Universal bounds on operator dimensions from the average null energy condition

Abstract

We show that the average null energy condition implies novel lower bounds on the scaling dimensions of highly-chiral primary operators in four-dimensional conformal field theories. Denoting the spin of an operator by a pair of integers (k,k) specifying the transformations under chiral su(2) rotations, we explicitly demonstrate these new bounds for operators transforming in (k, 0) and (k, 1) representations for sufficiently large k. Based on these calculations, along with intuition from free field theory, we conjecture that in any unitary conformal field theory, primary local operators of spin (k,k) and scaling dimension ∆ satisfy ∆ ≥ max{k,k}. If |k −k| > 4, this is stronger than the unitarity bound.

Implications of ANEC for SCFTs in four dimensions

“…5 up to j = 10 3 . Although the bound initially agrees with the conjecture of [6], it departs from it for j 21 and follows a different pattern which is well fitted by the expression ∆ = 1 2 j + 1 + δ 1 15 (13j + 42). It would be tempting to assign a meaning to the kink at j ∼ 21, but the explanation might simply reside in the fact that, going to large values, the integer nature of j becomes less and less important and new solutions for D 1 become available.…”

“…We find that for certain classes of operators the unitarity bounds of [8] cannot be saturated. Just as in [6], our results follow from a careful analysis of three-point functions of the schematic type OT µν O with O a conformal primary and O its conjugate. The difference with the nonsupersymmetric case is that here such conformal three-point functions are encoded in superconformal three-point functions involving the Ferrara-Zumino multiplet [9].…”

“…Following [6,23] we define the state |ψ of (1.1) by acting with some operator O(x, η, η) on the CFT vacuum |0 and taking the Fourier transform in order to give the state a definite momentum, 18 which for our purposes we can set to q µ = (1, 0). Then we multiply by (x + ) 2 /16 and send x + → ∞ to simplify the computations.…”

“…The restrictions imposed by SO(2) invariance imply that only certain terms can appear, i.e. The integrals have been computed explicitly for some values of j in [6]. Here we provide a general formula, whose proof can be found in Appendix B:…”

“…Let us briefly review the results obtained in [6] and prove a few results for generic values of j. First let us consider conformal primaries in the ( 1 2 j, 0) Lorentz representation.…”