The Weizs\"acker-Williams distribution of linearly polarized gluons (and its fluctuations) at small x

The conventional and linearly polarized Weizs"acker-Williams gluon distributions at small x are defined from the two-point function of the gluon field in light-cone gauge. They appear in the cross section for dijet production in deep inelastic scattering at high energy. We determine these functions in the small-x limit from solutions of the JIMWLK evolution equations and show that they exhibit approximate geometric scaling. Also, we discuss the functional distributions of these WW gluon distributions over the JIMWLK ensemble at rapidity $Y\sim 1\mathrm{/}{\alpha }_s$. These are determined by a 2d Liouville action for the logarithm of the covariant gauge function $g^2 $\mathrm{t}\mathrm{r}$,A^+(q)A^+(-q)$. For transverse momenta on the order of the saturation scale we observe large variations across configurations (evolution trajectories) of the linearly polarized distribution up to several times its average, and even to negative values.