Abelian versus Non-Abelian Higgs Model in Three Dimensions

We study the phase structure of the abelian Higgs model in three dimensions based on perturbation theory and a set of gauge independent gap equations for Higgs boson and vector boson masses. Contrary to the non-abelian Higgs model, the vector boson mass vanishes in the symmetric phase. In the Higgs phase the gap equations yield masses consistent with perturbation theory. The phase transition is first-order for small values of the scalar self-coupling λ, where the employed loop expansion is applicable. The “free-energy functional” of the Ginzburg-Landau theory of superconductivity is given by the action of the abelian Higgs model in three dimensions. Its phase structure has first been analyzed by Halperin, Lubensky and Ma [1]. For a type-I superconductor, where the scalar self-coupling λ is small compared to the gauge coupling g, the phase transition from the normal “symmetric” phase to the superconducting “Higgs” phase is weakly first-order. The case of a type-II superconductor, where λ/g is large, is more complicated and has been studied by various methods, in particular the ǫ-expansion and renormalization group techniques [1]. The three-dimensional abelian Higgs model also describes the corresponding fourdimensional theory at high temperatures. As a model for the cosmological electroweak phase transition, this case was studied by Kirzhnits and Linde [2], who also found a firstorder transition from the symmetric phase to the Higgs phase for λ/g ≪ 1. In recent years the abelian Higgs model at high temperatures has been studied in more detail [3, 4] using resummed perturbation theory, and the effective potential has been determined to order g, λ by a complete two-loop calculation [5]. In the electroweak phase transition non-perturbative effects are expected to be important, at least for large values of λ/g. They are related to the infrared behaviour of the non-abelian SU(2) Higgs model in three dimensions. So far, the nature of the symmetric phase and the order of the phase transition for large λ/g have not been firmly established. In a recent paper [6] we have studied some non-perturbative aspects of the SU(2) Higgs model by means of gap equations. Complementing the mass resummation by a vertex resummation a gauge independent set of gap equations was obtained for Higgs boson and vector boson masses, defined on the respective mass shells. The analysis led to the conclusion that the symmetric phase is again a Higgs phase, just with different parameters. The first-order phase transition, found for λ/g < 1, changes to a crossover at a critical scalar coupling λc, whose value is correlated with the magnitude of the vector boson mass in the symmetric phase. In this letter we apply the same resummation method to the abelian Higgs model in three dimensions. Due to the absence of gauge boson self-couplings the abelian Higgs model does not suffer from the same infrared problems as the non-abelian theory. It may therefore serve as a testing ground for the method employed in [6]. Much work has been done on the compact and non-compact versions of the abelian Higgs model on the lattice. Monte Carlo simulations provide evidence for a phase transition from a Higgs phase to a symmetric Coulomb phase with zero-mass photon for all values of λ/g [8]. For a review and references, see [7].