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Quantum chromodynamics (QCD) is an $SU(3)$ gauge theory that exhibits confinement.

Which properties of QCD lead to confinement? Is it the gauge symmetry, the coupling strength, or something else? What would be a minimal field theory of confinement?

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  • $\begingroup$ Are you asking what the simplest field theory that exhibits confinement is? $\endgroup$
    – user1504
    Oct 21 '16 at 10:49
  • $\begingroup$ @user1504, yes, that is correct. $\endgroup$
    – leongz
    Oct 23 '16 at 20:11
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    $\begingroup$ Not sure if this is really what you want, but: QED in 1+2d with massless fermion (aka, the Schwinger Model) exhibits confinement. In 4d, the only understood cases are supersymmetric. $\endgroup$
    – user1504
    Oct 24 '16 at 19:57
  • $\begingroup$ @user1504, thanks. Could you provide an example of the supersymmetric cases? $\endgroup$
    – leongz
    Oct 24 '16 at 20:44
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    $\begingroup$ I have in mind Seiberg & Witten's solution of N=2 supersymmetric Yang-Mills. $\endgroup$
    – user1504
    Oct 24 '16 at 21:47
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The quickest conventional prototype of a confining theory is an SU(2) nonabelian gauge theory with just the three gauge fields and no matter coupled to them. It is renormalizable, and, like all quantum nonabelian gauge theories, its β-function, for small coupling is $$ \beta(g) = \frac{\partial g}{\partial \log(\mu)} = -\left(\frac{11}{3}C_2(G)-\frac{1}{3}n_sT(R_s)-\frac{4}{3}n_f T(R_f)\right)\frac{g^3}{16\pi^2}~, $$ where $C_2(G)$ (=2 for G =SU(2), here) is the positive Casimir invariant of the group, and T(R) is another (positive) Casimir invariant defined by Tr$(T^a_RT^b_R) = T(R)\delta^{ab}$ for generators $T^{a,b}_R$ of the Lie algebra in the respective representations R of the matter in the system, fermions and scalars.

That means that, for small couplings and no matter, the coupling increases with descending energy μ (long distances), due to quantum interactions. If this trend persists for strong coupling, (which it does, as confirmed by numerical lattice computer experiments), the coupling grows without bound and thus confines. A mass gap indicates there are no massless gluons around, anymore.

Takeaway: So, confinement is visibly undergirded by this infrared slavery trend, effected by the nonlinear self-couplings of the gauge particles (gluons). You need a nonabelian gauge group for that, so U(1) of EM won't do, but SU(2) suffices. Matter always works against the trend, so leave it out: drop the second and third term in the parenthesis above. You have the 3-vector gauge bosons of the simplest original theory invented in 1954 by Yang-and Mills.

There is a plethora of pictures illustrating the quantum anti-screening (how the effective charge diminishes as we get closer to the source), but most physicists have learned to distrust such. There are also simpler but less familiar theories, such as nonlinear σ-models in 2 spacetime dimensions, but their formal simplicity and seat-of-the pants asymptotic freedom demonstrations are mooted by subtler infrared ambiguities, fit for endless theoretical discussion.

An ironclad mathematical proof of confinement, of course, is still an outstanding Millenium Prize Problem.

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