Temperature dependence of transport coefficients of QCD in high-energy heavy-ion collisions

Using our developed new relativistic viscous hydrodynamics code, we investigate the temperature dependence of shear and bulk viscosities from comparison with the ALICE data: single-particle spectra and collective flows of Pb+Pb $\sqrt{s_{NN}}=2.76$ TeV collisions at the Large Hadron Collider. We find that from the comprehensive analyses of centrality dependence of single-particle spectra and collective flows we can extract detailed information on the quark-gluon plasma bulk property, without the information being smeared by the final state interactions.

Figure 1

The temperature dependence of the shear viscosity to entropy density ratio $\eta /s$ (a) and the bulk viscosity to entropy density ratio $\zeta /s$ (b).

Figure 2

The pseudorapidity distributions of charged hadrons in 0–$5\%$, 10–$20\%$, 30–$40\%$, and 50–$60\%$ centralities together with the ALICE data (open circles) [69, 70]. In the top panel, the pseudorapidity distributions were obtained in the case of $\eta /s=0.08$, 0.17, and 0.24 at vanishing bulk viscosity (a). The middle panel shows the computational results with shear and bulk viscosities (b). In the bottom panel, temperature dependence of $\eta /s$ is included (c). See the text for details.

Figure 3

The $p_T$ distributions for ${\pi }^{+}$ (a), $K^{+}$ (b), and $p$ (c) in 0–5%, 10–20 %, 30–40%, and 50–60% centralities, together with the ALICE data (the open circles) [71]. The orange dashed line, the black solid line, and the blue dashed-dotted line stand for $\eta /s=0.08$, 0.17, and 0.24, respectively.

Figure 4

The time evolution of the spatial averaged radial flow $v_T=\sqrt{v_x^2+v_y^2}$ of fluid cells whose temperatures are $T>150$ MeV at ${\eta }_s=0$ in 0–5% centrality. The values are taken from the average over 20 events. The solid line, the dashed line, and the dashed-dotted line stand for $\eta /s=0.17$ with $N=110, \eta /s=0.08$ with $N=110$, and $\eta /s=0.08$ with $N=116$, respectively.

Figure 5

The longitudinal flow $v_{{\eta }_s}$ as a function of ${\eta }_s$ ($x=y=0$ fm) at $\tau =7$ fm in 0–5% centrality. The solid line stands for $\eta /s=0.17$ and the dashed line stands for $\eta /s=0.08$. Both of them are computed with the normalization $N=110$ for one event.

Figure 6

The elliptic and triangular flows of charged hadrons as a function of $p_T$ in 0–$5\%$ [(a) for $v_2$ and (c) for $v_3$] and 30–$40\%$ centralities [(b) for $v_2$ and (d) for $v_3$], together with the ALICE data (the open circles) [72]. The orange dashed line, black solid line, and the blue dashed-dotted line stand for $\eta /s=0.08$, 0.17, and 0.24, respectively.

Figure 7

The $p_T$ distributions for ${\pi }^{+}$ (a), $K^{+}$ (b), and $p$ (c) in 0–$5\%$, 10–$20\%$, 30–$40\%$, and 50–$60\%$ centralities, together with the ALICE data (the open circles) [71]. The black dashed line, the green solid line, and the yellow dashed-dotted line stand for vanishing bulk viscosity, $b=40$, and $b=60$, respectively.

Figure 8

The time evolution of the spatial averaged radial flow $v_T=\sqrt{v_x^2+v_y^2}$ of fluid cells whose temperatures are $T>150$ MeV at ${\eta }_s=0$ in 0–5% centrality. The values are taken from the average over 20 events. The red solid line, the green dashed line, and the blue dashed-dotted line stand for $\zeta /s=0$ with $N=110, b=40$ with $N=110$, and $b=40$ with $N=94$, respectively. For all cases we fix the shear viscosity to $\eta /s=0.17$.

Figure 9

The $v_x$ (a) and temperature (b) distributions as a function of $x$ fm ($y=0$ fm and ${\eta }_s=0$) at $\tau =7$ fm in 0–5% centrality. The red solid line stands for $\eta /s=0.17$ with vanishing bulk viscosity and the green dashed line stands for $\eta /s=0.17$ with $b=40$. Both of them are computed with the normalization $N=110$ for one event.

Figure 10

The profile functions in 0–5% centrality. The red solid line, the green dashed line, and the blue dashed-dotted line stand for $\zeta /s=0$ with $N=110, b=40$ with $N=110$, and $b=40$ with $N=94$, respectively. All calculations are done with $\eta /s=0.17$. The values are taken the average over ten events.

Figure 11

The elliptic and triangular flows of charged hadrons as a function of $p_T$ in 0–$5\%$ [(a) for $v_2$ and (c) for $v_3$] and 30–$40\%$ [(b) for $v_2$ and (d) for $v_3$] centralities, together with the ALICE data (the open circles) [72]. The black dashed line, the green solid line, and the dashed-dotted line stand for $\eta /s=0.17$ without bulk viscosity, $\eta /s=0.17$ with $b=40$, and $\eta /s=0.17$ with $b=60$, respectively.

Figure 12

The $p_T$ distributions for ${\pi }^{+}$ (a), $K^{+}$ (b), and $p$ (c) in 0–$5\%$, 10–$20\%$, 30–$40\%$, and 50–$60\%$ centralities together with the ALICE data (the open circles) [71]. The red dashed line, the blue dashed-dotted line, and the purple dotted line stand for (f) $c_1=10,c_2=0.7$, (g) $c_1=0,c_2=0.7$, and (h) $c_1=10,c_2=0.7$, respectively. See the details in the text.

Figure 13

The mean $p_T$ of ${\pi }^{+}$ (open circles), $K^{+}$ (open triangles), and $p$ (open squares) as a function of centrality, together with the ALICE data [71]. We evaluate the mean $p_T$ for the whole region of calculated $p_T, p_T>0$ GeV.

Figure 14

The time evolution of the spatial averaged radial flow $v_T=\sqrt{v_x^2+v_y^2}$ of fluid cells whose temperatures are $T>150$ MeV at ${\eta }_s=0$ in 0–5% centrality, for cases (d), (f), (g), and (h). The values are taken from the average over 20 events.

Figure 15

The elliptic and triangular flows of charged hadrons as a function of $p_T$ in 0–$5\%$ [(a) for $v_2$ and (c) for $v_3$] and 30–$40\%$ [(b) for $v_2$ and (d) for $v_3$] centralities, together with the ALICE data (the open circles) [72]. The red dashed line, the blue dashed, and purple dotted line stand for (f) $c_1=10,c_2=0.7$, (g) $c_1=0,c_2=0.7$, and (h) $c_1=10,c_2=0.7$, respectively.

Figure 16

The integrated $v_2$ and $v_3$ of charged hadrons as a function of centrality in the cases of (b) $\eta /s=0.17$ (the black solid line), (d) $\eta /s=0.17,b=40$ (the green solid line), (f) $c_1=10,c_2=0.7$ (the red dashed line), (g) $c_1=0,c_2=0.7$ (the blue dashed-dotted line), and (h) $c_1=10,c_2=0$ (the purple dotted line). The open circles and squares stand for the $v_2$ and $v_3$ of the ALICE data, respectively [72].

Figure 17

Calculated results of integrated $v_2$ as a function of pseudorapidity in centralities 0–5% (a), 10–20% (b), 30–40% (c), and 50–60% (d), in the cases of (b) $\eta /s=0.17$ (the black solid line), (d) $\eta /s=0.17,b=40$ (the green solid line), (f) $c_1=10,c_2=0.7$ (the red dashed line), (g) $c_1=0,c_2=0.7$ (the blue dashed-dotted line), and (h) $c_1=10,c_2=0$ (the purple dotted line), together with the ALICE data (the open circles) [75].

Figure 18

Calculated results of integrated $v_3$ as a function of pseudorapidity in centralities 0–$5\%$ (a), 10–20% (b), 30–$40\%$ (c), and 50–60% (d), in the cases of (b) $\eta /s=0.17$ (the black solid line) (d) $\eta /s=0.17,b=40$ (the green solid line), (f) $c_1=10,c_2=0.7$ (the red dashed line), (g) $c_1=0,c_2=0.7$ (the blue dashed-dotted line), and (h) $c_1=10,c_2=0$ (the purple dotted line), together with the ALICE data [75].

Figure 19

The $p_T$ distributions for ${\pi }^{+}, K^{+}$, and $p$ in 0–$5\%$ (a) and 30–$40\%$ (b) centralities, together with the ALICE data (the open circles) [71]. The red solid line, the green dashed line, and the blue dashed-dotted line stand for computed $p_T$ spectra with rescattering, without rescattering (only resonance decay), and without UrQMD (just from hydrodynamic evolution), respectively.

Figure 20

The mean $p_T$ for ${\pi }^{+}, K^{+}$, and $p$ with rescattering (the red solid circles), without rescattering (the green solid triangles), and without UrQMD (the blue solid squares) in 0–5% (a) and 30–40% centralities (b), together with the ALICE data (the open circles) [71]. We evaluate the mean $p_T$ in the region of $p_T>0$ GeV.

Figure 21

The elliptic flow of charged hadrons with rescattering (the red solid line), without rescattering (the blue dashed line), and without UrQMD (the blue dashed-dotted line) as a function of $p_T$ in 0–5% (a) and 30–40% (b) centralities, together with the ALICE data [72].

Figure 22

The integrated $v_2$ as a function of pseudorapidity in centralities 0–$5\%$ (a) and 30–$40\%$ (b), together with the ALICE data [75]. The red solid line stands for $v_2$ with rescattering, the green dashed line stands for $v_2$ without rescattering, and the blue dashed-dotted line stands for $v_2$ without UrQMD.