Gauge Theories of Quark Confinement in 2+1 Dim: A Comparative Study of SU(2) Yang-Mills and Yang-Mills-Higgs Systems

Abstract

We summarize some recent work on the confinement problem in 2+1 dim. We first review the Debye-Hückel screening mechanism of confinement in the Yang-Mills-Higgs system due to Polyakov. The effective field theory is a very weakly coupled sine-Gordon theory with a very small mass gap. A vacuum of adjoint monopoles does indeed lead to a correct screening theory in the sense that only SU(2) representations with non-trivial Z₂ charge are confined, unlike compact QED₃ where all representations are confined. Using the Polyakov model and D’Hoker’s Monte-Carlo results as a guide we construct the vacuum of the pure SU (2) gauge theory in terms of adjoint monopoles. The effective field theory is a strongly coupled sine-Gordon theory which is consistent with a mass gap M ≈ 2g and predicts the exact relation $\sigma = \frac{1}{8}{{g}^{2}}M$. The values M ≈ 2g² and $\sigma \approx \frac{1}{4}{{g}^{4}}$ are consistent with the Monte-Carlo numbers. The mechanism of quark confinement in the pure gauge theory is the one proposed by Wilson and Kogut and Susskind.