Examination of directed flow as a signature of the softest point of the equation of state in QCD matter

We analyze the directed flow of protons and pions in high-energy heavy-ion collisions in the incident energy range from $\sqrt{s_{NN}}=7.7$ to 27 GeV within a microscopic transport model. Standard hadronic transport approaches do not describe the collapse of directed flow below $\sqrt{s_{NN}}\simeq 20$ GeV. By contrast, a model that simulates effects of a softening of the equation of state descibes well the behavior of directed flow data recently obtained by the STAR Collaboration [Phys. Rev. Lett. 112, 162301 (2014)]. We give a detailed analysis of how directed flow is generated. Particularly, we found that softening of the effective equation of state at the overlapping region of two nuclei, i.e., the reaction stages where the system reaches high baryon density state, is needed to explain the observed collapse of proton directed flow within a hadronic transport approach.

Figure 1

Directed flows of protons and pions in mid-central Au+Au collisions (10–40%) at $\sqrt{s_{NN}}=7.7\text{–}27$ GeV from the JAM cascade mode (dashed lines) and the JAM cascade with attractive orbits (solid lines) in comparison with the STAR data [6].

Figure 2

Same as in Fig. 1, but for central collisions (0–10%).

Figure 3

Directed flows of protons (left) and pions (right) in mid-central Au+Au collisions (10-40)% at $\sqrt{s_{NN}}=11.5$ GeV. In the JAM calculations, momentum dependent hadronic mean-field potentials are included (JAM/MS). The dotted lines correspond to the result from the standard JAM/MS model, while the solid lines are for JAM/MS with attractive orbit results. Symbols show STAR data [6].

Figure 4

Effective EoS extracted from the time evolution of simulations in Au+Au collisions at $\sqrt{s_{NN}}=7.7$ and 27 GeV. Full (open) circles and full (open) diamonds represent the pressures $P$ from standard JAM (JAM with attractive orbits) at 7.7 and 27 GeV, respectively. Triangles and boxes represent the transverse part of the pressure $P_{\perp }$ for the JAM standard simulation at 7.7 and 27 GeV, respectively. The dashed and sold lines represent the EoS from hadron gas and the EoS with a first-order phase transition used in Ref. [45].

Figure 5

Time evolution of sign weighted $v_1^{*}$ integrated over midrapidity $\left|y\right|<1$ for baryons in semicentral Au+Au collisions for $\sqrt{s_{NN}}=7.7,11.5,19.6,27$ GeV from (a) standard JAM and (b) JAM with attractive orbits simulations. Time evolutions of normalized net-baryon density ${\rho }_B/{\rho }_0$ and energy density $e$ from the standard JAM calculations are shown in the panels (c) and (d), respectively. Net-baryon density and energy density are averaged over a cylindrical volume of transverse radius 3 fm and longitudinal distance of 1 fm centered at the origin. Those particles which have not interacted yet are not included in the calculations of $v_1^{*},{\rho }_B$, and $e$.

Figure 6

Directed flows of protons and pions in mid-central Au+Au collisions (10-40%). The lines correspond to the results from the JAM cascade model with only baryon-baryon collisions included.

Figure 7

Same as in Fig. 5, but JAM cascade simulations with only baryon-baryon collisions.

Figure 8

Rapidity distribution of directed flow $v_1$ for protons and pions in semicentral Au+Au collisions for $\sqrt{s_{NN}}=11.5$ GeV from JAM simulation in which attractive orbits are imposed at different time intervals; the solid line corresponds to the simulation with attractive orbits for time $t<1$ fm/$c$, the dashed line for $1<t<2$ fm/$c$, and the dotted line for $t>2$ fm/$c$.

Figure 9

Time evolution of sign weighted $v_1^{*}$ for baryons (upper panel) and effective EoS (lower panel) in Au+Au collisions at $\sqrt{s_{NN}}=11.5$ GeV from JAM simulations in which attractive orbits are imposed at different time intervals; the circles correspond to the simulation with attractive orbits for time $t<1$ fm/$c$, the squares for $1<t<2$ fm/$c$, and the triangles for $t>2$ fm/$c$.

Figure 10

Elliptic flows of charged hadrons in mid-central Au+Au collisions (10–40%). The dotted lines correspond to the result from the standard JAM cascade model, while the solid lines are for the JAM cascade with attractive orbit results. Symbols show STAR data [48].

Figure 11

Equation of state implemented into JAM simulations. The sold line represents the first-order phase transition EoS, and the dotted line represents the EoS from lattice QCD [53]. The circles and squares are the results obtained by the JAM simulations in semicentral Au+Au collisions at $\sqrt{s_{NN}}=11.5$ GeV, which are almost same as the inputs. Effective EoS obtained from JAM standard simulation is also displayed by the triangles.

Figure 12

Rapidity distribution of directed flow $v_1$ for protons and pions in semicentral Au+Au collisions for $\sqrt{s_{NN}}=11.5$ GeV from JAM simulations with different EoS. The solid line presents the JAM result with the EoS with first-order phase transition, and the dashed line presents the JAM result with the crossover EoS. The dotted line is for the standard JAM result.