On Local Dilatation Invariance

TL;DR

This paper critically examines the differences between local Weyl scaling invariance and local dilatation invariance, highlighting the restrictive conditions for achieving the latter and implications for matter field couplings.

Contribution

It clarifies the conditions under which local dilatation invariance can be realized and distinguishes it from local Weyl invariance in field theories.

Findings

Actions invariant under local Weyl scalings are straightforward to construct.

Invariant Abelian vector kinetic terms require spontaneous symmetry breaking.

Couplings of matter fields with vector fields depend on symmetry realization.

Abstract

The relationship between local Weyl scaling invariant models and local dilatation invariant actions is critically scrutinized. While actions invariant under local Weyl scalings can be constructed in a straightforward manner, actions invariant under local dilatation transformations can only be achieved in a very restrictive case. The invariant couplings of matter fields to an Abelian vector field carrying a non-trivial scaling weight can be easily built, but an invariant Abelian vector kinetic term can only be realized when the local scale symmetry is spontaneously broken.