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As I understand it color confinement comes from the fact that as the distance between two color charges increases the color potential energy increases, instead of decreasing, and the energy needed to pull two quarks apart is the same as the energy needed to create two new quarks. The way the color potential energy between two color charges is related to the fact that gluons themselves have color charge. For the electric potential energy between two electric charges the relationship between distance and potential energy depends on the number of dimensions, and for the spacetime curvature around a massive body the relationship between the distance from the massive body and the spacetime curvature depends on the number of dimensions, but I'm not sure if the relationship between distance and color potential energy depends on the number of dimensions?

Would color confinement apply in $n+1$ dimensions, in which $n>3$, or could color charges be free particles in higher dimensions?

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Confinement is a non-perturbative phenomenon, not visible in a small-coupling expansion, so non-perturbative methods are needed in order to address this question. One relatively well-developed non-perturbative method uses numerical calculations in which continuous spacetime is replaced by a discrete lattice. Lattice calculations are easier when fermions (quarks) are not included, and they are also easier when the number of colors is two (gauge group SU(2)) instead of three. Probably for these reasons, published results are relatively abundant for QCD without quarks and with only two colors, including some results for five-dimensional spacetime.

This answer cites some theoretical evidence regarding the fate of confinement in higher dimensions, but it doesn't explain the underlying reason. That would be a tall order, because the reason for confinement even in the most important case of four dimensions is still not completely understood, as reviewed in Greensite (2011), An Introduction to the Confinement Problem.

Lattice results for 5-d QCD without quarks

To extract predictions that are relevant to continuous spacetime from models formulated on a discrete lattice, the model's parameters are tuned to make the correlation length much larger than the lattice spacing — nominally infinitely larger. Such a divergence of the correlation length occurs near second-order phase transitions. According to the review [1], numerical studies of five-dimensional QCD with two colors and without quarks shows a first-order phase transition separating a confinement phase from a deconfined (Coulomb) phase. (See figure 2 in [1].) In other words, according to this numerical evidence, higher-dimensional QCD exhibits both confinement and non-confinement, at least without quarks, depending on the value of the coupling constant. However, the higher-dimensional theory doesn't necessarily have a continuum limit. According to page 11 in [2],

...the phase diagram of the $d = 5$ SU(2) Yang–Mills theory on the lattice does not contain a second order phase transition or a critical point where a five-dimensional continuum theory can be defined non-perturbatively...

In the context of a small-coupling expansion, higher-dimensional QCD is non-renormalizable (in the power-counting sense), suggesting that it might not have a continuum limit [2]. The small-coupling expansion may not be a reliable guide for that question, but this suggestion is at least consistent with the numerical evidence.

The paper [3], which claims to be the first lattice study of five-dimensional gauge theory with three colors (gauge group SU(3) but still without quarks), finds a similar structure: both a confined phase and a deconfined phase, separated from each other by a first-order transition (no continuum limit).

The question of the existence of a five-dimensional continuum limit is not yet settled, though. The paper [3] says,

The existence of the second order critical end point even for the SU(2) gauge theory is still under investigation..., and we need the large lattice data to show it.

The effect of dynamical quarks

What happens to this picture when quarks are included? I don't know of any lattice studies of higher-dimensional QCD with dynamical quarks, but the small-coupling expansion in four-dimensional QCD indicates that asymptotic freedom disappears when the number of quark flavors is sufficiently large. If the loss of asymptotic freedom entails a loss of confinement (?), then this indicates that adding more quarks to the theory decreases the chances that the theory is confining. That's a pretty loose argument, but it suggests that the existence of a confining phase in QCD without quarks is at least a necessary condition for the existence of a confining phase with quarks. In this sense, the lattice evidence cited above isn't completely irrelevant to the question; but as far as I know, a definitive answer to the question is not yet available.


References:

[1] "Extra-dimensional models on the lattice," https://arxiv.org/abs/1605.04341

[2] "Lattice Simulations of 10d Yang-Mills toroidally compactified to 1d, 2d and 4d," https://arxiv.org/abs/1612.06395

[3] "Phase structure of pure SU(3) lattice gauge theory in 5 dimensions," https://arxiv.org/abs/1403.6277

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