Yang–Mills existence and mass gap: a 4D quantum SU(N) gauge theory with Δ > 0
Constructive QFT has not produced any interacting four-dimensional theory and Balaban's programme stopped short of d = 4, so difficulty 93 sits just below Riemann (96) and just above the unforced Navier–Stokes problem (92). Application value 17 is indirect (rigorous renormalization technology), above abc (13) but well below RH (30). Leverage 52 is moderate because physics already uses the theory through lattice QCD and few theorems are conditional on it.
The problem
Build a precise 4D quantum theory of the strong force and prove its particles have a minimum mass.
- 1Strong force described by Yang-Mills theory
- 2Physics predicts a minimum particle mass
- 3Maths cannot yet define the theory in 4D
- 4Construct it, then prove the gap
The task. For each compact simple gauge group (for instance ), construct a non-trivial quantum Yang–Mills theory on four-dimensional spacetime that satisfies the or axioms, and prove it has a : a number such that every state other than the vacuum has energy at least above it.
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Step 2
- Harmonic analysisTheory
- Step 3
- Mathematical quantum field theoryTheory · host
- Then
- Yang–Mills existence and mass gap: a 4D quantum SU(N) gauge theory with Δ > 0The problem
State of the art
Open in four dimensions. In two dimensions the Yang–Mills measure has been constructed; in three dimensions Chandra, Chevyrev, Hairer and Shen (2022, Inventiones 2024) built a state space of gauge orbits and local-in-time stochastic dynamics for the renormalized Yang–Mills–Higgs equation, but no full axiomatic theory is known in d = 3 or 4. On the lattice, Shen, Zhu and Zhu (2022) proved a unique infinite-volume measure and a positive mass gap for SU(N) and SO(N) gauge theory at strong coupling with 't Hooft scaling; Nissim (arXiv, October 2025, preprint) extended this to U(N). Chatterjee (2020) gave a rigorous link between unbroken center symmetry and confinement. These results hold at fixed lattice spacing and strong coupling, while the continuum theory needs weak coupling at short distances. Balaban's 1980s multiscale analysis handled ultraviolet stability in part but was never completed in d = 4. OpenAI's 6 October 2026 manuscript release (github.com/openai/math) claims a construction of the 2D asymptotically free O(3) sigma model with a mass gap and Lean files; it is unrefereed and does not address four-dimensional gauge theory. No claimed solution of the Clay problem is accepted, and Clay lists it as unsolved. Physical evidence is strong: lattice QCD shows a mass gap, and asymptotic freedom (Gross–Wilczek, Politzer 1973) is established at the physics level.
Realistic applications
None for engineering in the near term. A rigorous construction would put the Standard Model's strong sector on a mathematical footing and would supply constructive-QFT technology (rigorous renormalization, gauge-orbit state spaces) for other theories. Rigorous bounds could in principle calibrate non-perturbative lattice methods.
What counts as solved
A construction of 4D Yang–Mills theory for every compact simple G satisfying the OS or Wightman axioms, with a proof of Δ > 0, accepted by experts after refereeing. Results for d = 2, 3, lattice theories at strong coupling, or large-N limits do not count.
How it gets verified
Expert reviewOnly expert review of the argument is possible today.
The Clay problem needs a rigorous construction of 4D quantum Yang-Mills theory satisfying the OS or Wightman axioms, which has no formal statement in Mathlib or Formal Conjectures today. It cannot be formalised yet because the axiomatic framework (Euclidean field measures, reflection positivity) is not built. Experts referee the construction and the proof of Delta > 0.
Claims under verification
+ add a claimNo claimed results on the Verification board for this problem yet.
Where the work is
Constructing the continuum 4D theory is a problem in mathematical physics. Lattice computation supplies evidence and guidance but no proof.
Bottlenecks
ranked by votes- #1Theory4D continuum limit (UV control)
Constructive QFT controls super-renormalizable models with d ≤ 3. In 4D the coupling flows only logarithmically, so infinitely many scales must be controlled. Balaban's gauge-invariant renormalization group was never completed for d = 4.
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Progress
- Apr 26, 2022Strong-coupling lattice mass gap with 't Hooft scalingsource
Shen–Zhu–Zhu prove a positive gap for SU(N) and SO(N).
- Jan 10, 20223D Yang–Mills–Higgs state space and dynamicssource
Chandra–Chevyrev–Hairer–Shen build the gauge-orbit state space and local dynamics.
- Jun 1, 1973Asymptotic freedomsource
Gross–Wilczek and Politzer show non-abelian gauge theories become free at short distances (date approximate).