Evolution of charge fluctuations and correlations in the hydrodynamic stage of heavy ion collisions

Charge fluctuations for a baryon-neutral quark-gluon plasma have been calculated in lattice gauge theory. These fluctuations provide a well-posed rigorous representation of the quark chemistry of the vacuum for temperatures above $T_c\gtrsim 155\mathrm{MeV}$. Due to the finite lifetime and spatial extent of the fireball created in relativistic heavy ion collisions, charge-charge correlations can only equilibrate for small volumes due to the finite time required to transport charge. This constraint leads to charge correlations at finite relative position that evolve with time. The source and evolution of such correlations is determined by the evolution of the charge fluctuation and the diffusion constant for light quarks. Here, calculations are presented for the evolution of such correlations superimposed onto hydrodynamic simulations. Results are similar to preliminary measurements from STAR, but significant discrepancies remain.

Figure 1

The off-diagonal components of ${\chi }_{ab}$ vanish at high temperature as quarks evolve independently, but at low temperatures the combination of multiple quarks within a single hadron drives strong off-diagonal elements. For a fixed amount of entropy, an expanding volume element strongly increases the amount of charge as it reaches the hadronization region, again due to the fact that multiple charges occupy an individual hadron, even though the entropy per hadron in the hadron phase is similar to the entropy per quark in the plasma phase. For regions to the right of the dashed lines, lattice calculations were used, while to the left of the dashed lines are displayed the results for a hadron gas. The intermediate region used an interpolation between the two calculations, $155<T<175\mathrm{MeV}$.

Figure 2

The source function that generates the correlation function $C_{ab}^{\prime }\left(\mathrm{\Delta }\eta \right)$ is shown as a function of proper time $\tau$. The function is found by convoluting the susceptibility from lattice calculations with the hydrodynamic evolution as prescribed in Eq. (22). Multiplied by the bin width, $0.5\mathrm{fm}/c$, the values give the number of produced charge pairs within the time step that sample the evolution. The initial spike of the source function describes the correlation created at initial thermalization, ${\tau }_0$.

Figure 3

The correlation $C_{ab}^{\prime }$ as a function of relative rapidity at breakup. The $uu$, $dd$, and $ss$ components have an extended range because they originate from the original creation of charge at ${\tau }_0$. The off-diagonal elements, $ud$, $ud$, and $ds$ arise from the creation of mesons at hadronization. The reduction of strangeness as the system falls below $T_c$ explains the rise in $C_{ss}^{\prime }$ at small $\mathrm{\Delta }\eta$.

Figure 4

The kernel $K_{h{h}^{\prime };ab}$ translates correlations indexed by charge, $a,b=u,d,s$, to those indexed by hadron species, $h,h^{\prime }=\pi ,K,p$. After convoluting with the susceptibility ${\chi }_{ab}$ or the source function $S_{ab}$, due to the isospin symmetry of $\chi$ and $S$, only four independent combinations of $K_{h{h}^{\prime };ab}$ contribute, and are shown above. The right-side panel (b) differs from (a) by including the effects of decays as described in Eq. (27). Feed down decays are especially important for spreading the effect of correlations involving strange quarks amongst all hadron species. These values assumed an interface temperature of 155 MeV.

Figure 5

Hadronic correlations in coordinate space immediately after hadronization.

Figure 6

The susceptibility, ${\chi }_{ab}\left(T\right)$, from Fig. 1, and the source function $S_{ab}\left(\tau \right)$, from Fig. 2, are convoluted with the kernel $K_{h{h}^{\prime };ab}$ to define $S_{h{h}^{\prime }}$ and illustrate the temperatures and times at which correlations of specific hadron species are seeded.

Figure 7

Calculations for default parameters are shown with and without corrections for STAR acceptance. Applying the filter indeed brings the balance function into the neighborhood of preliminary measurements from STAR for central Au+Au collisions at ${\sqrt{s}}_{nn}=200\mathrm{GeV}$. Experimental results are reproduced at the $\approx 20\%$ level. The limited acceptance of STAR significantly restricts the insight one might possibly gain from a broader-acceptance device.

Figure 8

The left-side panels (a)–(f) show charge balance function calculations where the decoupling temperature is varied. For low temperatures, the fall in the baryon yield is accompanied by a negative contribution to the source function for $S_{aa}$ at late times. This then provides a dip in the balance function at small relative rapidity. Varying the width of the initial correlations, ${\sigma }_0$, broadens the correlations $C_{aa}^{\prime }$ at early times as seen in the central panels (g)–(l). This mainly affects the balance functions involving protons and kaons. Finally, in the right-side panels (m)–(r), increasing the diffusion constant also more strongly broadens the correlations driven by early sources, such as $B_{KK}$ and $B_{pp}$.

Figure 9

Spectra for the simple parametrization of the hydrodynamic history, described Eq. (A8), are calculated from a nearly analytic formula, Eq. (A11), and from reading the history from a mesh and applying the algorithm described here. The nearly analytic (lines) and the spectra calculated from a discretized mesh (points) agree well.