Molecular dynamics description of an expanding $q$/$\overline{q}$ plasma with the Nambu–Jona-Lasinio model and applications to heavy ion collisions at energies available at the BNL Relativistic Heavy Ion Collider and the CERN Large Hadron Collider

We present a relativistic molecular dynamics approach based on the Nambu–Jona-Lasinio Lagrangian. We derive the relativistic time evolution equations for an expanding plasma, discuss the hadronization cross section, and explain how they act in such a scenario. We present in detail how one can transform the time evolution equation to a simulation program and apply this program to study the expansion of a plasma created in experiments at the Relativistic Heavy Ion Collider and the Large Hadron Collider. We present first results on the centrality dependence of $v_2$ and of the transverse momentum spectra of pions and kaons and discuss in detail the hadronization mechanism.

Figure 1

Masses of $u$ and $s$ quarks as a function of $T$.

Figure 2

Effective interaction between two quarks in the random phase approximation (RPA).

Figure 3

The quark-antiquark polarization propagator for pseudoscalar coupling [41].

Figure 4

The meson propagator corresponding to the RPA sum.

Figure 5

Masses of $\pi$ (a) and $K$ (b) mesons as a function of $T$ with the NJL model.

Figure 6

Feynman diagrams for elastic $q\overline{q}$ scattering.

Figure 7

Elastic cross section for the different channels as a function of $\sqrt{s}$ for a temperature close to $T_{\text{Mott}}$ and at $\mu =0$ [41].

Figure 8

Feynman diagrams for hadronization.

Figure 9

Example of an inelastic (right) [20] $q\overline{q}$ cross section as a function of $\sqrt{s}$.

Figure 10

Standard algorithm for molecular dynamics calculations.

Figure 11

Distribution of temperature $T$ as a function of the normalized radius $r/r_0$ for $T_0=350$ MeV.

Figure 12

Distribution of temperature $T$ (in MeV) in the transverse plane $x,y$.

Figure 13

Dependence of the masses on the temperature and on the chemical potential of the environment in the NJL model. We display the masses of $u$ (a) and $s$ (b) quarks, as well as those of $\pi$ (c) and $K$ (d) mesons.

Figure 14

Local temperature at $r=0$ as a function of the distance $r$ and for a different number of fellow particles which all have a distance $r$ to the considered particle.

Figure 15

Collision between two particles in a medium.

Figure 16

Decay of a meson in a medium.

Figure 17

Total number of collisions for the same initial condition in a box simulation as a function of the time step $\Delta \tau$ for constant cross sections of 1 mb (a) and 5 mb (b).

Figure 18

Distributions of the mass (a), momentum (b), temperature (c), and scalar density (d) for light and strange quarks as a function of the distance $r$ from the center and for central collisions.

Figure 19

Energy density distribution (in GeV/fm${}^3$) in the transverse plane $x,y$ for EPOS [47] for a RHIC collision at $b=6$ fm. Colored areas are QGP bubbles.

Figure 20

Number of initial and final particles as a function of the impact parameter $b$.

Figure 21

Evolution of the total energy, mass, and momentum of the system (a), and the variation of the total energy (b) for a central RHIC collision as a function of time.

Figure 22

$v_2$ compared to experimental data from the PHOBOS experiment [48] as a function of the impact parameter $b$.

Figure 23

Time evolution of the elliptic flow $v_2$ for $b=6$ fm (a), and the comparison with PHSD calculations for similar initial conditions [49] (b).

Figure 24

The $dN/2\pi {\mathbf{p}}_{\mathrm{T}}d{\mathbf{p}}_{\mathrm{T}}$ spectrum for $b=4$ fm, and the results of a hydrodynamical calculation and of the experiment for similar conditions (centrality 0$\%$–5$\%$) [51].

Figure 25

($t,\mathbf{r}$) distributions for inelastic (a) and elastic (b) collisions at $b=0$ fm for RHIC conditions.

Figure 26

($t,\mathbf{r}$) distributions for inelastic collisions at $b=9$ fm for LHC conditions.

Figure 27

$\sqrt{s}$ distribution for inelastic collisions (a), and the distribution of the temperature at the production points, $T$, for pions and kaons (b).

Figure 28

$dN/dt$ for our simulations at $b=0$ fm and RHIC conditions (a) and $b=9$ and LHC conditions (b) and for PHSD for $b=0$ fm and RHIC conditions (c) [49].