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 ∆ϕ, and use a certain class of analytic functionals to show that the OPE coefficient squared {c}_{phi phi {T}^{mu nu}}^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}_{phi phi 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.