Dilaton in scalar QFT: a no-go theorem in $4-\varepsilon$ and $3-\varepsilon$ dimensions

TL;DR

This paper proves that in certain scalar quantum field theories near four and three dimensions, spontaneous scale symmetry breaking cannot produce an exactly massless dilaton due to fundamental consistency constraints.

Contribution

It demonstrates a no-go theorem showing the incompatibility of spontaneous scale symmetry breaking with an exactly massless dilaton in $4- ext{and} ext{ }3- ext{dimensional}$ scalar QFTs.

Findings

Spontaneous scale symmetry breaking can occur classically but not with an exactly massless dilaton.

Exact conformal symmetry at the Wilson-Fisher fixed point prevents a massless dilaton.

The no-go theorem applies to both $4- ext{and} ext{ }3- ext{dimensional}$ scalar theories.

Abstract

Spontaneous scale invariance breaking and the associated Goldstone boson, the dilaton, is investigated in renormalizable, unitary, interacting non-supersymmetric scalar field theories in $4-\varepsilon$ dimensions. At leading order it is possible to construct models which give rise to spontaneous scale invariance breaking classically and indeed a massless dilaton can be identified. Beyond leading order, in order to have no anomalous scale symmetry breaking in QFT, the models need to be defined at a Wilson-Fisher fixed point with exact conformal symmetry. It is shown that this requirement on the couplings is incompatible with having the type of flat direction which would be necessary for an exactly massless dilaton. As a result spontaneous scale symmetry breaking and an exactly massless dilaton can not occur in renormalizable, unitary $4-\varepsilon$ dimensional scalar QFT. The arguments apply to $\phi^6$ theory in $3-\varepsilon$ dimensions as well.