  Weyl-invariant extensions of three-dimensional New Massive Gravity, generic
n-dimensional Quadratic Curvature Gravity theories and three-dimensional
Born-Infeld gravity theory are analyzed in details. As required by
Weyl-invariance, the actions of these gauge theories do not contain any
dimensionful parameter; therefore the local symmetry is spontaneously broken in
(Anti) de Sitter vacua in analogy with the Standard Model Higgs mechanism.
About the flat vacuum, symmetry breaking mechanism is more complicated: The
conformal symmetry is radiatively broken (at two loop level in 3-dimensions and
at one-loop level in 4-dimensions) a la Coleman-Weinberg mechanism and hence
the dimensionful parameters come from dimensional transmutation in the quantum
field theory. In the broken phases, save for New Massive Gravity, the theories
generically propagate with a unitary (tachyon and ghost-free) massless tensor,
massive (or massless) vector and massless scalar particles for the particular
intervals of the dimensionless parameters. For New Massive Gravity, there is a
massive Fierz-Pauli-type graviton. Finally, it is shown that n-dimensional
Weyl-invariant Einstein-Gauss-Bonnet theory is the only unitary higher
dimensional Weyl-invariant Quadratic Curvature Gravity theory.
