Universal Bounds on Operator Dimensions from the Average Null Energy Condition

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 $\mathfrak{s}\mathfrak{u}(2)$ 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 $\mathrm{\Delta }$ satisfy $\Delta \geq $\text{max}${k,\bar{k}}.$If$|k-\bar{k}| > 4$, this is stronger than the unitarity bound.