TMD's at High Density

Transverse-momentum-dependent (TMD) parton distribution functions encode information about the spin and momentum structure of partons inside a hadron. Understanding this spin-orbit structure is essential to resolve the "Proton Spin Puzzle": the experimental observation that the total spin of the proton is not accounted for by the measured polarization of its quarks or gluons. In general, TMD's are nonperturbative hadronic matrix elements that can only be modeled or studied phenomenologically. But when applied to a hadronic system at high density, many features of TMD's become amenable to first-principles calculation. A hadronic system at high density generates an intrinsic momentum scale, known as the saturation momentum $Q_s$. At sufficiently high densities, this intrinsic momentum becomes a hard scale, dynamically cutting off the infrared gluon field of the dense system. In this limit, the dominant degrees of freedom are classical gluon fields, which can be dressed by subsequent quantum evolution. A particularly useful high-density system to study is a heavy nucleus, since it introduces a tunable parameter in its mass number $A$. In this talk, I will discuss the calculation of TMD's in the high density limit. First I will present the calculation of the TMD's of an unpolarized heavy nucleus in the quasi-classical approximation. This calculation reveals the unique role played by orbital angular momentum in the dense system, which leads to testable predictions through the mixing of various TMD's. Then I will discuss the leading-logarithmic quantum evolution corrections to the TMD's, both in $Q^2$ and in $x$. The $Q^2$ and unpolarized $x$ evolution are both well-known in the literature, but much less is known about the polarized distributions at small $x$. I will present our recently-derived evolution equations for the helicity distribution at small $x$ and discuss some of the unique features of this polarized evolution.