Z flux-line lattices and self-dual equations in the standard model.

We derive gauge covariant self-dual equations for the SU(2)×U(1)Y theory of electroweak interactions and show that they admit solutions describing a periodic lattice of Z-strings. 1 E-mail addresses: [email protected] , [email protected] Solutions to the classical equations of motion of the Standard Model have recently attracted a considerable attention after the observation made by Vachaspati [1] that the embedding [2] of the NielsenOlesen [3] string into the Standard Model can be classically stable in a certain range of parameters. Further studies [4] have revealed that the region of classical stability does not overlap with the one compatible with experimental data. It has been suggested [5] that the cause of this instability might be traced back to the phenomenon of W-condensation [6] but the search of stable W-dressed strings has given negative results [7]. In their original investigations on electroweak magnetism, Ambjørn and Olesen [6] have solved the electroweak equations showing the existence of classically stable configurations representing a condensate of W bosons. Classical stability is ensured since these configurations satisfy a set of first order differential equations (self-dual equations) which arise from the requirement that they saturate a Bogomol’nyi like bound [8] for the energy. The solutions found in Ref. [6] carry no Z flux. The reason for this is that the self-dual equations were derived in the unitary gauge which excludes from the beginning the existence of Z-strings. In this work we present a gauge covariant expression of the self-dual equations and show that there exist boundary conditions which lead to configurations carrying Z flux. We will consider static configurations which are axially symmetric along the z-axis and such that physical observables are periodic in the x− y plane. The energy functional for a cell C of periodicity with area A is then given by E = ∫