Yang-Mills Mass Gap
Prove that quantum Yang-Mills theory exists in four dimensions and has a mass gap; simulations support it, but no mathematical proof exists.
Open in the interactive tree →Physicists use quantum Yang-Mills theory to describe the strong force, and experiments and computer simulations indicate that it has a mass gap: its lightest particles have positive mass. The Clay problem asks for a mathematically rigorous four-dimensional quantum theory and a proof of that gap. Clay prize: one million US dollars.
As of October 2026
October 2026: unsolved; Clay says progress will need fundamental new ideas in both physics and mathematics. Rigorous constructions exist only in lower dimensions or with added fields: in 2D a Markov process for the stochastic Yang-Mills heat flow (Chandra, Chevyrev, Hairer and Shen, 2022), in 3D local-in-time solutions for Yang-Mills-Higgs with possible blow-up (Inventiones 2024), and Cao and Chatterjee's heat-flow solutions from random initial data (2021). Chatterjee said in a 15 October 2025 Harvard lecture that recent progress had rejuvenated the quest, while the constructive-field-theory programme has not reached its original goal. On the numerical side, a March 2026 preprint reports normalizing flows that cut the variance of glueball correlators in SU(3) Yang-Mills by factors of 10-60.
What is missing
- A rigorous four-dimensional quantum Yang-Mills measure: the 2D and 3D results do not cover pure Yang-Mills in four dimensions
- A proof that the constructed theory has a positive gap above its lowest energy state
- Control of the continuum limit of lattice gauge theory with rigorous error bounds
- A way to turn numerical glueball masses into a proof
Becomes possible once solved
- A mathematically sound foundation for the Standard Model's strong-force sector
- A rigorous explanation of the mass gap behind quark confinement
- Rigorous tools for constructing other quantum field theories
Open steps
- Rigorous 4D Yang-Mills measure Low AI leverageConstruct the continuum four-dimensional Yang-Mills measure from the lattice with controlled renormalization, extending the 2D and 3D stochastic-quantisation results.
- Prove a mass gap in a rigorous model Low AI leverageProve a positive spectral gap with bounds uniform in the lattice spacing, or in a lower-dimensional or modified continuum model, as a stepping stone to four dimensions.
- Precise lattice glueball masses High AI leverageCompute the glueball spectrum of pure SU(3) Yang-Mills with controlled continuum extrapolation, using learned samplers that cut variance.
- Computer-assisted gap certificates Medium AI leverageCombine tensor-network or renormalization-group numerics with interval arithmetic to get rigorous gap bounds in simplified lattice gauge models.
Where AI could help
Low AI leverage. A proof needs a new construction idea; AI helps with Lean formalization, lattice computation and checking lemmas, with no AI result on this problem.
- Formalize the 2D and 3D constructions in Lean to expose what breaks in 4D
- Generate and test candidate estimates and proof strategies
- Train samplers that make lattice gauge computations faster and more precise
- Survey constructive field theory literature for overlooked techniques
Shown so far
- In March 2026, a preprint (revised August 2026) reported that normalizing-flow models cut the variance of glueball correlation functions in SU(3) Yang-Mills by factors of 10-60 (preprint, not peer reviewed). source
Prerequisites
- Calculus (Analysis)1684
- Electromagnetism1831-1865
- Quantum Mechanics1925
- Standard Model of Particles1973