Effective shear and bulk viscosities for anisotropic flow

We evaluate the viscous damping of anisotropic flow in heavy-ion collisions for arbitrary temperature-dependent shear and bulk viscosities. We show that the damping is solely determined by effective shear and bulk viscosities, which are weighted averages over the temperature. We determine the relevant weights for nucleus-nucleus collisions at $\sqrt{s_{NN}}=5.02$ TeV and 200 GeV, corresponding to the maximum Large Hadron Collider (LHC) and Relativistic Heavy Ion Collider (RHIC) energies, by running ideal and viscous hydrodynamic simulations. The effective shear viscosity is driven by temperatures below 210 MeV at RHIC and below 280 MeV at the LHC, with the largest contributions coming from the lowest temperatures, just above freeze-out. The effective bulk viscosity is driven by somewhat higher temperatures, corresponding to earlier stages of the collision. We show that at a fixed collision energy, the effective viscosity is independent of centrality and system size, to the same extent as the mean transverse momentum of outgoing hadrons. The variation of viscous damping is determined by Reynolds number scaling.

Figure 1

Effect of a temperature-dependent shear ($\eta$) or bulk ($\zeta$) viscosity on $v_2$ and $v_3$ in central Pb $+$ Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV. Shear corresponds to the left panels (a) and (c), and bulk to the right panels (b) and (d). The upper panels display the $(\eta /s)$ and $(\zeta /s)$ profiles used in our calculation. They are defined by Eq. (6), with $\sigma =16$ MeV, ${(\eta /s)}_{max}=0.04$ (a) and ${(\zeta /s)}_{max}=0.02$ (b), and each curve corresponds to a different value of $T_0$. The vertical lines indicate the freeze-out temperature $T_f=156.5$ MeV. The symbols in panels (c) and (d) display the corresponding values of ${\mathrm{\Delta }}_n$, defined by Eq. (1), as a function of $T_0$. Lines are fits using Eqs. (2) and (8).

Figure 2

Initial entropy density profile used in our ideal hydrodynamic calculation, defined by Eq. (7), with parameters tuned to match Pb $+$ Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV. (a) 0–5% centrality window (Secs. 3, 4, 5): $s_0=438{\mathrm{fm}}^{-3}$, $R_0=4.18$ fm, ${\varepsilon }_2=0.085$, ${\varepsilon }_3=0.075$. (b) 20–30% centrality window (Sec. 6): $s_0=337{\mathrm{fm}}^{-3}$, $R_0=2.97$ fm, ${\varepsilon }_2=0.35$, ${\varepsilon }_3=0.12$. The profile is identical for the viscous hydrodynamic calculation, except for the overall normalization (see text).

Figure 3

Weights $w_2^{\zeta }\left(T\right)$ (full line), $w_3^{\zeta }\left(T\right)$ (dashed line), $w_2^{\eta }\left(T\right)$ (dot-dashed line), and $w_3^{\eta }\left(T\right)$ (dotted line), defining the effective viscosities (3) at $\sqrt{s_{NN}}=5.02$ TeV. They are obtained by fitting the hydrodynamic results in Fig. 1 using Eq. (8). The shaded boxes to the left are meant to represent the discrete part $w_f\delta (T-T_f)$: Their area is $|w_f|$ (see numbers in Table 1), and their vertical placement is arbitrary.

Figure 4

(a) Various $(\eta /s)\left(T\right)$ profiles which are used to test Eq. (2). They are defined by $(\eta /s)\left(T\right)=C$ (full lines), $(\eta /s)\left(T\right)=CT/T_0$ (dashed lines), or $(\eta /s)\left(T\right)=C{T}_0/T$ (dash-dotted lines), with $T_0=225$ MeV, and $C=0.08$, 0.16, or 0.24. Panels (b) and (c) display ${\mathrm{\Delta }}_n$ defined by Eq. (1), as a function of ${\mathrm{\Delta }}_n$ defined by Eq. (4). Each symbol (squares, circles, triangles) in panels (b) and (c) corresponds to one of the profiles in panel (a). Full symbols correspond to the $0\text{–}5\%$ centrality window (Sec. 5), and full lines are the diagonals $y=x$. Gray diamonds represent the prediction for $20\text{–}30\%$ centrality from Reynolds number scaling, defined by Eq. (14). Open symbols correspond to the value calculated numerically in the $20\text{–}30\%$ centrality window (Sec. 6).

Figure 5

Same as Fig. 4 for bulk viscosity alone, and shear $+$ bulk. The $(\zeta /s)\left(T\right)$ profiles are represented in panel (a). They are defined by $(\zeta /s)\left(T\right)=C$ (full lines), $(\zeta /s)\left(T\right)=CT/T_0$ (dashed lines), with $T_0=225$ MeV, and $C=0.04$, 0.08, or 0.16. The dash-dotted line is the Duke parametrization [53]. Squares, circles, and triangles in panels (b) and (c) correspond to the profiles in panel (a). The stars represent a calculation done with a constant shear viscosity over entropy $\eta /s=0.04$ (we choose a small value so that nonlinear effects are negligible), on top of the Duke parametrization of bulk viscosity. As in Fig. 4, gray diamonds represent the prediction for $20\text{–}30\%$ centrality from Reynolds number scaling, defined by Eq. (14).

Figure 6

Same as Fig. 1, for the collision energy $\sqrt{s_{NN}}=200$ GeV.

Figure 7

Same as Fig. 3 for $\sqrt{s_{NN}}=200$ GeV.