Transverse spin in the light-ray OPE

We study a product of null-integrated local operators $ {\mathcal{O}}_1 $ and $ {\mathcal{O}}_2 $ on the same null plane in a CFT. Such null-integrated operators transform like primaries in a fictitious d − 2 dimensional CFT in the directions transverse to the null integrals. We give a complete description of the OPE in these transverse directions. The terms with low transverse spin are light-ray operators with spin J$_{1}$ + J$_{2}$− 1. The terms with higher transverse spin are primary descendants of light-ray operators with higher spins J$_{1}$ + J$_{2}$− 1 + n, constructed using special conformally-invariant differential operators that appear precisely in the kinematics of the light-ray OPE. As an example, the OPE between average null energy operators contains light-ray operators with spin 3 (as described by Hofman and Maldacena), but also novel terms with spin 5, 7, 9, etc. These new terms are important for describing energy two-point correlators in non-rotationally-symmetric states, and for computing multi-point energy correlators. We check our formulas in a non-rotationally-symmetric energy correlator in $ \mathcal{N} $ = 4 SYM, finding perfect agreement.