Event-by-event simulation of the three-dimensional hydrodynamic evolution from flux tube initial conditions in ultrarelativistic heavy ion collisions

K. Werner, Iu. Karpenko, T. Pierog, M. Bleicher, and K. Mikhailov

Phys. Rev. C 82, 044904 – Published 22 October 2010

Abstract

We present a sophisticated treatment of the hydrodynamic evolution of ultrarelativistic heavy ion collisions, based on the following features: initial conditions obtained from a flux tube approach, compatible with the string model and the color glass condensate picture; an event-by-event procedure, taking into the account the highly irregular space structure of single events, being experimentally visible via so-called ridge structures in two-particle correlations; the use of an efficient code for solving the hydrodynamic equations in $3 + 1$ dimensions, including the conservation of baryon number, strangeness, and electric charge; the employment of a realistic equation of state, compatible with lattice gauge results; the use of a complete hadron resonance table, making our calculations compatible with the results from statistical models; and a hadronic cascade procedure after hadronization from the thermal matter at an early time.

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Received 12 April 2010

DOI:https://doi.org/10.1103/PhysRevC.82.044904

Authors & Affiliations

K. Werner¹, Iu. Karpenko^1,2, T. Pierog³, M. Bleicher⁴, and K. Mikhailov⁵

¹SUBATECH, University of Nantes – IN2P3/CNRS– EMN, Nantes, France
²Bogolyubov Institute for Theoretical Physics, Kiev 143, 03680, Ukraine
³Forschungszentrum Karlsruhe, Institut fuer Kernphysik, Karlsruhe, Germany
⁴Frankfurt Institute for Advanced Studies (FIAS), Johann Wolfgang Goethe Universitaet, Frankfurt am Main, Germany
⁵Institute for Theoretical and Experimental Physics, Moscow 117218, Russia

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Vol. 82, Iss. 4 — October 2010

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Images

Figure 1
Macroscopic flux tubes (three in this example) made out of many individual ones of variable length.Reuse & Permissions
Figure 2
The energy density over $T^{4}$ as a function of the temperature $T$ . The dotted line indicates the “hadronization temperature” (i.e., the end of the thermal phase) when “matter” is transformed into hadrons.Reuse & Permissions
Figure 3
Particle ratios (hadron yields to $π^{+}$ yields) from our model calculations (thick horizontal line) as compared to the statistical model [10] (thin horizontal line) and to data [45, 46, 47] (points).Reuse & Permissions
Figure 4
Elementary interaction in the EPOS model.Reuse & Permissions
Figure 5
Elastic (upper plot) and inelastic (lower plot) “rescattering” of a ladder parton.Reuse & Permissions
Figure 6
Flux tube (string) at a given proper time. The picture is schematic in the sense that the string extends well into the transverse dimension, correctly taken into account in the calculations. The quantity $X$ is a four-vector.Reuse & Permissions
Figure 7
Flux tube with transverse kink in $I R^{3}$ space. The kink leads to transversely moving string regions (transverse arrow).Reuse & Permissions
Figure 8
Broken flux tube with transverse kink in $I R^{3}$ space. The string segments close to the kink give rise to transversely moving hadrons, constituting a jet (arrows).Reuse & Permissions
Figure 9
String segment at a given proper time. The picture is schematic in the sense that the string extends well into the transverse dimension, correctly taken into account in the calculations.Reuse & Permissions
Figure 10
Energy density versus temperature for our equation of state X3F (full line), compared to the lattice data [67] (points) and some other EoS choices, see text.Reuse & Permissions
Figure 11
Pressure versus temperature, for our equation of state X3F (full line), compared to the lattice data [67] (points) and some other EoS choices, see text.Reuse & Permissions
Figure 12
Dihadron $Δ η - Δ ϕ$ correlation in a central Au-Au collision at 200 GeV, as obtained from an event-by-event treatment of the hydrodynamical evolution based on random flux tube initial conditions. Trigger particles have transverse momenta between 3 and 4 GeV/c and associated particles have transverse momenta between 2 GeV/c and the $p_{t}$ of the trigger.Reuse & Permissions
Figure 13
Initial energy density in a central Au-Au collision at 200 GeV at a space-time rapidity $η_{s} = 0$ (upper) and $η_{s} = 1.5$ (lower).Reuse & Permissions
Figure 14
Energy density (upper panel) and radial flow velocity (lower panel) at a proper time $τ = 2.6$ fm/c, at a space-time rapidity $η_{s} = 0$ (left) and $η_{s} = 1.5$ (right).Reuse & Permissions
Figure 15
Energy density (upper panel) and radial flow velocity (lower panel) at a proper time $τ = 4.6$ fm/c, at a space-time rapidity $η_{s} = 0$ (left) and $η_{s} = 1.5$ (right).Reuse & Permissions
Figure 16
Schematic view of the translational invariance of the initial energy density (a), leading to a corresponding invariance of the transverse flow (b). We use the term “invariance” in the sense of a similarity transform: same shape, but different magnitude. The magnitude of the energy density at large $η_{s}$ is, of course, smaller than the one at $η_{s} = 0$ .Reuse & Permissions
Figure 17
Projection of the positions of nucleon-nucleon scattering to the transverse ( $x, y$ ) plane, from a simulation of a semiperipheral ( $b = 8$ fm/c) Au-Au event at 200 GeV.Reuse & Permissions
Figure 18
Schematic view of the projection of the positions of nucleon-nucleon scattering to the transverse ( $x, y$ ) plane, which defines “possible transverse positions” of the flux tubes, indicated by the thin lines. The actual flux tubes fluctuate concerning their longitudinal positions; a possible realization is shown by the thick lines.Reuse & Permissions
Figure 19
Centrality dependence of the ratio of $v_{2}$ over eccentricity. Points are data [74], the different curves refer to the full calculation: hydro and cascade (full line), only elastic hadronic scatterings (dotted), and no hadronic cascade at all (dashed). The thin solid line—above all others—refers to the hydrodynamic calculation till final freeze-out at 130 MeV.Reuse & Permissions
Figure 20
Centrality dependence of the ratio of $v_{2}$ over eccentricity, for a full calculation, hydro and hadronic cascade, for a (nonrealistic) first-order transition EoS (dashed-dotted line) compared to the crossover EoS, the default case (full line, same as the one in Fig. 19). Points are data [74].Reuse & Permissions
Figure 21
Pseudorapidity distributions of the elliptical flow $v_{2}$ for minimum bias events (upper left) and different centrality classes in Au-Au collisions at 200 GeV. Points are data [79], the different curves refer to the full calculation: hydro and cascade (full thick line), only elastic hadronic scatterings (dotted), no hadronic cascade at all (dashed), and hydrodynamic calculation till final freeze-out at 130 MeV (thin line).Reuse & Permissions
Figure 22
The transverse momentum dependence of $v_{2}$ for pions (circles, full lines), kaons (squares, dashed lines), and protons (triangles, dotted lines) for minimum bias events in minimum bias Au-Au collisions at 200 GeV. The symbols refer to data [80, 81], the lines to our full calculations.Reuse & Permissions
Figure 23
The transverse momentum dependence of $v_{2}$ for pions (circles, full lines), kaons (squares, dashed lines), and protons (triangles, dotted lines) for the 20–60 $%$ most central events in Au-Au collisions at 200 GeV. The symbols refer to data [82], the lines to our full calculations.Reuse & Permissions
Figure 24
Initial energy density as a function of the radius $r$ for azimuthal angles $ϕ = 0$ and $ϕ = π / 2$ , from six randomly chosen flux tube initial conditions (full thin line: $ϕ = 0$ , dotted thin line: $ϕ = π / 2$ ) and from Glauber initial conditions (full line: $ϕ = 0$ , dashed line: $ϕ = π / 2$ ), for a semiperipheral Au-Au collision.Reuse & Permissions
Figure 25
Initial energy density as a function of the radius $r$ for azimuthal angles $ϕ = 0$ and $ϕ = π / 2$ , from six randomly chosen flux tube initial conditions (full thin line: $ϕ = 0$ , dotted thin line: $ϕ = π / 2$ ) and from Glauber initial conditions (full line: $ϕ = 0$ , dashed line: $ϕ = π / 2$ ), for a semiperipheral Au-Au collision.Reuse & Permissions
Figure 26
Production of pions in Au-Au collisions at 200 GeV. Upper panel: transverse momentum spectra for central collisions at different rapidities (from top to bottom: 0, $2$ , $3$ ). The lower curves are scaled by factors of 1/2 and 1/4, for better visibility. Middle panels: transverse momentum (mass) distributions at rapidity zero for different centrality classes: from top to bottom: the 0–5 $%$ , the 20–30 $%$ , and the 40–50 $%$ most central collisions. Lower panel: the centrality dependence of the integrated yields for charged particles and pions. The symbols refer to data [85, 86, 87, 88], the full lines to our full calculations, the dotted lines to the calculations without hadronic cascade.Reuse & Permissions
Figure 27
Same as Fig. 26, but for kaons.Reuse & Permissions
Figure 28
Production of $λ$ ’s (left) and anti- $λ$ ’s (right) in Au-Au collisions at 200 GeV. Upper panel: transverse momentum distributions at rapidity zero for different centrality classes: from top to bottom: the 0–5 $%$ , the 20–30 $%$ , and the 40–50 $%$ most central collisions. The lower curves are scaled by factors of 1/2, 1/4, and 1/8 for better visibility. Lower panel: the centrality dependence of the integrated yields. The symbols refer to data [47], the full lines to our full calculations, the dotted lines to the calculations without hadronic cascade. The thin line refers to a hydrodynamic calculation till final freeze-out at 130 MeV.Reuse & Permissions
Figure 29
Same as Fig. 28, but for $Ξ$ and $Ξ̄$ .Reuse & Permissions
Figure 30
Same as Fig. 29, but comparing the calculation without hadronic cascade (dotted) with the one with only elastic hadronic rescattering (full thin line).Reuse & Permissions
Figure 31
Transverse momentum spectra of protons (left) and antiprotons (right) in Au-Au collisions at 200 GeV. Upper panel: spectra for central collisions at different rapidities (from top to bottom: 0, $2$ , $3$ ). The lower curves are scaled by factors of 1/2 and 1/4, for better visibility. Middle and lower panels: transverse momentum (mass) distributions at rapidity zero for different centrality classes: from top to bottom: the 0–5 $%$ , the 20–30 $%$ , and the 40–50 $%$ most central collisions. The symbols refer to data [47, 85, 87], the full lines to our full calculations, the dotted lines to the calculations without hadronic cascade.Reuse & Permissions
Figure 32
Femtoscopic radii $R_{out}$ , $R_{side}$ , and $R_{long}$ , as well as $λ$ , for $π^{+}$ – $π^{+}$ pairs as a function of $m_{T}$ for different centralities (0–5 $%$ most central, 5–10 $%$ most central, and so on). The full lines are the full calculations (including hadronic cascade), the stars data [98].Reuse & Permissions
Figure 33
The source functions for $π^{+}$ – $π^{+}$ pairs as obtained from our simulations, for three different centralities (0–5 $%$ most central, 10–20 $%$ most central, and 30–50 $%$ most central), representing the distribution of the space separation of the emission points of the pairs in the “out”-“side”-“long” coordinate system in the longitudinal comoving frame. The different curves per plot correspond to the different $k_{T}$ bins, see text.Reuse & Permissions
Figure 34
The mean transverse momentum component $p_{x}$ of $π^{+}$ as a function of the $x$ coordinate of the emission point. Also shown is the number of produced $π^{+}$ as a function of $x$ . The different curves refer to different centralities: $0 - 5 % =$ full line, $10 - 20 % =$ dashed, $30 - 50 % =$ dotted.Reuse & Permissions
Figure 35
Radial flow effect on $m_{t}$ dependence of femtoscopic radii.Reuse & Permissions
Figure 36
Same as Fig. 34, but for the calculation without hadronic cascade.Reuse & Permissions
Figure 37
Same as Fig. 34, but for the full thermal scenario (freeze-out at 130 MeV).Reuse & Permissions
Figure 38
The source function for $π^{+}$ – $π^{+}$ pairs considering the “out” coordinate for the three scenarios: complete calculation, with hadronic cascade (full line), calculation without hadronic cascade and therefore final hadronization at 166 MeV (dashed), and full thermal scenario with hydrodynamic evolution till the final freeze-out at 130 MeV (dotted).Reuse & Permissions
Figure 39
Same as Fig. 33, but for a calculation without hadronic cascade.Reuse & Permissions
Figure 40
Same as Fig. 33, but for a calculation where the hydro evolution is continued till freeze-out at 130 MeV (being the final freeze-out, no cascade afterward).Reuse & Permissions
Figure 41
Same as Fig. 32, but the calculations are done without hadronic cascade (full line) or with a hydrodynamic evolution through the hadronic phase with freeze-out at 130 MeV (dashed line).Reuse & Permissions

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