On Local Dilatation Invariance
  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. 
T.E. Clark and S.T. Love
hep-th