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I've heard that the strong force doesn't decrease in strength with increasing distance, and that's why quarks must be confined within hadrons. But could there be, say, a single quark out there, so that the universe would be colour neutral without it, but with it, it's not? Would such a quark have a "strong field" that would extend through the entire universe?

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No, confinement means that such a state cannot exist, or more precisely, it cannot have a finite energy/mass. If such a colored state had a finite energy, it would mean that far enough from the colored particle, the quantum fields very closely approach the vacuum state. But if that's so, you could always combine two such objects of opposite colors. The total energy would be pretty much inevitably close to the sum of the energies of each, and you would get a "stretched meson", a pair of oppositely charged, widely separated, colored particles, in a direct conflict with confinement.

One may see why such an object is impossible in various more explicit ways. For example, close enough to the particle, the QCD gauge potentials would go like $1/r$, just like in the Coulomb case. But these "gluon" gauge fields are color-charged as well because QCD is non-Abelian, so they induce additional, increasing densities of the QCD field strengths.

Already if the color charge is uncancelled within the region comparable to the nuclear radii, the energy contribution just from this nuclear-radius-sized vicinity of the colored particle will greatly exceed the QCD scale. Due to positive energy theorems etc., this positive mass/energy can't be cancelled by anything that is even further from the colored source. So the non-existence of objects like yours is already decided at distances shorter than or comparable to the confinement length scale.

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