Collins-Soper-Sterman Hamiltonian: High energy evolution of rapidity dependent observables

We consider evolution of observables which depend on a small but fixed value of longitudinal momentum fraction x, to high rapidity, such that η>ln 1/x. We show that this evolution is not given by the Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) (or Balitsky-Kovchegov) equation. We derive the evolution Hamiltonian - HCSS-JIMWLK which generates this evolution in the cases of dilute and dense projectile wave function. The two limits yield identical results for HCSS-JIMWLK. We show that the resulting evolution for the gluon Transverse-Momentum-Dependent is identical to the (double logarithmic) perturbative Collins-Soper-Sterman evolution equation in the longitudinal resolution parameter at a fixed and very large transverse resolution.