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

Abstract

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-$x$ Weizs\"acker-Williams (WW) and dipole gluon distributions of Metz and Zhou, we find $n_{h,{\rm DP}}^{\rm max}=2n_g^{\rm max}\sim2\alpha_s^{-3/2}$ 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 $c_h^{\rm num}\simeq6.6$--$7.1$ for representative EIC scales.