Charged and strange hadron elliptic flow in $\mathrm{Cu}+\mathrm{Cu}$ collisions at $\sqrt{s_{\mathit{NN}}}=62.4$ and 200 GeV

We present the results of an elliptic flow, $v_2$, analysis of $\mathrm{Cu}+\mathrm{Cu}$ collisions recorded with the solenoidal tracker detector (STAR) at the BNL Relativistic Heavy Ion Collider at $\sqrt{s_{\mathit{NN}}}=62.4$ and $200$ GeV. Elliptic flow as a function of transverse momentum, $v_2$($p_T$), is reported for different collision centralities for charged hadrons $h^{\pm }$ and strangeness-ontaining hadrons $K_S^0$, $\Lambda$, $\Xi$, and $\varphi$ in the midrapidity region $\left|\eta \right|<1.0$. Significant reduction in systematic uncertainty of the measurement due to nonflow effects has been achieved by correlating particles at midrapidity, $\left|\eta \right|<1.0$, with those at forward rapidity, $2.5<\left|\eta \right|<4.0$. We also present azimuthal correlations in $p+p$ collisions at $\sqrt{s}=200$ GeV to help in estimating nonflow effects. To study the system-size dependence of elliptic flow, we present a detailed comparison with previously published results from $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{\mathit{NN}}}=200$ GeV. We observe that $v_2$($p_T$) of strange hadrons has similar scaling properties as were first observed in $\mathrm{Au}+\mathrm{Au}$ collisions, that is, (i) at low transverse momenta, $p_T<2\hspace{0.5em}\mathrm{GeV}/c$, $v_2$ scales with transverse kinetic energy, $m_T-m$, and (ii) at intermediate $p_T$, $2<p_T<4\hspace{0.5em}\mathrm{GeV}/c$, it scales with the number of constituent quarks, $n_q$. We have found that ideal hydrodynamic calculations fail to reproduce the centrality dependence of $v_2$($p_T$) for $K_S^0$ and $\Lambda$. Eccentricity scaled $v_2$ values, $v_2/\varepsilon$, are larger in more central collisions, suggesting stronger collective flow develops in more central collisions. The comparison with $\mathrm{Au}+\mathrm{Au}$ collisions, which go further in density, shows that $v_2/\varepsilon$ depends on the system size, that is, the number of participants $N_{\mathrm{part}}$. This indicates that the ideal hydrodynamic limit is not reached in $\mathrm{Cu}+\mathrm{Cu}$ collisions, presumably because the assumption of thermalization is not attained.

Figure 1

Invariant mass distributions for (a) $K_S^0$ ($1.2<p_T<1.4\hspace{0.5em}\mathrm{GeV}/c$), (b) $\varphi$ ($1.0<p_T<2.0\hspace{0.5em}\mathrm{GeV}/c$), (c) $\Lambda$ ($1.4<p_T<1.6\hspace{0.5em}\mathrm{GeV}/c$), and (d) $\Xi$ ($1.25<p_T<1.75\hspace{0.5em}\mathrm{GeV}/c$) in $\sqrt{s_{\mathit{NN}}}=200$ GeV $\mathrm{Cu}+\mathrm{Cu}$ $60\%$ most central collisions. The solid curves represent the fits to the invariant mass distributions: Gaussians plus fourth-order polynomials for $K_S^0$, $\Lambda$, and $\Xi$, and Breit-Wigner plus a linear function for $\varphi$. The dotted curves are the estimated backgrounds: the fourth-order polynomials for $K_S^0$ and $\Lambda$, a linear function for $\varphi$, and a rotation method described in the text for $\Xi$. For clarity, the invariant mass distributions for $K_S^0$, $\Lambda$, $\varphi$, and $\Xi$ are scaled by 1/50 000, 1/130 000, 1/5000 and 1/8000, respectively. The error bars are shown only for the statistical uncertainties.

Figure 2

Charged hadron azimuthal correlations as a function of $p_T$ in $\sqrt{s_{\mathit{NN}}}=200$ GeV $60\%$ most central $\mathrm{Cu}+\mathrm{Cu}$ collisions (solid squares) compared to those from $\sqrt{s_{\mathit{NN}}}=200$ GeV $p+p$ collisions (open squares). Flow vector calculated from (a) TPC tracks and (b) FTPC tracks. The error bars are shown only for the statistical uncertainties.

Figure 3

(a) Charged hadron $v_2$($p_T$) in $\sqrt{s_{\mathit{NN}}}=200$ GeV $0\%$–$60\%$ $\mathrm{Cu}+\mathrm{Cu}$ collisions. Open circles, solid circles, open squares, and solid squares represent the results of $v_2$ as function of $p_T$ measured by the TPC flow vector ($v_2\{\mathrm{TPC}\}$), the FTPC flow vector ($v_2\{\mathrm{FTPC}\}$), and the TPC and FTPC flow vectors with subtracting the azimuthal correlations in $p+p$ collisions ($v_2\{\mathit{AA}-\mathit{pp},\mathrm{TPC}\}$, $v_2\{\mathit{AA}-\mathit{pp},\mathrm{FTPC}\}$). (b) The ratio of the results for the various methods described in panel (a). The error bars are shown only for the statistical uncertainties.

Figure 4

Charged hadron $v_2\{\mathrm{FTPC}\}$ (solid circles) and $v_2\{\mathit{AA}-\mathit{pp},\mathrm{FTPC}\}$ (open circles) as a function of $p_T$ in $\sqrt{s_{\mathit{NN}}}=200$ GeV $\mathrm{Cu}+\mathrm{Cu}$ collisions for centrality bins: (a) $50\%$–$60\%$, (b) $40\%$–$50\%$, (c) $30\%$-$40\%$, (d) $20\%$–$30\%$, (e) $10\%$–$20\%$, and (f) $0\%$–$10\%$. The percentages refer to fractions of most central events. The error bars are shown only for the statistical uncertainties.

Figure 5

Ratios of $v_2\{\mathit{AA}-\mathit{pp},\mathrm{FTPC}\}/{\mathrm{v}}_2\{\mathrm{FTPC}\}$ for charged hadrons as a function of $p_T$ in $\sqrt{s_{\mathit{NN}}}=200$ GeV $\mathrm{Cu}+\mathrm{Cu}$ collisions for centrality bins: (a) $50\%$–$60\%$, (b) $40\%$–$50\%$, (c) $30\%$–$40\%$, (d) $20\%$–$30\%$, (e) $10\%$–$20\%$, and (f) $0\%$–$10\%$. The percentages refer to fractions of most central events.

Figure 6

Charged hadron $v_2$ integrated over $p_T$ and $\eta$ vs centrality for the various methods described in the text in $\sqrt{s_{\mathit{NN}}}=200$ and 62.4 GeV $\mathrm{Cu}+\mathrm{Cu}$ collisions. The error bars are shown only for the statistical uncertainties.

Figure 7

Charged hadron $v_2$ as a function of $p_T$ for $50\%$–$60\%$ (solid circles), $40\%$–$50\%$ (solid squares), $30\%$–$40\%$ (solid triangles), $20\%$–$30\%$ (open circles), $10\%$–$20\%$ (open squares), and $0\%$–$10\%$ (open triangles) in $\sqrt{s_{\mathit{NN}}}=200$ GeV and 62.4 GeV $\mathrm{Cu}+\mathrm{Cu}$ collisions. The error bars are shown only for the statistical uncertainties.

Figure 8

(a) Charged hadron $v_2$ as a function of $p_T$ in $\mathrm{Cu}+\mathrm{Cu}$ collisions. The results from $\sqrt{s_{\mathit{NN}}}=200$ GeV and 62.4 GeV are presented by open symbols and solid symbols, respectively. (b) Ratios of the $v_2$($p_T$) from $\sqrt{s_{\mathit{NN}}}=62.4$ GeV to 200 GeV. The error bars are shown only for the statistical uncertainties.

Figure 9

$v_2$ of $K_S^0$ (open circles), $\Lambda$ (solid squares), $\Xi$ (solid triangles), and $\varphi$ (open stars) as a function of $p_T$ for (a1) $0\%$–$60\%$, (a2) $0\%$–$20\%$, and (a3) $20\%$–$60\%$ and as a function of $m_T-m$ for (b1) $0\%$–$60\%$, (b2) $0\%$–$20\%$, and (b3) $20\%$–$60\%$. For comparison, the results from ideal hydrodynamic calculations [49, 50] are also shown. At a given $p_T$, from top to bottom, the curves represent $\pi$, $K$, $p$, $\varphi$, $\Lambda$, $\Xi$, and $\Omega$. When $p_T$ is converted to $m_T-m$, this mass hierarchy is reversed in the model results. All data are from $\sqrt{s_{\mathit{NN}}}=200$ GeV $\mathrm{Cu}+\mathrm{Cu}$ collisions. The error bars are shown only for the statistical uncertainties.

Figure 10

Same as Fig. 9(a1) and 9(b1), but expanded for the low $p_T$ and $m_T-m$ regions. The data points with large errors have not been plotted. At a given $p_T$, from top to bottom, the curves represent the ideal hydrodynamic calculations for $\pi$, $K$, $p$, $\varphi$, $\Lambda$, $\Xi ,$ and $\Omega$ [49, 50]. When $p_T$ is converted to $m_T-m$, this mass hierarchy is reversed in the model results. All data are from $0\%$ to $60\%$ $\mathrm{Cu}+\mathrm{Cu}$ collisions at $\sqrt{s_{\mathit{NN}}}=200$ GeV. The error bars are shown only for the statistical uncertainties.

Figure 11

$v_2/n_q$ versus $p_T/n_q$ [panels (a1)–(a3)] and ($m_T-m$)$/n_q$ [panels (b1)–(b3)], where $n_q$ is the number of constituent quarks in the hadron. The parametrization Eq. (13) fitted to the data is shown as the dashed curves. All data are from $\sqrt{s_{\mathit{NN}}}=200$ GeV $\mathrm{Cu}+\mathrm{Cu}$ collisions. The error bars are shown only for the statistical uncertainties.

Figure 12

$v_2$ scaled by participant eccentricity as a function of $p_T$ in $\sqrt{s_{\mathit{NN}}}=200$ and 62.4 GeV $\mathrm{Cu}+\mathrm{Cu}$ collisions. The error bars are shown only for the statistical uncertainties.

Figure 13

Centrality dependence of $v_2$ scaled by number of quarks and participant eccentricity [$v_2/$($n_q{\varepsilon }_{\mathrm{part}}\{2\}$)] for $K_S^0$ (left) and $\Lambda$ (right) as a function of ($m_T-m$)$/n_q$ in $0\%$–$10\%$, $10\%$–$40\%$, and $40\%$–$80\%$ $\mathrm{Au}+\mathrm{Au}$ collisions (open symbols) [31] and $0\%$–$20\%$ and $20\%$–$60\%$ $\mathrm{Cu}+\mathrm{Cu}$ collisions (solid symbols) at $\sqrt{s_{\mathit{NN}}}=200$ GeV. Curves are the results of $n_q$ scaling fits from Eq. (13) normalized by ${\varepsilon }_{\mathrm{part}}\{2\}$ to combined $K_S^0$ and $\Lambda$ for five centrality bins. At a given $p_T$, from top to bottom, the curves show a decreasing trend as $N_{\mathrm{part}}$ decreases. The error bars are shown only for the statistical uncertainties.

Figure 14

Number of quarks and participant eccentricity scaled $v_2$ [$v_2/$($n_q{\varepsilon }_{\mathrm{part}}\{2\}$)] of identified particles as a function of ($m_T-m$)$/n_q$ in $0\%$–$80\%$ $\mathrm{Au}+\mathrm{Au}$ collisions (open symbols) [31] and $0\%$–$60\%$ $\mathrm{Cu}+\mathrm{Cu}$ collisions (solid symbols) at $\sqrt{s_{\mathit{NN}}}=200$ GeV. Circles, squares, and triangles represent the data for $K_S^0$, $\Lambda$, and $\Xi$, respectively. The error bars are shown only for the statistical uncertainties.