Is nonlocality consistent with scale invariance? For sure I'm excluding gravity at first step, the question is that if nonlocality is compatible with scale invariance. At the classical and quantum levels for field theory in Minkowski spacetime.
Then what about the case of gravity?
 A: Everyone knows the difference between a local operator and a nonlocal operator. But the interesting question is what it means for a theory to be nonlocal. This has been given some poor definitions in the past, e.g. by people who say that everything quantum mechanical is nonlocal because of entanglement.
A much better definition is that nonlocal field theories are the ones that fail to have a local conserved current for continuous symmetries. Any QFT with a nonlocal Lagrangian will be an example. One of the simplest is
\begin{equation}
S = \int d^dx \int d^dy \frac{\phi(x)\phi(y)}{|x - y|^{2(d - \Delta_\phi)}}. \quad (1)
\end{equation}
Since this is a well defined theory (it obeys Wick's theorem), it makes perfect sense to call the $y$ integral a nonlocal Lagrangian. It also clearly has no local stress tensor which means energy is conserved only globally... not through a continuity equation. The last thing to notice about (1) is that is is scale invariant (it is in fact the so called generalized free CFT). So the answer to the question is yes.
