Spinodal instabilities of baryon-rich quark matter in heavy ion collisions

Using the test-particle method to solve the transport equation derived from the Nambu-Jona-Lasino (NJL) model, we study how phase separation occurs in expanding quark matter like that in a heavy ion collision. To test our method, we first investigate the growth rates of unstable modes of quark matter in a static cubic box and find them to agree with the analytical results that were previously obtained using the linear response theory. In this case, we also study the higher-order scaled density moments in the quark matter, which have values of 1 for a uniform density distribution or a distribution where the nonzero density regions all have same value, and they are found to increase with time and saturate at values significantly larger than 1 after the phase separation. The skewness of the quark number event-by-event distribution in a small subvolume of the system is also found to increase, but this feature disappears if the subvolume is large. For the expanding quark matter, two cases are considered: one using a blast-wave model for the initial conditions and the other using initial conditions from a multiphase transport (AMPT) model. In both cases, we find that the expansion of the quark matter is slowed down by the presence of a first-order phase transition. Also, density clumps appear in the system and the momentum distribution of partons becomes anisotropic, which can be characterized by large scale density moments and nonvanishing anisotropic elliptic and quadrupolar flows, respectively. The large density fluctuations further lead to an enhancement in the dilepton yield. In the case with the AMPT initial conditions, the presence of a first-order phase transition also results in a narrower distribution of partons in rapidity. These effects of density fluctuations can be regarded as possible signals for a first-order phase transition that occurs in the baryon-rich quark matter formed in relativistic heavy ion collisions.

Figure 1

Time evolution of an unstable density mode of wave number $k=0.31{\mathrm{fm}}^{-1}$.

Figure 2

Growth rates extracted from numerically solving the Boltzmann equation for unstable modes of wave numbers 0.13, 0.16, 0.21, 0.31, and $0.63{\mathrm{fm}}^{-1}$ for quark matter of density $\rho =0.7{\mathrm{fm}}^{-3}$ and temperature $T=45$ MeV. Analytical results from the linearized Boltzmann equation of Ref. [6] are shown by solid and dashed curves for the cases with and without the collision term, respectively.

Figure 3

Time evolution of density distribution for a single event in quark matter of temperature $T=20$ MeV and net quark density $n_q=0.5{\mathrm{fm}}^{-3}$ for the cases of $G_V=0$ (left column) and $G_V=G_S$ (right column).

Figure 4

Cross sectional view of density distribution for a single event on the $z=0$ plane at $t=40\mathrm{fm}/c$ for the case $G_V=0$ with a first-order phase transition.

Figure 5

Demonstration of the phase separation in the phase diagram.

Figure 6

Time evolution of the density-density correlation function, averaged over 1000 events, in quark matter of temperature $T=20$ MeV and average net quark density $n_q=0.5{\mathrm{fm}}^{-3}$ inside the spinodal region.

Figure 7

Time evolution of the scaled density moments, obtained by averaging over 1000 events, in quark matter of temperature $T=20$ MeV and average net quark density $n_q=0.5{\mathrm{fm}}^{-3}$ inside the spinodal region.

Figure 8

Time evolution of the event-by-event distribution of the number of quarks in a subvolume of size 0.6 ${\mathrm{fm}}^3$ (upper panel) and 30 ${\mathrm{fm}}^3$ (lower panel) for quark matter of temperature $T=20$ MeV and average net quark density $n_q=0.5{\mathrm{fm}}^{-3}$ inside the spinodal region. The total number of events is 1000.

Figure 9

Phase trajectory of the central cell of expanding quark matter obtained by averaging over 100 events for the two cases with (solid line) and without (dashed line) a first-order phase transition using the blast wave initial conditions. The spinodal region in the case of $G_V=0$ is shown by the gray color, and it disappears for $G_V=G_S$. The region where quark matter can be bound in the case of $G_V=0$ is shown by red color.

Figure 10

Time evolution of the density of the central cell of expanding quark matter obtained by averaging over 100 events for the two cases with (solid line) and without (dashed line) a first-order phase transition.

Figure 11

Density distributions of expanding quark matter for a single event on the $z=0$ plane at $t=20$ fm/$c$ for the case with a first-order phase transition (left panel) and at $t=10$ fm/$c$ for the case without a first-order phase transition (right panel).

Figure 12

Scaled density moments obtained by averaging over 100 events as functions of time for the cases with (black lines) and without (red lines) a first-order phase transition.

Figure 13

Final anisotropic flow coefficients $v_2$ (upper panel) and $v_4$ (lower panel) distributions for 100 events of expanding quark matter with the same blast wave initial conditions.

Figure 14

Dilepton yield, averaged over 100 events, as a function of the invariant mass $M$ for the cases with (solid line) and without (dashed line) a first-order phase transition in expanding quark matter with the blast wave initial conditions.

Figure 15

Same as Fig. 9 for the phase trajectories of the central part of expanding quark matter obtained by averaging over 100 events for the cases with (solid line) and without (dashed line) a first-order phase transition using the initial parton distribution from the AMPT model that is smeared in the longitudinal direction as described in the text.

Figure 16

Time evolution of the density distributions in a single event in central Au+Au collisions at $\sqrt{s_{NN}}=2.5\mathrm{GeV}$ using smeared initial conditions from the AMPT for the cases with (left column) and without (right column) a first-order phase transition.

Figure 17

Scaled density moments averaged over 100 events as functions of time for the cases with (black lines) and without (red lines) a first-order phase transition.

Figure 18

Final rapidity distribution of quarks averaged over 100 events (upper panel) and its event-by-event variance (lower panel) for the cases with (solid curve) and without (dashed curve) a first-order phase transition from expanding quark matter using the AMPT initial conditions.

Figure 19

Dilepton yield averaged over 100 events as a function of invariant mass $\sqrt{s}$ for the cases with (solid curve) and without (dashed curve) a first-order phase transition from expanding quark matter using the AMPT initial conditions.