Scale without conformal symmetry in hydrodynamics

Scale without conformal symmetry corresponds to an inhomogeneous conservation equation for the virial current sourced by the trace of the energy-momentum tensor. Fluids that are just scale-invariant differ qualitatively from their conformal counterparts, and generic dissipation effects relax the hydrodynamic response over sufficiently long time scales. Remarkably, this holds true already at the ideal order. Spontaneously broken scale symmetry does not, in general, add any new mode to the hydrodynamic sector.