Scale versus conformal invariance at the IR fixed point of quantum gravity

We examine the question of scale vs. conformal invariance for the linearized Einstein-Hilbert action, which describes the IR fixed point of quantum gravity. In $D = 4$, although the action is not conformally invariant in the usual sense, we explicitly show that the theory is a conformal field theory at the level of correlation functions. In higher dimensions, we show that the theory is scale but not conformally invariant, but can be embedded into a larger non-unitary conformal field theory, analogous to what has been found for Maxwell theory in $D>4$. We give evidence that similar statements are true for all free higher spin theories.