Implications of ANEC for SCFTs in four dimensions

Abstract

We explore consequences of the Averaged Null Energy Condition (ANEC) for scaling dimensions ∆ of operators in four-dimensional N = 1 superconformal field theories. We show that in many cases the ANEC bounds are stronger than the corresponding unitarity bounds on ∆. We analyze in detail chiral operators in the ( 1 2 j, 0) Lorentz representation and prove that the ANEC implies the lower bound ∆ 3 2 j, which is stronger than the corresponding unitarity bound for j > 1. We also derive ANEC bounds on ( 1 2 j, 0) operators obeying other possible shortening conditions, as well as general ( 1 2 j, 0) operators not obeying any shortening condition. In both cases we find that they are typically stronger than the corresponding unitarity bounds. Finally, we elucidate operator-dimension constraints that follow from our N = 1 results for multiplets of N = 2, 4 superconformal theories in four dimensions. By recasting the ANEC as a convex optimization problem and using standard semidefinite programming methods we are able to improve on previous analyses in the literature pertaining to the nonsupersymmetric case.

Differential operators for superconformal correlation functions

“…Using the formalism presented here 5 we expand a three-point function of an exotic primary X , its conjugate X and the stress tensor J into N = 1 multiplets. We then show that the resulting correlators do not satisfy the Ward identities computed in [48] and are therefore inconsistent. The only assumption is the existence of a conserved stress tensor, i.e.…”

“…(5.9) 19 In the basis of [48] the coefficients A1,2 are translated into the coefficients C1,...,10 as follows: This immediately implies that there are no solutions for j > 0 and thus the exotic primaries cannot couple consistently with the stress tensor and must be absent from any local theory.…”

“…It suffices to extract the contribution from the Ferrara-Zumino multiplet J(z 2 ) inside of J (z 2 ). Then we can use the results of [48] to apply the Ward identities. We will keep denoting as X the N = 1 primary appearing as the lowest order in the exotic operator multiplet.…”

“…The ANEC [49][50][51] states the positivity of integrated expectation values of the stress tensor. There are non-supersymmetric results only in four dimensions for spin (j, 0) and (j, 1) [52] and N = 1 supersymmetric results for general multiplets with spin (j, 0) [48], which impose lower bounds on the conformal dimension, sometimes stronger than unitarity. As of now, there is no intuition on the shape of these bounds for more general Lorentz representations or for extended supersymmetry and thus more investigations are needed.…”

Selected Topics in Analytic Conformal Bootstrap: A Guided Journey

“…Due to the intrinsic Lorentzian nature of these bounds, an approach to find and study them is through lightcone approaches applied to four-point correlators and imposing analyticity. These ideas have been introduced in [363], and other developments in [364][365][366][367][368][369].…”