Yang–Mills existence and mass gap

Yang–Mills theory, the nonabelian gauge theory introduced in 1954 [YangMills1954IsotopicSpin], underlies the Standard Model of particle physics, yet its four-dimensional quantum version has never been constructed as a mathematical object. The problem, one of the Clay Millennium Prize problems posed in 2000 [JaffeWitten2000YangMills], asks for every compact simple gauge group $G$ for Euclidean Schwinger functions on $\mathbb R^4$ satisfying the Osterwalder–Schrader axioms, whose reconstructed relativistic theory is nontrivial and has a mass gap: the energy–momentum spectrum consists of the vacuum together with states of mass at least some $\Delta>0$.

Substantial evidence supports both existence and the gap. Lattice gauge theory provides a nonperturbative regularization with convincing numerics, perturbation theory describes the short-distance regime, and constructive field theory has produced interacting models in lower dimensions. What is missing is the continuum limit itself: no interacting four-dimensional Yang–Mills measure satisfying the axioms has been constructed, let alone one with a proven spectral gap [Clay2026YangMillsStatus].

The problem is fully open. A resolution requires a nonperturbative construction of the continuum theory and, beyond that, a proof that its spectrum has a strictly positive gap above the vacuum.