Treading on the cut: Semi inclusive observables at high energy

We develop the formalizm for calculating semi inclusive observables at high energy in the JIMWLK/KLWMIJ approach. We consider several examples including diffractive processes, elastic and inclusive over the target degrees of freedom, scattering with fixed total transverse momentum transfer and inclusive gluon production. We discuss evolution of these observables with respect to various rapidity variables involved in their definitions (total rapidity, rapidity gap, width of diffractive interval etc.). We also discuss the dipole model limit of these observables and derive closed simple (as opposed to functional) differential equations in this approximation. We point out that there are some interesting differences between the full JIMWLK/KLWMIJ evolution and the dipole model evolution of diffractive cross section. In particular we show that in the dipole approximation the target does not diffract beyond the valence rapidity interval, consistently with the intuition suggested by the Pomeron fan diagramms. On the other hand such diffractive processes do exist in the full JIMWLK/KLWMIJ approach, albeit suppressed by the factor 1/N_c^2.