Is color confinement detected? I'm a graduate student studying QFT. I'm quite interested that is color confinement detected or proved? (both directly and indirectly) Or it is just an assumption?
 A: Here is an experimentalist's answer.
Color confinement is a theoretical concept  arising from the plethora of experimental observations that are summed up theoretically in the Standard Model. We have no free quarks or gluons, we do have quark jets and gluon jets. So confinement as predicted by the ${\rm SU}(3)\times {\rm SU}(2)\times \rm U(1)$ SM is consistent with all the existing data.
One has to keep in mind that a theory applying to experimental data can be falsified, or  can be found consistent with the data; but consistent is not proof, it is a temporary validation.
A theory of course has axioms and mathematical proofs, so a theorist should answer whether the theory of QCD allows unconfined manifestations of color. These should be in phase spaces not explored by present experiments.
A: Color confinement is still an unsolved problem in particle physics (see https://arxiv.org/abs/2111.06183). Indeed, the confinement phenomenon in QCD cannot be accommodated within the standard framework of quantum field theory (see https://arxiv.org/abs/hep-ph/0105142).
Our experimental data are consistent with color confinement being a true property of QCD. It would be interesting to know if color confinement is a mathematical consequence of QCD, being this a nonabelian Yang-Mills theory associated with the compact $\rm SU(3)$ group. But such a proof does not yet exist.
Kugo and Ojima (1979) derived a criterion for color confinement, by assuming (1) a manifestly Lorentz covariant operator formalism on an indefinite metric state space and (2) the existence of the nilpotent BRST symmetry under the Lorentz covariant gauge fixing condition. If the Kugo-Ojima criterion is satisfied, then no colored object can be observed (this equivalent to color confinement). However, it has not yet been proved that the Kugo-Ojima criterion is a necessary consequence of QCD, although it is an attractive statement because quark confinement and gluon confinement immediately follow as special cases of color confinement once color confinement.
