Saturation of Elliptic Flow and the Transport Opacity of the Gluon Plasma at RHIC

TL;DR

The paper investigates the development of elliptic flow and jet-quenching signatures in RHIC-style heavy-ion collisions by solving the covariant Boltzmann transport equation for a dense gluon plasma using the MPC parton cascade. It shows that the transport opacity $χ=\sigma_t\langle \int dz\, ρ\rangle$ largely governs $v_2(p_\perp)$ and gluon spectra, with high opacities needed to match STAR data under purely elastic $2\to2$ scattering; achieving this either requires unrealistically large initial gluon densities or extremely large effective cross sections. The study also demonstrates that, without inelastic processes, the high-$p_\perp$ behavior and hadronization effects cannot simultaneously reproduce the observed elliptic flow saturation and spectra, highlighting the need to include inelastic energy loss mechanisms. Overall, the work establishes transport opacity as a central parameter in heavy-ion dynamics and provides quantitative baselines for future, more complete treatments that incorporate inelastic interactions and realistic hadronization.

Abstract

Differential elliptic flow and particle spectra are calculated taking into account the finite transport opacity of the gluon plasma produced in Au+Au at Ecm ~ 130 A GeV at RHIC. Covariant numerical solutions of the ultrarelativistic Boltzmann equation are obtained using the MPC parton cascade technique. For typical pQCD (~3 mb) elastic cross sections, extreme initial gluon densities, dN/deta ~ 15000, are required to reproduce the elliptic flow saturation pattern reported by STAR. However, we show that the solutions depend mainly on the transport opacity, $χ=\int dz σ_tρ_g$, and thus the data can also be reproduced with dN/deta ~ 1000, but with extreme elastic parton cross sections, \~45 mb. We demonstrate that the spectra and elliptic flow are dominated by numerical artifacts unless parton subdivisions ~100-1000 are applied to retain Lorentz covariance for RHIC initial conditions.

Figures (17)

• Figure 1: Strong dependence of the gluon elliptic flow on parton subdivision as a function of $p_\perp$ is shown for Au+Au at $\sqrt{s}=130A$ GeV with $b=8$ fm. Solutions for transport opacity $\chi = 9.74^{A)}$ (see Table \ref{['Table:chi']}), and particle subdivisions $\ell=1$, 5, 50, 225, and 450 are shown.
• Figure 2: Strong dependence of the final gluon $p_\perp$ spectra on parton subdivision is shown for Au+Au at $\sqrt{s}=130A$ GeV with $b=8$ fm. Solutions for transport opacity $\chi = 9.74^{A)}$ (see Table \ref{['Table:chi']}), and particle subdivisions $\ell=1$, 5, 50, 225, and 450 are shown as in Fig. \ref{['Figure:v2_vs_l']}. The spectra are normalized here to $dN(0)/d\eta = 210$.
• Figure 3: The very weak dependence of the elliptic flow and gluon spectra quenching on the angular distribution of the parton cross section is shown. See Table \ref{['Table:chi']} for the simulation parameters corresponding to each curve. The solutions are seen to depend mainly on the transport opacity.
• Figure 4: Gluon elliptic flow as a function of $p_\perp$ for Au+Au at $\sqrt{s}=130A$ GeV with impact parameters $b=2$, 4, 6, 8, 10, and 12 fm is shown for transport opacities $\chi=7.90^{C)}$, 6.98$^{C)}$, 5.68$^{C)}$, 3.90$^{C)}$, 1.98$^{C)}$, and 0.52$^{C)}$. STAR dataSTARv2 below 2 GeV/$c$ are shown. Preliminary STAR dataSnellings suggest that $v_2\sim 0.15-0.17$ may saturate in the $2<p_T<4$ GeV/$c$ range.
• Figure 5: Same as Fig. \ref{['Figure:v2_g40mb']} except that solutions for transport opacities $\chi=19.4^{A)}$, 17.2$^{A)}$, 14.1$^{A)}$, 9.74$^{A)}$, 5.00$^{A)}$, and 1.32$^{A)}$ are shown. STAR data STARv2 below 2 GeV/$c$ are shown. Preliminary STAR dataSnellings suggest that $v_2\sim 0.15-0.17$ may saturate in the $2<p_T<4$ GeV/$c$ range.