Abstract
In this paper, we analyze the constraints imposed by unitarity and crossing symmetry on conformal theories in large dimensions. In particular, we show that in a unitary conformal theory in large dimension D, the four-point function of identical scalar operators ϕ with scaling dimension ∆ϕ such that ∆ϕ/D < 3/4, is necessarily that of the generalized free field theory. This result follows only from crossing symmetry and unitarity. In particular, we do not impose the existence of a conserved spin two operator (stress tensor). We also present an argument to extend the applicability of this result to a larger range of conformal dimensions, namely to ∆ϕ/D < 1. This extension requires some reasonable assumptions about the spectrum of light operators. Together, these results suggest that if there is a non-trivial conformal theory in large dimensions, not necessarily having a stress tensor, then its relevant operators must be exponentially weakly coupled with the rest.
Highlights
We show that in a unitary conformal theory in large dimension D, the four-point function of identical scalar operators φ with scaling dimension ∆φ such that ∆φ/D < 3/4, is necessarily that of the generalized free field theory
This extension requires some reasonable assumptions about the spectrum of light operators. These results suggest that if there is a non-trivial conformal theory in large dimensions, not necessarily having a stress tensor, its relevant operators must be exponentially weakly coupled with the rest
In generalized free field theories” (GFFTs) a higher point correlation function is defined as the sum of products of two-point function i.e. as a Wick contraction
Summary
In this subsection we give a quick overview of the paper, highlighting the structure of our argument. We define ∆ = δD and spin as = ωD With this scaling, the correlator which is expressed as the sum over conformal blocks can be approximated by an integral over conformal blocks multiplied by OPE coefficient “density”. The correlator which is expressed as the sum over conformal blocks can be approximated by an integral over conformal blocks multiplied by OPE coefficient “density” We argue that this integral is of Laplace type and can be performed by saddle point approximation. Appendix A gives explicit formulas for conformal blocks in our large D scaling limit. In appendix C we present a detailed analysis of the constraints of unitarity and crossing
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
