Wigner, Husimi and GTMD distributions in the Color Glass Condensate

TL;DR

This work investigates gluon phase-space distributions in the small-$x$ saturation regime by solving the BK equation with impact-parameter dependence, leveraging an SO(3) symmetry to simplify the numerics. It computes the gluon Wigner distribution and its elliptic component, then derives the Husimi distribution and generalized TMDs (GTMDs), revealing a common saturation-scale peak at $k\sim Q_s$ and a small elliptic component that remains measurable. The Husimi distribution is found to be positive definite, enabling a probabilistic interpretation, while the Wigner and GTMD require modeling of confinement effects to handle large-distance tails. The results point to diffractive dijet production in DIS as a potential probe of these phase-space structures and outline directions for incorporating higher-order BK corrections and finite-$N_c$ effects in future work.

Abstract

We study the phase space distributions of gluons inside a nucleon/nucleus in the small-$x$ regime including the gluon saturation effect. This can be done by using the relation between the gluon Wigner distribution and the dipole S-matrix at small-$x$, the latter satisfies the Balitsky-Kovchegov (BK) equation. By efficiently solving the BK equation with impact parameter dependence, we compute the Wigner, Husimi and generalized TMD (GTMD) distributions in the saturation regime. We also investigate the elliptic angular dependence of these distributions which has been recently shown to be measurable in DIS experiments.

Figures (10)

• Figure 1: Left: Dipole S-matrix $S_Y(d^2)$ as a function of $d^2$ up to $Y=10$. Right: $T_Y=1-S_Y$ as a function of $\ln r$ at $b=1$ and $\cos(\phi_b- \phi_r)=0$.
• Figure 2: The angular-independent part of the Wigner distribution $xW_0$ in the $(k,b)$ plane. Left: $Y=5$; Right: $Y=10$.
• Figure 3: The elliptic Wigner distribution $xW_1$ in the $(k,b)$ plane. Left: $Y=5$; Right: $Y=10$.
• Figure 4: The $k$-distribution of $W_0$ and $W_1$ at fixed $b=1$.
• Figure 5: The Husimi distribution at $Y=8$ (left) and $Y=10$ (right).