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Confinement in a Higgs Model on $R^{3}\times S^{1}$

We determine the phase structure of an SU(2) gauge theory with an adjoint scalar on $R^{3}\times S^{1}$ using semiclassical methods. There are two global symmetries: a $Z(2)_{H}$ symmetry associated with the Higgs field and a $Z(2)_{C}$ center symmetry. We analyze the order of the deconfining phase transition when different deformation terms are used. After finding order parameters for the global symmetries, we show that there are four distinct phases: a deconfined phase, a confined phase, a Higgs phase, and a mixed confined phase. The mixed confined phase occurs where one might expect a phase in which there is both confinement and the Higgs mechanism, but the behavior of the order parameters distinguishes the two phases. In the mixed confined phase, the $Z(2)_{C}\times Z(2)_{H}$ global symmetry breaks spontaneously to a Z(2) subgroup that acts non-trivially on both the scalar field and the Polyakov loop. We find explicitly the BPS and KK monopole solutions of the Euclidean field equations in the BPS limit. In the mixed phase, a linear combination of $ϕ$ and $A_{4}$ enters into the monopole solutions. In all four phases, Wilson loops orthogonal to the compact direction are expected to show area-law behavior. We show that this confining behavior can be attributed to a dilute monopole gas in a broad region that includes portions of all four phases. A duality argument similar to that applied recently [Poppitz and Unsal, 2011] to the Seiberg-Witten model on $R^3 \times S^1$ shows that the monopole gas picture, arrived at using Euclidean instanton methods, can be interpreted as a gas of finite-energy dyons.