Spontaneously Broken Spacetime Symmetries and the Role of Inessential Goldstones

TL;DR

This paper investigates the elimination of inessential Goldstone bosons in spontaneously broken spacetime symmetries, clarifying conditions for inverse Higgs constraints and their implications for effective field theories.

Contribution

It provides new criteria for applying inverse Higgs constraints, explores generalizations involving integrating out inessential Goldstones, and analyzes mappings between different non-linear realizations.

Findings

Inverse Higgs constraints require specific structure constant conditions.

Generalized inverse Higgs constraints can produce equivalent effective theories.

Mappings between different coset parametrizations may not preserve inverse Higgs constraints.

Abstract

In contrast to internal symmetries, there is no general proof that the coset construction for spontaneously broken spacetime symmetries leads to universal dynamics. One key difference lies in the role of Goldstone bosons, which for spacetime symmetries includes a subset which are inessential for the non-linear realisation and hence can be eliminated. In this paper we address two important issues that arise when eliminating inessential Goldstones. The first concerns the elimination itself, which is often performed by imposing so-called inverse Higgs constraints. Contrary to claims in the literature, there are a series of conditions on the structure constants which must be satisfied to employ the inverse Higgs phenomenon, and we discuss which parametrisation of the coset element is the most effective in this regard. We also consider generalisations of the standard inverse Higgs constraints, which can include integrating out inessential Goldstones at low energies, and prove that under certain assumptions these give rise to identical effective field theories for the essential Goldstones. Secondly, we consider mappings between non-linear realisations that differ both in the coset element and the algebra basis. While these can always be related to each other by a point transformation, remarkably, the inverse Higgs constraints are not necessarily mapped onto each other under this transformation. We discuss the physical implications of this non-mapping, with a particular emphasis on the coset space corresponding to the spontaneous breaking of the Anti-De Sitter isometries by a Minkowski probe brane.