Light-ray moments as endpoint contributions to modular Hamiltonians

Abstract We consider excited states in a CFT, obtained by applying a weak unitary perturbation to the vacuum. The perturbation is generated by the integral of a local operator J (n) of modular weight n over a spacelike surface passing through x = 0. For |n| ≥ 2 the modular Hamiltonian associated with a division of space at x = 0 picks up an endpoint contribution, sensitive to the details of the perturbation (including the shape of the spacelike surface) at x = 0. The endpoint contribution is a sum of light-ray moments of the perturbing operator J (n) and its descendants. For perturbations on null planes only moments of J (n) itself contribute