Maximum phase-space density of linearly polarized gluon TMDs in the saturation region

We calculate the Sudakov-limited maximum phase-space density associated with the linearly polarized gluon TMD coefficient $h_1^{\perp g}$ in the saturation region. Using Mueller’s occupancy argument together with the small-$x$ Weizsäcker-Williams (WW) and dipole gluon distributions of Metz and Zhou, we find $n_{h,\mathrm{D}\mathrm{P}}^{\mathrm{m}\mathrm{a}\mathrm{x}}=2n_g^{\mathrm{m}\mathrm{a}\mathrm{x}}\sim 2{\alpha }_s^{-3\mathrm{/}2}$ for the dipole distribution in the same phase-space normalization. This dipole result is a process-dependent TMD proxy, not a literal gluon number density. For the WW distribution, the deep-saturation tensor coefficient lacks the logarithmic enhancement needed for the Mueller saddle, so the maximum is pushed toward the saturation boundary. We also perform a numerical Collins-Soper evolution study and find that the $J_2$ Bessel weight in the tensor TMD definition reduces the resolved peak, giving $c_h^{\mathrm{n}\mathrm{u}\mathrm{m}}\simeq 6.6$–$7.1$ for representative EIC scales.