Radiative energy loss and $v_2$ spectra for viscous hydrodynamics

This work investigates the first correction to the equilibrium phase-space distribution and its effects on spectra and elliptic flow in heavy-ion collisions. We show that the departure from equilibrium on the freeze-out surface is the largest part of the viscous corrections to $v_2(p_T)$. However, the momentum dependence of the departure from equilibrium is not known a priori, and it is probably not proportional to $p_T^2$ as has been assumed in hydrodynamic simulations. At high momentum in weakly coupled plasmas, it is determined by the rate of radiative energy loss and is proportional to $p_T^{3/2}$. The weaker $p_T$ dependence leads to straighter $v_2(p_T)$ curves at the same value of viscosity. Furthermore, the departure from equilibrium is generally species dependent. A species-dependent equilibration rate, with baryons equilibrating faster than mesons, can explain “constituent quark scaling” without invoking coalescence models.

Figure 1

Typical results for $v_2(p_T)$ from a viscous hydrodynamic model employing the quadratic Ansatz. The run parameters are $\eta /s=0.08$, $T_{\mathrm{frzout}}=140$ MeV, and $p=1/3\epsilon$. Further details are in Appendix .

Figure 2

Left: $v_2(p_T)$ using the linear or quadratic Ansätze for the distribution function. Right: Integrated $v_2$ vs centrality showing independence from the precise form of the viscous correction. Run parameters can be found in Fig. 1.

Figure 3

The points are from the numerical solution of the Boltzmann equation for pure glue at leading order (LO) and without $1\leftrightarrow 2$ processes (i.e., collisional energy loss only). The lines are $\chi \propto p^{2-\alpha }$ for $\alpha =2$, 1.6, 1.38, and 1, going from top to bottom.

Figure 4

$v_2(p_T)$ for a perturbative gluon gas at leading order. The linear and quadratic Ansätze are shown for comparison. Run parameters can be found in Fig. 1.

Figure 5

Curves at lower momentum are $\chi (p/T)$ for a perturbative gluon gas at leading order for three values of the nonperturbative parameter $\eta /s$. The curves at higher momentum show the asymptotic forms of $\chi$ for three values of the non-perturbative parameter $\mathrm{q̂}$. There must be a consistency between $\eta /s$ and $\mathrm{q̂}$ in order that the curves merge at intermediate momentum.

Figure 6

Off-equilibrium correction for the case of a perturbative two-flavor QGP evaluated at leading order. The inset figure shows the ratio of the quark to gluon correction which asymptotically approaches ${\chi }_{\mathrm{quark}}/{\chi }_{\mathrm{gluon}}\approx 1.7$.

Figure 7

Left: Elliptic flow of quarks and gluons. Right: Both $v_2$ and $p_T$ scaled by $n=3$, 2 for gluons and quarks, respectively. Run parameters can be found in Fig. 1.

Figure 8

Elliptic flow of $K_S$ mesons and $\Lambda$ baryons from viscous hydrodynamics with radiative or quadratic Ansätze. The run parameters are $\eta /s=0.16$, $T_{\mathrm{frzout}}=150$ MeV, and lattice EOS. Further details are in Appendix . The data are from the STAR Collaboration [6].

Figure 9

Left: $v_2$ of $K_S$ and $\Lambda$. Right: Constituent quark scaling of $v_2(p_T)$. Run parameters can be found in Fig. 8. The data are from the STAR Collaboration [6] and are plotted in $(M_T-m)/n$ as suggested by the PHENIX Collaboration [7]. (Recent PHENIX data [48, 49] clearly deviate from constituent quark scaling above $(M_T-m)/n\simeq 1$ GeV.)

Figure 10

Off-equilibrium correction for the case of collisional energy loss. The points are from the numerical solution of the linearized Boltzmann equation, and the curve is the asymptotic form, Eq. (B12). Specifically the curve is a one-parameter fit to the $N_c=3$ form, $(24\pi /9)\hspace{0.5em}{\mathrm{p̃}}^2/\log (\mathrm{p̃}/C)$, with fit parameter $C^{-1}=1.3$.

Figure 11

Available phase space for the collision integrals in Eq. (B13). The band shows the dominant region of the integration in a leading $\log (p/T)$ approximation.