Radial and elliptic flow in Pb $+$ Pb collisions at energies available at the CERN Large Hadron Collider from viscous hydrodynamics

A comprehensive viscous hydrodynamic fit of spectra and elliptic flow for charged hadrons and identified pions and protons from $\mathrm{Au}+\mathrm{Au}$ collisions of all centralities measured at the Relativistic Heavy Ion Collider (RHIC) is performed and used as the basis for predicting the analogous observables for $\mathrm{Pb}+\mathrm{Pb}$ collisions at the Large Hadron Collider (LHC) at $\sqrt{s}=2.76$ and 5.5$A$ TeV. Comparison with recent measurements of the elliptic flow of charged hadrons by the ALICE experiment shows that the model slightly overpredicts the data if the same (constant) specific shear viscosity $\eta /s$ is assumed at both collision energies. In spite of differences in our assumptions for the equation of state, the freeze-out temperature, the chemical composition at freeze-out, and the starting time for the hydrodynamic evolution, our results agree remarkably well with those of Luzum [Phys. Rev. C 83, 044911 (2011)], indicating robustness of the hydrodynamic model extrapolations. Future measurements of the centrality and transverse momentum dependence of spectra and elliptic flow for identified hadrons predicted here will further test the model and shed light on possible variations of the quark-gluon transport coefficients between RHIC and LHC energies.

Figure 1

Centrality dependence of the charged hadron multiplicity per unit pseudorapidity, $d{N}_{\mathrm{ch}}/d\eta /(N_{\mathrm{part}}/2)$, as a function of $N_{\mathrm{part}}$, in $200A$ GeV $\mathrm{Au}+\mathrm{Au}$ collisions at the RHIC (bottom panel) and in $(2.76–5.5)A$ TeV $\mathrm{Pb}+\mathrm{Pb}$ collisions at the LHC (top panel). Experimental data are from the PHOBOS Collaboration [34] for $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s}=200A$ GeV and from the ALICE Collaboration [39] for $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s}=2.76A$ TeV. The lines are from the MC-KLN model (see text). For $\mathrm{Au}+\mathrm{Au}$ at the RHIC the MC-KLN model was normalized to the measured multiplicity in the 0%–5% centrality bin; at the LHC, the lines bounding the shaded region were normalized to $d{N}_{\mathrm{ch}}/d\eta =1548$ and 1972 (or $d{N}_{\mathrm{ch}}/dy=1800$ and 2280) at 0%–5% centrality, respectively.

Figure 2

$p_T$ spectra of all charged hadrons (a), positive pions (b), and protons (c) for 200$A$ GeV $\mathrm{Au}+\mathrm{Au}$ collisions of different centralities as indicated. The symbols show data from the STAR ([41, 43, 44], $\times$) and PHENIX ([42, 45], $+$) experiments. The lines are results from the viscous hydrodynamic model for constant $\eta /s=0.20$ and MC-KLN initial conditions. See text for other model parameters.

Figure 3

Differential elliptic flow $v_2\left(p_T\right)$ for charged hadrons from 200$A$ GeV $\mathrm{Au}+\mathrm{Au}$ collisions of different centralities as indicated. Open symbols are experimental data from the STAR experiment for $v_2\left\{4\right\}\left(p_T\right)$ [46, 47]; lines with filled symbols of the same shape are the corresponding hydrodynamic fits with the same model as in Fig. 1. For the $40\%$–$50\%$ centrality bin, data and theory are vertically offset by 0.1 for better visibility.

Figure 4

$p_T$ spectra of all charged hadrons, positive pions, and protons for minimum bias 200$A$ GeV $\mathrm{Au}+\mathrm{Au}$ (thin red lines and data points) and $(2.76–5.5)A$ TeV $\mathrm{Pb}+\mathrm{Pb}$ collisions (black lines with shaded area). The RHIC data are from the PHENIX experiment [45]. The shaded bands for the LHC predictions are limited at the bottom (top) by lines for $\sqrt{s}=2.76\left(5.5\right)A$ TeV, corresponding to $d{N}_{\mathrm{ch}}/dy=1800$ (2280) [$d{N}_{\mathrm{ch}}/d\eta =1548\left(1972\right)$]. In the calculations the same constant $\eta /s=0.2$ is assumed at all shown collision energies.

Figure 5

$p_T$-integrated elliptic flow of charged hadrons for 200$A$ GeV $\mathrm{Au}+\mathrm{Au}$ collisions at the RHIC (open symbols: STAR data [46, 47]; lower red line: the result from viscous hydrodynamics) and for 2.76 $A$ TeV $\mathrm{Pb}+\mathrm{Pb}$ collisions at the LHC (filled symbols: ALICE data [1]; upper magenta line: the viscous hydrodynamic prediction). In both experiment and theory the differential elliptic flow $v_2\left(p_T\right)$ (see Figs. 3 and 7) was integrated over the range $0.15<p_T<2$ GeV/$c$ for $\mathrm{Au}+\mathrm{Au}$ at the RHIC and over $p_T>0.2$ GeV/$c$ for $\mathrm{Pb}+\mathrm{Pb}$ at the LHC.

Figure 6

$p_T$ spectra per unit pseudorapidity for charged hadrons (a) and per unit rapidity for pions (b) and protons (c) for $\mathrm{Pb}+\mathrm{Pb}$ collisions at the LHC. The lower and upper ends of the shaded bands represent viscous hydrodynamic predictions for $\sqrt{s}=2.76$ and $5.5A$ TeV (corresponding to $d{N}_{\mathrm{ch}}/d\eta =1548$ and 1972, or $d{N}_{\mathrm{ch}}/dy=1800$ and 2280), respectively. Experimental data in panel (a) are from the ALICE experiment [50].

Figure 7

Differential elliptic flow $v_2\left(p_T\right)$ for unidentified charged hadrons (a) and identified pions (b) and protons (c), for $\mathrm{Pb}+\mathrm{Pb}$ collisions of different centralities at the LHC. Experimental data for charged hadron $v_2\left(p_T\right)$, denoted by solid symbols, are from the ALICE experiment [1]; they should be compared with theoretical lines carrying open symbols of the same shape and color. The shaded bands show the variation of the hydrodynamic predictions with collision energy between $\sqrt{s}=2.76$ and 5.5 $A$ TeV (corresponding to $d{N}_{\mathrm{ch}}/dy=1800$ and 2280, respectively). The lines corresponding to the lower collision energy ($\sqrt{s}=2.76$ $A$ TeV) define the lower end of the shaded regions at $p_T=3$ GeV/$c$.

Figure 8

Differential elliptic flow $v_2\left(p_T\right)$ for all charged hadrons (a) and identified pions and protons (b), for minimum bias 200$A$ GeV $\mathrm{Au}+\mathrm{Au}$ collisions at the RHIC and $(2.76–5.5)A$ TeV $\mathrm{Pb}+\mathrm{Pb}$ collisions at the LHC. Experimental data for $v_2\left\{2\right\}$ from $\mathrm{Au}+\mathrm{Au}$ collisions at the RHIC are from the STAR experiment [54]. Solid lines are viscous hydrodynamic results for 200$A$ GeV $\mathrm{Au}+\mathrm{Au}$ collisions with the same hydrodynamic parameters as in Figs. 1, 2, 3, 4; note their disagreement with the $v_2\left\{2\right\}$ data shown here (in contrast to their excellent agreement with $v_2\left\{4\right\}$ data shown in Fig. 3). The shaded bands are LHC predictions and show the variation of the theoretical expectations for $\mathrm{Pb}+\mathrm{Pb}$ collisions at collision energies ranging from $\sqrt{s}=2.76$ to $5.5A$ TeV (corresponding to $d{N}_{\mathrm{ch}}/dy=1800$ and 2280, respectively). As in Fig. 7, the lines defining the lower end of the shaded region at $p_T=3$ GeV/$c$ correspond to the lower LHC energy $\sqrt{s}=2.76A$ TeV.

Figure 9

Comparison of the differential elliptic flow $v_2\left(p_T\right)$ for $200A$ GeV $\mathrm{Au}+\mathrm{Au}$ collisions at the RHIC (dashed lines) and $2.76A$ TeV $\mathrm{Pb}+\mathrm{Pb}$ collisions at the LHC (solid lines), at $10\%$–$20\%$ (a and b) and $40\%$–$50\%$ (c and d) centrality, for a variety of different hadron species. Note the slightly negative elliptic flow for the heavy $\Omega$ hyperons at low $p_T$.

Figure 10

Eccentricity-scaled elliptic flow $v_2/\overline{\varepsilon }$ as a function of the charged hadron multiplicity density per unit overlap area $S$ from viscous hydrodynamic calculations at $\sqrt{s}=0.2,2.76,$ and 5.5 $A$ TeV (corresponding to $d{N}_{\mathrm{ch}}/dy=814$ ($\mathrm{Au}+\mathrm{Au}$) and 1800 and 2280 ($\mathrm{Pb}+\mathrm{Pb}$), respectively). Each line corresponds to one collision system at fixed collision energy but different collision centralities. (From right to left, the symbols correspond to 0%–5%, 5%–10%, 10%–15%, 15%–20%, 20%–30%, 30%–40%, 40%–50%, 50%–60%, 60%–70%, and 70%–80% centrality.) The four panels show $v_2\left(\eta \right)/\overline{\varepsilon }$ vs $(1/S)(d{N}_{\mathrm{ch}}/d\eta )$ (where $\eta$ is pseudorapidity) (a and c) and $v_2\left(y\right)/\overline{\varepsilon }$ vs $(1/S)(d{N}_{\mathrm{ch}}/dy)$ (where $y$ is rapidity) (b and d), with $\overline{\varepsilon }$ and $S$ evaluated with the participant-plane averaged energy density $\overline{e}({\mathit{r}}_{\perp },{\tau }_0)$ as a weight function (a and b) (default option; see Sec. 2) or (for comparison with other work) with the corresponding entropy density $\overline{s}({\mathit{r}}_{\perp },{\tau }_0)$ as a weight (c and d). The RHIC curves for $\eta /s=0.16$ (black squares) illustrate the effect of changing the value of the specific shear viscosity by 0.04. The LHC calculations are done for $\eta /s=0.20$ as obtained from the hydrodynamic fit to RHIC data.

Figure 11

Three temperature-dependent parametrizations $(\eta /s)\left(T\right)$ studied in this section. In all cases, $\eta /s=0.2$ for $T<T_{\mathrm{chem}}=165$ MeV.

Figure 12

Final charged multiplicity per wounded nucleon pair as a function of number of participant nucleons in $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s}=2.76A$ TeV, for different functional forms of $(\eta /s)\left(T\right)$ and initial conditions for the shear stress tensor ${\pi }^{\mu \nu }$ (see text).

Figure 13

Charged hadron transverse momentum spectra (top) and differential elliptic flow (bottom) for $2.76A$ TeV $\mathrm{Pb}+\mathrm{Pb}$ collisions at $20\%$–$30\%$ centrality, for different models for the temperature dependence of $\eta /s$ and different initial conditions for ${\pi }^{\mu \nu }$ [Navier-Stokes (“NS”) or 0 (“Zero”)]. The ALICE data in the bottom panel are from Ref. [1].