Examination of the directed flow puzzle in heavy-ion collisions

Recent STAR data for the directed flow of protons, antiprotons, and charged pions obtained within the beam energy scan program are analyzed within the parton-hadron-string-dynamics (PHSD and HSD) transport models and a 3-fluid hydrodynamics approach. Both versions of the kinetic approach, HSD and PHSD, are used to clarify the role of partonic degrees of freedom. The PHSD results, simulating a partonic phase and its coexistence with a hadronic one, are roughly consistent with data. The hydrodynamic results are obtained for two equations of state (EoS), a pure hadronic EoS and an EoS with a crossover type transition. The latter case is favored by the STAR experimental data. Special attention is paid to the description of antiproton directed flow based on the balance of $p\overline{p}$ annihilation and the inverse processes for $p\overline{p}$ pair creation from multimeson interactions. Generally, the semiqualitative agreement between the measured data and the model results supports the idea of a crossover type of quark-hadron transition that softens the nuclear EoS but shows no indication of a first-order phase transition.

Figure 1

Particle abundance at midrapidity calculated for central collisions $b=$ 2 fm in the HSD (dashed lines) and PHSD (solid lines) models. The experimental data are from a compilation of Ref. [51] complemented by recent data from the STAR Collaboration [52] and the latest update of the compilation of NA49 results [53, 54].

Figure 2

The directed flow $v_1\left(y\right)$ for protons, antiprotons as well as negative pions from 10 to 40% central $\mathrm{Au}+\mathrm{Au}$ collisions at different collision energies from $\sqrt{s_{\mathit{NN}}}=$ 7.7 to 200 GeV from HSD (dashed lines) and PHSD (solid lines). Experimental data are from the STAR Collaboration [39].

Figure 3

The beam energy dependence of the directed flow slope at midrapidity for protons, antiprotons, and charged pions from semicentral $\mathrm{Au}+\mathrm{Au}$ collisions. The shaded band corresponds to the UrQMD results as cited in Ref. [39]. The experimental data are from the STAR Collaboration [39] along with results of prior experiments using comparable cuts [12, 59].

Figure 4

Excitation function of the antiproton slope calculated in the PHSD model with (dotted line) and without (solid line) including fluctuations of the reaction plane. The dotted line corresponds to a use of the cubic equation (2) for the slope calculation.

Figure 5

Snapshots of the baryon energy density distribution in the PHSD model at the time $t=$ 3 and 6 fm/$c$ for $\mathrm{Au}+\mathrm{Au}$ collisions and $\sqrt{s_{\mathit{NN}}}=$ 11.5 GeV. The energy density scale is given on the right side in GeV/fm${}^3$. The solid curves display parton density levels for 0.6 and 0.01 partons/fm${}^3$. The arrows show the local velocity of baryonic matter (in relative units).

Figure 6

Evolution of the average momentum projection on the reaction plane for protons, pions, and quarks at the shifted rapidity $y=0.25\pm 0.05$. The results are given for $8.7\times {10}^4$ PHSD events of $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{\mathit{NN}}}=$ 11.5 GeV.

Figure 7

Particle abundance at midrapidity calculated for central collisions $(b=$ 2 fm) in the 3FD model with an EoS for a pure hadronic phase (dashed lines) and for the case of a crossover transition (solid lines). The experimental data are the same as in Fig. 1.

Figure 8

Rapidity distributions of the directed flow for protons, antiprotons, and positive and negative pions from 10 to 40% central $\mathrm{Au}+\mathrm{Au}$ collisions at different collision energies calculated within the 3FD model. The experimental data are from the STAR Collaboration [39]. The dashed lines correspond to a hadronic EoS while the solid lines stand for a crossover transition.

Figure 9

The beam energy dependence of the directed flow slope at midrapidity for protons. The lines are calculated within the 3FD model with a hadronic (dotted lines) and a crossover (solid lines) EoS. For comparison the results of calculations in other collective models are taken from Ref. [40]. The experimental data are from the STAR measurements [39] and prior experiments with comparable acceptance cuts [12, 59, 73]. Note the different scales as compared to Fig. 3.

Figure 10

Rapidity distribution of nucleons (dashed lines) and all hadrons (solid lines) for the average over many events and for a single event in PHSD. The shaded strips correspond to the spectator region.

Figure 11

Particle distribution in center-of-mass rapidity fluctuation in PHSD. The smooth curve is the Gaussian approximation (7).