The Effect of Shear Viscosity on Spectra, Elliptic Flow, and HBT Radii

TL;DR

The paper analyzes how shear viscosity modifies the thermal distribution and hydrodynamic evolution in relativistic heavy-ion collisions. It derives the first-order viscous correction to the distribution function and propagates this through a blast-wave framework to compute spectra, elliptic flow, and HBT radii. Key findings show that viscous corrections become significant for $p_T$ around 1.5–2 GeV in the blast-wave setup, that elliptic flow is suppressed unless the sound attenuation length is small, and that the longitudinal HBT radius is notably reduced and its $m_T$ scaling broken. The results delineate the finite domain where hydrodynamics is applicable and highlight the necessity of full viscous simulations with dynamical $η/s$ to faithfully describe RHIC data.

Abstract

I calculate the first correction to the thermal distribution function of an expanding gas due to shear viscosity. With this modified distribution function I estimate viscous corrections to spectra, elliptic flow, and HBT radii in hydrodynamic simulations of heavy ion collisions using the blast wave model. For reasonable values of the shear viscosity, viscous corrections become of order one when the transverse momentum of the particle is larger than 1.7 GeV. This places a bound on the $p_{T}$ range accessible to hydrodynamics for this observable. Shear corrections to elliptic flow cause $v_{2}(p_{T})$ to veer below the ideal results for $p_{T} \approx 0.9$ GeV. Shear corrections to the longitudinal HBT radius $R^{2}_{L}$ are large and negative. The reduction of $R_{L}^2$ can be traced to the reduction of the longitudinal pressure. Viscous corrections cause the longitudinal radius to deviate from the $\frac{1}{\sqrt{m_T}}$ scaling which is observed in the data and which is predicted by ideal hydrodynamics. The correction to the sideward radius $R^{2}_{S}$ is small. The correction to the outward radius $R^{2}_{O}$ is also negative and tends to make $R_{O}/R_{S} \approx 1$.

Figures (5)

• Figure 1: (a) The $p_z$ distribution of particles with coordinate-space rapidity $\eta_s=0$, with and without viscous corrections. (b) The $z$ distribution of particles with momentum-space rapidity $y=0$, with and without viscous corrections. The curves are drawn for a Bjorken expansion without transverse flow at $\tau_o=7\,\hbox{fm}$ for a Boltzmann gas with temperature, $T=160\,\hbox{MeV}$, $m=140\,\hbox{MeV}$. The transverse momentum is fixed, $p_T=400\,\hbox{MeV}$. The viscous correction is linearly proportional to $\Gamma_s/\tau_o$.
• Figure 2: The solid line shows the ratio between the viscous correction ($\delta\, dN\equiv \frac{dN^{(1)}}{d^2p_T dy}$) and the ideal spectrum ($dN^{(0)} \equiv \frac{dN^{(0)}}{d^2p_T dy}$). The dashed line shows the Bjorken result without transverse flow given in Eq. \ref{['spectra2']}. The band indicates where the hydrodynamic description of the $p_T$ spectrum in the blast wave model can not be reliably calculated. The viscous correction is linearly proportional to $\Gamma_s/\tau_o$.
• Figure 3: Elliptic flow $v_2$ as a function of $p_T$ for different values of $\Gamma_s/\tau_{o}$. The data points are four particle cummulant data from the STAR collaboration v2-star1. Only statistical errors are shown. The difference between the ideal and viscous curves is linearly proportional to $\Gamma_s/\tau_o$.
• Figure 4: (a) Ideal blast wave fit to the experimental HBT radii $R_O$, $R_S$, and $R_L$ shown in (b) as a function of transverse momentum $K_T$. The solid symbols are from the STAR collaboration HBT-star and the open symbols are from the PHENIX collaboration HBT-phenix. For clarity, the experimental points have been slightly shifted horizontally.
• Figure 5: (a) Viscous correction $\delta R^2$ for $R_O$, $R_S$, and $R_L$ relative to ideal blast wave HBT radii $(R^2)^{(0)}$ . (b) The HBT radii $R_{O}$, $R_{S}$, and $R_L$ including the viscous correction. The viscous correction is linearly proportional to $\Gamma_s/\tau_{o}$.