Mean-field effects on collective flow in high-energy heavy-ion collisions at 2–158A GeV energies

Collective flows in heavy-ion collisions from AGS [$(2–11)A$ GeV] to SPS [$(40,158)A$ GeV] energies are investigated in a nonequilibrium transport model with the nuclear mean field (MF). Sideward $\langle p_x\rangle$, directed $v_1$, and elliptic flows $v_2$ are systematically studied with different assumptions for the nuclear equation of state (EOS). We find that the momentum dependence on the nuclear MF is important for the understanding of the proton collective flows at AGS and SPS energies. Calculated results with momentum-dependent MF qualitatively reproduce the experimental data of proton sideward, directed, and elliptic flows in a incident energy range of $(2–158)A$ GeV.

Figure 1

Left, momentum dependence of the single-particle potentials, Eq. (6), for momentum-dependent hard (MH) and soft (MS) and momentum-independent hard (H) and soft (S) potentials are compared with the real part of the global Dirac optical potential [38]. Right, density dependence of total energy per nucleon in Eq. (4) for momentum-dependent (MH, MS) and -independent (H, S) potentials.

Figure 2

Sideward flows $\langle p_x\rangle$ of protons in mid-central Au+Au collisions at $(2–8)A$ GeV are compared with the AGS-E895 data [20]. Curves show the calculated results of cascade with momentum-dependent hard or soft mean field (MH or MS, left-hand panels), cascade with momentum-independent mean field (H or S, right-hand panels), and cascade without mean field (CS). The experimental data are shown in both the left and the right panels.

Figure 3

Comparison of calculated sideward flow $\langle p_x\rangle$ of protons (left) and pions (right) in $11A$ GeV Au+Au mid-central collisions to AGS-E877 data [18]. The meaning of the curves is the same as in Fig. 2.

Figure 4

Transverse momentum dependence of the elliptic flow $v_2$ for protons in Au+Au mid-central collisions at $(2,4,6)A$ GeV are compared with AGS-E895 data [20]. The meaning of the curves is the same as in Fig. 2.

Figure 5

Proton directed flows $v_1$ as a function of rapidity in mid-central Pb+Pb collisions at $E_{\mathrm{inc}}=40A$ GeV (left panel) and $158A$ GeV (right panel) in comparison with SPS-NA49 data [24]. Curves show the calculated results of cascade with momentum-dependent hard or soft mean field (MH or MS), cascade with momentum-independent mean field (H or S), and cascade without mean field (CS).

Figure 6

Proton directed flows $v_1$ as a function of transverse momentum in mid-central Pb+Pb collisions at $E_{\mathrm{inc}}=40A$ GeV (left panel) and $158A$ GeV (right panel) are compared with SPS-NA49 data [24]. The meaning of the curves is the same as Fig. 5.

Figure 7

Pion directed flows $v_1$ as a function of rapidity in mid-central Pb+Pb collisions at $E_{\mathrm{inc}}=40A$ GeV (left panel) and $158A$ GeV (right panel) are compared with SPS-NA49 data [24]. The meaning of the curves is the same as Fig. 5.

Figure 8

Pion directed flows $v_1$ as a function of transverse momentum for $0<y<1.5$ in Pb+Pb collisions at $E_{\mathrm{inc}}=40A$ GeV (left panel) and $158A$ GeV (right panel) are compared with SPS-NA49 data [24]. The meaning of the curves is the same as Fig. 5.

Figure 9

Proton elliptic flows $v_2$ as a function of rapidity in mid-central Pb+Pb collisions at $40A$ GeV (left) and $158A$ GeV (right). Lines show the calculated results of cascade with momentum dependent hard or soft mean field (MH or MS), cascade with momentum-independent mean field (H or S), and cascade without mean field (CS). Experimental data are taken from SPS-NA49 [24].

Figure 10

Proton elliptic flows $v_2$ as a function of transverse momentum. The meaning of the curves is the same as Fig. 9.

Figure 11

Pion elliptic flows $v_2$ as a function of rapidity. The meaning of the curves is the same as Fig. 9.

Figure 12

Pion elliptic flow $v_2$ as a function of transverse momentum. The meaning of the curves is the same as Fig. 9.

Figure 13

Incident energy dependence of proton elliptic flow at mid-rapidities in mid-central heavy-ion collisions from 1A GeV to 158A GeV. Dotted, bold-solid, dashed, and dotted-dashed and thin-solid lines show the results of cascade with MH, MS, H, S mean-field, and cascade without mean-field (CS), respectively. The experimental data are taken from LBL-EOS, AGS-E895, E877 from Ref. 19, and SPS-NA49 from Ref. 24.

Figure 14

Sideward (directed) flows of AGS (left) and SPS (right) energies are compared with different assumptions for the mean fields. The momentum-dependent soft (MS) mean-field has been used in these calculations. The first term in the parenthesis means that all baryons (B) or only nucleons (N) are included for the MF. The second term (0.05/0.1/0.5) denotes simulation time step size dt (fm/c).

Figure 15

Elliptic flows are compared with different assumptions for the mean fields. The meaning of the curves is the same as Fig. 14.