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