Thermalization of gluons in ultrarelativistic heavy ion collisions by including three-body interactions in a parton cascade

Zhe Xu and Carsten Greiner

Phys. Rev. C 71, 064901 – Published 2 June 2005

Abstract

We develop a new 3 + 1 dimensional Monte Carlo cascade solving the kinetic on-shell Boltzmann equations for partons including the inelastic $g g \leftrightarrow g g g$ pQCD processes. The back reaction channel is treated—for the first time—fully consistently within this scheme. An extended stochastic method is used to solve the collision integral. The frame dependence and convergency are studied for a fixed tube with thermal initial conditions. The detailed numerical analysis shows that the stochastic method is fully covariant and that convergency is achieved more efficiently than within a standard geometrical formulation of the collision term, especially for high gluon interaction rates. The cascade is then applied to simulate parton evolution and to investigate thermalization of gluons for a central Au+Au collision at RHIC energy. For this study the initial conditions are assumed to be generated by independent minijets with $p_{T} > p_{0} = 2$ GeV. With that choice it is demonstrated that overall kinetic equilibration is driven mainly by the inelastic processes and is achieved on a scale of 1 fm/c. The further evolution of the expanding gluonic matter in the central region then shows almost an ideal hydrodynamical behavior. In addition, full chemical equilibration of the gluons follows on a longer time scale of about 3 fm/c.

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Received 21 July 2004

DOI:https://doi.org/10.1103/PhysRevC.71.064901

Authors & Affiliations

Zhe Xu^1,2,* and Carsten Greiner^2,†

¹Institut für Theoretische Physik, Justus-Liebig-Universität Giessen, D-35392 Giessen, Germany
²Institut für Theoretische Physik, Johann Wolfgang Goethe Universität Frankfurt, D-60054 Frankfurt am Main, Germany

^*Email address: Zhe.Xu@theo.physik.uni-giessen.de
^†Email address: carsten.greiner@th.physik.uni-frankfurt.de

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Vol. 71, Iss. 6 — June 2005

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Figure 1
Energy distribution at final time ( $3 \to 2$ fm/c) of a system consisting of $N = 2000$ massless particles in a fixed box. The initial energy distribution is set to be a delta function at 6 GeV. The size of the box is 5 fm × 5 fm × 5 fm. We here apply the geometrical collision algorithm. The collisions are taken as isotropic and the total cross section is fixed to be a constant $σ_{22} = 10$ mb. The dotted line denotes the analytical result of temperature $σ_{22} = 10$ GeV. The numerical distribution is obtained from an ensemble of 50 independent realizations.Reuse & Permissions
Figure 2
Collision rates for given particle densities. The size of the box is 5 fm × 5 fm × 5 fm. We apply here the geometrical collision algorithm. The collisions are isotropic and the total cross section is fixed to a constant $σ_{22} = 10$ mb. The particle system is taken initially as thermal with a temperature of $T = 1$ GeV. The solid line shows the expected relationship between collision rate and particle density: $R = n σ_{22}$ . The solid squares show the calculated collision rates without test particles ( $N_{test} = 1$ ) and the open squares show the results with 50 test particles per real particle ( $N_{test} = 50$ ).Reuse & Permissions
Figure 3
Energy distributions from box calculations. The thin histogram shows the same distribution as in Fig. 1. The thick histogram shows the result with a smaller total cross section of $σ_{22} = 0.1$ mb.Reuse & Permissions
Figure 4
Time evolution of the momentum anisotropy from box calculations. The initial condition and parameters are set to be the same as in Fig. 1. The dotted (dashed) curve shows the results obtained by employing the geometrical method without test particles (with 50 test particles per real particle). The solid curve shows the result obtained by employing the stochastic method with ten test particles per real particle.Reuse & Permissions
Figure 5
Energy distribution from box calculations. The initial conditions and parameters are set to be the same as in Fig. 1. We apply here the stochastic collision algorithm. The box is divided into equal cells. The length of a cell is 1 fm.Reuse & Permissions
Figure 6
Collision rates for given particle densities. The initial conditions and parameters are set to be the same as in Fig. 2. We apply here the stochastic collision algorithm. The cell configuration is the same as in Fig. 5.Reuse & Permissions
Figure 7
Time evolution of momentum anisotropy from box calculations. The initial conditions and parameters are set to be the same as in Fig. 1 (or Fig. 5). The stochastic method is used here. The cell configuration is the same as in Fig. 5. The curves show the results with different number of test particles.Reuse & Permissions
Figure 8
Time evolution of the particle density from box calculations. The initial conditions and parameters are set to be the same as in Fig. 1 (or Fig. 5). We consider isotropic inelastic collisions ( $σ_{22} = 0.1$ ) with a constant cross section of $σ_{23} = 10$ mb and employ the stochastic collision algorithm. The cell configuration is the same as in Fig. 5. The dotted line denotes the estimate using a simple time relaxation approximation.Reuse & Permissions
Figure 9
Energy distribution from the same calculations as in Fig. 8. The histogram shows the numerical result. The dotted line shows the analytical expectation and the dashed line shows the analytical distribution (the same as in Fig. 5) if the particle number would be conserved.Reuse & Permissions
Figure 10
Time evolution of the fugacity $[n (t)] / n_{eq}$ versus the momentum anisotropy from the same calculation as in Fig. 8.Reuse & Permissions
Figure 11
Gluon collision rates and thermally averaged $〈 v_{rel} σ 〉$ as function of temperature. The solid, dashed, dotted, and dash-dotted line show the temperature dependence for $g g \to g g g, g g \to g g, gq \to gq$ , and $g g \to q \bar{q}$ transitions, respectively. We consider here two quark flavors and employ a constant coupling $α_{s} = 0.3$ (for the cross sections and the screening masses).Reuse & Permissions
Figure 12
Time evolution of the gluon and quark number in box calculations. We consider here gluons and quarks with two flavors as parton species. Collision processes are the elementary two-body parton-parton scatterings and three-body processes $g g \leftrightarrow g g g$ in leading order of perturbative QCD. The coupling is assumed to be a constant of $α_{s} = 0.3$ . The initial momentum distribution of particles is taken from the minijets production in central rapidity interval $y \in (- 0.5 : 0.5)$ in a nucleon-nucleon collision at RHIC energy $\sqrt{s} = 200$ GeV. The initial particles are gluons and distributed homogenously in the box. The size of the box is 3 fm × 3 fm × 3 fm and the box is divided into equal cells. The length of a cell is 1 fm. The initial gluon number is set to be 500. The results are obtained from an average over 60 runs.Reuse & Permissions
Figure 13
Energy distributions at different times from the same calculation as in Fig. 12.Reuse & Permissions
Figure 14
Time evolution of the momentum anisotropy for gluons and quarks from the same calculation as in Fig. 12.Reuse & Permissions
Figure 15
Time evolution of the temperature for gluons and quarks from the same calculation as in Fig. 12.Reuse & Permissions
Figure 16
Time evolution of the fugacity for gluons and quarks from the same calculation as in Fig. 12.Reuse & Permissions
Figure 17
Time evolution of the screening mass for gluons and quarks from the same calculation as in Fig. 12.Reuse & Permissions
Figure 18
Time evolution of the collision rates for gluons in the different channels. The results are obtained from the same calculation as in Fig. 12.Reuse & Permissions
Figure 19
One dimensional expansion in a tube. The lab frame is labeled by $X, Y$ , and Z, the boosted frame by $X^{'}, Y^{'}$ , and $Z^{'}$ which is moving with a velocity of β relative to the lab frame.Reuse & Permissions
Figure 20
Space-time rapidity distributions at different times ( $t = 0.11, 0.13, 0.16$ , and $0.2$ fm/c from histogram with smallest amplitude to histogram with largest amplitude) from a simulation of one dimensional expansion in a tube. We consider a thermal and boost-invariant initial condition for evolving particles: Particles are produced initially on a hyperbola of $τ_{0} = 0.1$ fm/c and are distributed homogenously within a space-time rapidity interval $η \in [- 3 : 3], d N / d η (τ_{0}) = 1748$ , which is depicted by the dotted straight line. The initial temperature is set to be $T (τ_{0}) = 2.6$ GeV. The radius of the tube is $R = 5$ fm. We consider $2 \leftrightarrow 2$ collisions with isotropic cross section and a constant total cross section of $σ_{22} = 10$ mb. The stochastic method is used in the simulation. The η bins of the cell configuration are set to be $Δ η_{c} = 0.2$ . No test particles ( $N_{test} = 1$ ) are used in the simulation. The distributions are obtained by an average over $10^{4}$ independent realizations.Reuse & Permissions
Figure 21
Space-time rapidity distributions at time $t = 0.2$ fm/c in tube calculations. The initial condition and collision cross section are the same as in Fig. 20. The stochastic method is employed in the simulations. The result showing structure with larger(smaller) period is obtained from the simulation with $t = 0.2$ ( $0.1$ ). In the simulation with $Δ η_{c} = 0.1$ we use two test particles per real particle in order to achieve the same statistics in each cell as that in the simulation with $Δ η_{c} = 0.2$ and $N_{test} = 1$ . The histogram, which is nearly constant, is obtained from the simulation with improved moving cell configuration of $Δ η_{c} = 0.2$ and $N_{test} = 1$ . The dotted line shows the initial distribution $d N / d η = 1748$ . All the distributions are received by an average over $10^{4}$ independent realizations.Reuse & Permissions
Figure 22
Time evolution of the particle density, energy density, and temperature extracted in the central space-time rapidity region $η \in [- 0.5 : 0.5]$ from simulations of one dimensional expansion in the lab and boosted frame of a tube. The geometrical method is employed in the simulations. The initial condition and collision cross section are the same as in Fig. 20. No test particles ( $N_{test} = 1$ ) are used. Only $2 \leftrightarrow 2$ processes are included. The results are obtained by an anerage over 20 independent realizations. The thin lines indicate time evolutions of the quantities in the hydrodynamical limit.Reuse & Permissions
Figure 23
Time evolution of the particle density, energy density and temperature extracted in the central space-time rapidity region $η \in [- 0.5 : 0.5]$ from simulations employing the stochastic method in the lab and boosted frame of a tube. The initial condition and collision cross section are the same as in Fig. 20. No test particles ( $N_{test} = 1$ ) are used. Only $2 \leftrightarrow 2$ processes are implemented. We apply the moving cell configuration with $Δ η_{c} = 0.2$ . The results are obtained by an average over 20 independent realizations.Reuse & Permissions
Figure 24
Collision rate in the central space-time rapidity region for various particle densities experienced during the expansion. The results are extracted from the same simulations performed for the extractions of $n (t)$ and $ε (t)$ in Figs. 22 and 23. The solid line shows the analytical expectation.Reuse & Permissions
Figure 25
Averaged difference in space-time rapidity of colliding particles extracted in the central space-time rapidity region for various particle densities experienced during the expansion. The results are extracted from the same simulations performed for the extractions of $ε (t)$ and $ε (t)$ in Figs. 22 and 23. The labeling of the symbols is identical to that of Fig. 24.Reuse & Permissions
Figure 26
Particle distributions versus space-time rapidity at the proper time $ε (t)$ and $1.0$ fm/c extracted from simulations employing the geometrical and stochastic method in the lab and boosted frame. The initial condition and collision cross section (and cell configuration) are the same as in Fig. 22 (and in Fig. 23). In order to compare the distributions in the same physical regions directly, we have shifted the distributions in the boosted frame by $- η_{0} = - 2$ . Except that the distributions extracted from the simulations in the boosted frame using the stochastic method are obtained by an average over ten independent realizations, all other distributions are obtained from 20 independent realizations. The thin solid lines indicate the initial distribution $d N / d η (τ_{0}) = 1748$ . The cut at $η = 4$ in the distributions at $τ = 1.0$ fm/c for the expansion in the boosted frame is due to the fact that the end time of the simulation in the boosted frame is $t^{'} = 210$ fm/c and thus particles with η being greater than 6 (or 4 after the shift) have smaller proper time than 1 fm/c.Reuse & Permissions
Figure 27
Particle distributions versus momentum rapidity at the proper time $τ = 0.2$ and $1.0$ fm/c extracted from simulations employing the geometrical and stochastic method in the lab and boosted frame. The results are obtained from the same simulations performed for the extractions of $d N / d η (τ)$ in Fig. 26. The thin solid curves indicate the initial distribution at $τ_{0} = 0.1$ fm/c.Reuse & Permissions
Figure 28
Distributions of the transverse momentum per unit rapidity at $y = 0$ at $τ = 1.0$ and $4.0$ fm/c (from upper to lower histogram) in a simulation employing the geometrical method in the lab frame. The initial condition and collision cross section are the same as in Fig. 22. The results are obtained by an average over 20 independent realizations. The solid lines show the analytical distributions (54) with the temperatures read off from Fig. 22.Reuse & Permissions
Figure 29
Distributions of the transverse momentum per unit rapidity at $y = 0$ at $τ = 0.2, 1.0$ , and $4.0$ fm/c (from upper to lowest histogram) in a simulation employing the stochastic method in the lab frame. The initial condition, collision cross section, and cell configuration are the same as in Fig. 23. The results are obtained by an average over 20 independent realizations. The solid lines show the analytical distributions (54) with the temperatures read off from Fig. 23.Reuse & Permissions
Figure 30
Proper time evolution of the transverse energy per unit momentum rapidity at $y = 0$ in the simulations employing the geometrical and stochastic method in the lab and boosted frame. The initial condition and collision cross section (and cell configuration) are the same as in Fig. 22 (and in Fig. 23). The results are obtained by an average over 20 independent realizations. The thin solid line shows the analytical evolution in the hydrodynamical limit.Reuse & Permissions
Figure 31
Relative frame dependence of the particle density, energy density, and temperature in the simulations employing the geometrical method. The initial condition and collision cross section are the same as in Fig. 22. The results are obtained by averaging $20, 2$ , and 20 independent realizations for increasing test particles $N_{test} = 1, 4$ , and 25, respectively.Reuse & Permissions
Figure 32
Comparison of the space-time rapidity distribution with $N_{test} = 25$ with the distribution without test particles at $τ = 0.2$ fm/c. The distributions are extracted from the simulations employing the geometrical method in the lab frame by averaging 20 independent realizations. The initial condition and the collision cross section are the same as in Fig. 22. The thin solid line indicates the initial distribution $τ = 0.2$ within $η \in [- 3 : 3]$ .Reuse & Permissions
Figure 33
Convergency of temperature in the simulations in the lab frame with increasing test particles. The initial condition and collision cross section (and cell configuration) are the same as in Fig. 22 (and in Fig. 23). The results in the simulations employing the geometrical method are obtained by averaging $20, 2, 20, 5, 5$ , and 5 independent realizations for $N_{test} = 1, 4, 25, 100, 400$ , and 900, respectively. The results in the simulations employing the stochastic method are obtained by averaging $20, 10, 2$ , and 1 independent realizations for $N_{test} = 1, 2, 4$ , and 8, respectively.Reuse & Permissions
Figure 34
Collision rate and averaged difference in space-time rapidity of colliding particles. The results are extracted in the central space-time rapidity region $η \in [- 0.5 : 0.5]$ for various particle densities experienced during the expansion. The simulations are the same as performed in the upper panel of Fig. 33 when discussing the convergency of the temperature.Reuse & Permissions
Figure 35
Time evolution of the particle density, energy density, and temperature extracted in the central space-time rapidity region $η \in [- 0.5 : 0.5]$ from the simulations employing the stochastic method in the lab and boosted frame. The initial condition and cell configuration are the same as in Fig. 23. No test particles ( $N_{test} = 1$ ) are used. $2 \leftrightarrow 2$ as well as $2 \leftrightarrow 3$ processes are included in the simulations. We consider isotropic collisions with constant cross sections of $σ_{22} = 5$ mb and $σ_{23} = 2.5$ mb. The results are obtained by an average over ten independent realizations. The thin solid lines indicate time evolutions in the hydrodynamical limit.Reuse & Permissions
Figure 36
Time evolution of fugacity extracted from the same simulations performed for the extraction of $n (t)$ and $ε (t)$ in Fig. 35.Reuse & Permissions
Figure 37
Collision rate in the central region for various particle densities experienced during the expansion. The results are extracted from the same simulations performed for the extractions of $n (t)$ and $ε (t)$ in the lab frame in Fig. 35. The solid squares, solid circles, and open circles depict, respectively, the collision rates for $2 \leftrightarrow 2, 2 \to 3$ , and $3 \to 2$ transitions. The solid and dotted line show the analytical expectations.Reuse & Permissions
Figure 38
Particle distributions versus space-time rapidity and momentum rapidity at the proper time $τ = 0.2$ and $1.0$ fm/c, extracted from the simulations employing the stochastic method in the lab and boosted frame. The initial condition, collision cross section, and cell configuration are the same as in Fig. 35. The distributions extracted in the lab(boosted) frame are obtained by averaging 20(6) independent realizations. The thin solid lines indicate the initial distributions at $τ_{0} = 0.1$ fm/c.Reuse & Permissions
Figure 39
Convergence of energy density in the simulations in the lab frame with increasing test particles. The cascade simulations are performed employing the stochastic algorithm. The dotted lines depict the results with $σ_{22} = 5$ mb and $σ_{22} = 5$ mb, while the thin solid lines depict the results with purely elastic collisions and $σ_{22} = 10$ mb. The results are obtained by averaging $20, 10, 2$ , and 1 independent realizations for $N_{test} = 1, 2, 4$ , and 8, respectively. The thin dashed lines show the hydrodynamical limit.Reuse & Permissions
Figure 40
Gluon number distribution versus space-time rapidity at the time $t = 0.2, 0.5, 1.0, 2.0, 3.0$ , and $4.0$ fm/c during the expansion in a real, fully 3D central Au+Au collision at the maximal RHIC energy.Reuse & Permissions
Figure 41
Gluon number distribution versus momentum rapidity at the time $t = 0.2, 0.5, 1.0, 2.0, 3.0$ , and $4.0$ fm/c during the expansion.Reuse & Permissions
Figure 42
Momentum rapidity distributions of the transverse energy (upper panel) and the total energy (lower panel) of gluons at the time $t = 0.2, 0.5, 1.0, 2.0, 3.0$ , and $4.0$ fm/c during the expansion.Reuse & Permissions
Figure 43
Time evolution of the gluon density and energy density in the central region: radial transverse extension $x_{T} < 1.5$ fm and $η \in [- 0.5 : 0.5]$ for a central Au+Au collision at the maximal RHIC energy. The dotted curves denote the ideal hydrodynamical limit with a fixed intercept at time $t = 0.5$ fm/c.Reuse & Permissions
Figure 44
Transverse momentum spectrum in the central region at different times ( $t = 0.2, 0.5, 1, 2, 3$ , and 4 fm/c from second upper to lowest histogram) during the expansion. The most-upper and bold-folded histogram with a lower cutoff at $t = 0.2, 0.5, 1, 2, 3$ GeV denotes the spectrum of the primary gluons (minijets).Reuse & Permissions
Figure 45
Time evolution of the momentum anisotropy extracted in the central region. The solid curve shows the result from the simulation with full dynamics, while the dashed curve shows the result from the simulation with only elastic scatterings. The dotted curve depicts the result from the simulation with isotropic elastic collisions and with (unrealistic) large cross section of $σ = 30$ mb.Reuse & Permissions
Figure 46
Time evolution of the effective temperature (upper panel) and the exponent describing the cooling of the system (lower panel) in the central region. The curves are arranged in the same way as in Fig. 45.Reuse & Permissions
Figure 47
Time evolution of the gluon fugacity extracted in the central region.Reuse & Permissions
Figure 48
Time evolution of the averaged cross section and the averaged transport cross section in the central region. The solid and dashed (short-dashed and short-dotted) curves depict the averaged cross sections (transport cross sections) for the $g g \to g g$ and $g g \to g g g$ processes, respectively.Reuse & Permissions
Figure 49
Angular distribution of the scattering processes $g g \to g g$ (solid curve) and $g g \to g g g$ for a representative situation during the gluon evolution. $θ_{3}$ denotes the scattering angle of the radiated gluon and its radiation partner has the angle $θ_{3}$ . The distributions are computed with the parameters $m_{D}^{2} / s = 0.05$ and $λ_{g} \sqrt{s} = 4$ extracted in the central region at an intermediate time during the evolution.Reuse & Permissions
Figure 50
Transverse momentum spectrum in the central space-time rapidity slice ( $η \in [- 0.5 : 0.5]$ and all gluons in the transverse plan are counted for) at different times ( $t = 0.2, 0.5, 1.0, 2.0, 3.0$ , and $4.0$ fm/c from second upper to lowest histogram). The most-upper and bold-folded histogram with a lower cutoff at $p_{T} = 2$ GeV denotes the spectrum of the primary gluons (minijets).Reuse & Permissions
Figure 51
Transverse momentum spectrum in the central region extracted from the simulation with only elastic collisions at different times. The most-upper and bold-folded histogram with a lower cutoff at $p_{T} = 2$ GeV denotes the spectrum of the primary gluons (minijets). According to the increase of the population of the soft gluons below 2 GeV, the other histrograms present the spectrum at times $0.2, 0.5, 1.0, 2.0, 3.0$ , and $4.0$ fm/c, respectively.Reuse & Permissions
Figure 52
Transverse momentum spectrum in the central region extracted from the simulation with isotropic elastic scatterings and a large cross section of $σ = 30$ mb at different times ( $t = 0.2, 0.5, 1.0, 2.0, 3.0$ , and $4.0$ fm/c from second upper to lowest histogram). The most-upper and bold-folded histogram with a lower cutoff at $p_{T} = 2$ GeV denotes the spectrum of the primary gluons (minijets).Reuse & Permissions
Figure 53
Time evolution of the transverse energy per unit momentum rapidity at midrapidity. The curves are arranged in the same way as in Fig. 45.Reuse & Permissions
Figure 54
Time evolution of the number (left panel) and energy (right panel) density extracted in the central region from the simulation with $dx = d y = 0.25$ fm and $N_{test} = 960$ by the dotted lines, compared with the results with the default settings $dx = d y = 0.5$ fm and $N_{test} = 240$ , depicted by the solid lines.Reuse & Permissions
Figure 55
Probability distribution of difference in “collision times” within the geometrical collision algorithm. In the calculations a thermal system is assumed and the cross section is set to be a constant.Reuse & Permissions

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