Emergent Local Phase-Space Scaling in Small-x Gluon Evolution

Abstract

Geometric scaling is a central output of nonlinear small-$x$ evolution, but it is less clear whether the same dynamics fixes a probability distribution in transverse phase space. Using fixed-coupling impact-parameter BK evolution in the $SO(3)$-symmetric construction, we build a normalized gluon Husimi phase-space distribution and resolve it with a local coarse graining whose ultraviolet boundary follows $Q_s(Y,b)$. The main result is a distribution-level one: after this $Q_s$-adaptive resolution, the conditional momentum distributions collapse as functions of $k/Q_s(Y,b)$. The conditional entropy then grows with unit slope relative to $\langle\ln Q_s^2\rangle$, as the integrated consequence of that collapse and the two-dimensional momentum measure. Fixed laboratory cutoffs do not show this law, while dense-rapidity, cutoff-window, box-size, regulator-shape, and Husimi-resolution scans keep the $Q_s$-adaptive result stable in the controlled window. Within this fixed-coupling $SO(3)$-BK setting, the result identifies a local phase-space scaling structure of the gluon Husimi distribution rather than a universal law for unregulated global entropy.