Exponentially localised instantons in a hierarchy of Higgs models

Summary

Introduction

The systematic construction of hierarchies of Abelian (in d=2) and non-Abelian (in d>2) Higgs models in d dimensions which support finite action and topologically stable lump solutions, was reviewed in Ref.[l]. Very briefly, the method involves the construction of a hierarcy of Yang-Mills (YM) models in all even dimensions supporting such lump solutions, and then subjecting these to dimensional reduction where the residual systems are the hierarchies of Higgs models in question.

In this article we shall investigate in more detail, the asymptotic properties of the lumps of the Higgs models. These are very different from the asymptotic properties of the even (higher) dimensional YM hierarchies whose connectionfields have a pure-gauge type of behaviour at infinity, their lumps are localised with a power behaviour, and in those cases where these systems are scale invariant this localisation exhibits a further scale arbitrariness. By contrast, the lumps of the hierarchies of Higgs models are in general exponentially localised to an absolute scale. This property is potentially very important from the viewpoint of physical applications and hence is highlighted in the title of this article.

The material is presented in two sections below. In the first, we present the hierarchy of Higgs models in d dimensions in a formal way and examine particular asymptotic properties of the lumps by examining the fields in the (Dirac) string-gauge.