Bound on the central charge of CFTs in large dimension

Abstract

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 $\phi$ with scaling dimension $\Delta_{\phi}$, and use a certain class of analytic functionals to show that the OPE coefficient squared $c^2_{\phi \phi T^{\mu \nu}}$ must be exponentially small in $D$. For this to hold, we need to make a mild assumption about the nature of the spectrum below $2\Delta_\phi$. Our argument is robust and can be applied to any OPE coefficient squared $c^2_{\phi \phi O}$ with $\Delta_O< 2\Delta_\phi$. This suggests that conformal field theories in large dimensions (if they exist) must be exponentially close to generalized free field theories.