Causal viscous hydrodynamics in 2+1 dimensions for relativistic heavy-ion collisions

TL;DR

This work develops and applies a (2+1)-dimensional causal viscous hydrodynamics framework (VISH2+1) based on Israel-Stewart theory to Cu+Cu collisions, using η/s fixed at the AdS/CFT lower bound. It shows that shear viscosity delays longitudinal cooling and accelerates transverse expansion, extending the QGP lifetime and producing flatter final spectra, while strongly suppressing elliptic flow v2 in non-central collisions. The study analyzes how δf corrections and the relaxation time τπ influence observables and underscores model-dependence and IS equation ambiguities that complicate extracting η/s from data. Overall, the results support the near-perfect fluid picture of the QGP but highlight the need for refined inputs and potential matching to microscopic dynamics for robust, quantitative conclusions.

Abstract

We explore the effects of shear viscosity on the hydrodynamic evolution and final hadron spectra of Cu+Cu collisions at ultrarelativistic collision energies, using the newly developed (2+1)-dimensional viscous hydrodynamic code VISH2+1. Based on the causal Israel-Stewart formalism, this code describes the transverse evolution of longitudinally boost-invariant systems without azimuthal symmetry around the beam direction. Shear viscosity is shown to decelerate the longitudinal and accelerate the transverse hydrodynamic expansion. For fixed initial conditions, this leads to a longer quark-gluon plasma (QGP) lifetime, larger radial flow in the final state, and flatter transverse momentum spectra for the emitted hadrons compared to ideal fluid dynamic simulations. We find that the elliptic flow coefficient v_2 is particularly sensitive to shear viscosity: even the lowest value allowed by the AdS/CFT conjecture, eta/s=1/4pi, suppresses v_2 enough to have significant consequences for the phenomenology of heavy-ion collisions at the Relativistic Heavy Ion Collider. A comparison between our numerical results and earlier analytic estimates of viscous effects within a blast-wave model parametrization of the expanding fireball at freeze-out reveals that the full dynamical theory leads to much tighter constraints for the specific shear viscosity eta/s, thereby supporting the notion that the quark-gluon plasma created at RHIC exhibits almost ``perfect fluidity''.

Figures (25)

• Figure 1: The equations of state EOS Q (dashed line) and SM-EOS Q ("modified EOS Q", solid line).
• Figure 2: (Color online) Time evolution of the hydrodynamic source terms (\ref{['S00']}-\ref{['S02']}), averaged over the transverse plane, for central Cu+Cu collisions, calculated with EOS I in the left panel and with SM-EOS Q in the right panel. The smaller insets blow up the vertical scale to show more detail. The dashed blue lines are for ideal hydrodynamics with $e_0{\,=\,}30$ GeV/fm$^3$ and $\tau_0{\,=\,}0.6$ fm/$c$. Solid red lines show results from viscous hydrodynamics with identical initial conditions and $\frac{\eta}{s}{\,=\,}\frac{1}{4\pi}{\,\approx\,}0.08$, $\tau_\pi{\,=\,}\frac{3\eta}{sT}{\,\approx\,}0.24\left(\frac{200\,\mathrm{MeV}}{T}\right)$ fm/$c$. The positive source terms drive the transverse expansion while the negative ones affect the longitudinal expansion.
• Figure 3: (Color online) Time evolution of the local temperature in central Cu+Cu collisions, calculated with EOS I (left) and SM-EOS Q (right), for the center of the fireball ($r{\,=\,}0$, upper set of curves) and a point near the edge ($r{\,=\,}9$ fm, lower set of curves). Same parameters as in Fig. \ref{['source']}. See text for discussion.
• Figure 4: (Color online) Time evolution of the local entropy density for central Cu+Cu collisions, calculated with EOS I (left) and SM-EOS Q (right), for the center of the fireball ($r{\,=\,}0$, upper set of curves) and a point at $r{\,=\,}3$ fm (lower set of curves). Same parameters and color coding as in Fig. \ref{['localT']}. See text for discussion.
• Figure 5: (Color online) Surfaces of constant temperature $T$ and constant transverse flow velocity $v_\perp$ for central Cu+Cu collisions, evolved with EOS I (left panel) and SM-EOS Q (right panel). In each panel, results from viscous hydrodynamics in the left half are directly compared with the corresponding ideal fluid evolution in the right half. (The thin isotherm contours in the right halves of each panel are reflected from the left halves, for easier comparison.) The lines of constant $v_\perp$ are spaced by intervals of 0.1, from the inside outward, as indicated by the numbers near the top of the figures. The right panel contains two isotherms for $T_c{\,=\,}164$ MeV, one separating the mixed phase (MP) from the QGP at energy density $e_\mathrm{Q}{\,=\,}1.6$ GeV/fm$^3$, the other separating it from the hadron resonance gas (HRG) at energy density $e_\mathrm{H}{\,=\,}0.45$ GeV/fm$^3$. See text for discussion.