Centrality dependence of charged hadron and strange hadron elliptic flow from $\sqrt{s_{\mathit{NN}}}=200$ GeV Au+Au collisions

We present STAR results on the elliptic flow $v_2$ of charged hadrons, strange and multistrange particles from $\sqrt{s_{\mathit{NN}}}=200$ GeV Au+Au collisions at the BNL Relativistic Heavy Ion Collider (RHIC). The detailed study of the centrality dependence of $v_2$ over a broad transverse momentum range is presented. Comparisons of different analysis methods are made in order to estimate systematic uncertainties. To discuss the nonflow effect, we have performed the first analysis of $v_2$ with the Lee-Yang zero method for $K_S^0$ and Λ. In the relatively low $p_T$ region, $p_T⩽2\hspace{1em}\mathrm{GeV}/c$, a scaling with $m_T-m$ is observed for identified hadrons in each centrality bin studied. However, we do not observe $v_2(p_T)$ scaled by the participant eccentricity to be independent of centrality. At higher $p_T,2⩽p_T⩽6\hspace{1em}\mathrm{GeV}/c,v_2$ scales with quark number for all hadrons studied. For the multistrange hadron Ω, which does not suffer appreciable hadronic interactions, the values of $v_2$ are consistent with both $m_T-m$ scaling at low $p_T$ and number-of-quark scaling at intermediate $p_T$. As a function of collision centrality, an increase of $p_T$-integrated $v_2$ scaled by the participant eccentricity has been observed, indicating a stronger collective flow in more central Au+Au collisions.

Figure 1

Plots (a1)–(d1) represent the invariant mass distributions for $K_S^0(1.4⩽p_T⩽1.6\hspace{1em}\mathrm{GeV}/c),\Lambda (1.4⩽p_T⩽1.6\hspace{1em}\mathrm{GeV}/c),\Xi (2.3⩽p_T⩽2.6\hspace{1em}\mathrm{GeV}/c)$, and $\Omega (2.5⩽p_T\hspace{1em}⩽3.0\hspace{1em}\mathrm{GeV}/c)$, respectively, from $\sqrt{s_{\mathit{NN}}}=200$ GeV minimum bias (0–80%) Au+Au collisions. The dashed lines are the background distributions. The corresponding data for the $v_2$ distributions are shown in plots (a2)–(d2) as open circles. The thick-dashed, thin-dashed, and the dot-dashed lines represent the relative contributions of $v_2(\mathrm{Sig}),v_2(\mathrm{Bg})$, and $v_2(\mathrm{Sig}+\mathrm{Bg})$, respectively. For clarity, the invariant mass plots for $K_S^0,\Lambda ,\Xi ,\hspace{1em}\mathrm{and}\hspace{1em}\Omega$, are scaled by 1/50 000, 1/170 000, 1/2.5, and 1/3, respectively. The error bars are shown only for the statistical uncertainties.

Figure 2

Examples of the modulus of the second harmonic Lee-Yang zero generating functions plotted as a function of the imaginary axis coordinate $r$. The sum generating function is shown in (a) and the product generating function in (b). The vertical arrows indicate the positions of the first minimum, called $r_0$. Note that in (b) the horizontal scale does not go out as far because the calculations were terminated. All data are from $\sqrt{s_{\mathit{NN}}}=200$ GeV Au+Au collisions.

Figure 3

$v_2$ for charged hadrons from the Lee-Yang zero product generating function (solid circles) and from the event plane method (open circles), as a function of pseudorapidity. Both sets of data have been averaged over $p_T$ from 0.15 to 2.0 GeV/$c$ and centrality from 10 to 40% of $\sqrt{s_{\mathit{NN}}}=200$ GeV Au+Au collisions. For comparison, the PHOBOS data (10–40%) [35] (crosses) are also shown. The error bars are shown only for the statistical uncertainties.

Figure 4

(a) $v_2$ as a function of $p_T$ for charged hadrons with $\left|\eta \right|\hspace{1em}<1.0$ in 10–40% Au+Au collisions, at $\sqrt{s_{\mathit{NN}}}=200$ GeV, from the event plane method (open circles), four-particle cumulant method (solid squares), and Lee-Yang zero method (solid circles) with the sum generating function. (b) Ratios to the polynomial fit to $v_2${EP} for $v_2\{4\}/v_2${EP} and $v_2${LYZ}/$v_2${EP} as a function of transverse momentum. Error bars are shown only for the statistical uncertainties.

Figure 5

(a) $p_T$-integrated charged hadron $v_2$ in the TPC as a function of geometrical cross section. Shown are the event plane method ($v_2${EP}) (open circles), Lee-Yang zero method with sum generating function (solid circles), Lee-Yang zero method with product generating function (open stars), and 4-particle cumulant method ($v_2\{4\}$) (solid squares). For the TPC, $\left|\eta \right|<1.0$ was used, except for Lee-Yang zero method with Product Generating Function where the η limit went to 1.3. (b) $v_2$ divided by $v_2${EP}. All data are from $\sqrt{s_{\mathit{NN}}}=200$ GeV Au+Au collisions. Error bars are shown only for the statistical uncertainties.

Figure 6

$v_2$ as a function of $p_T$ for 10–40% centrality using event plane method (open circles), Lee-Yang zero method with sum generating function (solid circles), and η-subevent method (open crosses) for (a) Λ and (b) $K_S^0$. All data are from $\sqrt{s_{\mathit{NN}}}=200$ GeV Au+Au collisions. The ratios $v_2${LYZ}$/v_2${EP} and $v_2\{\eta \}/v_2${EP} are shown for (c) Λ and (d) $K_S^0$. $v_2${LYZ}$/v_2${EP} ratios for charged hadrons are shown as shaded bands. Error bars are shown only for the statistical uncertainties.

Figure 7

$v_2$ of $K_S^0$ (open circles), Λ (open squares), Ξ (filled triangles), and Ω (filled circles) as a function of $p_T$ for (a) 0–80%, (b) 40–80%, (c) 10–40%, and (d) 0–10% in Au+Au collisions at $\sqrt{s_{\mathit{NN}}}=200$ GeV. The error bars represent statistical uncertainties. The bands on the data points represent systematic uncertainties as discussed in the text. For comparison, pion (stars) and proton (filled squares) results are shown in (a). The systematic uncertainty of nonflow for $K_S^0$ and Λ for 10–40% (c) is plotted as a shaded band near 0. For comparison, results from ideal hydrodynamic calculations [36, 37] are shown: at a given $p_T$, from top to bottom, the lines represent the results for $\pi ,K,p,\Lambda ,\Xi$, and Ω.

Figure 8

Same as Fig. 7, but as a function of $m_T-m.$

Figure 9

Number-of-quark scaled $v_2(v_2/n_q)$ of identified particles vs $p_T/n_q$ in $(\mathrm{GeV}/c)$ (left column) and $(m_T-m)/n_q$ in $(\mathrm{GeV}/c^2)$ (right column). Dot-dot-dashed lines are the results of sixth-order polynomial fits to $K_S^0,\Lambda ,\Xi$, and Ω. The ratios of the data points over the fit are shown in panels (c) and (d) for $K_S^0,\pi ,p,\Lambda$, and Ξ, and in panels (e) and (f) for Ω and ϕ [16]. The error bars are shown only for the statistical uncertainties. Ideal hydrodynamic calculations for $\pi ,K,\Lambda ,\Xi$, and Ω are presented by solid lines, dashed lines, dot-dashed lines, dot-long-dashed lines, and dot-dot-dot-dashed lines, respectively. The ratio of hydrodynamic calculations over the fit are also shown for comparison in panels (c) and (d). The data are from minimum bias (0–80%) Au+Au collisions at $\sqrt{s_{\mathit{NN}}}=200$ GeV.

Figure 10

Centrality dependence of the number-of-quark scaled $v_2\hspace{1em}(v_2/n_q)$ of identified particles vs $p_T/n_q$ in $(\mathrm{GeV}/c)$ (left column) and ($m_T-m)/n_q$ in $(\mathrm{GeV}/c^2)$ (right column). The error bars are shown only for the statistical uncertainties. The sixth-order polynomial fits are shown as dot-dot-dashed-lines. Ideal hydrodynamic curves [37] are also plotted. All data are from $\sqrt{s_{\mathit{NN}}}=200$ GeV Au+Au collisions.

Figure 11

Centrality dependence of the $v_2/n_q$ ratio to a common polynomial fit vs $p_T/n_q$ in $(\mathrm{GeV}/c)$ (left column) and ($m_T-m)/n_q$ in $(\mathrm{GeV}/c^2)$ (right column). The error bars are shown only for the statistical uncertainties. Ideal hydrodynamic calculations over the same fit are also shown for comparison. Ideal hydrodynamic calculations for $\pi ,K,\Lambda ,\Xi$ and Ω are presented by solid lines, dashed lines, dot-dashed lines, dot-long-dashed lines, and dot-dot-dot-dashed lines, respectively. All data are from $\sqrt{s_{\mathit{NN}}}=200$ GeV Au+Au collisions.

Figure 12

$v_2$ scaled by the number of quarks $n_q$ and participant eccentricity ${\varepsilon }_{\mathrm{part}}\hspace{1em}[v_2/(n_q{\varepsilon }_{\mathrm{part}})]$ of identified particles (particle+antiparticle) vs (a) the scaled $p_T/n_q$ and (b) $(m_T-m)/n_q$ for three centrality bins. For comparison, ideal hydrodynamic model calculations [37] are shown as lines in (b). In (c), data from (b) are scaled by the integrated $v_2$ of each particle, instead of ${\varepsilon }_{\mathrm{part}}$. In (d), data from (b) are scaled by the integrated $v_2$ of all charged hadrons. The insert in (d) expands the low $m_T$ region. Error bars are shown only for statistical uncertainties. All data are from $\sqrt{s_{\mathit{NN}}}=200$ GeV Au+Au collisions.

Figure 13

Centrality dependence of $v_2/{\varepsilon }_{\mathrm{part}}$ vs number of participants for charged hadrons (crosses), $K_S^0$ (circles), ϕ (stars) [16], $\Lambda +\overline{\Lambda }$ (squares), and $\Xi +{\overline{\Xi }}^{+}$ (triangles). Both unidentified charged hadron and identified hadron $v_2$ were analyzed with the standard event plane method. All data are from $\sqrt{s_{\mathit{NN}}}=200$ GeV Au+Au collisions. The data points are displaced slightly horizontally for clarity. The statistical and systematic uncertainties are shown as bars and brackets, respectively. Ideal hydrodynamic model calculations are also shown as dashed lines [37, 46] for, from top to bottom, $\Omega ,\Xi ,\Lambda ,p,K$, and π.

Figure 14

$p_T$ dependence of the eccentricity-scaled $v_2$ ratios of (a) 62.4 GeV and (b) 17.2 GeV [43] over 200 GeV data for $K_S^0$ (circles) and $\Lambda +\overline{\Lambda }$ (squares). As indicated in (b), the SPS data points are from 5–25% and the RHIC data points are from 10–40%. Error bars are statistical only.

Figure 15

Centrality dependence of $v_2/(n_q{\varepsilon }_{\mathrm{part}})$ for identified hadrons (particle+antiparticle) from (a) 62.4 [51] and (b) 200 GeV Au+Au collisions. Error bars are statistical only. For the 10–40% centrality, the systematic uncertainty should be the same as in Fig. 7.