Swampland conditions for higher derivative couplings from CFT

There are effective field theories that cannot be embedded in any UV complete theory. We consider scalar theories, with and without dynamical gravity, $D$-dimensional anti-de Sitter (AdS) spacetime large radius derive precise bounds (analytically) on the coupling constants of higher derivative interactions $\phi^2\Box^k\phi^2$ by only requiring dual CFT obeys standard conformal bootstrap axioms. In particular, we show all such constants, for even $k\ge 2$, must satisfy positivity, monotonicity, log-convexity conditions absence gravity. Inclusion gravity affects constraints involving $\phi^2\Box^2\phi^2$ interaction which now can have a negative constant. Our setup is Lorentzian four-point correlator Regge limit. also utilize this to multiple scalars. argue similar analysis should impose nontrivial graviton scattering amplitude AdS.