Hadron spectra and elliptic flow for 200 $A$ GeV $\text{Au}+\text{Au}$ collisions from viscous hydrodynamics coupled to a Boltzmann cascade

It is shown that the recently developed hybrid code VISHNU, which couples a relativistic viscous fluid dynamical description of the quark-gluon plasma (QGP) with a microscopic Boltzmann cascade for the late hadronic rescattering stage, yields an excellent description of charged and identified hadron spectra and elliptic flow measured in 200 $A$ GeV $\text{Au}+\text{Au}$ collisions at the Relativistic Heavy-Ion Collider (RHIC). Using initial conditions that incorporate event-by-event fluctuations in the initial shape and orientation of the collision fireball and values $\eta /s$ for the specific shear viscosity of the quark-gluon plasma that were recently extracted from the measured centrality dependence of the eccentricity-scaled, $p_T$-integrated charged hadron elliptic flow $v_{2,\mathrm{ch}}/\varepsilon$, we obtain universally good agreement between theory and experiment for the $p_T$ spectra and differential elliptic flow $v_2(p_T)$ for both pions and protons at all collision centralities.

Figure 1

$p_T$ spectra of pions (left) and protons (right) for 200 $A$ GeV $\text{Au}+\text{Au}$ collisions of different centralities as indicated. Data from the STAR ($\times$, [15, 16, 17]) and PHENIX ($+$, [18]) experiments are compared with VISHNU calculations using MC-Glauber (dashed lines) or MC-KLN initial conditions (solid lines) and different values $\eta /s$ for the QGP shear viscosity as indicated. Different $\eta /s$ values are associated with different starting times ${\tau }_0$ for the hydrodynamic evolution as discussed in the text. The STAR and PHENIX proton data shown in the right column are feeddown corrected by removing protons from weak hyperon decays [15, 18]. Where necessary, PHENIX yields from neighboring narrower centrality bins were averaged to obtain data in the wider centrality bins used by the STAR Collaboration.

Figure 2

Eccentricity-scaled elliptic flow $v_2/\varepsilon$ as function of $p_T$ for charged hadrons from 200 $A$ GeV $\text{Au}+\text{Au}$ collisions at different centralities. The experimental data (solid symbols) are $v_2\{\mathrm{EP}\}$ measurements from the STAR experiment [19], scaled by $\langle {\varepsilon }_{\mathrm{part}}^{\alpha }\rangle {}^{1/\alpha }$ from the Monte Carlo Glauber model (left column) and the MC-KLN model (right column), respectively. $\alpha$ depends on the event-plane resolution $R$ and varies from one centrality bin to the next (see Table ). Dashed and solid lines with open symbols are results from VISHNU for two different values of $(\eta /s){}_{\mathrm{QGP}}$ (0.08 and 0.16 for the MC-Glauber calculations, 0.16 and 0.24 for the MC-KLN calculations). The theoretical lines show the ratio $\langle v_2\rangle /{\mathrm{\varepsilon ̄}}_{\mathrm{part}}$ where $\langle \dots \rangle$ denotes an average over events, and ${\mathrm{\varepsilon ̄}}_{\mathrm{part}}$ is the eccentricity of the smooth average initial entropy density. Different symbols denote different collision centralities as indicated.

Figure 3

Same as Fig. 2, but with $v_2\{\mathrm{EP}\}$ data from the PHENIX Collaboration [25] instead of STAR data. See Table for the effective eccentricities $\langle {\varepsilon }_{\mathrm{part}}^{\alpha }\rangle {}^{1/\alpha }$ used for each centrality bin.

Figure 4

Comparison of the STAR [19] and PHENIX $v_2\{\mathrm{EP}\}$ data [25] used in Figs. 2 and 3. Both data sets are scaled by the effective eccentricity $\varepsilon \equiv \langle {\varepsilon }_{\mathrm{part}}^{\alpha }\rangle {}^{1/\alpha }$, where $\alpha$ depends on the event-plane resolution $R$ [21] and thus varies with centrality and from experiment to experiment (see Table ).

Figure 5

Same as Fig. 2 but for identified pions (left column) and protons (right column). Solid symbols denote measurements of $v_2\{2\}/\langle {\varepsilon }_{\mathrm{part}}^2\rangle {}^{1/2}$ from the STAR experiment [19], solid and dashed lines with open symbols show $\langle v_2\rangle /{\mathrm{\varepsilon ̄}}_{\mathrm{part}}$ from VISHNU calculations with $(\eta /s){}_{\mathrm{QGP}}=0.08$ and 0.16, respectively, using MC-Glauber (top row) and MC-KLN (bottom row) initial conditions. Different symbols denote different collision centralities as indicated.

Figure 6

Integrated charged hadron elliptic flow as a function of collision centrality from the PHENIX [44] and STAR [19, 43] experiments are compared with VISHNU calculations using participant plane (PP) averaged (a) and reaction plane (RP) averaged (b) initial conditions from the MC-KLN and MC-Glauber models and $(\eta /s){}_{\mathrm{QGP}}$ values as indicated. In the STAR data and the calculations $v_2$ was integrated over the range 0.15 GeV/$c<p_T<2$GeV/$c$; the PHENIX data were integrated over 0.2GeV/$c<p_T<8$GeV/$c$.

Figure 7

Eccentricity ${\mathrm{\varepsilon ̄}}_{\mathrm{part}}$ (a) and transverse area $S$ (b) of the participant-plane averaged initial entropy density distribution from the MC-Glauber model, for smearing areas ${\sigma }_s=42$ and 4.2 mb, respectively. (See also Fig. 4 in Ref. [22] for comparison.)

Figure 8

(a) Centrality dependence of the charged hadron rapidity density per participant pair $({\mathit{dN}}_{\mathrm{ch}}/\mathit{dy})/(N_{\mathrm{part}}/2)$. Experimental data are from STAR [17] and PHOBOS [45], using ${\mathit{dN}}_{\mathrm{ch}}/\mathit{dy}=1.16\hspace{0.5em}{\mathit{dN}}_{\mathrm{ch}}/d\eta$ for the PHOBOS data. Theoretical lines are explained in the text. (b) Eccentricity-scaled elliptic flow $v_2/\varepsilon$ as function of multiplicity density $(1/S)({\mathit{dN}}_{\mathrm{ch}}/\mathit{dy})$, for different values of $(\eta /s){}_{\mathrm{QGP}}$, using participant-plane (PP) and reaction-plane (RP) averaged initial entropy density profiles from the MC-KLN and MC-Glauber models, normalized to $({\mathit{dN}}_{\mathrm{ch}}/\mathit{dy}){}_{\max }=810$ in the most central ($0\%-5\%$) $\text{Au}+\text{Au}$ collisions. Insensitivity to the smearing area ${\sigma }_s$ in the MC-Glauber models is also shown.

Figure 9

Comparison of the universal theoretical curves for $v_2[\eta /s]/\varepsilon$ vs $(1/S)({\mathit{dN}}_{\mathrm{ch}}/\mathit{dy})$ from the MC-Glauber model with ${\sigma }_s=42$ mb [1] with experimental data for $\langle v_2\rangle$ [21], normalized by the eccentricity $\langle {\varepsilon }_{\mathrm{part}}\rangle$ and transverse fireball area $S$ of the initial profile from the participant-plane averaged MC-Glauber model with standard (${\sigma }_s=42$ mb) and reduced (${\sigma }_s=4.2$ mb) smearing areas.