Effect of Multiple Higgs Fields on the Phase Structure of the SU(2)-Higgs Model

The SU(2)-Higgs model, with a single Higgs field in the fundamental representation and a quartic self-interaction, has a Higgs region and a confinement region which are analytically connected in the parameter space of the theory; these regions thus represent a single phase. The effect of multiple Higgs fields on this phase structure is examined via Monte Carlo lattice simulations. For the case of N ≥ 2 identical Higgs fields, there is no remaining analytic connection between the Higgs and confinement regions, at least when Lagrangian terms that directly couple different Higgs flavours are omitted. An explanation of this result in terms of enhancement from overlapping phase transitions is explored for N = 2 by introducing an asymmetry in the hopping parameters of the Higgs fields. It is found that an enhancement of the phase transitions can still occur for a moderate (10%) asymmetry in the resulting hopping parameters. The phase structure of the basic SU(2)-Higgs model (i.e. an SU(2) gauge theory coupled to a single scalar field with a quartic self-interaction and a quadratic term) is clearly of direct relevance to the Higgs sector of the Standard Model, where discussions of spontaneous symmetry breaking and the transition from a symmetric phase to a Higgs phase are paramount. The addition of extra Higgs fields occurs in a variety of extensions to the Standard Model, including the minimal supersymmetric Standard Model (see e.g. Ref. [1]). In the case of a single Higgs, the SU(2)-Higgs model has three parameters in its continuum formulation (Higgs self-coupling, gauge coupling, and Higgs quadratic term); in the lattice formulation of these models the three parameters become the Higgs self-coupling λ, the gauge coupling β, and the hopping parameter κ. For a single Higgs field in the fundamental representation, there exist theoretical arguments that there must be an analytic connection in (β, κ, λ) parameter space between the Higgs and confinement regions of the theory [2]. This behaviour is manifested in lattice simulations as a phase-transition line that begins at (β = ∞, κ∞ > 0) and terminates at a point (β ≥ 0, κ > κ∞), with a general progression of this termination point to smaller values of (β, κ) with decreasing λ [3, 4]. The resulting analytic connection between the Higgs and confinement regions (located in the corner of parameter space toward larger λ, smaller β, and larger κ) is consistent with Elitzur’s theorem [5] which says that a local gauge symmetry cannot break spontaneously. For a recent discussion of the breaking of global subgroups within the SU(2)Higgs model and their connection to the phase diagram, see Ref. [6]. We note in passing that Ref. [6] also mentions the SU(2)-Higgs model where the Higgs field is in the adjoint representation rather than the fundamental; in that case the theoretical arguments of [2] do not apply and the center symmetry can be broken spontaneously. Many basic properties of the phase structure for a single Higgs field in the fundamental representation are well understood, though there is no consensus on a detailed understanding of the nature of the phase transition (PT) across the entire parameter space. For small λ, the PT is demonstrably of first order Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, SK, S7N 5E2, Canada Department of Physics and Astronomy, York University, Toronto, ON, M3J 1P3, Canada Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, SK, S7N 5E2, Canada 1 [7] but the PT strength weakens with increasing λ, complicating the classification of the PT. The most effective approaches used to address this issue are searches for bimodal (“two-peaked”) distributions and lattice scaling dependence on the resulting energy gap [8, 9, 10, 11] or scaling effects within specific heats (susceptibilities) [9, 10, 11, 12, 13]. For example, Ref. [10] concludes that for λ ∼ 1 the PT is first-order for β at and slightly above the terminal point of the phase line, but is unable to ascertain the existence of a tricritical point where the order of the PT would change; it would be called a critical line in (β, κ, λ) space. Since the PT decreases in strength with increasing λ, the λ = ∞ case presents the greatest challenge in the classification of the PT. Early work [8] suggested a weak first-order PT, but a recent study with large lattices presents evidence for a smooth crossover [13]. Furthermore, the λ = ∞ model augmented with an additional interaction leads to a line of first-order PTs which decrease in strength as the additional coupling approaches zero [11]. In this paper we study the phase structure of the SU(2)-Higgs model with multiple Higgs fields in the fundamental representation. For simplicity, we will omit Lagrangian terms which would mix more than one flavour of Higgs. We find that the phase transition line extends all the way to β = 0 from β = ∞ for two or more Higgs fields regardless of the numerical value of λ, in marked contrast to the case of a single Higgs field. We show that this enhancement of the PT is associated with overlapping PTs, as has also been reported for the multi-Higgs three-dimensional U(1) theory [14]. The continuum SU(2)-Higgs action in Euclideanized space-time is Sc = ∫