Bound on the central charge of CFTs in large dimension

In this paper, we use crossing symmetry and unitarity constraints to put a lower bound on the central charge of conformal field theories in large space-time dimensions D . Specifically, we work with the four-point function of identical scalars ϕ with scaling dimension ∆ ϕ , and use a certain class of analytic functionals to show that the OPE coefficient squared c_ϕϕT^μν^2 must be exponentially small in D . For this to hold, we need to make a certain assumption about the nature of the spectrum below 2∆ ϕ . Our argument is robust and can be applied to any OPE coefficient squared c_ϕϕ O^2 with ∆ O < 2∆ ϕ . This suggests that conformal field theories in large dimensions (if they exist) must be exponentially close to generalized free field theories.