete named axiom set;
3. a reconstruction of a positive-energy Lorentzian theory;
4. a nontriviality criterion that excludes a free realization of the intended theory, with the observable choice explicit; and
5. for the Yang–Mills existence-and-mass-gap problem, a positive gap and the intended gauge-invariant short-distance structure.

A proof should separately state which group, vacuum sector, limits and observables it covers. Uniqueness across all regulator trajectories, confinement and asymptotic completeness should be added only when proved. A valid construction for one group or one controlled trajectory must not be rejected merely for lacking these stronger claims.

A negative resolution would require a theorem whose hypotheses cover the proposed universality class and whose conclusion rules out all non-Gaussian limits—not an extrapolation from the nearest-neighbor scalar theorem.

Research connections

The scalar capstone follows a stable regulated model through Gaussian integration, scattering, renormalization and effective matching. Return to its final stage to identify the exact extra claim that a continuum construction would require: finite-order control of the amplitude does not supply convergence and reconstruction of the complete theory.

The wider context is mathematical and constructive QFT. Euclidean lattice and continuum extrapolation develops regulator-to-continuum inference, while functional equations and functional RG studies candidate fixed points. Comparisons must keep measures, selected correlators, formal expansions and reconstructed theories distinct.

Literature selection and cutoff

The searches were carried out on 7 September 2026, using the existing scalar and constructive-gauge references as seeds. Literal queries included "Construction of YM4 with an infrared cutoff" pdf, "Marginal triviality" "corrigendum" Duminil IHES, "four dimensional" "constructive" "nontrivial" "2026" -site:qft.org, and "nontrivial" "four-dimensional" Grosse Wulkenhaar reflection positivity arxiv. Publisher, author, arXiv, Clay and bibliographic records were inspected; references and related-paper results supplied further candidates. The search was targeted, without a database-wide date filter or a claim of exhaustive coverage.

The scalar theorem statements, their printed endpoints and the complete one-page correction were inspected, as were the stated Higgs theorem and Moyal model definitions. Bałaban and the infrared-regulated Yang–Mills result are used at the level of their publisher abstracts; this page does not claim an independent verification of those full proofs. These review depths limit the strength of the assessment.

Recent construction claims were also screened. Terin’s July 2026 abstract concerns a regulated Fundamental Modular Region construction and is not used as evidence of ultraviolet removal Terin 2026, arXiv v1, Abstract. Kirk’s version 3 claims a substantially stronger $SU(2)$ continuum result Kirk 2026, version 3, record description and notes. Following that record’s successor link identified version 4, dated 2 August 2026, which claims a construction for each fixed compact connected simple gauge group. Its stated scope is a local curvature-jet observable sector, with group-dependent estimates and a directed continuum limit; it does not claim uniformity across groups or uniqueness outside that construction Kirk 2026, version 4, record description and notes. Both descriptions were inspected; neither full proof has been assessed here. They remain unresolved candidates, neither accepted theorems nor refuted claims in this comparison. The finite search therefore does not support an assertion that every proposed construction has failed.

References

• Aizenman, Michael, and Hugo Duminil-Copin. “Marginal Triviality of the Scaling Limits of Critical 4D Ising and $\phi^4_4$ Models.” Annals of Mathematics 194 (2021): 163–235. DOI; Open PDF, arXiv v4. Corrigendum, Annals of Mathematics 199 (2024): 479. DOI; Open PDF, complete correction in public first-page preview.
• Bałaban, Tadeusz. “Renormalization Group Approach to Lattice Gauge Field Theories. I. Generation of Effective Actions in a Small Field Approximation and a Coupling Constant Renormalization in Four Dimensions.” Communications in Mathematical Physics 109 (1987): 249–301. DOI.
• Chatterjee, Sourav. “A Scaling Limit of $SU(2)$ Lattice Yang–Mills–Higgs Theory.” Probability and Mathematical Physics 7 (2026): 339–381. DOI; Open PDF, arXiv v4.
• Clay Mathematics Institute. “Yang–Mills and the Mass Gap.” Millennium Prize Problems, checked 7 September 2026. Official problem page.
• Grosse, Harald, Alexander Hock, and Raimar Wulkenhaar. “Solution of the Self-Dual $\Phi^4$ QFT-Model on Four-Dimensional Moyal Space.” Journal of High Energy Physics 2020, 081. DOI; Open PDF, arXiv v1, 2019.
• Jaffe, Arthur, and Edward Witten. “Quantum Yang–Mills Theory.” Official Millennium problem statement, undated PDF with its own printed pagination. Open PDF.
• Kirk, Harold D. “Mass Gap and Nontriviality for Four-Dimensional $SU(2)$ Yang–Mills Theory via Karcher Blocking and Exact Haar-Pivot Matching.” Preprint, version 3, 21 July 2026; description inspected, full proof not reviewed. DOI. Successor: “Yang-Mills Existence, Mass Gap and Nontriviality for Fixed Compact Simple Gauge Groups via Karcher Blocking and Exact Haar-Pivot Matching,” version 4, 2 August 2026; description inspected, full proof not reviewed. DOI.
• Magnen, Jacques, Vincent Rivasseau, and Roland Sénéor. “Construction of $YM_4$ with an Infrared Cutoff.” Communications in Mathematical Physics 155 (1993): 325–383. DOI.
• Terin, Rodrigo Carmo. “A Regulated Zero-Temperature Construction of the Fundamental Modular Region in Pure Yang–Mills Theory and QCD.” arXiv:2607.27896v1 (2026); abstract screened. Preprint.