Instability of a Nielsen-Olesen vortex embedded in the electroweak theory

Abstract

The stability of an abelian (Nielsen-Olesen) vortex embedded in the electroweak theory against W production is investigated in a gauge defined by the condition of a single-component Higgs field. The model is characterized by the parameters β = $\left( {\frac{{M_H }}{{M_Z }}} \right)^2$and γ = cost²Ø_w where Ø_w is the weak mixing angle. It is shown that the equations for W's in the background of the Nielsen-Olesen vortex have no solutions in the linear approximation. A necessary condition for the nonlinear equations to have a solution in the region of parameter space where the abelian vortex is classically unstable is that the W's be produced in a state of angular momentum m such that 0 > m > -2n. The integer n is defined by the phase of the Higgs field, exp(_⇌ It is shown that, in the region of parameter space (β, γ) where the nonlinear equations have a solution with energy lower than that of the abelian vortex, this vortex is a saddle point of the energy in the space of classical field configurations. Solutions for a set of values of the parameters β and y in this region were obtained numerically for the case -m = n = 1. The possibility of existence of a stationary state for n = 1 with W's in the state m = -1 was investigated. The boundary conditions for the Euler-Lagrange equations required to make the energy finite cannot be satisfied at r = 0. For these values of n and m the possibility of a finite-energy stationary state defined in terms of distributions is discussed.