Multiparticle azimuthal correlations for extracting event-by-event elliptic and triangular flow in Au + Au collisions at $\sqrt{s_{NN}}=200$ GeV

We present measurements of elliptic and triangular azimuthal anisotropy of charged particles detected at forward rapidity $1<\left|\eta \right|<3$ in Au + Au collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV, as a function of centrality. The multiparticle cumulant technique is used to obtain the elliptic flow coefficients $v_2\left\{2\right\},v_2\left\{4\right\},v_2\left\{6\right\}$, and $v_2\left\{8\right\}$, and triangular flow coefficients $v_3\left\{2\right\}$ and $v_3\left\{4\right\}$. Using the small-variance limit, we estimate the mean and variance of the event-by-event $v_2$ distribution from $v_2\left\{2\right\}$ and $v_2\left\{4\right\}$. In a complementary analysis, we also use a folding procedure to study the distributions of $v_2$ and $v_3$ directly, extracting both the mean and variance. Implications for initial geometrical fluctuations and their translation into the final-state momentum distributions are discussed.

Figure 1

Tracking efficiency and acceptance in the FVTX as a function of $p_T$. At $p_T=0.3$ GeV/$c$ the efficiency is 75% of its asymptotic value.

Figure 2

Tracking efficiency in the FVTX as a function of $\eta$ for two different $z$-vertex selections.

Figure 3

Example distribution of $Q^{\mathrm{north}}-Q^{\mathrm{south}}$ for the $n=2$ case corresponding to Au + Au collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV and centrality 20–30%. (a) The two-dimensional distribution. (b) The projection onto $Q_x$. (c) The projection onto $Q_y$. Shown for (b) and (c) are the extracted Gaussian widths ${\delta }_{2SE}$—the ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ values of the fits are 1.02 and 1.18 for (b) and (c), respectively.

Figure 4

Example distribution of $Q$ for the (a), (b) elliptic $n=2$ case and (c), (d) triangular $n=3$ case (bottom). (a) and (c) show the two-dimensional distribution. (b) and (d) show the one-dimensional distribution $\left|Q\right|$. The distribution corresponds to Au + Au collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV and centrality 20–30%. Also shown in (b), (d) is the best fit for this run Bessel-Gaussian truth distribution and the corresponding forward folded result (i.e., pushing the truth distribution through the response matrix).

Figure 5

Two-dimensional color plot of ${\chi }^2-{\chi }_{\min }^2$ values as a function of the two Bessel-Gaussian parameters, $v_n^{\mathrm{RP}}$ and ${\delta }_{v_n}$ for the 20%–30% centrality bin. (a) shows the second harmonic and (b) shows the third harmonic. Only ${\chi }^2$ values up to ${\chi }_{\min }^2+25$ are shown.

Figure 6

Centrality dependence of (a) $v_2\left\{2\right\}$ and (b) $v_2\left\{4\right\}$. (a) The red points indicate no pseudorapidity gap whereas the magenta points indicate a pseudorapidity gap of $\left|\mathrm{\Delta }\eta \right|>2.0$. (b) The black points indicate $v_2\left\{4\right\}$ with no pseudorapidity gap, the blue points indicate a two-subevent method with $\left|\mathrm{\Delta }\eta \right|>2.0$ but where some short-range pairs are allowed, and the red points indicate a two-subevent method with $\left|\mathrm{\Delta }\eta \right|>2.0$ where no short-range pairs are allowed.

Figure 7

Multiparticle $v_2$ as a function of centrality in Au + Au collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV. The magenta open diamonds indicate the $v_2\left\{SP\right\}$, the blue open squares indicate $v_2\left\{4\right\}$, the black open circles indicate $v_2\left\{6\right\}$, and the green filled diamonds indicate $v_2\left\{8\right\}$.

Figure 8

Cumulant method estimate of ${\sigma }_{v_2}/\langle v_2\rangle$ as a function of centrality in Au + Au collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV. The data are shown as black open squares. The same calculation as done in data is done in ampt, shown as a solid green line. Calculations of ${\sigma }_{{\varepsilon }_2}/\langle {\varepsilon }_2\rangle$ performed in the Monte Carlo Glauber model are shown as blue lines. The solid blue line is the Monte Carlo Glauber calculation done using the same estimate as the data, the dashed blue line is the direct calculation of the moments of the MC Glauber ${\varepsilon }_2$ distribution.

Figure 9

Centrality dependence of $v_3\{2,|\mathrm{\Delta }\eta |>2\}$ in Au + Au collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV, shown as magenta diamonds. The systematic uncertainty is indicated as a shaded magenta band. Also shown as black squares are $\sqrt{\langle v_3^2\rangle }$ as determined from the folding analysis, which is shown in the next section.

Figure 10

Centrality dependence of $c_3\left\{4\right\}$ for Au + Au collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV. (a) Calculations using both arms: $c_3\left\{4\right\}$ (black circles), $c_3{\left\{4\right\}}_{ab|ab}$ (blue diamonds), $c_3{\left\{4\right\}}_{aa|bb}$ (red squares), and comparison to STAR [36] (black stars). (b) Comparison of $c_3\left\{4\right\}$ determined using both arms (open symbols) and a single arm (closed symbols). Note that the open black circles are the same in (a) and (b).

Figure 11

Folding results for (a) $v_2$, (b) ${\sigma }_{v_2}$, and (c) ${\sigma }_{v_2}/\langle v_2\rangle$. The black lines above and below the points indicate the systematic uncertainties. The red (green) boxes indicate the statistical uncertainties at the 68.27% (95.45%) confidence level. In the case of $\langle v_2\rangle$, the statistical uncertainties at the 68.27% confidence level are too small to be seen, and the uncertainties at the 95.45% confidence level are visible but noticeably smaller than the marker size. Shown as blue squares are the same quantities as determined using the cumulant based calculation—these points are slightly offset in the $x$ coordinate to improve visibility.

Figure 12

Folding results for (a) $\langle v_3\rangle$, (b) ${\sigma }_{v_3}$, and (c) ${\sigma }_{v_3}/\langle v_3\rangle$. The black lines above and below the points indicate the systematic uncertainties. The red (green) boxes indicate the statistical uncertainties at the 68.27% (95.45%) confidence level. The ${\sigma }_{v_3}/\langle v_3\rangle$ values are all $\approx 0.52$, the apparent limiting value of this quantity for the Bessel-Gaussian distribution.

Figure 13

The observed ratio of the mean to the standard deviation (i.e., ${\sigma }_{v_n}/\langle v_n\rangle )$ of the Bessel-Gaussian as a function of the ratio of the two free parameters ${\delta }_{v_n}/v_n^{\mathrm{RP}}$. For values of ${\delta }_{v_n}>v_n^{\mathrm{RP}}$, the observed ${\sigma }_{v_n}/\langle v_n\rangle$ saturates at 0.52.

Figure 14

The observed ratio of the standard deviation to the mean (i.e., ${\sigma }_{v_n}/\langle v_n\rangle )$ for $n=2$ and $n=3$ as a function of centrality in Pb + Pb collisions at $\sqrt{s_{{}_{NN}}}=2.76$ TeV [10, 11]. Also shown are ${\sigma }_{{\varepsilon }_3}/{\varepsilon }_3$ values from Monte Carlo Glauber estimated using the small-variance limit cumulant technique (shown as the dashed line) and the direct calculation from the moments (shown as the solid line).

Figure 15

(a) Response matrix for test unfolding problem; (b) Selected right-singular values of the response matrix; (c) Picard plot for inverse problem $\widehat{A}{\vec{Q}}_2^{\mathrm{true}}={\vec{Q}}_2^{\mathrm{obs}}$, see text for details; (d) True, smeared, and unfolded $Q_2$ as determined using SVD with attenuation factors.

Figure 16

(a) Data distribution for $Q_2^{\mathrm{meas}}$ in Au + Au collisions at $\sqrt{s_{{}_{NN}}}=200$ GeV in the 20%–30% centrality class. Also shown is an assumed Bessel-Gaussian truth distribution and its resolution smeared result. (b) Data divided by the resolution smeared solution showing a good agreement within statistical uncertainties. (c) Picard plot for inverse problem with data $\widehat{A}{\vec{Q}}_2^{\mathrm{true}}={\vec{Q}}_2^{\mathrm{obs}}$, see text for details;