Estimation of the shear viscosity at finite net-baryon density from $A+A$ collision data at $\sqrt{s_{\mathit{\text{NN}}}}=7.7–200$ GeV

Hybrid approaches based on relativistic hydrodynamics and transport theory have been successfully applied for many years for the dynamical description of heavy-ion collisions at ultrarelativistic energies. In this work a new viscous hybrid model employing the hadron transport approach UrQMD for the early and late nonequilibrium stages of the reaction, and 3+1 dimensional viscous hydrodynamics for the hot and dense quark-gluon plasma stage, is introduced. This approach includes the equation of motion for finite baryon number and employs an equation of state with finite net-baryon density to allow for calculations in a large range of beam energies. The parameter space of the model is explored and constrained by comparison with the experimental data for bulk observables from Super Proton Synchrotron and the phase I beam energy scan at Relativistic Heavy Ion Collider. The favored parameter values depend on energy but allow extraction of the effective value of the shear viscosity coefficient over entropy density ratio $\eta /s$ in the fluid phase for the whole energy region under investigation. The estimated value of $\eta /s$ increases with decreasing collision energy, which may indicate that $\eta /s$ of the quark-gluon plasma depends on baryochemical potential ${\mu }_B$.

Figure 1

The earliest possible starting time of the hydrodynamic evolution as a function of $\sqrt{s_{\mathit{\text{NN}}}}$ according to Eq. (1).

Figure 2

An example of a fluctuating (single-event) initial energy density profile in the transverse plane at $\eta =0$. The profile is obtained with $R_{\perp }=R_{\eta }=1$ fm Gaussian smearing and corresponds to a 20–30% Au-Au collision at $\sqrt{s_{\mathit{\text{NN}}}}=39$ GeV.

Figure 3

Transverse momentum spectra of negative pions, positive and negative kaons in $E_{\mathrm{lab}}=40$ A GeV ($\sqrt{s_{\mathit{\text{NN}}}}=8.8$ GeV) central Pb-Pb collisions. The experimental data from the NA49 Collaboration [29] are compared to the hybrid model calculations with $\eta /s=0$ (dashed lines) and $\eta /s=0.2$ (solid lines) in the hydrodynamic phase. The results from UrQMD model with no intermediate hydro phase (dubbed as “pure UrQMD”) are shown with dotted lines.

Figure 4

Pion and kaon $dN/dy$ in $E_{\mathrm{lab}}=40$ A GeV ($\sqrt{s_{\mathit{\text{NN}}}}=8.8$ GeV) central Pb-Pb collisions (top) and charged hadron $dN/d\eta$ distributions at $\sqrt{s_{\mathit{\text{NN}}}}=19.6$, 39, 62.4, and 200 GeV central Au-Au collisions (bottom). The experimental data from the NA49 [29] and the PHOBOS Collaborations [34] are compared to the hybrid model calculations with $\eta /s=0$ (dashed lines) and $\eta /s=0.2$ (solid lines) in the hydrodynamic phase.

Figure 5

$p_T$ integrated ($0.2<p_T<2.0$ GeV and $\left|\eta \right|<1$) elliptic ($v_2$) and triangular ($v_3$) flows of all charged hadrons in 20–30% central Au-Au collisions as a function of collision energy, calculated with the event-plane method. The elliptic and triangular flow data is from the STAR Collaboration [10, 40]. The solid line depicts the calculation with $\eta /s=0.2$, the dashed line shows the calculation with $\eta /s=0$, whereas the dotted line corresponds to the “pure” UrQMD calculation with no intermediate hydrodynamic stage.

Figure 6

Parameter dependence of the total yield at midrapidity (top) and the effective temperature of pion, kaon, and proton $p_T$ spectra (bottom) in 0–5% central Au-Au collisions at $\sqrt{s_{\mathit{\text{NN}}}}=19.6$ GeV.

Figure 7

Parameter dependence of $p_T$ integrated elliptic flow $v_2$ of charged hadrons in 20–30% central Au-Au collisions at $\sqrt{s_{\mathit{\text{NN}}}}=19.6$ GeV. The experimental value of the elliptic flow is shown with a solid red (gray) line for comparison.

Figure 8

Pseudorapidity distributions of charged hadrons (top) in Au-Au collisions at $\sqrt{s_{\mathit{\text{NN}}}}=19.6$, 39, 62.4, and 200 GeV energies, and rapidity distributions of identified hadrons in Pb-Pb collisions at $E_{\mathrm{lab}}=158$ and 40 A GeV ($\sqrt{s_{\mathit{\text{NN}}}}=17.6$ and 8.8 GeV) energies (middle and bottom panels, respectively). The calculations were done using the collision energy-dependent parameters listed in Table 2. The data are from the PHOBOS [34] and the NA49 [29] Collaborations.

Figure 9

$p_T$ spectra of identified hadrons in Au-Au collisions at $\sqrt{s_{\mathit{\text{NN}}}}=62.4$ GeV energy (top) and in Pb-Pb collisions at $E_{\mathrm{lab}}=158$ and 40 A GeV ($\sqrt{s_{\mathit{\text{NN}}}}=17.6$ and 8.8 GeV) energies (middle and bottom panels, respectively). The model calculations were carried out using the collision energy-dependent parameters listed in Table 2, and the data are from the PHOBOS and NA49 Collaborations [29, 35, 46].

Figure 10

$p_T$ integrated elliptic and triangular flow coefficients $v_2$ and $v_3$ as a function of collision energy. Both the experimental and calculated coefficients were evaluated using the event-plane method. The calculation was done using the collision energy-dependent parameters listed in Table 2, and the data are from the STAR Collaboration [10, 40].

Figure 11

$p_T$ integrated elliptic flow coefficient $v_2$ in $\sqrt{s_{\mathit{\text{NN}}}}=39$ GeV Au-Au collisions as function of centrality. Both the experimental and calculated $v_2$ was evaluated using the event plane method. The calculation was done using the collision energy-dependent parameters listed in Table 2, and the data are from the STAR collaboration [10].

Figure 12

Effective values of shear viscosity over entropy density $\eta /s$ used to describe the experimental data at different collision energies as shown in Table 2. The green (gray) band represents an estimate of uncertainty in $\eta /s$ resulting from the allowed variation of model parameters around their optimal values.

Figure 13

$p_T$ integrated elliptic and triangular flow coefficients $v_2$ and $v_3$ as a function of collision energy. Solid red (gray) line represents the results from Fig. 10 obtained using collision energy-dependent $\eta /s$. Dashed (blue) and dotted (green) lines correspond to collision independent $\eta T/w=0.08$ and $\eta /s=0.08$, respectively. In all three cases the other model parameters were taken to depend on the collision energy as shown in Table 2. The experimental data are from the STAR Collaboration [10, 40].