Embedded Defects

TL;DR

This paper develops a method for embedding classical solutions, including topological defects, into field theories with complex symmetry groups, and analyzes their stability, especially in the context of the electroweak model.

Contribution

It introduces a prescription for embedding defects in theories with non-simple symmetry groups and examines the stability of electroweak strings and monopoles.

Findings

The electroweak model contains Z and W strings, with the W strings forming a one-parameter family.

Many embedded defects are unstable without bound states or condensates.

Z strings are unstable at a Weinberg angle of π/4 for all Higgs masses.

Abstract

We give a prescription for embedding classical solutions and, in particular, topological defects in field theories which are invariant under symmetry groups that are not necessarily simple. After providing examples of embedded defects in field theories based on simple groups, we consider the electroweak model and show that it contains the $Z$ string and a one parameter family of strings called the $W(\alpha )$ string. It is argued that, although the members of this family are gauge equivalent when considered in isolation, each member should be considered distinct when multi-string solutions are considered. We then turn to the issue of stability of embedded defects and demonstrate the instability of a large class of such solutions in the absence of bound states or condensates. The $Z$ string is shown to be unstable when the Weinberg angle ($\theta_w$) is $\pi /4$ for all values of the Higgs mass. The $W$ strings are also shown to be unstable for a large range of parameters. Embedded monopoles suffer from the Brandt-Neri-Coleman instability. A simple physical understanding of this instability is provided in terms of the phenomenon of W-condensation. Finally, we connect the electroweak string solutions to the sphaleron: ``twisted'' loops of W string and finite segments of W and Z strings collapse into the sphaleron configuration, at least, for small values of $\theta_w$.