Calculating TMDs of a Large Nucleus: Quasi-Classical Approximation and Quantum Evolution

TL;DR

The paper develops a first-principles framework to compute transverse-m momentum-dependent distributions ($TMDs$) in dense QCD systems using saturation theory. It introduces a quasi-classical factorization that expresses the nuclear TMD correlator as a convolution of a nuclear Wigner distribution, nucleon TMDs, and Wilson-line staples, and shows how spin-orbit coupling induces mixing between nuclear and nucleon TMDs, e.g., $f_1^A$ mixing with $f_{1T}^{\bot N}$ and $h_1^{\bot A}$ mixing with nucleon transversity/pretzelosity. The work provides explicit quasi-classical results for the unpolarized nucleus, $f_1^A$ and $h_1^{\bot A}$, and analyzes their evolution: large-$x$ evolution yields a Sudakov form factor consistent with CSS, while small-$x$ evolution proceeds via BK/JIMWLK for unpolarized TMDs and a Reggeon-based framework for polarized TMDs. This approach offers a controlled path to global TMD fits in dense systems and connects TMD phenomenology to the proton spin problem via potential low-$x$ spin contributions and GTMD extensions.

Abstract

We set up a formalism for calculating transverse-momentum-dependent parton distribution functions (TMDs) 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.

Figures (14)

• Figure 1: Quasi-classical factorization of semi-inclusive deep inelastic scattering on a heavy nucleus.
• Figure 2: SIDIS cross section as a square of the scattering amplitude explicitly illustrating the $x^-$-ordering of the nucleons in the nucleus. The solid vertical line denotes the final-state cut.
• Figure 3: Schematic decomposition of the spin sum occurring in \ref{['e:QCFact3']}. Summing over the 4 independent components of the $( 2 \times 2 )$ matrices $[\phi]_{\lambda \lambda'} , \, [W]_{\lambda' \lambda}$ corresponds to summing over the 4 possible intermediate polarizations of the nucleons: $U$ = unpolarized, $L$ = longitudinally polarized, $T^j$ = transversely polarized in the $\hat{j}$ direction.
• Figure 4: Spin structure of the unpolarized nucleus. An unpolarized nucleus can give rise to unpolarized nucleons or transversely-polarized nucleons, which contribute to the quark distribution through various nucleonic TMDs. Interestingly there is no contribution from longitudinally-polarized nucleons in the unpolarized nucleus \ref{['e:Wig9']}.
• Figure 5: An example of quantum evolution corrections to the SIDIS process and for corresponding quark TMDs.