Relating measurable correlations in heavy ion collisions to bulk properties of equilibrated QCD matter

To compare theoretical calculations of thermal fluctuations of conserved quantities, such as charge susceptibilities or specific heat, to experimentally measured correlations and fluctuations in heavy ion collisions, one must confront the reality of changing conditions within the collision environment and transport of conserved quantities within the finite duration of the expansion and dissolution of the reaction. In previous work, fluctuations of conserved charges from lattice calculations, where charge is allowed to fluctuate within the designated volume consistent with the grand canonical ensemble, was linked to correlations in heavy ion collisions, which accounted for the finite time with which to transport absolutely conserved quantities. In this case details of the correlations were related to the evolution of the susceptibility. In this work, this paradigm is extended to compare fluctuations of momentum or energy to transverse energy correlations that can be measured in heavy ion collisions. The sensitivity of these correlations to the equation of state, viscosity, and diffusion is illustrated by considering simple models without transverse expansion. Only correlations in relative spatial rapidity are discussed here, but the prospects for extending these ideas to realistic calculations and for making realistic connections with experiment are discussed.

Figure 1

The source for generating correlations, see Eq. (16), is driven by $S\sim -{\partial }_{\tau }{\chi }_{ab}/s$. For charge correlations $\chi /s$ is shown in the upper panel from lattice gauge theory [22, 23] for the up-up, strange-strange, and up-strange charge fluctuations. The three filled points show susceptibilities for a hadron gas at $T=165$ MeV. Assuming a purely one-dimensional Bjorken expansion, temperature vs time graphs were generated by using the lattice equation of state. The lower panel shows the same susceptibilities plotted against time. The source function for up-up correlations has an initial surge corresponding to $\chi /s$ reaching its equilibrated alue at $\tau \approx 1\mathrm{fm}/c$. After that time there is a steady rise of ${\chi }_{uu}/s$ corresponding to a steady source function with a strengthening as one approaches $T_c$. In contrast the strange-strange susceptibility slightly weakens as one cools and the later-stage source function has the opposite sign.

Figure 2

The temperature vs time is shown for three equations of state in the lower panel where the matter undergoes a one-dimensional Bjorken expansion and reaches a temperature $T=155$ MeV at $\tau =12\mathrm{fm}/c$. The lattice equation of state is softer than that of a gas of massless partons, where the speed of sound is a constant $c_s^2=1/3$, and is stiffer than a bag-model equation of state, which features a strong first-order phase transition. The upper panel shows $TdT/d\tau$, which drives the source function for transverse momentum correlations, as seen in Eq. (28). The stiffer equations of state have significantly stronger sources throughout the evolution, which should lead to stronger correlations of transverse momentum at lower relative rapidity.

Figure 3

Transverse momentum correlations at a time ${\tau }_f=12\mathrm{fm}/c$ for a boost-invariant model where the temperature reached 155 MeV at ${\tau }_f$ and earlier temperatures were determined by entropy conservation, $s\sim 1/\tau$, combined with the lattice equation of state. The correlation spreads diffusively with the diffusion constant being determined by the shear viscosity, Eq. (27), and the source coming from Eq. (28). Larger viscosities broaden both the correlation from the initial state (lower blue dashed line) and that from the source function, $S_{xx}(\tau >{\tau }_0)$ with ${\tau }_0=1\mathrm{fm}/c$ (upper green dashed line). Results are also affected by the unknown width of the initial correlation at ${\tau }_0$. Results are shown for both narrow and broad initial widths, ${\sigma }_0=0.5,2.0$, and for three viscosities, $4\pi {\eta }_s/s=1,2$, and 4. The black lines indicate the net correlation.

Figure 4

Transverse momentum fluctuations for three equations of state. Results from a realistic lattice equation of state, a stiff equation of state with $c_s^2=1/3$, and from a bag model equation of state with a first-order phase transition differ significantly. The stiffer equations of state reach higher initial temperatures, and given that all three have fixed final temperatures, have stronger sources which provide sharper maxima near $\mathrm{\Delta }\eta =0$.

Figure 5

The transverse energy correlator, $C_{TT}^{\prime }$ at $\tau =12\mathrm{fm}/c$ due to point-like sources at ${\tau }_0=1\mathrm{fm}/c$ and propagating due to ideal hydrodynamics. Transverse expansion is neglected. The left-side panels show the correlation due to a source $S_{TT}$, with a strength that integrates to unity, whereas the right-side panels present the correlation from a source $S_{LL}$, also with unit strength. The upper and lower panels show the same results, but in the upper panels the scale is decreased and the numerical solutions are represented by symbols. Numerical calculations were performed by using the explicit equations of motion of the correlator from Eq. (59) and are represented by the green squares or blue circles in the upper panels and by lines in the lower panels. The correlator was also calculated from the explicit analytic solutions of the Green's functions, which are then convoluted with one another according to Eq. (65) and are represented by red lines in the upper panels. For the numerical solutions the source function pulse is represented by a finite-width Gaussian, which leads to a small discrepancy with the analytic method which assumes $\delta$-function sources. The sharp peaks come from the contribution where ${\eta }_1$ and ${\eta }_2$ come from the forward- and/or backward-moving pulses.

Figure 6

Correlations for the same conditions as in Fig. 5 were calculated with the effects of viscosity. The small viscosity, ${\eta }_s=s/4\pi$, spread the contribution from the $\delta$-function pulses, which for the previous calculation had appeared at $\eta =0$ and $\eta =2cln(\tau /{\tau }_0)$, noted by the thin vertical dashed line. The upper line (solid green) shows the correlation of the transverse energy due to point-like source for the transverse energy, $S_{TT}$, at ${\tau }_0$, whereas the lower line (dashed blue) provides the correlation from a point-like source of the longitudinal momentum.

Figure 7

Transverse energy correlations, scaled by the entropy per unit rapidity, as driven by the continuous source for $\tau >1\mathrm{fm}/c$, according to the self-consistent evolution equations for the correlations, Eq. (70). Results are shown for both the lattice equations of state (left-side panels) and for a constant sound speed (right-side panels). The softer lattice equation leads to stronger negative correlations at small relative spatial rapidity. Correlations are displayed for a final temperature of 155 MeV, assuming that temperature was reached at $12\mathrm{fm}/c$ (upper panels) to represent a more central collision, or at $6\mathrm{fm}/c$ (lower panels) to represent a less central collision. For the more central case, correlations extend to large relative rapidity.

Figure 8

Source functions for transverse energy correlations $S_{TT}$ and for transverse momentum correlations $S_{LL}$ are displayed as a function of time in the upper panels. The values are divided by the entropy per unit rapidity so that the values are independent of the transverse size. The transverse energy correlations $C_{TT}$ are driven by the previous history of both $S_{TT}$ and $S_{LL}$ due to the equations of hydrodynamics which allow sound waves to be generate by either fluctuations in energy or momentum. The left-side panels show the results using the lattice equation of state, whereas the right-side panels show results for a constant speed of sound. The function $S_{LL}$ is scaled by ${\left(c\tau \right)}^{-2}$ due to the fact that it is defined with an extra factor of ${\tau }^2$ and is divided by $c^2$ because the strength of the energy and momentum in sound pulses varies by a factor $c$. The lattice equation of state has much stronger negative contributions at later times, which explains the stronger dips seen in the correlations of Fig. 7. The difference between $T^2$ and between $T^2/c^2$, illustrated in the lower panels, for the two equations of state drive the source functions, as described in Table 1.

Figure 9

The solid green line shows Gaussian correlations coming from charge from source functions that have moved diffusively with a Gaussian spread of $\sigma =1$. Assuming a large number of sources, each contributing a with the same strength and width, we adjust the strength of the $N$ sources so that the resulting correlation in relative rapidity intersects the $\mathrm{\Delta }\eta$ axis at unity. Including the noise by mixing contributions between different sources adds noise to the correlation. One such calculation gives noise as described in Eq. (83). Averaging 10 such calculations is represented by the dashed red lines in the upper panels. The noise is highly correlated, so the averaging of 10 calculations is repeated several times to illustrate the variation of the curves due to a finite number of runs. In the lower panel 100 calculations are averaged, and the curves begin to reliably approach the correct answer. To obtain accuracy of better than 2%, roughly 1000 calculations, each with its own hydrodynamic evolution, are required.