Bound on the central charge of CFTs in large dimension

Abstract

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 ϕϕ T μν 2 $$ {c}_{\phi \phi {T}^{\mu \nu}}^2 $$ 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 ϕϕO 2 $$ {c}_{\phi \phi O}^2 $$ 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.

Keywords

• Field Theories in Higher Dimensions
• Scale and Conformal Symmetries