Scaling Properties of Out-of-equilibrium Hydrodynamic Fluctuations near the QCD Critical Point

In a heavy ion experiment the system travels along trajectories of constant n/s in absence of dissipation. Ideally one can tune the beam energy so that the system goes exactly through the QCD critical point (CP), in which case the equilibrium two point correlator ${〈\delta (n/s)\delta (n/s)〉}_{eq}$ [5] of the thermal fluctuations $\delta (n/s)$ diverges at the CP while the actual value of this correlator $〈\delta (n/s)\delta (n/s)〉$ falls out of equilibrium and consequently remains finite. In this paper we first show that this out-of-equilibrium dynamics exhibits universal scaling properties known as Kibble-Zurek scaling. The relevant physical scale is called Kibble-Zurek scale and it is a function of a small dimensionless parameter $ϵ\equiv {\tau }_o/{\tau }_Q$, where ${\tau }_o$ is the microscopic mean-free-time and ${\tau }_Q$ is the macroscopic total expansion time. Realistically the trajectory always misses the CP, the equilibrium correlator is then cut off by a new scale which we call the crossing scale. The crossing scale is a function of another small dimensionless parameter ${\Delta }_s\equiv (n_c/s_c)(s/n-s_c/n_c)$ that's controlled by the beam energy. We then show from the scaling properties of Ising EOS that the equilibrium correlator scales universally with the crossing scale. The paper is concluded by a quantitative analysis on how the out-of-equilibrium KZ dynamics competes with missing the CP since they both limit the growth of the correlator $〈\delta (n/s)\delta (n/s)〉$ near the CP.