Transverse momentum dependent parton distributions at small $x$

Abstract: Transverse-momentum-dependent (TMD) parton distributions provide a three-dimensional description of the momentum structure of hadrons. They are probed in proton-proton and proton-nucleus collisions at the LHC and RHIC, and are a central part of the physics program of the future Electron-Ion Collider (EIC). While TMD factorization is well established at moderate Bjorken $x$, the small-$x$ regime requires a different treatment, since large logarithms of $1/x$ must be resummed and non-linear effects due to high gluon densities become important. The Color Glass Condensate (CGC) effective theory provides a natural framework to describe TMDs in this saturation regime and to connect them with standard TMD factorization. This chapter reviews quark and gluon TMDs at small $x$, covering operator definitions, gauge-link structures, and Wilson-line correlators in the CGC for unpolarized and polarized distributions. We discuss factorization for inclusive and diffractive observables, highlighting the role of process dependence and the emergence of a limited set of Wilson-line operators. We then summarize the evolution of these distributions, namely the resummation of logarithms of $1/x$ by the Balitsky-Kovchegov and JIMWLK equations, and of Sudakov logarithms of the hard scale through Collins-Soper-Sterman evolution. We also review TMDs beyond the eikonal approximation, including the sub-eikonal Wilson-line operators that give access to helicity and other spin-dependent distributions, their double-logarithmic small-$x$ evolution and the resulting asymptotics, and the sub-eikonal corrections to the eikonal distributions, which provide a first step toward the moderate-$x$ regime. Finally, we discuss phenomenological implications for present and future experiments, with emphasis on the EIC, and outline the theoretical challenges toward a unified description of TMDs.