Scaling properties at freeze-out in relativistic heavy-ion collisions

Identified charged pion, kaon, and proton spectra are used to explore the system size dependence of bulk freeze-out properties in $\mathrm{Cu}+\mathrm{Cu}$ collisions at $\sqrt{s_{\mathit{NN}}}=200$ and 62.4 GeV. The data are studied with hydrodynamically motivated blast-wave and statistical model frameworks in order to characterize the freeze-out properties of the system. The dependence of freeze-out parameters on beam energy and collision centrality is discussed. Using the existing results from Au $+$ Au and $\mathit{pp}$ collisions, the dependence of freeze-out parameters on the system size is also explored. This multidimensional systematic study furthers our understanding of the QCD phase diagram revealing the importance of the initial geometrical overlap of the colliding ions. The analysis of $\mathrm{Cu}+\mathrm{Cu}$ collisions expands the system size dependence studies from $\mathrm{Au}+\mathrm{Au}$ data with detailed measurements in the smaller system. The systematic trends of the bulk freeze-out properties of charged particles is studied with respect to the total charged particle multiplicity at midrapidity, exploring the influence of initial state effects.

Figure 1

The left panel shows the truncated mean ionization energy loss ($\langle \mathit{dE}/\mathit{dx}\rangle$) in the TPC as a function of transverse momentum for positively charged tracks from 200 GeV $\mathrm{Cu}+\mathrm{Cu}$ collisions. The right panel shows $Z(\pi )$, the logarithm of the measured $\langle \mathit{dE}/\mathit{dx}\rangle$ divided by the theoretical expectation for energy loss of charged pions, for $0.40<p_T<0.45$ GeV/$c$. Also shown is an example four-Gaussian fit that is used to extract the raw yields for different species.

Figure 2

Estimated fraction of background protons in the raw proton sample as function of transverse momentum, for the most central $\sqrt{s_{\mathit{NN}}}=200$ GeV $\mathrm{Cu}+\mathrm{Cu}$ collisions. No strong centrality or energy dependence for this correction was observed for all $\mathrm{Cu}+\mathrm{Cu}$ data available.

Figure 3

The top row shows negatively charged pion (leftmost column), kaon (center), and antiproton (right) spectra from $\mathrm{Cu}+\mathrm{Cu}$ collisions at $\sqrt{s_{\mathit{NN}}}=200$ GeV. Six centrality classes are shown as dark (central $0\%$–$10\%$) to light ($50\%$–$60\%$) shades. Spectra for 62.4 GeV $\mathrm{Cu}+\mathrm{Cu}$ are shown on the bottom row. Statistical and systematic errors [which do not exceed $7\%$ (${\pi }^{-}$, $\overline{p}$) and $13\%$ ($K^{-}$)] are smaller than the symbol size.

Figure 4

The top row shows positively charged pion (leftmost column), kaon (center), and proton (right) spectra from $\mathrm{Cu}+\mathrm{Cu}$ collisions at $\sqrt{s_{\mathit{NN}}}=200$ GeV. Six centrality classes are shown as dark (central $0\%$–$10\%$) to light ($50\%$–$60\%$) shades. Spectra for 62.4 GeV $\mathrm{Cu}+\mathrm{Cu}$ are shown on the bottom row. Statistical and systematic errors [which do not exceed $7\%$ (${\pi }^{+}$,$p$) and $13\%$ ($K^{+}$)] are smaller than the symbol size.

Figure 5

Comparison between the $10\%$ most central $\mathrm{Cu}+\mathrm{Cu}$ collision data (symbols) at $\sqrt{s_{\mathit{NN}}}=200$ GeV and the corresponding blast-wave-model fit (dashed line) to ${\pi }^{+}$ (left), ${\mathrm{K}}^{+}$ (center), and proton (right) spectra—note that the fit is performed simultaneously across species. The pion data points below $p_T=0.5$ GeV/$c$ were not included in the blast-wave fits to reduce the effect of resonance decays. A Bose-Einstein fit to the pion spectra over the entire fiducial range is also shown. The lower panels illustrate the quality of the fits by showing the difference between the measured points and the fit expressed as the number of standard deviations.

Figure 6

Mean transverse momentum as a function of charged hadron multiplicity at midrapidity for pions, kaons, and antiprotons. For comparison, the mean-$p_T$ values for $\mathrm{Au}+\mathrm{Au}$ data are shown as bands. Filled (open) symbols or bands depict data at $\sqrt{s_{\mathit{NN}}}=200$ GeV (62.4 GeV). Error bars represent statistical and systematic uncertainties added in quadrature.

Figure 7

Integrated yields at midrapidity for pions, kaons, and antiprotons as a function of the charged particle density (${\mathit{dN}}_{\mathrm{ch}}/d\eta$), which is used as a measure of centrality. For comparison, $\mathrm{Au}+\mathrm{Au}$ data are shown as bands. Filled (open) points or bands depict data at $\sqrt{s_{\mathit{NN}}}=200$ GeV (62.4 GeV). Error bars represent statistical and systematic uncertainties added in quadrature.

Figure 8

Integrated particle yield ratios at $\sqrt{s_{\mathit{NN}}}=200$ GeV (closed symbols) and 62.4 GeV (open) for $\mathrm{Cu}+\mathrm{Cu}$ (black) and $\mathrm{Au}+\mathrm{Au}$ collisions (gray bands) versus ${\mathit{dN}}_{\mathrm{ch}}/d\eta$ at midrapidity. Error bars represent statistical and systematic uncertainties added in quadrature.

Figure 9

Enhancement factors for negatively charged pions (left), kaons (center), and antiprotons (right) as function of $N_{\mathrm{part}}$ in $\sqrt{s_{\mathit{NN}}}=200$ $\mathrm{Cu}+\mathrm{Cu}$ and $\mathrm{Au}+\mathrm{Au}$ collisions. Error bars represent statistical and systematic uncertainties on the A $+$ A measurements added in quadrature. The shaded bands depict model uncertainties on number of participants calculation. The bands on the left show uncertainties from the $\mathit{pp}$ measurements that are correlated for all data points.

Figure 10

Comparison of kinetic freeze-out properties obtained from fits to $\mathrm{Cu}+\mathrm{Cu}$ (symbols) and $\mathrm{Au}+\mathrm{Au}$ (bands) collision data at $\sqrt{s_{\mathit{NN}}}=200$ (closed symbols or bands) and 62.4 GeV (open). The kinetic freeze-out temperature $T_{\mathrm{kin}}$ is shown versus flow velocity $\beta$ and multiplicity in panels (a) and (b), respectively (more central collisions are to the right side of each plot). Panel (b) also shows the multiplicity dependence of the chemical freeze-out temperature $T_{\mathrm{ch}}$ (square symbols). Panel (c) shows the multiplicity dependence of the average radial flow velocity.

Figure 11

Comparison of the spectral shape between $\mathrm{Cu}+\mathrm{Cu}$ and $\mathrm{Au}+\mathrm{Au}$ data at $\sqrt{s_{\mathit{NN}}}=200$ GeV. Centrality classes are chosen with a similar average charged hadron multiplicity at midrapidity. Pion (left), kaon (center), and antiproton (right) spectra are shown for $10\%$–$20\%$ central ($40\%$–$50\%$) $\mathrm{Cu}+\mathrm{Cu}$ (symbols) compared to $40\%$–$50\%$ midperipheral ($60\%$–$70\%$) $\mathrm{Au}+\mathrm{Au}$ (lines).

Figure 12

The upper panel shows statistical-model-fit predictions (gray lines) for the measured particle ratios (circles) from central 200-GeV $\mathrm{Cu}+\mathrm{Cu}$ collisions. The lower panel illustrates the fit quality by showing the difference between the measured data and the model prediction in terms of the number of standard deviations ($N_{\sigma }$) determined by systematic (data) uncertainty.

Figure 13

Baryon and strangeness chemical potentials ${\mu }_{\mathrm{B}}$ and ${\mu }_{\mathrm{S}}$ as a function of ${\mathit{dN}}_{\mathrm{ch}}/d\eta$ for 200 and 62.4 GeV in $\mathrm{Cu}+\mathrm{Cu}$ (symbols) and $\mathrm{Au}+\mathrm{Au}$ collisions (bands).

Figure 14

Left: chemical freeze-out temperature $T_{\mathrm{ch}}$ as function of the baryon chemical potential ${\mu }_{\mathrm{B}}$ derived for central $\mathrm{Au}+\mathrm{Au}$ ($0\%$–$5\%$ for 200 and 62.4 GeV [7] and $0\%$–$10\%$ for 9.2 GeV [16]) and $\mathrm{Cu}+\mathrm{Cu}$ ($0\%$–$10\%$) collisions. For comparison, results for minimum-bias $\mathit{pp}$ collisions at 200 GeV are also shown along with additional heavy-ion data points compiled for lower collision energies [22]. The dashed line represents a common fit to all available heavy-ion data described in the text. Right: strangeness suppression factor ${\gamma }_{\mathrm{S}}$ as a function of ${\mathit{dN}}_{\mathrm{ch}}/d\eta$ for 200 and 62.4 GeV in $\mathrm{Cu}+\mathrm{Cu}$ (symbols) and $\mathrm{Au}+\mathrm{Au}$ collisions (bands).