On formulating and proving confinement and the mass gap problem in lattice QCD with propabilistic methods:
Sourav Chatterjee, Yang-Mills for probabilists, in: Probability and Analysis in Interacting Physical Systems, PROMS 283 (2019) Springer (arXiv:1803.01950, doi:10.1007/978-3-030-15338-0)
Sourav Chatterjee, A probabilistic mechanism for quark confinement, Comm. Math. Phys. 2020 (arXiv:2006.16229)
The confinement of quarks is one of the enduring mysteries ofmodern physics. $[\ldots]$ In spite of many decades of research, physically relevant quantum gauge theories have not yet been con-structed in a rigorous mathematical sense. $[$ non-perturbatively, that is $]$ $[\ldots]$ Perhaps the most important example is four-dimensional SU(3)-lattice gauge theory. If one can show that this theory has a mass gap at all values of the coupling strength, that would explain why particles known as glue-balls in the theory of strong interactions have mass. All such questions remain open.
The second big open question is the problem of quark confinement. Quarks are the constituents of various elementary particles, such as protons and neutrons. It is an enduring mystery why quarks are never observed freely in nature. The problem of quark confinement has received enormous attention in the physics literature, but the current consensus seems to be that a satisfactory theoretical explanation does not exist.