${\cal N}=4$ Supergravity with Local Scaling Symmetry in Four Dimensions

We construct the most general four-dimensional ${\cal N}=4$ supergravity coupled to an arbitrary number $n$ of vector multiplets in which the global scaling symmetry is gauged, in addition to a subgroup of $\text{SL}(2,\mathbb{R}) \times \text{SO}(6,n)$. The various gaugings are parametrized by an embedding tensor built out of $2 \binom{n+6}{3}+4(n+6)$ parameters that satisfy a specific set of quadratic consistency constraints, to which we provide an explicit solution. We also derive the local supersymmetry transformation rules and the equations of motion for the four-dimensional ${\cal N}=4$ matter-coupled supergravity with local scaling symmetry. Such supergravity theories do not possess an action, since the scaling symmetry is only an on-shell symmetry of the corresponding ungauged theories.