Yang–Mills existence and mass gap

Construct four-dimensional quantum Yang–Mills theory for every compact simple gauge group and prove that it has a positive mass gap.

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1	# Yang–Mills existence and mass gap
2
3	Yang–Mills theory is the quantum field theory at the heart of the standard model; with gauge group SU(3), coupled to quarks, it describes the strong nuclear force. Physics requires that every excitation of the vacuum carry at least some fixed positive energy, a "mass gap", even though classical Yang–Mills waves are massless. The task is to construct the four-dimensional quantum theory as a rigorous mathematical object and prove that it has this gap.
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5	## Problem
6
7	Let $G$ be a compact simple Lie group. Classically, a Yang–Mills field is a connection $A$ with values in the Lie algebra of $G$, with curvature $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+[A_\mu,A_\nu]$ and action proportional to $\int_{\mathbb{R}^4}\operatorname{Tr}F_{\mu\nu}F^{\mu\nu}\,d^4x$.
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9	Problem (Clay Mathematics Institute, stated by Jaffe and Witten). Prove that for every compact simple gauge group $G$, a non-trivial quantum Yang–Mills theory exists on $\mathbb{R}^4$ and has a mass gap $\Delta>0$.
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11	Existence means a quantum field theory satisfying axioms at least as strong as the Wightman or Osterwalder–Schrader axioms, with local field operators corresponding to gauge-invariant polynomials in the curvature such as $\operatorname{Tr}F_{ij}F_{kl}(x)$, whose correlation functions agree at short distances with asymptotic freedom and perturbative renormalization. The mass gap means that the Hamiltonian $H$, with $H\Omega=0$ for the vacuum $\Omega$, has no spectrum in $(0,\Delta)$. A consequence is exponential clustering: for local operators with $\langle\Omega,\mathcal{O}\Omega\rangle=0$ and any $C<\Delta$,
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13	$$\lvert\langle\Omega,\mathcal{O}(\vec{x})\,\mathcal{O}(\vec{y})\,\Omega\rangle\rvert\le e^{-C\lvert\vec{x}-\vec{y}\rvert}\quad\text{for large }\lvert\vec{x}-\vec{y}\rvert .$$
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15	A limit obtained by compactness alone does not count unless its properties, such as the axioms and the gap, are also established [1].
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17	## Background
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19	The mass gap is what makes the strong force short-ranged. Experiments and computer simulations strongly support it, but it is invisible in the classical equations. The Clay Mathematics Institute posed the problem in 2000 as one of its seven Millennium Prize Problems, with the official description by Arthur Jaffe and Edward Witten [1]. A solution would give four-dimensional gauge theory the rigorous footing that constructive field theory has reached for interacting models in two and three dimensions [1].
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21	## What is known
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23	- Lattice gauge theory makes the functional integral finite-dimensional and keeps reflection positivity; what is missing is control of the limits of zero lattice spacing and infinite volume [1].
24	- Balaban's renormalization-group analysis on a torus gives estimates uniform in the lattice spacing in three dimensions (1985) and, using asymptotic freedom, in four (1987 onward) [2, 3]. It has yet to reach expectations of gauge-invariant observables, and a mass gap uniform in the volume needs new ideas [1].
25	- Magnen, Rivasseau and Sénéor constructed four-dimensional Yang–Mills with an infrared cutoff by continuum methods (1993) [4].
26	- On the lattice, confinement and related properties were proved at strong coupling (small $\beta$) by Osterwalder and Seiler in 1978 [6]. Recent stochastic-analysis methods give a mass gap at strong coupling that is uniform in $N$ for $\mathrm{SU}(N)$ with 't Hooft scaling [5]. The continuum limit in four dimensions, however, is believed to require weak coupling, $\beta\to\infty$ [6].
27	- Claimed proofs keep appearing, and none has been recognised: the Clay Mathematics Institute lists the problem as unsolved as of October 2026. A 2025 preprint claiming a construction for SU(3) was withdrawn by arXiv's moderators ([arXiv:2506.00284](https://arxiv.org/abs/2506.00284)). In 2026 Faizal and Shabir claimed a reflection-positive construction of four-dimensional SU($N$) Yang–Mills with a mass gap and confinement, published in Fortschritte der Physik ([doi:10.1002/prop.70097](https://doi.org/10.1002/prop.70097)).
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29	## What would count as progress
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31	- The full statement: a construction satisfying the axioms above, with a proof of the mass gap.
32	- The lattice version posed by Chatterjee: for four-dimensional lattice gauge theory with a non-abelian group, show that plaquette correlations decay exponentially at every inverse coupling $\beta>0$, with a correlation length that tends to infinity as $\beta\to\infty$ [6]. Any proof beyond the strong-coupling regime, with explicit constants, would be new.
33	- The continuum limit of three-dimensional lattice Yang–Mills for gauge-invariant observables such as Wilson loops, or a mass gap in three dimensions. Stochastic quantization already gives local-in-time dynamics for three-dimensional Yang–Mills–Higgs ([Chandra, Chevyrev, Hairer and Shen, 2024](https://doi.org/10.1007/s00222-024-01264-2)).
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35	## References
36
37	1. A. Jaffe and E. Witten, "Quantum Yang–Mills theory", official problem description, Clay Mathematics Institute (2000). [PDF](https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf)
38	2. T. Balaban, "Ultraviolet stability of three-dimensional lattice pure gauge field theories", Communications in Mathematical Physics 102, 255–275 (1985). [doi:10.1007/BF01229380](https://doi.org/10.1007/BF01229380)
39	3. T. Balaban, "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, 249–301 (1987). [doi:10.1007/BF01215223](https://doi.org/10.1007/BF01215223)
40	4. J. Magnen, V. Rivasseau and R. Sénéor, "Construction of YM₄ with an infrared cutoff", Communications in Mathematical Physics 155, 325–383 (1993). [doi:10.1007/BF02097397](https://doi.org/10.1007/BF02097397)
41	5. H. Shen, R. Zhu and X. Zhu, "A stochastic analysis approach to lattice Yang–Mills at strong coupling", Communications in Mathematical Physics 400, 805–851 (2023). [doi:10.1007/s00220-022-04609-1](https://doi.org/10.1007/s00220-022-04609-1)
42	6. S. Chatterjee, "Yang–Mills for probabilists", survey (2018). [arXiv:1803.01950](https://arxiv.org/abs/1803.01950)
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44	## Source
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46	Imported from [Clay Millennium Prize Problems: Yang–Mills and the mass gap](https://www.claymath.org/millennium/yang-mills-the-maths-gap/), curated by the Clay Mathematics Institute. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.