Evolution of the transverse-momentum dependent gluon distribution at small $x$

Abstract: Using the colour dipole picture for photon-nucleus interactions at small $x$ together with the Color Glass Condensate (CGC) effective theory, we demonstrate that the next-to-leading (NLO) order corrections to the cross-section for the inclusive production of a pair of hard jets encode not only the JIMWLK evolution with decreasing $x$, but also the DGLAP evolution of the gluon distribution function and the CSS evolution of the gluon transverse momentum dependent (TMD) distribution. The emergent CSS equation takes the form of a rate equation describing the evolution of the dijet distribution in the transverse momentum imbalance $K_\perp$ when increasing the dijet relative momentum $P_\perp$. All three types of evolution become important when both $P_\perp$ and $K_\perp$ are much larger than the nuclear saturation momentum $Q_s(x)$ and we propose a framework which encompasses all of them. The solution to the JIMWLK equation provides the source term for the DGLAP evolution with increasing $K_\perp$, which in turn generates the initial condition for the CSS evolution with increasing $P_\perp$.