On multivortices in the electroweak theory I: Existence of periodic solutions

Abstract

In this paper we consider the bosonic sector of the electroweak theory. It has been shown in the work of Ambjorn and Olesen that when the Higgs mass equals to the mass of the Z boson, the model in two dimensions subject to the 't Hooft periodic boundary condition may be reduced to a BogomoΓnyi system and that the solutions of the system are vortices in a "dual superconductor". We shall prove using a constrained variational reformulation of the problem the existence of such vortices. Our conditions for the existence of solutions are necessary and sufficient when the vortex number N = 1,2.

On a System of Nonlinear Elliptic Equations Arising in Theoretical Physics

“…We will use (31) to show that the two first terms on the right-hand side of (30) are strong enough to achieve the desired coercivity inequality. In fact, by virtue of (31), we see that…”

“…When M=T, (50) gives rise to a solution of the model with a doubly-periodic lattice structure. Such a situation corresponds to the so-called vortex condensation phenomenon [1,2,3,10,25,30,33]. When M=R 2 , (50) gives rise to a highly localized vortex-like soliton solution which is often interpreted as representing a multiple particle system at equilibrium [8,20,7,26,28,35].…”

Selfdual Maxwell-Chern-Simons Vortices

“…We also recall [49] and [36] for other interesting results about Liouville's systems. Concerning self-dual electroweak vortices, we mention the work in [120], [27] and [28] in the planar case, and [119], [11] in the periodic case where sharp existence results are established for vortex number N = 1, 2, 3, 4. The analysis of the case N > 4 is still incomplete and in this direction a first contribution is contained in [9] that aims to extend the analysis of [43], [44] to "singular" Liouville-type equations in presence of Dirac measures.…”

Selfgravitating electroweak strings

“…Satisfactory results are available also in case we neglect the effect of gravity, and take = G = 0 in (1.11). In this case the resulting (2 × 2) system has been treated in [11,7] to yield various classes of planar Electroweak vortex-like configurations, while Electroweak periodic vortices have been established in [10,5]. It is the main goal of this paper to show that, if…”