Real-time evolution of non-Gaussian cumulants in the QCD critical regime

We derive a coupled set of equations that describe the nonequilibrium evolution of cumulants of critical fluctuations for spacetime trajectories on the crossover side of the QCD phase diagram. In particular, novel expressions are obtained for the nonequilibrium evolution of non-Gaussian skewness and kurtosis cumulants. UBy utilizing a simple model of the spacetime evolution of a heavy-ion collision, we demonstrate that, depending on the relaxation rate of critical fluctuations, skewness and kurtosis can differ significantly in magnitude as well as in sign from equilibrium expectations. Memory effects are important and shown to persist even for trajectories that skirt the edge of the critical regime. We use phenomenologically motivated parametrizations of freeze-out curves and of the beam-energy dependence of the net baryon chemical potential to explore the implications of our model study for the critical-point search in heavy-ion collisions.

Figure 1

A sketch of the crossover side of the scaling regime in the $r,h$ plane. We define the scaling regime with the criterion ${\xi }_{\min }<{\xi }_{\mathrm{eq}}<{\xi }_{\text{max}}$. To be specific, we take ${\xi }_{\text{max}}/{\xi }_{min}=3$. The solid curve delineates the boundary of the critical regime. The dotted curves show where the equilibrium value of the kurtosis $K$ changes sign. The trajectory A (see text) is shown with the black dashed vertical line.

Figure 2

(a) The evolution of nonequilibrium mean $M/M_A$, (b) effective correlation length $\xi /{\xi }_{\min }$, (c) skewness $S/S_A$, and (d) kurtosis $K/K_A$ as a function of $(T_c-T)/\mathrm{\Delta }T$ along trajectory A (cf. Fig. 1). Results for ${\tau }_{\text{rel}}/{\tau }_I=0.005, 0.02, 0.05, 0.2$ are shown in solid red, dotted blue, single-dot-dashed green, and double-dot-dashed orange curves, respectively. The dashed curves plot the corresponding equilibrium values. All results are normalized by the corresponding equilibrium value at the endpoint trajectory A (cf. Fig. 1). The dashed vertical lines illustrate $T$ at the point where the a freeze-out curve intersects with trajectory A. (From left to right, freeze-out curves are of type I, II, III, respectively. For a discussion, see text in Sec. 4b.)

Figure 3

Contour plot of equilibrium (top) and nonequilibrium skewness (S) with $\tau /{\tau }_I=0.05$ (middle) and $\tau /{\tau }_I=0.02$ (bottom). The $S>0$ region is shown in red and the $S<0$ region is shown in blue.

Figure 4

Contour plot of equilibrium (top) and nonequilibrium kurtosis (K) with $\tau /{\tau }_I=0.05$ (middle) and $\tau /{\tau }_I=0.02$ (bottom). The $K>0$ region is shown in red and the $K<0$ region is shown in blue.

Figure 5

A sketch of the crossover side of the scaling regime in the $T-\mu$ plane. The dotted curves show where the equilibrium value of the kurtosis $K$ changes sign. The trajectory A (see text) is shown with the black dashed vertical line. Different types of freeze-out curves are shown in red (upper), blue (middle) and green (lower) dashed curves, corresponding to types I, II, III, respectively (see text).

Figure 6

The nonequilibrium values of the effective correlation length ${\xi }_F$ on the freeze-curves as a function of $\mu$. The results corresponding to freeze-out curves of type I, II, and III (as shown in Fig. 5) are shown in panels (a), (b), and (c), respectively. Nonequilibrium cumulants with ${\tau }_{\text{rel}}/{\tau }_I=0.005, 0.02, 0.05, 0.2$ are represented by solid red, dotted blue, single-dot-dashed green, and double-dot-dashed orange curves, respectively. The dashed curves plot the corresponding equilibrium values—where not visible, they fully overlap with the solid red curve. All results are normalized by the corresponding equilibrium value at the endpoint of trajectory A, as shown in Fig. 5.

Figure 7

The nonequilibrium values of the non-Gaussian cumulant $S_F$ on the freeze-curves as a function of $\mu$. The results corresponding to freeze-out curves of type I, II, and III (as shown in Fig. 5) are shown in panels (a), (b), and (c), respectively. Nonequilibrium cumulant results for ${\tau }_{\text{rel}}/{\tau }_I=0.005, 0.02, 0.05, 0.2$ are displayed with solid red, dotted blue, single-dot-dashed green, and double-dot-dashed orange curves, respectively. The dashed curves plot the corresponding equilibrium values. All results are normalized by the corresponding equilibrium value at the endpoint trajectory A, as shown in Fig. 5.

Figure 8

The nonequilibrium values of the non-Gaussian cumulant $K_F$ on the freeze-curves as a function of $\mu$. The results corresponding to freeze-out curves of type I, II, and III (as shown in Fig. 5) are shown in panels (a), (b), and (c), respectively. Nonequilibrium cumulant results for ${\tau }_{\text{rel}}=0.005, 0.02, 0.05, 0.2{\tau }_I$ are displayed with solid red, dotted blue, single -dot-dashed green, and double-dot-dashed orange curves, respectively. The dashes curves plot the corresponding equilibrium values. All results are normalized by the corresponding equilibrium value at the endpoint trajectory A, as shown in Fig. 5.

Figure 9

The $\sqrt{s}$ dependence [assuming Eq. (4.2)] of nonequilibrium non-Gaussian cumulants on the freeze curves (FCs) with different choices of the relative position of freeze-out curves and ${\tau }_{\text{rel}}$. Top panel shows skewness $S_F$ vs $\sqrt{s}$, bottom panel shows kurtosis $K_F$ vs $\sqrt{s}$. The results obtained from the combination, i.e., (FC I, ${\tau }_I/{\tau }_{\text{rel}}=0.02$), (FC II, ${\tau }_I/{\tau }_{\text{rel}}=0.05$), (FC III, ${\tau }_I/{\tau }_{\text{rel}}=0.2$), are displayed in solid red, dotted blue, and single-dot-dashed green curves, respectively.