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 fi eld theories. Denoting the spin of an operator by a pair of integers (k, (k) over bar) 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 fi eld theory, we conjecture that in any unitary conformal fi eld theory, primary local operators of spin (k, (k) over bar) and scaling dimension Delta satisfy Delta >= max {k,k}. If vertical bar k - k vertical bar > 4, this is stronger than the unitarity bound.