Effective kinetic description of the expanding overoccupied glasma

We report on a numerical study of the Boltzmann equation including $2\leftrightarrow 2$ scatterings of gluons and quarks in an overoccupied glasma undergoing longitudinal expansion. We find that when a cascade of gluon number to the infrared occurs, corresponding to an infrared enhancement analogous to a transient Bose-Einstein condensate, gluon distributions qualitatively reproduce the results of classical-statistical simulations for the expanding glasma. These include key features of the distributions that are not anticipated in the “bottom-up” thermalization scenario. We also find that quark distributions, like those of gluons, satisfy self-similar scaling distributions in the overoccupied glasma. We discuss the implications of these results for a deeper understanding of the self-similarity and universality of parton distributions in the glasma.

Figure 1

The time evolution of the Debye mass squared for different values of the coupling. The three curves lie on top of each other. Except at early times, the Debye mass squared decreases as $1/\tau$.

Figure 2

The time evolution of the transverse momentum distribution evaluated at $p_z=0$. Left panel: original distribution. Right panel: rescaled distribution. For $p_{\perp }\gtrsim Q_s$, the scaling behavior shown in Eq. (50) is well satisfied.

Figure 3

The second moment of the transverse distribution, $p_{\perp }^2\tilde{f}(\tilde{\tau },p_{\perp },0)$, at different times. Left panel: original distribution. Right panel: rescaled distribution. The typical transverse momentum scale stays at $\sim Q_s$.

Figure 4

The time evolution of the longitudinal momentum distribution evaluated at $p_{\perp }=Q_s$. Left panel: original distribution. Right panel: rescaled distribution.

Figure 5

The time evolution of the transverse hard scale (left panel) and the longitudinal hard scale (right panel). Several values of the coupling are compared. The occupancy measure is fixed to the value $n_0=0.1$. In the right figure, the free streaming behavior ${\mathrm{\Lambda }}_L^2(\tau )\sim (Q_s\tau {)}^{-2}$ and the classical scaling behavior $\sim (Q_s\tau {)}^{-2/3}$ are indicated by gray and black dashed lines, respectively.

Figure 6

Time evolution of the bulk anisotropy for several values of the coupling constant. A significant deviation from the temporal power law $(Q_s\tilde{\tau }{)}^{-2/3}$ is observed even for weak couplings.

Figure 7

Time evolution of the longitudinal hard scale. Several values of the anisotropy parameter ${\xi }_0$ are compared for the initial distribution in Eq. (41). Box IC denotes the initial distribution given by Eq. (53).

Figure 8

Time evolution of the Coulomb logarithm.

Figure 9

The time evolution of the longitudinal hard scale. The result for the case where the time dependence of the Coulomb logarithm is taken into account is compared to the result where logarithm is fixed to its initial value.

Figure 10

The time evolution of the quark number density multiplied by time. Different values of the quark initial occupancy $F_0$ are compared.

Figure 11

The second moment of the transverse momentum distribution for quarks, $p_{\perp }^{2}F(\tilde{\tau },p_{\perp },0)$, for different times. Left panel: original distribution. Right panel: rescaled distribution. The typical transverse momentum scale remains centered at $\sim Q_s$.

Figure 12

The time evolution of the longitudinal momentum distribution for quarks evaluated at $p_{\perp }=Q_s$. Left panel: original distribution. Right panel: rescaled distribution.

Figure 13

Time evolution of the longitudinal hard scale for quarks. Three different values of the initial quark occupancy $F_0$ are compared. The results for $F_0=0.5$ and $F_0=1$ are indistinguishable.

Figure 14

Time evolution of the longitudinal hard scale. A comparison of the scales for gluons and that for quarks is shown.

Figure 15

Time evolution of the pressure ratio for gluons, ${\mathcal{P}}_{L,g}/{\mathcal{P}}_{T,g}$, and that for quarks, ${\mathcal{P}}_{L,q}/{\mathcal{P}}_{T,q}$.

Figure 16

The ratio of the longitudinal pressure to the transverse pressure as a function of time for the coupling $g=0.5$. Three different values of the initial quark occupancy are compared.

Figure 17

The ratio of the quark number and the gluon number as a function of time for the coupling $g=0.5$. Three different values of the quark initial occupancy are compared. The black dashed line denotes the value expected in the thermal equilibrium.

Figure 18

The infrared flow (b1) as a function of time. Several values of the minimum momentum $p_{\min }$ are compared.

Figure 19

Time evolution of the particle number density in the BEC. Results for several values of the minimum momentum $p_{\min }$ are compared. The result is insensitive to the value of $p_{\min }$ as long as it is sufficiently small.

Figure 20

The number of particles in the BEC as a function of time for different values of the coupling.

Figure 21

The transverse momentum distribution $\tilde{f}(\tilde{\tau },p_{\perp },p_z=0)$ (red lines) and the longitudinal momentum distribution $\tilde{f}(\tilde{\tau },p_{\perp }=0,p_z)$ (blue dashed lines) at $Q_s\tilde{\tau }=1.5$, 2.0, 2.5, 10. The time at which condensation occurs is $Q_s{\tilde{\tau }}_c\simeq 2.08$.