Restricted Weyl Symmetry and Spontaneous Symmetry Breakdown of Conformal Symmetry

TL;DR

This paper explores how restricted Weyl symmetry in scalar-tensor gravity leads to spontaneous breaking of conformal symmetry, resulting in a non-independent Nambu-Goldstone boson related to the dilaton, within a Poincaré invariant phase.

Contribution

It demonstrates the connection between restricted Weyl symmetry and spontaneous conformal symmetry breaking, revealing the nature of the Nambu-Goldstone boson in this context.

Findings

Spontaneous symmetry breaking of global scale symmetry occurs with a non-zero scalar vacuum expectation value.

Broken conformal symmetry induces a Nambu-Goldstone boson expressed as a derivative of the dilaton.

The theory maintains Poincaré invariance despite symmetry breaking.

Abstract

We elucidate the relation between the restricted Weyl symmetry and spontaneous symmetry breakdown of conformal symmetry. Using a scalar-tensor gravity, we show that the restricted Weyl symmetry leads to spontaneous symmetry breakdown of a global scale symmetry when the vacuum expectation value of a scalar field takes a non-zero value. It is then shown that this spontaneous symmetry breakdown induces spontaneous symmetry breakdown of special conformal symmetry in a flat Minkowski space-time, but the resultant Nambu-Goldstone boson is not an independent physical mode but expressed in terms of the derivative of the dilaton which is the Nambu-Goldstone boson of the global scale symmetry. In other words, the theories which are invariant under the general coordinate transformation and the restricted Weyl transformation exhibit a Nambu-Goldstone phase where both special conformal transformation and dilatation are spontaneously broken while preserving the Poincar\'{e} symmetry.