Scalings of Elliptic Flow for a Fluid at Finite Shear Viscosity

Abstract

Within a parton cascade approach we investigate the scaling of the differential elliptic flow $v_2(p_T)$ with eccentricity $\epsilon_x$ and system size and its sensitivity to finite shear viscosity. We present calculations for shear viscosity to entropy density ratio $\eta/s$ in the range from $1/4\pi$ up to $1/\pi$, finding that the $v_2$ saturation value varies by about a factor 2. Scaling of $v_2(p_T)/\epsilon_x$ is seen also for finite $\eta/s$ which indicates that it does not prove a perfect hydrodynamical behavior, but is compatible with a plasma at finite $\eta/s$. Introducing a suitable freeze-out condition, we see a significant reduction of $v_2(p_T)$ especially at intermediate $p_T$ and for more peripheral collisions. This causes a breaking of the scaling for both $v_2(p_T)$ and the $p_T-$averaged $v_2$, while keeping the scaling of $v_2(p_T)/\la v_2\ra$. This is in better agreement with the experimental observations and shows as a first indication that the $\eta/s$ should be significantly lower than the pQCD estimates. We finally point out the necessity to include the hadronization via coalescence for a definite evaluation of $\eta/s$ from intermediate $p_T$ data.