Beam-energy-dependent two-pion interferometry and the freeze-out eccentricity of pions measured in heavy ion collisions at the STAR detector

We present results of analyses of two-pion interferometry in $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{\mathit{NN}}}=7.7$, 11.5, 19.6, 27, 39, 62.4, and 200 GeV measured in the STAR detector as part of the BNL Relativistic Heavy Ion Collider Beam Energy Scan program. The extracted correlation lengths (Hanbury-Brown–Twiss radii) are studied as a function of beam energy, azimuthal angle relative to the reaction plane, centrality, and transverse mass ($m_T$) of the particles. The azimuthal analysis allows extraction of the eccentricity of the entire fireball at kinetic freeze-out. The energy dependence of this observable is expected to be sensitive to changes in the equation of state. A new global fit method is studied as an alternate method to directly measure the parameters in the azimuthal analysis. The eccentricity shows a monotonic decrease with beam energy that is qualitatively consistent with the trend from all model predictions and quantitatively consistent with a hadronic transport model.

Figure 1

The energy loss in the TPC, $dE/dx$. The colored region highlights the pions selected for this analysis. The gaps in the colored region at $\left|p\right|\approx 0.2\text{GeV}/c$ are caused by the cut to eliminate electrons from the analysis in the region where the electron and pion bands overlap. This example is from $0\%–5\%$ central, 27-GeV $\mathrm{Au}+\mathrm{Au}$ collisions.

Figure 2

Two-dimensional projections of a correlation function in the $q_{\text{o}}\text{−}{q}_{\text{s}}$, $q_{\text{s}}\text{−}{q}_{\text{l}}$, and $q_{\text{o}}\text{−}{q}_{\text{l}}$ planes for like-sign pions at midrapidity in $20\%–30\%$ central, 27-GeV collisions with $0.15<k_T<0.6\text{GeV}/c$. All scales are in $\text{GeV}/c$. In each case the third component is projected over $\pm 0.03\text{GeV}/c$. The emission angles relative to the event plane are within $\pm 22.5^{\circ }$ of the bin centers indicated along the right side. The tilt in the $q_{\text{o}}\text{−}{q}_{\text{s}}$ plane is clearly visible. Contour lines represent projections of the corresponding fit.

Figure 3

The event-plane resolutions for $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{\mathit{NN}}}=7.7$, 11.5, 19.6, 27, 39, 62.4, and 200 GeV as a function of collision centrality. The resolutions, computed using the TPC ($\left|\eta \right|<1$), enter into both the correction algorithm and the global fit method. Statistical errors are smaller than the symbols.

Figure 4

Examples of the angular oscillations of the HBT radii relative to the event plane from $20\%–30\%$ central, 19.6-GeV $\mathrm{Au}+\mathrm{Au}$ collisions for $0.15<k_T<0.6\text{GeV}/c$. Open circles show the radii before correction for finite-bin-width and event-plane resolution. Open cross symbols demonstrate that correcting these effects increases the oscillation amplitude. The corrected and uncorrected results are obtained with the HHLW fit method (see text) before and after the correction algorithm (Sec. 5b) is applied. The points at $0^{\circ }$ are repeated on the plot at ${180}^{\circ }$ for clarity. The solid bands show the Fourier decomposition directly extracted using a global fit (Sec. 5c) to all four angular bins; the width of the bands represent $1\sigma$ uncertainties from the fit. The value of $\lambda$ is consistent for the two methods.

Figure 5

Sample fit projections onto the $q_{\mathrm{out}}$ (top row), $q_{\mathrm{side}}$ (middle row), and $q_{\mathrm{long}}$ (bottom row) axes for four angular bins relative to the reaction plane. Results from the HHLW fit method and the global fit method are shown for direct comparison. These projections are from results for $10\%–20\%$ central, 19.6-GeV $\mathrm{Au}+\mathrm{Au}$ collisions with $0.15<k_T<0.6\text{GeV}/c$.

Figure 6

Energy dependence of the HBT parameters for central $\mathrm{Au}+\mathrm{Au}$, $\mathrm{Pb}+\mathrm{Pb}$, and $\mathrm{Pb}+\mathrm{Au}$ collisions at midrapidity and $\langle k_T\rangle \approx 0.22\text{GeV}/c$ [26, 27, 28, 29, 30, 31, 36]. The text contains discussion about variations in centrality, $k_T$, and analysis techniques between experiments. Errors on NA44, NA49, WA98, CERES, and ALICE points include systematic errors. The systematic errors for STAR points at all energies (from Table 2) are of similar size to the error bar for 39 GeV, shown as a representative example. Errors on other results are statistical only, to emphasize the trend. For some experiments the $\lambda$ value was not specified.

Figure 7

(Top) The difference between the squared transverse HBT radii are plotted as a function of the collision energy for STAR and ALICE measurements of the most central heavy ion collisions. (Bottom) The ratio of the out and side HBT radii for STAR and ALICE are plotted for the same collisions. In both cases, statistical errors are shown by solid error bars. Systematic errors are shown only for the data at $m_T=0.33$ GeV ($m_T=0.38$ GeV) for STAR (ALICE); systematic errors are common for all $m_T$ cuts. The systematic errors are driven by two-track cuts that are common to all STAR energies and so are drawn only for the $\sqrt{s_{\mathit{NN}}}=62.4$-GeV data.

Figure 8

The $\langle m_T\rangle$ dependence of $R_{\mathrm{out}}$, $R_{\mathrm{side}}$, and $R_{\mathrm{long}}$ for each energy and multiple centralities. Errors are statistical only. For 7.7 and 11.5 GeV, the results for $60\%–70\%$ centrality are excluded owing to lack of statistics.

Figure 9

The dependence of the HBT radii on multiplicity, ${\langle d{N}_{\mathrm{ch}}/d\eta \rangle }^{1/3}$, for $\langle k_T\rangle \approx 0.22\text{GeV}/c$ (left) and $\langle k_T\rangle \approx 0.39\text{GeV}/c$ (right). Results are for $\mathrm{Au}+\mathrm{Au}$ collisions at STAR, $\mathrm{Pb}+\mathrm{Au}$ at CERES [28], $\mathrm{Pb}+\mathrm{Pb}$ at ALICE [36], and $\mathrm{Si}+\mathrm{A}$ at E802 [25]. Errors are statistical only.

Figure 10

The beam energy dependence of the volume, $V={\left(2\pi \right)}^{3/2}{R}_{\mathrm{side}}^2{R}_{\mathrm{long}}$, of the regions of homogeneity at kinetic freeze-out in central $\mathrm{Au}+\mathrm{Au}$, $\mathrm{Pb}+\mathrm{Pb}$, and $\mathrm{Pb}+\mathrm{Au}$ collisions with $\langle k_T\rangle \approx 0.22\text{GeV}/c$ [26, 27, 28, 29, 30, 31, 32, 36]. The systematic errors for STAR points at all energies (from Table 2) are of similar size to the error bar for 39 GeV, shown as a representative example. Errors on other results are statistical only, to emphasize the trend. The PHOBOS points are offset in $\sqrt{s_{\mathit{NN}}}$ for clarity. The text contains some discussion about variations in centrality, $\langle k_T\rangle$, and analysis techniques between different experiments.

Figure 11

The lifetime, $\tau$, of the system as a function of beam energy for central $\mathrm{Au}+\mathrm{Au}$ collisions assuming a temperature of $T=0.12$ GeV at kinetic freeze-out. Statistical uncertainties from the fits are smaller than the data points. For all experiments except E895, which did not report systematic uncertainties, error bars indicate systematic uncertainties, based on reported systematic uncertainty on $R_{\mathrm{long}}$. The line extrapolates between the lowest and highest energy. The text contains a discussion about variations in centrality and analysis techniques between different experiments. The yellow band demonstrates the effect on $\tau$ of varying the assumed temperature by $\pm 0.02$ GeV.

Figure 12

Centrality dependence of the Fourier coefficients that describe azimuthal oscillations of the HBT radii, at midrapidity ($-0.5<y<0.5$), in 7.7-GeV collisions with $\langle k_T\rangle \approx 0.31\text{GeV}/c$. Open symbols are results using separate Gaussian fits to each angular bin, the HHLW method. Solid circles represent results using a single global fit to all angular bins to directly extract the Fourier coefficients. Crosses directly compare the azimuthally integrated radii and the zeroth-order Fourier coefficients. Error bars include only statistical uncertainties. The $0\%–5\%$ and $5\%–10\%$ global fit points have been excluded.

Figure 13

Centrality dependence of the Fourier coefficients that describe azimuthal oscillations of the HBT radii, at backward ($-1<y<-0.5$), forward ($0.5<y<1$), and mid- ($-0.5<y<0.5$) rapidity, in 7.7-GeV collisions with $\langle k_T\rangle \approx 0.31\text{GeV}/c$ using the global fit method. Error bars include only statistical uncertainties. The $0\%–5\%$ and two $5\%–10\%$ points have been excluded.

Figure 14

Centrality dependence of the Fourier coefficients that describe azimuthal oscillations of the HBT radii, at midrapidity ($-0.5<y<0.5$), in 11.5-GeV collisions with $\langle k_T\rangle \approx 0.31\text{GeV}/c$. The symbols have the same meaning as in Fig. 12. Error bars include only statistical uncertainties. The $0\%–5\%$ global fit point is excluded.

Figure 15

Centrality dependence of the Fourier coefficients that describe azimuthal oscillations of the HBT radii, at backward ($-1<y<-0.5$), forward ($0.5<y<1$), and mid- ($-0.5<y<0.5$) rapidity, in 11.5-GeV collisions with $\langle k_T\rangle \approx 0.31\text{GeV}/c$ using the global fit method. Error bars include only statistical uncertainties. The $0\%–5\%$ and two $5\%–10\%$ points have been excluded.

Figure 16

Centrality dependence of the Fourier coefficients that describe azimuthal oscillations of the HBT radii, at midrapidity ($-0.5<y<0.5$), in 19.6-GeV collisions with $\langle k_T\rangle \approx 0.31\text{GeV}/c$. The symbols have the same meaning as in Fig. 12. Error bars include only statistical uncertainties. The $0\%–5\%$ global fit point is excluded.

Figure 17

Centrality dependence of the Fourier coefficients that describe azimuthal oscillations of the HBT radii, at backward ($-1<y<-0.5$), forward ($0.5<y<1$), and mid- ($-0.5<y<0.5$) rapidity, in 19.6-GeV collisions with $\langle k_T\rangle \approx 0.31\text{GeV}/c$ using the global fit method. Error bars include only statistical uncertainties. The $0\%–5\%$ global fit point is excluded.

Figure 18

Centrality dependence of the Fourier coefficients that describe azimuthal oscillations of the HBT radii, at midrapidity ($-0.5<y<0.5$), in 27-GeV collisions with $\langle k_T\rangle \approx 0.31\text{GeV}/c$. The symbols have the same meaning as in Fig. 12. Error bars include only statistical uncertainties. The $0\%–5\%$ global fit method point is excluded.

Figure 19

Centrality dependence of the Fourier coefficients that describe azimuthal oscillations of the HBT radii, at backward ($-1<y<-0.5$), forward ($0.5<y<1$), and mid- ($-0.5<y<0.5$) rapidity, in 27-GeV collisions with $\langle k_T\rangle \approx 0.31\text{GeV}/c$ using the global fit method. Error bars include only statistical uncertainties. The $0\%–5\%$ global fit point is excluded.

Figure 20

Centrality dependence of the Fourier coefficients that describe azimuthal oscillations of the HBT radii, at midrapidity ($-0.5<y<0.5$), in 39-GeV collisions with $\langle k_T\rangle \approx 0.31\text{GeV}/c$. The symbols have the same meaning as in Fig. 12. Error bars include only statistical uncertainties. The $0\%–5\%$ global fit method point is excluded.

Figure 21

Centrality dependence of the Fourier coefficients that describe azimuthal oscillations of the HBT radii, at backward ($-1<y<-0.5$), forward ($0.5<y<1$), and mid- ($-0.5<y<0.5$) rapidity, in 39-GeV collisions with $\langle k_T\rangle \approx 0.31\text{GeV}/c$ using the global fit method. Error bars include only statistical uncertainties. The $0\%–5\%$ global fit point is excluded.

Figure 22

Centrality dependence of the Fourier coefficients that describe azimuthal oscillations of the HBT radii, at midrapidity ($-0.5<y<0.5$), in 62.4-GeV collisions with $\langle k_T\rangle \approx 0.31\text{GeV}/c$. The symbols have the same meaning as in Fig. 12. Error bars include only statistical uncertainties. The $0\%–5\%$ global fit method point is excluded.

Figure 23

Centrality dependence of the Fourier coefficients that describe azimuthal oscillations of the HBT radii, at backward ($-1<y<-0.5$), forward ($0.5<y<1$) and mid ($-0.5<y<0.5$) rapidity, in 62.4 GeV collisions with $\langle k_T\rangle \approx 0.31\text{GeV}/c$ using the global fit method. Error bars include only statistical uncertainties. The $0\%–5\%$ global fit point is excluded.

Figure 24

Centrality dependence of the Fourier coefficients that describe azimuthal oscillations of the HBT radii, at midrapidity ($-0.5<y<0.5$), in 200-GeV collisions with $\langle k_T\rangle \approx 0.31\text{GeV}/c$. The symbols have the same meaning as in Fig. 12. Error bars include only statistical uncertainties. The $0\%–5\%$ global fit method point is excluded.

Figure 25

Centrality dependence of the Fourier coefficients that describe azimuthal oscillations of the HBT radii, at backward ($-1<y<-0.5$), forward ($0.5<y<1$), and mid- ($-0.5<y<0.5$) rapidity, in 200-GeV collisions with $\langle k_T\rangle \approx 0.31\text{GeV}/c$ using the global fit method. Error bars include only statistical uncertainties. The $0\%–5\%$ global fit point is excluded.

Figure 26

Beam energy dependence of the $R_{\text{ol},0}^2$ cross term for forward and backward rapidity with $\langle k_T\rangle \approx 0.31\text{GeV}/c$. The yellow band shows estimated systematic uncertainty.

Figure 27

The eccentricity of the collisions at kinetic freeze-out, ${\varepsilon }_{\mathrm{F}}$, as a function of initial eccentricity relative to the participant plane, ${\varepsilon }_{\text{PP}}$, at midrapidity. All results are for $\langle k_T\rangle \approx 0.31\text{GeV}/c$. Error bars include only statistical uncertainties. The line has a slope of 1, indicating no change in shape. Points further below the line evolve more to a round shape.

Figure 28

The dependence of the kinetic freeze-out eccentricity of pions on collision energy in midcentral $\mathrm{Au}+\mathrm{Au}$ collisions (E895, STAR) and $\mathrm{Pb}+\mathrm{Au}$ collisions (CERES) for three rapidity regions and with $\langle k_T\rangle \approx 0.31\text{GeV}/c$. For clarity, the points for forward and backward rapidity from STAR are offset slightly. Error bars include only statistical uncertainties. Several ($2+1$)-dimensional hydrodynamical models and UrQMD calculations are shown. Model centralities correspond to the data. The trend is consistent with a monotonic decrease in eccentricity with beam energy. Systematic measurement uncertainty on $\epsilon$ is about the size of the data points (0.005) and independent of $\sqrt{s_{\mathit{NN}}}$. This systematic uncertainty is significantly smaller than statistical uncertainties and so is not drawn, to reduce clutter.