Universal Bounds on Operator Dimensions from the Average Null Energy Condition

TL;DR

This work uses the average null energy condition in the conformal collider framework to derive universal lower bounds on the scaling dimensions of highly chiral primary operators in 4D CFTs. By solving conformal Ward identities and evaluating ANEC-induced inequalities for operators in $(k,0)$ and $(k,1)$ representations, the authors establish a real gap above the unitarity bound (e.g., $Δ \ge k$ for $(k,0)$ and, for certain $(k,1)$, $Δ$ strictly above the unitarity limit). They conjecture a general spectrum bound $Δ \ge \max\{k,\bar{k}\}$, stronger than unitarity when $|k-\bar{k}|>4$, and they show consistency with free-field spectra and explicit examples (such as $Δ(3,0)\ge 7/2$ and saturations for some $(7,1)$ free-field-like operators). The results illuminate how ANEC constraints shape the operator spectrum and hint at broader extensions to other dimensions and to superconformal theories. Overall, the paper provides a concrete, calculable mechanism to bound anomalous dimensions of chiral operators beyond traditional unitarity.

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 $(k,\bar{k})$ specifying the transformations under chiral $\frak{su}(2)$ 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 $(k,\bar{k})$ and scaling dimension $Δ$ satisfy $Δ\geq \text{max}\{k,\bar{k}\}.$ If $|k-\bar{k}| > 4$, this is stronger than the unitarity bound.

Figures (2)

• Figure 1: Summary of the conjectured bounds for $(k,0)$ representations when $k$ is even. The unitarity bound sets $\Delta \ge 1 + k/2$, and the average null energy condition strengthens this to $\Delta \ge k$. This bound is saturated by operators constructed from free fields.
• Figure 2: Summary of the conjectured bounds for $(k,0)$ representations when $k$ is odd. The unitarity bound sets $\Delta \ge 1 + k/2$, and the average null energy condition strengthens this to $\Delta \ge k$. There is a gap of size $1/2$ between this lower bound and the lowest dimension operator of this spin that can be constructed from free fields.