Harmonic decomposition of three-particle azimuthal correlations at energies available at the BNL Relativistic Heavy Ion Collider

We present measurements of three-particle correlations for various harmonics in Au+Au collisions at energies ranging from $\sqrt{s_{\mathrm{NN}}}=7.7$ to 200 GeV using the STAR detector. The quantity $\langle cos(m{\varphi }_1+n{\varphi }_2-(m+n){\varphi }_3)\rangle$, with $\varphi$ being the azimuthal angles of the particles is evaluated as a function of $\sqrt{s_{\mathrm{NN}}}$, collision centrality, transverse momentum, $p_T$, pseudorapidity difference, $\mathrm{\Delta }\eta$, and harmonics ($m$ and $n$). These data provide detailed information on global event properties such as the three-dimensional structure of the initial overlap region, the expansion dynamics of the matter produced in the collisions, and the transport properties of the medium. A strong dependence on $\mathrm{\Delta }\eta$ is observed for most harmonic combinations, which is consistent with breaking of longitudinal boost invariance. An interesting energy dependence is observed when one of the harmonics $m,n,$ or $m+n$ is equal to two, for which the correlators are dominated by the two-particle correlations relative to the second-harmonic event plane. These measurements can be used to constrain models of heavy-ion collisions over a wide range of temperature and baryon chemical potential.

Figure 1

The $\mathrm{\Delta }\eta$ dependence of $C_{1,1,2}$ scaled by $N_{\mathrm{part}}^2$ for nine centrality intervals with the three most central classes shown in the top panels and the three most peripheral in the bottom. The $N_{\mathrm{part}}$ values used for the corresponding centralities are 350.6, 298.6, 234.3, 167.6, 117.1, 78.3, 49.3, 28.2, and 15.7. In the panels on the left, $\mathrm{\Delta }\eta$ is taken between particles 1 and 2 while on the right it is between particles 1 and 3 (which is identical to 2 and 3 since $m=n=1$ for $C_{1,1,2}$). Data are from 200 GeV Au+Au collisions and for charged hadrons with $p_T>0.2\mathrm{GeV}/c, \left|\eta \right|<1$.

Figure 2

The $\mathrm{\Delta }\eta$ dependence of $C_{1,2,3}$ scaled by $N_{\mathrm{part}}^2$ for nine centrality intervals with the three most central classes shown in the top panels and the three most peripheral in the bottom. In the panels on the left, $\mathrm{\Delta }\eta$ is taken between particles 1 and 2 while on the right it is between particles 1 and 3. Data are from 200 GeV Au+Au collisions and for charged hadrons with $p_T>0.2\mathrm{GeV}/c, \left|\eta \right|<1$.

Figure 3

The $\mathrm{\Delta }\eta$ dependence of $C_{2,2,4}$ scaled by $N_{\mathrm{part}}^2$ for nine centrality intervals with the three most central classes shown in the top panels and the three most peripheral in the bottom. In the panels on the left, $\mathrm{\Delta }\eta$ is taken between particles 1 and 2 while on the right it is between particles 1 and 3 (which is identical to 2 and 3 since $m=n=2$ for $C_{2,2,4}$). Data are from 200 GeV Au+Au collisions and for charged hadrons with $p_T>0.2\mathrm{GeV}/c, \left|\eta \right|<1$.

Figure 4

The $\mathrm{\Delta }\eta$ dependence of $C_{2,3,5}$ scaled by $N_{\mathrm{part}}^2$ for nine centrality intervals with the three most central classes shown in the top panels and the three most peripheral in the bottom. In the panels on the left, $\mathrm{\Delta }\eta$ is taken between particles 1 and 2 while on the right it is between particles 2 and 3. Data are from 200 GeV Au+Au collisions and for charged hadrons with $p_T>0.2\mathrm{GeV}/c, \left|\eta \right|<1$.

Figure 5

The centrality dependence of the $C_{m,n,m+n}$ correlations scaled by $N_{\mathrm{part}}^2$ for charged hadrons with $p_T>0.2\mathrm{GeV}/c$ and $\left|\eta \right|<1$ from 200, 62.4, 39, and 27 GeV Au+Au collisions for $(m,n)=(1,1),(1,2),(1,3)$ (left), $(2,2),(2,3),(2,4)$ (center), and $(3,3),(3,4)$ (right). Systematic errors are shown as bands. All panels in the same row share the same scale but $C_{2,2,4}$ has been divided by a factor of 5 to fit on the panel. The labels in the top panels apply to all the panels in same column.

Figure 6

The same quantities as Fig. 5 but for the lower energy Au+Au collisions 19.6, 14.5, 11.5, and 7.7 GeV.

Figure 7

Three-particle azimuthal correlations $C_{1,1,2}$ scaled by $N_{\mathrm{part}}^2/p_{T,1}$ as a function of the first particles $p_T$ for 200 GeV Au+Au collisions for charged hadrons with $p_T>0.2\mathrm{GeV}/c$ and $\left|\eta \right|<1$. The top and bottom panels show the same quantity but for a different set of centrality intervals. Systematic errors are shown as solid lines enclosing the respective data points.

Figure 8

Three-particle azimuthal correlations $C_{1,2,3}$ scaled by $N_{\mathrm{part}}^2/p_T$ as a function of the $p_T$ using the $p_T$ of particle one (left panels) or of particle two (right panels) for 200 GeV Au+Au collisions. Data are for charged hadrons with $p_T>0.2\mathrm{GeV}/c$ and $\left|\eta \right|<1$. The top and bottom panels show the same quantity but for a different set of centrality intervals. Systematic errors are shown as solid lines enclosing the respective data points.

Figure 9

Three-particle azimuthal correlations $C_{2,2,4}$ scaled by $N_{\mathrm{part}}^2/p_{T,1}$ as a function of $p_{T,1}$ for 200 GeV Au+Au collisions. Data are for charged hadrons with $p_T>0.2\mathrm{GeV}/c$ and $\left|\eta \right|<1$. The top and bottom panels show the same quantity but for a different set of centrality intervals. Systematic errors are shown as solid lines enclosing the respective data points.

Figure 10

Three-particle azimuthal correlations $C_{2,3,5}$ scaled by $N_{\mathrm{part}}^2/p_T$ as a function of $p_T$ where the $p_T$ is taken for either particle one (left panels) or particle two (right panels) for 200 GeV Au+Au collisions. Data are for charged hadrons with $p_T>0.2\mathrm{GeV}/c$ and $\left|\eta \right|<1$. The top and bottom panels show the same quantity but for a different set of centrality intervals. Systematic errors are shown as solid lines enclosing the respective data points.

Figure 11

The centrality dependence of $C_{1,1,2}$ (left) and $C_{1,2,3}$ (right) scaled by $N_{\mathrm{part}}^2$ for a selection of energies.

Figure 12

The centrality dependence of $C_{2,2,4}$ (left) and $C_{2,3,5}$ (right) scaled by $N_{\mathrm{part}}^2$ for a selection of energies.

Figure 13

The $\sqrt{s_{{}_{\mathrm{NN}}}}$ dependence of $N_{\mathrm{part}}{C}_{m,n,m+n}/v_2$ for $(m,n)=(1,1)$ (top left), (1,2) (top right), (2,2) (bottom left) and (2,3) (bottom right) for three selected centrality intervals. In the bottom right panel, the lowest energy points for the $20–30\%$ and $30–40\%$ centrality intervals, having large uncertainties, are omitted for clarity. Statistical uncertainties are shown as vertical error bars while the systematic errors are shown as shaded regions or bands.

Figure 14

The $\sqrt{s_{{}_{\mathrm{NN}}}}$ dependence and centrality dependence of $N_{\mathrm{part}}{v}_1^2\left\{2\right\}$ (left) and $N_{\mathrm{part}}{v}_2^2\left\{2\right\}$ (right) after short-range correlations, predominantly from quantum and Coulomb effects, have been subtracted. For more details see Ref. [18]. The centrality intervals correspond to $0–5\%, 5–10\%, 10–20\%, 20–30\%, 30–40\%, 40–50\%, 50–60\%, 60–70\%$, and $70–80\%$. The $N_{\mathrm{part}}$ values used for the corresponding centralities are 350.6, 298.6, 234.3, 167.6, 117.1, 78.3, 49.3, 28.2, and 15.7 independent of energy.

Figure 15

The $\sqrt{s_{{}_{\mathrm{NN}}}}$ dependence and centrality dependence of $N_{\mathrm{part}}{v}_4^2\left\{2\right\}$ (left) and $N_{\mathrm{part}}{v}_5^2\left\{2\right\}$ (right) after short-range correlations, predominantly from quantum and Coulomb effects, have been subtracted. For more details see Ref. [18].