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 $\beta =(\frac{M_H}{M_Z}{)}^2$ and $\gamma ={\cos }^2{\theta }_{\mathrm{w}}$ where ${\theta }_{\mathrm{w}}$ 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, $\exp (in\phi )$. 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.

Comments: 16 pages, plain latex, talk given at the Gürsey Memorial Conference I on Strings and Symmetries, Istanbul, Turkey 6-10 June 1994, Preprint NORDITA 94/52P