Minimal field theory of confinement

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?

• Are you asking what the simplest field theory that exhibits confinement is? Oct 21 '16 at 10:49
• @user1504, yes, that is correct. Oct 23 '16 at 20:11
• 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. Oct 24 '16 at 19:57
• @user1504, thanks. Could you provide an example of the supersymmetric cases? Oct 24 '16 at 20:44
• I have in mind Seiberg & Witten's solution of N=2 supersymmetric Yang-Mills. Oct 24 '16 at 21:47

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.