Open problem, Mathematical physics, Analysis, posed 2000

Yang–Mills existence and mass gap

OpenMillennium

Prove that for any compact simple gauge group $G$ , a nontrivial quantum Yang–Mills theory exists on $\mathbb{R}^4$ satisfying the Wightman (or Osterwalder–Schrader) axioms of constructive quantum field theory, and that the spectrum of the Hamiltonian $H$ has a strictly positive mass gap $\Delta > 0$ — that is, $H$ has no spectrum in the open interval $(0, \Delta)$ above the vacuum energy $0$ .

Research frontier as of 2026

As of 2026 the four-dimensional Yang–Mills existence and mass gap problem remains open. On the ultraviolet (short-distance) side, Tadeusz Bałaban's 1980s renormalization-group program established uniform bounds on effective actions on a finite four-dimensional lattice torus as the lattice spacing $a \to 0$ , and modern stochastic quantization using regularity structures and paracontrolled calculus (Hairer, Chandra, Chevyrev, Shen) has constructed the local Langevin dynamics of two- and three-dimensional gauge fields cleanly. On the infrared (long-distance) side, Wilson's lattice gauge theory gives a rigorous proof of exponential decay of correlations (hence a mass gap) and the Wilson-loop area law in the strong-coupling regime (large bare coupling $g$ ). What is missing is a bridge between the two: taking the continuum limit $a \to 0$ requires sending the bare coupling $g(a) \to 0$ by asymptotic freedom, which exits the strong-coupling regime where cluster expansions converge, and no rigorous control of the infrared behavior across the weak-to-strong crossover in four dimensions has been achieved.

Best known results

Ultraviolet stability of four-dimensional lattice Yang–Mills on a finite torus via multiscale renormalization group (Bałaban, 1984–1989).
Rigorous proof of a positive mass gap and Wilson-loop area law in four-dimensional lattice Yang–Mills at strong bare coupling via cluster expansions (Osterwalder–Seiler, 1978).
Construction of the gauge-covariant Langevin (stochastic quantization) dynamics for Yang–Mills in 2D and 3D using regularity structures (Chandra–Chevyrev–Hairer–Shen, 2022–2024).

Tools and where they stop

Tool	Achieved	Where it stops
Lattice gauge theory and strong-coupling cluster expansions	Replaces the ill-defined continuum path integral by a finite-dimensional integral over group elements on lattice edges, and proves confinement and a positive mass gap when the bare coupling $g$ is large.	Taking the continuum limit $a \to 0$ in 4D requires sending $g(a) \to 0$ (asymptotic freedom), leaving the radius of convergence of the strong-coupling expansion.
Multiscale renormalization group (Bałaban's block-spin program)	Controls ultraviolet singularities as $a \to 0$ on a small finite four-dimensional volume, proving stability bounds independent of the lattice spacing.	Works only while the running coupling remains small (short physical scales); as the scale grows toward the confinement length, the running coupling becomes large and the perturbative renormalization step breaks down.
Stochastic quantization and regularity structures	Constructs the singular stochastic PDE (Parisi–Wu Langevin equation) for Yang–Mills measures in 2D and 3D in a gauge-covariant way.	In 4D the Yang–Mills equation is scaling-critical (not subcritical), where the theory of regularity structures hits its critical barrier, and global-in-time control yielding an invariant measure with a spectral gap remains out of reach.

Open questions

Can one construct a non-perturbative renormalization group flow that tracks four-dimensional lattice Yang–Mills all the way from weak coupling at the lattice scale to the strong-coupling regime at the confinement scale?
Can the continuum limit and mass gap be established first for simpler three-dimensional non-abelian Yang–Mills theory, where the coupling is super-renormalizable?

References

Chen Ning Yang, Robert L. Mills (1954). Conservation of isotopic spin and isotopic gauge invariance · DOI:10.1103/physrev.96.191
Arthur Jaffe, Edward Witten (Clay Mathematics Institute) (2000). Quantum Yang–Mills theory
Kenneth G. Wilson (1974). Confinement of quarks · DOI:10.1103/physrevd.10.2445
Sourav Chatterjee (2019). Yang–Mills for probabilists · arXiv:1803.01950