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Confinement is the physics phenomenon that quarks cannot be isolated. The color-charged quarks are confined with other quarks by the strong interaction to form pairs or triplets so that the net color is neutral. The force between quarks increases as the distance between them increases, so no quarks can be found individually.
The reasons for quark confinement are somewhat complicated; there is no analytic proof that quantum chromodynamics should be confining, but intuitively confinement is due to the force-carrying gluons having color charge. As two electrically-charged particles separate, the electric fields between them diminish quickly, allowing electrons to become unbound from nuclei. However, as two quarks separate, the gluon fields form narrow tubes (or strings) of color charge. Thus the force experienced by the quark remains constant regardless of its distance from the other quark. Since energy goes as force times distance, the total energy increases linearly with distance.
When two quarks become separated, as happens in particle accelerator collisions, at some point it is more energetically favorable for a new quark/anti-quark pair to "pop" out of the vacuum than to allow the quarks to separate further. As a result of this, when quarks are produced in particle accelerators, instead of seeing the individual quarks in detectors, scientists see "jets" of many color-neutral particles (mesons and baryons), clustered together. This process is called hadronization, fragmentation or string breaking, and is one of the least understood processes in particle physics.
The confining phase is usually defined by the behavior of the action of the Wilson loop, which is simply the path in spacetime traced out by a quark-antiquark pair created at one point and annihilated at another point. In a non-confining theory, the action of such a loop is proportional to its perimeter. However, in a confining theory, the action of the loop is instead proportional its area. Since the area will be proportional to the separation of the quark-antiquark pair, free quarks are suppressed. Mesons are allowed in such a picture, since a loop containing another loop in the opposite direction will have only a small area between the two loops.
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