Investigating the collision energy dependence of $\eta /s$ in the beam energy scan at the BNL Relativistic Heavy Ion Collider using Bayesian statistics

We determine the probability distributions of the shear viscosity over the entropy density ratio $\eta /s$ in the quark-gluon plasma formed in Au + Au collisions at $\sqrt{s_{NN}}=19.6,39$, and $62.4\mathrm{GeV}$, using Bayesian inference and Gaussian process emulators for a model-to-data statistical analysis that probes the full input parameter space of a transport + viscous hydrodynamics hybrid model. We find the most likely value of $\eta /s$ to be larger at smaller $\sqrt{s_{NN}}$, although the uncertainties still allow for a constant value between 0.10 and 0.15 for the investigated collision energy range.

Figure 1

The ratio of the Gaussian process prediction over the model result for positively charged pion yields, for the six test points at $\sqrt{s_{NN}}=19.6\mathrm{GeV}$. Top panel: Untrained emulator. Bottom panel: After emulator training.

Figure 2

Representation of the posterior probability distribution of the input parameters at $\sqrt{s_{NN}}=19.6\mathrm{GeV}$. Panels on a diagonal show one-dimensional projections of the posterior for each parameter axis, while the off-diagonal panels display projections on the different 2-D planes of the input parameter space. Dashed red lines indicate median values of the one-dimensional distributions.

Figure 3

Charged particle pseudorapidity distribution at $\sqrt{s_{NN}}=19.6\mathrm{GeV}$ for (0–6)% centrality (top) and (15–25)% centrality (bottom) from the simulations using the posterior peak values for model parameters. Blue boxes with whiskers represent the range of Gaussian process emulator predictions for 100 samples from the posterior. Black open squares are emulator predictions for the model output at distribution median values (red triangles). PHOBOS data from Ref. [28].

Figure 4

Centrality dependence of average charged-particle pseudorapidity $\langle \eta \rangle$ for $\eta >0.0$ at $\sqrt{s_{NN}}=19.6\mathrm{GeV}$. Blue box-whisker plots are GP estimates for samples from the posterior distribution. Black open squares are emulator predictions for the model output at distribution median values (red triangles). Dashed gray boxes represent the range of values from the training data. Experimental values are estimated from PHOBOS data [28].

Figure 5

Multiplicities of ${\pi }^{+}$ and $K^{+}$ at midrapidity as a function of centrality at $\sqrt{s_{NN}}=19.6\mathrm{GeV}$. Experimental data from Ref. [12].

Figure 6

Mean transverse momentum of ${\pi }^{+}$ (red up triangles) and $K^{+}$ (orange down triangles) as a function of centrality at $\sqrt{s_{NN}}=19.6\mathrm{GeV}$. Experimental data from Ref. [12].

Figure 7

Transverse momentum spectra of ${\pi }^{+}$ (red up triangles) and $K^{+}$ (orange down triangles) for (0–5)% centrality at $\sqrt{s_{NN}}=19.6\mathrm{GeV}$. Experimental data from Ref. [12].

Figure 8

$R_{\mathrm{out}}, R_{\mathrm{side}}$, and $R_{\mathrm{long}}$ of charged $\pi$ at $\langle k_T\rangle \approx 0.22$ GeV/$c$ at (0–5)% and (20–30)% centralities (red up triangles and orange down triangles, respectively) at $\sqrt{s_{NN}}=19.6\mathrm{GeV}$. Experimental data from Ref. [11].

Figure 9

$R_{\mathrm{out}}, R_{\mathrm{side}}$, and $R_{\mathrm{long}}$ of charged $\pi$ averaged over 0.15 $\mathrm{GeV}/c<k_T<$ 0.6 $\mathrm{GeV}/c$ at (0–5)% and (20–30)% centralities at $\sqrt{s_{NN}}=19.6\mathrm{GeV}$. Experimental data from Ref. [11].

Figure 10

Centrality dependence of charged particle $v_2\left\{2\right\}$ at $\sqrt{s_{NN}}=19.6\mathrm{GeV}$. Experimental data from Ref. [32].

Figure 11

Transverse momentum spectrum of $\mathrm{\Omega }$ at $\sqrt{s_{NN}}=19.6\mathrm{GeV}$. Experimental data from Ref. [33].

Figure 12

Representation of the posterior probability distribution of the input parameters at $\sqrt{s_{NN}}=39\mathrm{GeV}$. See caption of Fig. 2 for details.

Figure 13

Centrality dependence of positive pion and kaon yields at midrapidity at $\sqrt{s_{NN}}=39\mathrm{GeV}$. Experimental data from Ref. [12].

Figure 14

Centrality dependence of mean transverse momentum of ${\pi }^{+}$ (red up triangles) and $K^{+}$ (orange down triangles) at $\sqrt{s_{NN}}=39\mathrm{GeV}$. Experimental data from Ref. [12].

Figure 15

Transverse momentum of ${\pi }^{+}$ (red up triangles) and $K^{+}$ (orange down triangles) for (0–5)% centrality at $\sqrt{s_{NN}}=39\mathrm{GeV}$. Experimental data from Ref. [12].

Figure 16

$R_{\mathrm{out}}, R_{\mathrm{side}}$, and $R_{\mathrm{long}}$ of charged $\pi$ at $\langle k_T\rangle \approx 0.22$ GeV/$c$ at (0–5)% and (20–30)% centralities (red up triangles and orange down triangles, respectively) at $\sqrt{s_{NN}}=39\mathrm{GeV}$. Experimental data from Ref. [11].

Figure 17

$R_{\mathrm{out}}, R_{\mathrm{side}}$, and $R_{\mathrm{long}}$ of charged $\pi$ averaged over 0.15 $\mathrm{GeV}/c<k_T<$ 0.6 $\mathrm{GeV}/c$ at (0–5)% and (20–30)% centrality at $\sqrt{s_{NN}}=39\mathrm{GeV}$. Experimental data from Ref. [11].

Figure 18

Centrality dependence of charged particle $v_2\left\{2\right\}$ at $\sqrt{s_{NN}}=39\mathrm{GeV}$. Experimental data from Ref. [32].

Figure 19

Transverse momentum spectrum of $\mathrm{\Omega }$ at $\sqrt{s_{NN}}=39\mathrm{GeV}$. Experimental data from Ref. [33].

Figure 20

Representation of the posterior probability distribution of the input parameters at $\sqrt{s_{NN}}=62.4\mathrm{GeV}$. See caption of Fig. 2 for details.

Figure 21

Centrality dependence of charged particle multiplicity in $\left|\eta \right|<0.5$ at $\sqrt{s_{NN}}=62.4\mathrm{GeV}$. Experimental data from STAR [30].

Figure 22

Charged particle pseudorapidity distribution at $\sqrt{s_{NN}}=62.4\mathrm{GeV}$ for (0–3)% centrality (top) and (20–25)% centrality (bottom). PHOBOS data from Ref. [28].

Figure 23

Centrality dependence of average charged particle pseudorapidity $\langle \eta \rangle$ for $\eta >0.0$ at $\sqrt{s_{NN}}=62.4\mathrm{GeV}$. Experimental values are estimated from PHOBOS data [28].

Figure 24

Centrality dependence of kaon/pion ratio at $\sqrt{s_{NN}}=62.4\mathrm{GeV}$. Experimental data from STAR [30].

Figure 25

The centrality dependence of mean transverse momentum of ${\pi }^{-}$ (red up triangles) and $K^{+}$ (orange down triangles) at $\sqrt{s_{NN}}=62.4\mathrm{GeV}$. STAR data from Ref. [12].

Figure 26

The transverse momentum distributions of ${\pi }^{-}$ (red up triangles) and $K^{+}$ (orange down triangles) for (0–15)% centrality at $\sqrt{s_{NN}}=62.4\mathrm{GeV}$. PHOBOS data from Ref. [36].

Figure 27

$R_{\mathrm{out}}, R_{\mathrm{side}}$, and $R_{\mathrm{long}}$ of charged $\pi$ at $\langle k_T\rangle \approx 0.22$ GeV/$c$ at (0–5)% and (20–30)% centrality (red up triangles and orange down triangles, respectively) at $\sqrt{s_{NN}}=62.4\mathrm{GeV}$. Experimental data from Ref. [11].

Figure 28

$R_{\mathrm{out}}, R_{\mathrm{side}}$, and $R_{\mathrm{long}}$ of charged $\pi$ averaged over 0.15 $\mathrm{GeV}/c<k_T<$ 0.6 $\mathrm{GeV}/c$ at (0–5)% and (20–30)% centralities at $\sqrt{s_{NN}}=62.4\mathrm{GeV}$. Experimental data from Ref. [11].

Figure 29

Transverse momentum spectrum of $\mathrm{\Omega }$ (top) and proton (bottom) at $\sqrt{s_{NN}}=62.4\mathrm{GeV}$. Note that proton data were not used in the data calibration process. STAR data from Ref. [37] and PHOBOS data from Ref. [36].

Figure 30

Collision energy dependence of the effective shear viscosity over entropy density ratio $\eta /s$ during the hydrodynamical evolution. Error bars represent 90% confidence range around the median value (filled circles). Open symbols indicate the peak position of the distribution.

Figure 31

Collision energy dependence of the hydro starting time ${\tau }_0$. Error bars represent 90% confidence range around the median value (filled stars). Open symbols indicate the peak position of the distribution.

Figure 32

Collision energy dependence of the smearing factor $W_{\mathrm{trans}}$. Error bars represent 90% confidence range around the median value (filled triangles). Open symbols indicate the peak position of the distribution.

Figure 33

Collision energy dependence of the smearing factor $W_{\mathrm{long}}$. Error bars represent 90% confidence range around the median value (filled triangles). Open symbols indicate the peak position of the distribution.

Figure 34

Collision energy dependence of the switching energy density ${\epsilon }_{\mathrm{SW}}$. Error bars represent 90% confidence range around the median value (filled squares). Open symbols indicate the peak position of the distribution.

Figure 35

Collision energy dependence of the switching energy density ${\epsilon }_{\mathrm{SW}}$, if charged particle pseudorapidity distribution is excluded from the Bayesian analysis (red), compared to the result from full analysis (in gray).