Universal Bounds on Operator Dimensions from the Average Null Energy Condition

TL;DR

This paper derives new lower bounds on the scaling dimensions of highly-chiral primary operators in four-dimensional conformal field theories using the average null energy condition, proposing a universal bound related to operator spin.

Contribution

It establishes novel bounds on operator dimensions based on the average null energy condition and conjectures a universal inequality for all unitary conformal field theories.

Findings

New bounds for operators with large spin in 4D CFTs.

Conjecture of a universal dimension bound: Δ ≥ max{k,  k̄}.

Stronger bounds than unitarity bounds when |k - k̄| > 4.

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 $\Delta$ satisfy $\Delta \geq \text{max}\{k,\bar{k}\}.$ If $|k-\bar{k}| > 4$, this is stronger than the unitarity bound.