Precision studies of gluon saturation in DIS at small x: NLO corrections to SIDIS and double inclusive hadron production

We calculate the next to leading (NLO) corrections to single and double inclusive hadron production in Deep Inelastic Scattering (DIS) at small $x$ using the Color Glass Condensate formalism. It is shown that all UV and soft divergences cancel among various diagrams while the rapidity divergences lead to JIMWLK evolution of dipoles and quadrupoles with rapidity. Finally the collinear divergences are absorbed into parton-hadron fragmentation functions which lead to their DGLAP evolution with $Q^2$.