Confinement in 4D Yang–Mills theory

The target is confinement of unscreened external colour sources in pure gauge theory: for large separation $R$, the static potential should have a positive linear coefficient. A corresponding large-loop area law is $\langle W(C)\rangle\sim e^{-\sigma\,\mathrm{Area}(C)}$, after specifying renormalization and the loop limit. Nonzero $N$-ality matters: adjoint charges can be screened by gluons, and dynamical fundamental matter permits string breaking. An unqualified area law is therefore not a universal definition for every gauge theory with matter.

What is known: Strong-coupling lattice expansions establish area laws in their convergence regimes. Lattice Monte Carlo supports positive string tension in four-dimensional pure non-Abelian theories; this is numerical evidence, not a continuum proof. Rigorous Wilson-loop results in particular lattice gauge theories and large-$N$ settings provide additional benchmarks, whose gauge groups and coupling regimes must be retained.

Proving confinement in the continuum 4D theory presupposes (or must be done jointly with) the construction in QF1, and then requires showing the area law survives the weak-coupling continuum limit — precisely the regime where strong-coupling expansions fail. This is a long-horizon programme intertwined with QF1.

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.