In mathematical physics, the Yang–Mills existence and mass gap problem is an unsolved problem and one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute, which has offered a prize of US$1,000,000 to the one who solves it.
In physics, confinement of particles is such an important phenomenon that the Clay Mathematics Institute has even pledged an award of a million dollars to anyone who can give a convincing and exhaustive scientific explanation from a mathematical point of view. For example, the quarks are confined in pairs or threes by the strong interaction- the force which holds the nuclei of the atoms together- making up neutrons and protons.
NEW STUDY COORDINATED BY SISSA IN THE PAGES OF NATURE PHYSICS
A recent study at SISSA adds a new chapter to what we know about confinement. Using a relatively simple method, it has been shown how to determine whether, in a system with ferromagnetic characteristics, the emerging “particles” are subject to confinement. The study was published in Nature Physics, the prestigious journal in the Nature group.
In case you were not aware of this, finding a proof for confinement is one of the Millenium Problems by the Clay Mathematics Institute. You can find the (detailed) answer to your question in the official problem description by Arthur Jaffe and Edward Witten.
In short: proving confinement is essentially equivalent to showing that a quantum Yang-Mills theory exists and is equipped with a “mass gap”. The latter manifests itself in the fact that the lowest state in the spectrum of the theory cannot have an arbitrarily low energy, but can be found at some energy Δ>0Δ>0. Proving this means to formulate the theory in the framework of axiomatic quantum field theory and deduce systematically all of its properties.
Mass gap implies confinement
In order to understand why proving that the theory has a mass gap is equal to proving confinement, we first have to understand what confinement is. In technical language it means that all observable states of finite energy are singlets under transformations of the global colour SU(3)SU(3). In simple terms this means that all observable particles are colour-neutral. Since quarks and gluons themselves carry colour charge, this implies that they cannot propagate freely, but occur only in bound states, namely hadrons.
Proving that the states in the theory cannot have arbitrarily low energies, i.e. there is a mass gap, means that there are no free particles. This in turn means that there cannot be free massless gluons which would have no lower bound on their energy. Hence, a mass gap implies confinement.
Motivation
The existence of confinement, while phenomenologically well-established, is not fully understood on a purely theoretical level. Confinement is a low energy phenomenon and is as such not accessible by perturbative QCD. There exist various low energy effective theories such as chiral perturbation theory which, while giving good phenomenological descriptions of hadron physics, do not teach us much about the underlying mechanism. Lattice QCD, albeit good for certain qualitative and quantitative predictions, also does not allows us to prove something on a fundamental level. Furthermore, there is the AdS/CFT correspondence, which allows us to describe theories which are similar to QCD in many respects, but a description of QCD itself is not accessible at this point. To conclude: there are many open questions to answer before we have a full understanding of QCD.