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

arXiv comment23 pages, 8 figures

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