I see this all the time* that there still doesn't exist a mathematical proof for confinement. What does this really mean and how would a sketch of a proof look like?
What I mean by that second question is: what are the steps one needs to prove in order to "mathematically prove confinement"?
*See e.g. Scherer's "Introduction to Chiral Perturbation Theory" middle of page 7
The problem
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 $\Delta>0$. 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 $\text{SU}(3)$. 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.
Local non abelian YM theory( Quantum Field Theory) is a theory with quark as particle field and gluon as gauge field(theory of strong force). However, in all high energy collider experiments, quark structure of particle was observed but free quarks were never detected. Instead, they appear at low energy as form of neutron , proton or pion , so called confined field. This could not be shown mathematically as a consequence of YM equation.
