Driver and damping of the directed-flow response in heavy-ion collisions

Abstract

The initial-state geometry plays a crucial role in driving directed flow, while the dissipative response of the medium dampens it. Both of these factors influence how the directed-flow slope varies with system size. We developed a method to differentiate between these driving and damping effects on charged-hadron directed flow in O+O, Cu+Cu, Ru+Ru, Au+Au, and U+U collisions at $\sqrt{s_{\mathrm{NN}}} = 200$~GeV using an improved version of the string-melting AMPT model. We formulated three scaling observables based on the entropy density, the number of participants, and the mass number. A dimensionless ratio was constructed, revealing the threshold mass number $A \approx 35$ in central collisions from the entropy and participant scaling, indicating the onset of collective behavior. We constructed a kinetic-theory Knudsen-number ($\mathrm{Kn}$) map to analyze the contributions of the initial-state driver, which grows as $\mathrm{Kn}^κ$ with $κ\approx 2$, and a final-state viscous damping of the hydrodynamic response with characteristic scale $\mathrm{Kn}_0 \approx 0.27$ for the directed flow slope. This damping scale is found to be about a factor of 2.5 smaller than the $\mathrm{Kn}_0 \approx 0.7$ extracted from the elliptic flow. Furthermore, we determined the ratio of shear viscosity to entropy density, $η/s$, to be between 0.10 and 0.20, using an alternative method that does not rely on fitting flow harmonics.