Azimuthal anisotropy in Au+Au collisions at $\sqrt{s_{\mathit{NN}}}=200\mathrm{GeV}$

The results from the STAR Collaboration on directed flow ($v_1$), elliptic flow ($v_2$), and the fourth harmonic ($v_4$) in the anisotropic azimuthal distribution of particles from Au+Au collisions at $\sqrt{s_{\mathit{NN}}}=200\mathrm{GeV}$ are summarized and compared with results from other experiments and theoretical models. Results for identified particles are presented and fit with a blast-wave model. Different anisotropic flow analysis methods are compared and nonflow effects are extracted from the data. For $v_2$, scaling with the number of constituent quarks and parton coalescence are discussed. For $v_4$, scaling with $v_2^2$ and quark coalescence are discussed.

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Figure 1

(Color online) The event plane resolutions as a function of centrality for $v_{k2}\{{\mathrm{EP}}_2\}$.

Figure 2

(Color online) Directed flow of charged hadrons as a function of pseudorapidity. The measurements of $v_1\{{\mathrm{EP}}_1,{\mathrm{EP}}_2\}$ (circles; centrality 20–60%) agree with the published results of $v_1\{3\}$ (stars; centrality 10–70%).

Figure 3

(Color online) The product of $v_1^2$ and $v_2$. The shaded band is the mean value of this quantity with its error, averaged over centralities 20–60%. Because this quantity is positive, elliptic flow is measured to be in plane.

Figure 4

(Color online) Charged hadron $v_2$ vs. $p_t$ for the centrality bins (bottom to top) 5 to 10% and in steps of 10% starting at 10, 20, 30, 40, 50, 60, and 70 up to 80%. The solid lines are blast-wave fits.

Figure 5

(Color online) $v_2$ vs. $p_t$ for charged hadrons from 0–50% centrality collisions in comparison to data from PHOBOS [29]. The line is a polynomial fit to the STAR data. The gray error boxes represent the PHOBOS systematic errors. The bottom panel shows the ratio of the PHOBOS data to the polynomial fit.

Figure 6

(Color online) $v_2\{4\}$ vs. $p_t$ for identified particles in the 20–60% centrality range. The solid lines are hydrodynamic calculations [31].

Figure 7

(Color online) $v_2\{2\}$ vs. $p_t$ for charged pions, charged kaons, and antiprotons for the centrality bins (bottom to top) 5–10% and in steps of 10% starting at 10, 20, 30, 40, 50, 60, and 70 up to 80%. The solid lines are blast-wave fits.

Figure 8

(Color online) $v_2$ vs. $p_t$ for particles identified in the RICH detector from minimum bias collisions. The lines are hydrodynamic calculations [31] for pions (upper line), kaons (middle line), and protons (lower line).

Figure 9

(Color online) $v_2$ vs. $p_t$ for neutral and charged kaons for minimum bias collisions. The $K_S^0$ values are from Ref. [20]. The hydrodynamic model line is from Ref. [31]. The insert expands the low $p_t$ region to make the kaons from $\mathit{dE}/\mathit{dx}$ more visible.

Figure 10

(Color online) $v_2$ vs. $p_t$ for strange particles from minimum bias collisions. The STAR $K_S^0$ and $\Lambda +\overline{\Lambda }$ values are from Ref. [20]. The PHENIX data are from Ref. [32]. The hydro calculations are from Ref. [60].

Figure 11

(Color online) $v_2$ vs. $p_t$ for charged pions, charged kaons, and antiprotons from minimum bias collisions in comparison to similar data from PHENIX. The lines are polynomial fits to the STAR data. The bottom panel shows the ratio of the PHENIX data to the polynomial fits.

Figure 12

(Color online) Charged-hadron azimuthal correlations vs. $p_t$ in Au+Au collisions (squares) as a function of centrality (peripheral to central from left to right) compared to minimum bias azimuthal correlations in $p+p$ collisions (circles) and $d+\mathrm{Au}$ collisions (triangles). The Au+Au and $p+p$ data are from Ref. [6].

Figure 13

(Color online) The top panel shows the charged hadron scalar product vs. $p_t$ for different centrality classes in $d+\mathrm{Au}$ collisions, and minimum bias $p+p$ collisions. The Au+Au result is put there for a reference. The bottom panel shows the ratio of scalar product from $d+\mathrm{Au}$ collisions to minimum bias $p+p$ collisions for three different centrality classes.

Figure 14

(Color online) The ratio of the scalar product from the Au side ($-1.0<\eta <-0.5$) to that from deuteron side ($0.5<\eta <1.0)\hspace{1em}\mathrm{vs}.\hspace{1em}{p}_t$ from minimum bias collisions.

Figure 15

(Color online) Charged-hadron $v_2$ vs. η at $\sqrt{s_{\mathit{NN}}}=200$ GeV for STAR minimum bias (stars) and PHOBOS [29] midcentral (15–25%) centrality (circles). The open circles are PHOBOS data reflected about midrapidity.

Figure 16

(Color online) Charged hadron $v_2$ vs. η for 20–70% centrality collisions, including the FTPC regions. The open circles are $v_2\{2\}$, the filled squares are $v_2\{4\}$ and the stars are $v_2\{2\}$ for only FTPC particles, not using the main TPC particles.

Figure 17

(Color online) Charged hadron $v_2\{2\}$ vs. $p_t$ for centrality 20–70% in the FTPC ($2.5<\left|\eta \right|<4.0$) regions (stars) compared to $v_2\{4\}$ in the TPC + FTPCs (up-triangles). The down-triangles are Monte Carlo results fit to the up-triangles, and the solid circles include the FTPC momentum resolution.

Figure 18

(Color online) $v_2$ vs. $p_t$ for charged hadrons for different centrality bins. The standard reaction plane method is shown by open symbols and the modified reaction plane method by solid symbols.

Figure 19

(Color online) $v_2$ for the standard reaction plane method divided by $v_2$ for the modified reaction plane method vs. $p_t$ for charged hadrons in different centrality bins. Error bars are not shown as the same data set is used for both methods.

Figure 20

(Color online) $v_2$ vs. $p_t$ for charged hadrons from the modified reaction plane method (solid circles). Open circles (from Ref. [6], Fig. 2) are the two-particle cumulant results after subtracting the correlations measured in $p+p$ collisions. Error bars show statistical uncertainties only.

Figure 21

(Color online) $v_2$ scaled down by a factor of 2, and $v_4\{{\mathrm{EP}}_2\}$ vs. $p_t$ for charged hadrons from minimum bias events. Using a fit to the $v_2$ values, the lower solid line is the predicted $v_4$ needed to just remove the “peanut” waist (see text).

Figure 22

(Color online) Graphs of $v_n$ and $v_4/v_2^2$. The dashed lines are surface shell blast-wave fits with no ${\rho }_4$ or $s_4$ terms (see Sec. 6d) to the charged hadron $v_2$ minimum bias data. The resultant ratio $v_4/v_2^2$ is shown as the lower dashed line in the ratio graph (b). The solid lines are the fits with the addition of ${\rho }_4$ and $s_4$. The resultant ratio $v_4/v_2^2$ is shown as the solid curve in the ratio graph (b). The dotted line in the ratio graph (b) at 1.2 represents the average value of the data.

Figure 23

(Color online) A polar graph of the distribution $1+2v_2\cos (2\varphi )+2v_4\cos (4\varphi )$, where $\varphi$ is the azimuthal angle relative to the positive x axis. Plotted are the distributions for $v_2=16.5$% showing the waist, $v_4=3.8$% having a diamond shape, and both coefficients together.

Figure 24

(Color online) $v_4\{{\mathrm{EP}}_2\}$ vs. $p_t$ for charged hadrons for the 5–10% centrality bins (bottom to top) and in steps of 10% starting at 10, 20, 30, 40, 50, 60, and 70 up to 80%. The solid lines are blast wave fits.

Figure 25

(Color online) $v_4\{{\mathrm{EP}}_2\}$ and $v_2$ vs. $p_t$ for identified pions, antiprotons, $K_S^0$ , and $\Lambda +\overline{\Lambda }$ for minimum bias collisions. The dashed lines are at $1.2v_2^2$.

Figure 26

(Color online) The ratio $v_4/v_2^2$ vs. $p_t$ for identified pions. The dashed line is at $v_4/v_2^2=1.2$.

Figure 27

(Color online) Comparison of $v_2$ to $v_4\{{\mathrm{EP}}_2\}$ for charged hadrons from minimum bias collisions as a function of pseudorapidity. The fourth harmonic (squares) is consistent with zero at forward pseudorapidities but not at midrapidity. $v_2$ is shown by circles, scaled by a factor of 1/3 to fit on the plot. The larger data set available for the TPC only (triangles) confirms our measurement of $v_4\{{\mathrm{EP}}_2\}$ at midrapidities.

Figure 28

(Color online) High $p_t$ charged hadron $v_4\{3\}$ integrated for $3\le p_t\le 6/c\mathrm{GeV}$ vs. impact parameter b, compared to models of particle absorption: dashed curve is the hard shell, higher narrow band is Woods-Saxon, lower wider band is hard sphere. The bands have widths for absorption to match the observed range of yield suppression.

Figure 29

(Color online) Charged-hadron $v_2$ integrated over $p_t$ and η vs. centrality for the various methods described in the text. (b) The ratio of $v_2$ to the standard method $v_2$.

Figure 30

(Color online) Charged hadron $v_2\{2\},v_2\{4\}$, and $v_2\{6\}$ integrated values as a function of centrality.

Figure 31

(Color online) The nonflow parameter, $g_2$, as a function of centrality. The solid points are from the cumulant method. The open circles are from the q distribution method.

Figure 32

(Color online) (Upper panel) The ratio $v_2\{4\}/v_2\{2\}$ for charged hadrons as a function of centrality. The lines are a Monte Carlo Glauber model calculation of ${\varepsilon }_2\{4\}/{\varepsilon }_2\{2\}$. (Lower panel) The nonflow parameter, $g_2$, as a function of centrality. The lines are a Monte Carlo Glauber model calculation of $N_{\mathrm{WN}}(v_2/\varepsilon ){}^2({\varepsilon }_2^2\{2\}-{\varepsilon }_2^2\{4\})$. In both panels the solid lines assume nucleons, whereas the dotted lines assume quarks.

Figure 33

(Color online) (Top panel) Identified particle $v_2$ from minimum bias collisions. The vertical axis and horizontal axis have been scaled by the number of constituent quarks (n). Pions are not plotted. A polynomial curve is fit to the data. The possible systematic error is indicated by the gray band. (Bottom panel) The ratio of $v_2/n$ to the fitted curve.

Figure 34

(Color online) (Top panels) The $v_2$ of ${\pi }^{\pm },\overline{p},K_S^0$ , and $\Lambda +\overline{\Lambda }$ from three centrality bins (30–70%, 5–30%, and 0–5% of the collision cross section) scaled by the number of constituent quarks (n) vs. $p_t/n$. Polynomial curves are fit to the data excluding the pions. (Bottom panel) The ratios of $v_2/n$ to the fitted curves.

Figure 35

(Color online) $v_2(p_t)$ for charged $\pi ,K_S^0$ , $\overline{p}$, and $\Lambda +\overline{\Lambda }$ from minimum bias collisions. Hydrodynamic calculations [31, 60] are shown as dot-dashed lines. The possible systematic error is shown at the bottom.

Figure 36

(Color online) Charged π plots on the left and $\overline{p}$ plots on the right for $v_2$ for three centrality bins are shown as a function of $p_t$. The data are from centralities 40–50% (open triangles), 20–30% (open squares), and 0–5% (open circles). The corresponding results of a hydrodynamic calculation are shown as dot dashed lines, solid lines, and dashed lines, respectively. Plots on the top are for $\alpha =0$ and plots on the bottom are for $\alpha =0.02\hspace{1em}{\mathrm{fm}}^{-1}$. Here α determines the initial velocity kick for the hydrodynamic model calculation [61].

Figure 37

Schematic illustration of an elliptical subshell of the source. Here, the source is extended in the direction out of the event plane ($R_y>R_x$). Arrows represent the direction and magnitude of the flow boost. In this example, ${\rho }_2>0$. From Ref. [68].

Figure 38

(Color online) The blast-wave parameters ${\rho }_2,{\rho }_4,\varepsilon ,{\rho }_0$ and T plotted vs. centrality. The circles are for pions from a two-particle cumulant analysis, the squares for pions from a standard event-plane analysis, and the triangles for charged hadrons from a standard analysis. The lines are polynomial fits. In the middle panel the initial geometrical eccentricity is also plotted as a dashed line. The actual parameter values are available at http://www.star.bnl.gov/central/publications/.

Figure 39

(Color online) For centrality 20–30% we show $v_2\{2\}$ for pions (circles), $v_2$ for pions (squares), and $v_2$ for charged hadrons (triangles). The solid lines are blast-wave fits.