Researchers have calculated the maximum phase-space density associated with the linearly polarized gluon transverse momentum dependent (TMD) parton distribution function coefficient, $h_1^{\perp g}$, in the saturation region. This work is crucial for understanding the internal structure of hadrons, such as protons and neutrons, under conditions of high gluon density, where saturation effects become dominant. Gluon saturation is a phenomenon predicted by Quantum Chromodynamics (QCD) at low momentum fractions $x$, where the number of gluons inside the proton is so large that they begin to recombine, limiting their density.

For the calculation, Mueller's occupation argument was employed, combining it with the Weizsäcker-Williams (WW) and dipole gluon distributions at low $x$, proposed by Metz and Zhou. It was found that for the dipole distribution, the maximum phase-space density, $n_{h,{\rm DP}}^{\rm max}$, is approximately $2\alpha_s^{-3/2}$, which is twice the maximum gluon density $n_g^{\rm max}$ in the same phase-space normalization. It is important to note that this result for the dipole is an approximation of the process-dependent TMD, not a literal numerical density of gluons. In contrast, for the WW distribution, the tensorial coefficient in deep saturation lacks the necessary logarithmic increase for Mueller's saddle point, which shifts the maximum towards the saturation limit.

Additionally, the study included a numerical Collins-Soper evolution, revealing that the weight of the Bessel function $J_2$ in the definition of the tensorial TMD reduces the resolved peak. This yields numerical values for the coefficient $c_h^{\rm num}$ between 6.6 and 7.1 for representative scales of the future Electron-Ion Collider (EIC). These results are fundamental for interpreting experimental data to be obtained at the EIC and other deep inelastic scattering experiments, providing a more precise understanding of gluon dynamics in the saturation regime and polarization within nucleons.