Wigner, Husimi, and generalized transverse momentum dependent distributions in the color glass condensate

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 $x$, which latter satisfies the Balitsky-Kovchegov (BK) equation. By efficiently solving the BK equation with impact parameter dependence, we compute the Wigner, Husimi and generalized transverse momentum dependent 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.

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 $\cos ({\varphi }_b-{\varphi }_r)=0$.

Figure 2

The angular-independent part of the Wigner distribution $x{W}_0$ in the $(k,b)$ plane. Left: $Y=5$. Right: $Y=10$.

Figure 3

The elliptic Wigner distribution $x{W}_1$ 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).

Figure 6

The elliptic Husimi distribution at $Y=8$ (left) and $Y=10$ (right).

Figure 7

The angular independent part of the GTMD in the $(k,\mathrm{\Delta })$ plane at $Y=5$ (left) and $Y=10$ (right).

Figure 8

The elliptic GTMD in the $(k,\mathrm{\Delta })$ plane at $Y=5$ (left) and $Y=10$ (right).

Figure 9

The $Y$ evolution of the peak at $\mathrm{\Delta }=1$. Left: angular independent part. Right: Elliptic part.

Figure 10

The $Y$ evolution of the peak of $F_0$ at $\mathrm{\Delta }=2$ (left) and $\mathrm{\Delta }=3$ (right).