Higgs mechanism for new massive gravity and Weyl-invariant extensions of higher-derivative theories

New massive gravity provides a nonlinear extension of the Fierz-Pauli mass for gravitons in $2+1$ dimensions. Here we construct a Weyl-invariant version of this theory. When the Weyl symmetry is broken, the graviton gets a mass in analogy with the Higgs mechanism. In (anti)-de Sitter backgrounds, the symmetry can be broken spontaneously, but in flat backgrounds radiative corrections, at the two-loop level, break the Weyl symmetry à la Coleman-Weinberg mechanism. We also construct the Weyl-invariant extensions of some other higher-derivative models, such as the Gauss-Bonnet theory (which reduces to the Maxwell theory in three dimensions) and the Born-Infeld type gravities.