Instability of a Nielsen-Olesen Vortex Embedded in the Electroweak Theory: I. the Single-Component Higgs Gauge

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 \(\beta=({M_H\over M_Z})^2\) and \(\gamma=\cos^2\theta_{\rm w}\) where \(\theta_{\rm w}\) 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, \(\exp(in\varphi)\). Solutions for a set of values of the parameters \(\beta\) and \(\gamma\) 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.