Calculating TMDs of a large nucleus: Quasi-classical approximation and quantum evolution

We set up a formalism for calculating transverse-momentum-dependent parton distribution functions (TMDs) of a large nucleus using the tools of saturation physics. By generalizing the quasi-classical Glauber–Gribov–Mueller/McLerran–Venugopalan approximation to allow for the possibility of spin–orbit coupling, we show how any TMD can be calculated in the saturation framework. This can also be applied to the TMDs of a proton by modeling it as a large “nucleus.” To illustrate our technique, we calculate the quark TMDs of an unpolarized nucleus at large-x: the unpolarized quark distribution and the quark Boer–Mulders distribution. We observe that spin–orbit coupling leads to mixing between different TMDs of the nucleus and of the nucleons. We then consider the evolution of TMDs: at large-x, in the double-logarithmic approximation, we obtain the Sudakov form factor. At small-x the evolution of unpolarized-target quark TMDs is governed by BK/JIMWLK evolution, while the small-x evolution of polarized-target quark TMDs appears to be dominated by the QCD Reggeon.