Relativistic heavy-ion collisions within three-fluid hydrodynamics: Hadronic scenario^*

A three-fluid hydrodynamic model for simulating relativistic heavy-ion collisions is introduced. Along with two baryon-rich fluids, the new model considers the time-delayed evolution of a third, baryon-free (i.e., with zero net baryonic charge) fluid of newly produced particles. Its evolution is delayed because of a formation time τ, during which the baryon-free fluid neither thermalizes nor interacts with the baryon-rich fluids. After the formation it starts to interact with the baryon-rich fluids and quickly gets thermalized. Within this model with pure hadronic equation of state, a systematic analysis of various observables at incident energies between few and about 160A GeV has been done as well as a comparison with results of transport models. We have succeeded in reasonably reproducing a great body of experimental data in the incident energy range of $E_{\mathrm{lab}}\simeq$ (1–160)A GeV. The list includes proton and pion rapidity distributions, proton transverse-mass spectra, rapidity distributions of Λ and $\overline{\Lambda }$ hyperons, elliptic flow of protons and pions (with the exception of proton $v_2$ at 40A GeV), multiplicities of pions, positive kaons, ϕ mesons, hyperons, and antihyperons, including multistrange particles. This agreement is achieved on the expense of substantial enhancement of the interflow friction as compared to that estimated proceeding from hadronic free cross sections. However, we have also found out certain problems. The calculated yield of $K^{-}$ is approximately higher than that in the experiment by a factor of 1.5. We have also failed to describe directed transverse flow of protons and pion at $E_{\mathrm{lab}}\ge 40A$ GeV. This failure apparently indicates that the used EOS is too hard and thereby leaves room for a phase transition.

Figure 1

(Color online) Baryon-density dependence of the pressure at $T=0$. Shaded region is the constraint [47] derived from experimental data. Solid and dashed lines present the pressure $P(n_B,T=0)$ and that for the ideal hadronic gas, $P_{\text{gas}}(n_B,T=0)$ [cf. Eq. (35)], respectively.

Figure 2

(Color online) Fitted friction enhancement for the purely hadronic EOS [46] used in simulations, see Eq. (21).

Figure 3

(Color online) Rapidity spectra of protons (upper panel) and $(p-\overline{p})$ (lower panel) from central heavy-ion collisions. Bold solid lines and bold dashed line (upper panel) correspond to three-fluid hydrodynamic calculations at different impact parameters. Thin dashed-dotted and short-dashed lines are the appropriate UrQMD and HSD model results [48], respectively. Full symbols display measured experimental points, whereas the open ones are those reflected with respect to the midrapidity point. Experimental data are taken by Collaborations E802 [49], E877 [50], E917 [51], and NA49 [52, 53, 54]. The percentage shows the fraction of the total reaction cross section, corresponding to experimental selection of central events.

Figure 4

(Color online) Proton rapidity spectra at SIS and AGS energies for various impact parameters. Solid lines correspond to three-fluid calculations. The dashed line is that for deuteron production. For comparison the one-fluid result is shown by the dotted-dashed line at $E_{\text{lab}}=1.06A$ GeV. Experimental points at different energies are taken from [55] at $E_{\text{lab}}=1.06$A GeV, [56] at 2A and 4A GeV, and [51] at 6A, 8A, and 10.5A GeV. The percentage indicates the fraction of the total reaction cross section, corresponding to experimentally selected events.

Figure 5

(Color online) $(p-\overline{p})$ rapidity distribution from central Pb+Pb collisions at SPS energies. Solid lines are three-fluid calculations, whereas dashed-dotted and dashed lines are the corresponding predictions of the HSD and UrQMD models [48], respectively. NA49 experimental data are from Ref. 54 for $E_{\text{lab}}=40A$ and 80A GeV and from Refs. 52, 53, 54 for $E_{\text{lab}}=158A$ GeV. The percentage indicates the fraction of the total reaction cross section, corresponding to experimentally selected events.

Figure 6

(Color online) Transverse mass distributions of protons from Au(8A GeV)+Au collisions at $b=2$ fm and different rapidities in the c.m. system (left panel) and at the midrapidity and different impact parameters (right panel), cf. the right panel of Fig. 4. For clarity of representation, every next data set and the corresponding curve (from top to bottom) is multiplied by the additional factor 0.1. Experimental data are from Ref. 51.

Figure 7

(Color online) Transverse mass distributions of protons at the midrapidity from central Pb+Pb collisions at incident energies $E_{\text{lab}}=158A$, 80A, and 40A GeV and impact parameter $b=2.5$ fm, cf. Fig. 5. NA49 experimental data are taken from [52] (squares), [57] (open circles), and [57] (full circles).

Figure 8

(Color online) Pion rapidity spectra from central Au(10.5A GeV)+Au collisions. Bold solid and dashed lines represent three-fluid calculations at $b=2.5$ and 2.0 fm, respectively. Thin dashed-dotted and short-dashed lines correspond ${\pi }^{+}$ spectra from kinetic simulations within HSD and UrQMD models [48]. The ${\pi }^{-}$ (full squares and triangles) and ${\pi }^{+}$ (full circles) data are measured by E895 [56] and E877 [50] Collaborations. The percentage indicates the fraction of the total reaction cross section, corresponding to experimentally selected events. Open symbols are obtained by reflecting the full ones with respect to the midrapidity.

Figure 9

(Color online) Pion rapidity spectra from central collisions at the AGS (left panel) and SPS (right panel) energies. Experimental points are taken from [56] (AGS energies) and [58] (SPS energies). Notation is the same as in Fig. 8.

Figure 10

(Color online) Rapidity spectra of $\Lambda +{\Sigma }^0$ hyperons (left panel) and $\overline{\Lambda }+\overline{{\Sigma }^0}$ antihyperons (right panel) from central ($b=2.5$ fm) Pb+Pb collisions. Solid lines represent results of the three-fluid model. Contributions from the fireball fluid are shown by dashed lines. Preliminary experimental data are taken from Ref. 59. The percentage indicates the fraction of the total reaction cross section, corresponding to experimentally selected events.

Figure 11

(Color online) Antiproton rapidity spectrum from central Pb(158A GeV)+Pb collisions. Notation is the same as in Fig. 10. Experimental data are taken from Ref. 60.

Figure 12

(Color online) Transverse-momentum flow of nucleons and protons as a function of rapidity for semicentral Au+Au collisions at $E_{\text{lab}}=10.5$A GeV. The three-fluid calculations are presented for $b=4$ fm (solid line) and $b=6$ fm (dashed line). The data for protons (circles) and all baryons (triangles) are taken from Ref. 63. Full symbols correspond to measured data, open ones are those reflected with respect to the midrapidity point.

Figure 13

(Color online) Directed flows of protons (upper panels) and charged pions (lower panels) in semicentral Pb+Pb collisions at $E_{\text{lab}}=158A$ (right panels) and 40A (left panels) GeV as functions of rapidity. The three-fluid calculations are presented for $b=5.6$ fm (solid line) and $b=4$ fm (dashed line). The dot-dashed lines show contributions of the fireball fluid. Experimental data are taken from Ref. 64. These data were obtained by two different experimental procedures: the standard one [$v_{1,2}(\mathit{st})$] and the method of n-particle correlations [$v_{1,2}(n)$]. Full symbols correspond to measured data, whereas the open symbols are those reflected with respect to the mid rapidity. Updated data of the NA49 Collaboration [65][$v_{1,2}(\mathit{st})-98$] with acceptance 0.05 $<p_T<0.35$ GeV/c for pions and 0.6 $<p_T<2.0$ Gev/c for protons, are also shown.

Figure 14

(Color online) The same as in Fig. 13 but for the elliptic flow.

Figure 15

(Color online) Multiplicities for central Au+Au (at AGS energies) and Pb+Pb (at SPS energies) collisions as functions of the incident energy. Open triangles connected by solid lines represent the three-fluid results. The experimental data (full circles) are taken from Refs. 58, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85 for SPS energies and from Refs. 86, 87, 88, 89, 90 for AGS energies.

Figure 16

(Color online) The same as in Fig. 15 but for multiplicities of multistrange hyperons. The experimental data are taken from Refs. 80, 94, 95: full circles for hyperons and full squares for antihyperons.

Figure 17

(Color online) Temporal evolution of average baryon (upper panel) and energy (lower panel) densities for central Pb+Pb collisions within three-fluid model. The time is counted in the c.m. frame of the colliding nuclei.

Figure 18

(Color online) Time evolution of the entropy per baryon number in Pb+Pb central ($b=2.5$ fm) collisions at various incident energies $E_{\text{lab}}$. The total entropy is displayed by solid lines, while the entropy of baryon-rich subsystem—by dashed lines.

Figure 19

(Color online) Dynamical trajectories in the ($T,{\mu }_B$) plane for central Pb+Pb collisions ($b=2.5$ fm) at various incident energies. Numbers near the dynamical trajectories indicate the evolution time instants in the c.m. frame of the colliding nuclei. Bold parts of trajectories are related to approximately thermalized baryon-rich subsystem, whereas the thin ones—to yet nonequilibrium evolution. The light gray shaded region corresponds to the boundary of the phase transition from the hadronic phase to the QGP, as it was estimated in Ref. 97. Dotted line is the “experimental” freeze-out curve fitted to observed multiplicities in the approximation of the ideal gas model [98] under condition that the energy per hadron is 1 GeV. The star symbol is the critical end point calculated in Ref. 6.

Figure 20

(Color online) Invariant four-volume corresponding to conditions on $n_B^{\text{(noneq.)}}$ (solid lines) and on $n_B^{\text{(eq.)}}$ (long-dashed lines) for central Pb+Pb collisions as function of the incident energy. The dotted line on the lower panel is the four-volume $\Delta t\times \pi R^2\times 2R/{\gamma }_{c.m.}$ with $R=4$ fm, $\Delta t=3$ fm/c (see the text).