Freezing out fluctuations in $\text{Hydro}+$ near the QCD critical point (2024)

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  Figure 1

  Schematic view of a trajectory followed by an expanding cooling droplet of matter produced in a heavy ion collision on the QCD phase diagram in the vicinity of the critical point.

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  Figure 2

  The dependence of the correlation length $\xi$ on temperature for different trajectories of the fireball expansion (i.e., different ${\xi }_{\max }$).

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  Figure 3

  Geometric representation of Eq.(30). $x_{+}$ and $x_{-}$ are on the freeze-out surface; $x$ is the midpoint between them. The four-vector $\mathrm{\Delta }{x}_{\perp }$ (red) is perpendicular to the fluid four-velocity at the point $x$, $u(x)$, meaning that in the local fluid rest frame it is a four-vector with no time-component.

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  Figure 4

  Evolution of ${\varphi }_{\mathit{Q}}$ as a function of Bjorken time $\tau$, using model A and model H dynamics, corresponding to the relaxation rates given by Eqs.(10) and (5), respectively. We have taken ${\mathrm{\Gamma }}_0=1{\mathrm{fm}}^{-1}$, $D_0=1\mathrm{fm}$ and ${\xi }_{\max }=3\mathrm{fm}$ in both panels. The three solid curves in each figure correspond to different times $\tau$ as the boost-invariant, spatially hom*ogeneous, Bjorken fluid is expanding and cooling in the vicinity of a critical point. The temperatures are given by $T=235$, 160 and 140MeV, for times $\tau =1$, 4.6 and 8.8fm, respectively. The dashed curves represent the equilibrium values ${\overline{\varphi }}_{\mathit{Q}}$ for the corresponding temperatures (times). We have initialized the hydrodynamic solution and the fluctuations at ${\tau }_i=1\mathrm{fm}$: at that time ${\varphi }_{\mathit{Q}}={\overline{\varphi }}_{\mathit{Q}}$ at $T_i=235\mathrm{MeV}$. The dashed curves are highest at $\tau =4.6\mathrm{fm}$ because that is when the evolution trajectory was closest to the critical point; the fluctuations would be largest at that time if they were in equilibrium. We see that in model H the fluctuations (in our full, out-of-equilibrium, calculation) remain considerably enhanced at $\tau =8.8\mathrm{fm}$ over a range of nonzero values of $\mathit{Q}$. It is evident from the right plot that ${\varphi }_{\mathit{Q}}$ does not evolve at $\mathit{Q}=0$ in model H. This is a consequence of conservation laws. In both plots, at all times shown, ${\varphi }_{\mathit{Q}}$ and ${\overline{\varphi }}_{\mathit{Q}}$ are both normalized by their noncritical value (their value at a location far enough away from the critical point that $\xi ={\xi }_0$) at $\mathit{Q}=0$ in equilibrium, i.e., ${\overline{\varphi }}_{\mathbf{0}}^{\mathrm{nc}}=ZT{\xi }_0^2$.

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  Figure 5

  Normalized ${\varphi }_{\mathit{Q}}$ (a) and its inverse Fourier transform $\tilde{\varphi }$ (b)at freeze-out $T_f=140\mathrm{MeV}$ after evolution according to model H dynamics with two values of $D_0$. In the text, we explain the dependence of the shapes of the curves in both panels on $D_0$, and the consequences of the conservation laws on the shapes of these curves.

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  Figure 6

  Normalized proton multiplicity correlator $\tilde{C}(\mathrm{\Delta }y)$ for protons from Eq.(60) as a function of the rapidity gap $\mathrm{\Delta }y$ in the Bjorken scenario for two choices of the diffusion parameter $D_0$.

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  Figure 7

  The normalized fluctuation measure for protons, Eq.(36), as a function of ${\xi }_{\max }$, the maximum value of the equilibrium correlation length achieved along the system trajectory. Panels (a) and (b)correspond to different diffusion strengths, quantified by $D_0$, while red and blue curves correspond to different freeze-out temperatures. Panel (c)shows the result that would have been obtained under the assumption that fluctuations are in equilibrium at freeze-out.

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  Figure 8

  The space-time dependence of the temperature (represented by color) and flow velocity in the hydrodynamic simulation of the expanding cooling droplet of quark-gluon plasma. The magnitude of the radial flow at each space-time point is indicated by the tilt of the arrows. The dashed, dotted and solid black curves are the isothermal curves at $T=160\mathrm{MeV}$, 156MeV and 140MeV, respectively. Examples of fluid cell trajectories, or hydrodynamic flow lines, are illustrated by solid black lines tangential to local flow vectors.

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  Figure 9

  Hydro+ fluctuation measure ${\varphi }_{\mathit{Q}}$ along two hydrodynamic flow lines passing through $r=r_i$ at initial time $\tau ={\tau }_i$, with $r_i=0.7\mathrm{fm}$ (top four panels) and 5fm (bottom four panels). The four plots in the left (right) column are for $D_0=0.25\mathrm{fm}$ ($D_0=1\mathrm{fm}$), with ${\xi }_{\max }=1\mathrm{fm}$ and ${\xi }_{\max }=3\mathrm{fm}$ in alternating rows. The solid and dashed curves are, respectively, the ${\varphi }_{\mathit{Q}}$ and ${\overline{\varphi }}_{\mathit{Q}}$ (normalized to their values at $Q=0$ away from the critical point, where $\xi ={\xi }_0$) at three times $\tau$ indicated in the plot legends; the choice of $\tau$’s is explained in the text.

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  Figure 10

  The values of ${\varphi }_{\mathit{Q}}$ (suitably normalized) for three representative values of $Q$ (same for each column), and for values $D_0$ (same in top and bottom six panels) and ${\xi }_{\max }$ (same in alternating rows) as in Fig.9. The values of ${\varphi }_{\mathit{Q}}$ are taken along a fluid cell trajectory and plotted as a function of temperature, which is a monotonous function of time $\tau$ along the trajectory. The trajectory chosen for these plots begins at $r_i=r({\tau }_i)=1.8\mathrm{fm}$. The dashed and solid curves represent the equilibrium ${\overline{\varphi }}_{\mathit{Q}}$ and nonequilibrium ${\varphi }_{\mathit{Q}}$, respectively.

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  Figure 11

  The $\text{Hydro}+$ variable ${\varphi }_{\mathit{Q}}$ (normalized to its value at $Q=0$ away from the critical point, where $\xi ={\xi }_0$) at freeze-out evolved with two different diffusion parameters $D_0=0.25\mathrm{fm}$ (upper panels) and 1fm (lower panels) and ${\xi }_{\max }=3\mathrm{fm}$. The left (right) panels show results for evolution until the decreasing temperature has reached a higher (lower) freeze-out temperature. The blue, red and purple curves show the values of ${\varphi }_{\mathit{Q}}$ at different points on the freeze-out hypersurface, characterized by the radial coordinate $r$. The black dashed and dotted curves are the equilibrium curves at $T=T_f$ and $T=T_c$ respectively. The dashed brown curve is the (noncritical) equilibrium curves corresponding to $\xi ={\xi }_0$.

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  Figure 12

  $\tilde{\varphi }\times \mathrm{\Delta }{x}^2$, the measure of fluctuations of $\widehat{s}$ described by the correlator $⟨\delta \widehat{s}(x_{+})\delta \widehat{s}(x_{-})⟩$, at freeze-out as a function of the spatial separation between the points $\mathrm{\Delta }x\equiv |\mathrm{\Delta }{\mathit{x}}_{\perp }|$. In the calculations depicted in different panels, the ${\varphi }_{\mathit{Q}}$’s were evolved with two different $D_0$’s until freeze-out at two different $T_f$’s, with the inverse Fourier transform to obtain $\tilde{\varphi }({\mathit{x}}_{\perp })$ performed at $T_f$. In all panels, we have chosen a trajectory with ${\xi }_{\max }=3\mathrm{fm}$. The three $r$ values depicted via the colored curves correspond to three $r$ values on the freeze-out surface in the lab frame. The black dashed and dotted curves are the equilibrium curves at $T=T_f$ and $T=T_c$ respectively. The dashed brown curve is the (noncritical) equilibrium curve corresponding to $\xi ={\xi }_0$.

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  Figure 13

  Normalized measure of the fluctuations in proton multiplicity, ${\tilde{\omega }}_p=\frac{{\omega }_p}{{\omega }_p^{\mathrm{nc}}}$, as a function of the maximum equilibrium correlation length along the system trajectory, which is to say as a function of how closely the trajectory passes the critical point. As $D_0\to \infty$, the ${\tilde{\omega }}_p$’s approach their equilibrium values shown in panel (c).

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  Figure 14

  Normalized measure of the fluctuations in pion multiplicity, ${\tilde{\omega }}_{\pi }=\frac{{\omega }_{\pi }}{{\omega }_{\pi }^{\mathrm{nc}}}$, as a function of the maximum equilibrium correlation length along the system trajectory, which is to say as a function of how closely the trajectory passes the critical point. As $D_0\to \infty$, the ${\tilde{\omega }}_{\pi }$’s approach their equilibrium values shown in panel (c). The definition of the normalized measure of fluctuations $\tilde{\omega }$ is such that it is species-independent in equilibrium, meaning that panel (c)here is identical to panel (c)in Fig.13.

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  Figure 15

  Panel (a): Normalized $\varphi$ as a function of $Q$ evolved according to model H dynamics with two values of $D_0$, plotted at freeze-out $\tau ={\tau }_f$, corresponding to an equilibrium temperature of $T_f({\tau }_f)=140\mathrm{MeV}$. The solid and dashed curves were obtained from the full solution (b3) for ${\varphi }_{\mathbf{Q}}$ and its truncated polynomial expansion (b14) respectively. Panel (b): Normalized fluctuation measure observable (rapidity space correlator) for protons ${\tilde{C}}_p(\mathrm{\Delta }y)$ obtained with the full form (solid) and truncated form (dashed) of ${\varphi }_{\mathbf{Q}}$. The qualitative and even semiquantitative agreement between the same colored curves in the right plot indicates that the low-$Q$ modes contribute significantly to the variance of particle multiplicities. In obtaining these plots, ${\xi }_{\max }$ was set to 3fm and the fluctuations at ${\tau }_i=1\mathrm{fm}$ were initialized to their equilibrium value at $\tau ={\tau }_i=1\mathrm{fm}$ with $T_i({\tau }_i)=235\mathrm{MeV}$.

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  Figure 16

  $\text{Hydro}+$ fluctuation measure ${\varphi }_{\mathbf{Q}}$ evolved according to model A dynamics along two hydrodynamic flow lines passing through $r=r_i$ at initial time $\tau ={\tau }_i$, with $r_i=0.7\mathrm{fm}$ (top four panels) and 5fm (bottom four panels). Plots in the left (right) column are for ${\mathrm{\Gamma }}_0=1{\mathrm{fm}}^{-1}$ (${\mathrm{\Gamma }}_0=8{\mathrm{fm}}^{-1}$), with ${\xi }_{\max }=1\mathrm{fm}$ and ${\xi }_{\max }=3\mathrm{fm}$ in alternating rows. The solid (and dashed) curves are the ${\varphi }_{\mathit{Q}}$ (and ${\overline{\varphi }}_{\mathit{Q}}$), normalized to the zero mode of the noncritical fluctuations. The black, red and blue curves correspond to ${\varphi }_{\mathbf{Q}}$’s at the initial time ${\tau }_i$ and at the times when the equilibrium temperature reaches 160MeV and 140MeV respectively.

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  Figure 17

  The values of ${\varphi }_{\mathit{Q}}$ (suitably normalized) for three representative values of $Q$ (same for each column), and for two values ${\mathrm{\Gamma }}_0$ (same in top and bottom six panels) and ${\xi }_{\max }$ (same in alternating rows) as in Fig.16. The values of ${\varphi }_{\mathit{Q}}$ are taken along a fluid cell trajectory and plotted as a function of temperature, which is a monotonous function of time $\tau$ along the trajectory. The trajectory chosen for these plots begins at $r_i=r({\tau }_i)=1.8\mathrm{fm}$. The dashed and solid curves represent the equilibrium ${\overline{\varphi }}_{\mathit{Q}}$ and nonequilibrium ${\varphi }_{\mathit{Q}}$, respectively.

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  Figure 18

  The $\text{Hydro}+$ variable ${\varphi }_{\mathit{Q}}$ (normalized to its value at $Q=0$ away from the critical point, where $\xi ={\xi }_0$) at freeze-out evolved with ${\mathrm{\Gamma }}_0=1{\mathrm{fm}}^{-1}$ (upper panels) and $8{\mathrm{fm}}^{-1}$ (lower panels) and with ${\xi }_{\max }=3\mathrm{fm}$. The left (right) panels show results for evolution until the decreasing temperature has reached a higher (lower) freeze-out temperature. The blue, red and purple curves show the values of ${\varphi }_{\mathit{Q}}$ at different points on the freeze-out hypersurface, characterized by the radial coordinate $r$. The black dashed and dotted curves are the equilibrium curves at $T=T_f$ and $T=T_c$ respectively. The dashed brown curve is the (noncritical) equilibrium curve corresponding to $\xi ={\xi }_0$.

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  Figure 19

  $\tilde{\varphi }\times \mathrm{\Delta }{x}^2$, the measure of fluctuations of $\widehat{s}$ described by the correlator $⟨\delta \widehat{s}(x_{+})\delta \widehat{s}(x_{-})⟩$, at freeze-out as a function of the spatial separation between the points $\mathrm{\Delta }x\equiv |\mathrm{\Delta }{\mathit{x}}_{\perp }|$. In the calculations depicted in different panels, the ${\varphi }_{\mathit{Q}}$’s were evolved with two different ${\mathrm{\Gamma }}_0$’s until freeze-out at two different $T_f$’s, with the inverse Fourier transform to obtain $\tilde{\varphi }({\mathit{x}}_{\perp })$ performed at $T_f$. In all panels, we have chosen a trajectory with ${\xi }_{\max }=3\mathrm{fm}$. The three $r$ values depicted via the colored curves correspond to three $r$ values on the freeze-out surface in the lab frame. The black dashed and dotted curves are the equilibrium curves at $T=T_f$ and $T=T_c$ respectively. The dashed brown curve is the (noncritical) equilibrium curve corresponding to $\xi ={\xi }_0$.

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  Figure 20

  Normalized measure of the fluctuations in proton multiplicity, ${\tilde{\omega }}_p=\frac{{\omega }_p}{{\omega }_p^{\mathrm{nc}}}$, as a function of the maximum equilibrium correlation length along the system trajectory, which is to say as a function of how closely the trajectory passes the critical point. As ${\mathrm{\Gamma }}_0\to \infty$, the ${\tilde{\omega }}_p$’s approach their equilibrium values shown in panel (c).

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  Figure 21

  Normalized measure of the fluctuations in pion multiplicity, ${\tilde{\omega }}_{\pi }=\frac{{\omega }_{\pi }}{{\omega }_{\pi }^{\mathrm{nc}}}$, as a function of the maximum equilibrium correlation length along the system trajectory, which is to say as a function of how closely the trajectory passes the critical point. As ${\mathrm{\Gamma }}_0\to \infty$, the ${\tilde{\omega }}_{\pi }$’s approach their equilibrium values shown in panel (c).

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