Measurement of event-plane correlations in $\sqrt{s_{\mathit{NN}}}=2.76$ TeV lead-lead collisions with the ATLAS detector

A measurement of event-plane correlations involving two or three event planes of different order is presented as a function of centrality for 7 $\mu {\mathrm{b}}^{-1}$ $\mathrm{Pb}+\mathrm{Pb}$ collision data at $\sqrt{s_{{}_{\mathit{NN}}}}=2.76$ TeV, recorded by the ATLAS experiment at the Large Hadron Collider. Fourteen correlators are measured using a standard event-plane method and a scalar-product method, and the latter method is found to give a systematically larger correlation signal. Several different trends in the centrality dependence of these correlators are observed. These trends are not reproduced by predictions based on the Glauber model, which includes only the correlations from the collision geometry in the initial state. Calculations that include the final-state collective dynamics are able to describe qualitatively, and in some cases also quantitatively, the centrality dependence of the measured correlators. These observations suggest that both the fluctuations in the initial geometry and the nonlinear mixing between different harmonics in the final state are important for creating these correlations in momentum space.

Figure 1

Relative angle distributions between two raw EPs from ${\mathrm{ECalFCal}}_{\mathrm{N}}$ and ${\mathrm{ECalFCal}}_{\mathrm{P}}$ defined in Table 4 for the 20%–30% centrality interval for the foreground (open circles), background (open squares), and correlation function (solid circles) based on the EP method. The correlation functions give [via Eq. (12)] the two-plane correlators defined in Table 2. The $y$-axis scales are not the same for all panels.

Figure 2

Observed correlation signals based on the EP method, $\langle cosjk({\Psi }_n-{\Psi }_m)\rangle$, calculated from the correlation functions such as those in Fig. 1 as a function of $\langle N_{\mathrm{part}}\rangle$. The middle two panels in the top row have $j=2$ and $j=3$, while all other panels have $j=1$. The error bars and shaded bands indicate the statistical and systematic uncertainties, respectively.

Figure 3

The combined resolution factors based on the EP method for two-plane correlators, $\mathrm{Res}\left\{jk({\Psi }_n-{\Psi }_m)\right\}\equiv \mathrm{Res}\left\{jk{\Psi }_n\right\}\mathrm{Res}\left\{jk{\Psi }_m\right\}$, as a function of $\langle N_{\mathrm{part}}\rangle$. The middle two panels in the top row have $j=2$ and $j=3$, while all other panels have $j=1$. The error bars and shaded bands indicate the statistical and systematic uncertainties, respectively.

Figure 4

Relative angle distributions between three EPs from ${\mathrm{ECal}}_{\mathrm{N}}$, FCal, and ${\mathrm{ECal}}_{\mathrm{P}}$ defined in Table 4 for Type1 correlation in the 20%–30% centrality interval for foreground (open circles), background (open squares), and correlation function (solid circles) based on the EP method. The correlation functions give [via equations similar to Eq. (13)] the three-plane correlators defined in Table 3. The $y$ axis scales are not the same for all panels.

Figure 5

The $\langle N_{\mathrm{part}}\rangle$ dependence of the observed correlation signals (left panel), combined resolutions (middle panel), and corrected signals (right panel) based on the EP method for the three types of EP combinations for $2{\Phi }_2+3{\Phi }_3-5{\Phi }_5$ using the ATLAS calorimeters. The error bars and shaded bands indicate the statistical and systematic uncertainties, respectively.

Figure 6

The centrality dependence of eight two-plane correlators, $〈cos\left(\Sigma \Phi \right)〉$, with $\Sigma \Phi =jk({\Phi }_n-{\Phi }_m)$ obtained via the SP method (solid symbols) and the EP method (open symbols). The middle two panels in the top row have $j=2$ and $j=3$, respectively, while all other panels have $j=1$. The error bars and shaded bands indicate the statistical uncertainties and total systematic uncertainties, respectively. The expected correlations among PP angles ${\Phi }_n^{*}$ from a Glauber model are indicated by the solid curves for weighted case [Eq. (11)] and dashed lines for the unweighted case.

Figure 7

The centrality dependence of six three-plane correlators, $〈cos\left(\Sigma \Phi \right)〉$, with $\Sigma \Phi =c_{n}n{\Phi }_n+c_{m}m{\Phi }_m+c_{h}h{\Phi }_h$ obtained via the SP method (solid symbols) and the EP method (open symbols). The error bars and shaded bands indicate the statistical uncertainty and total systematic uncertainty, respectively. The expected correlations among PP angles ${\Phi }_n^{*}$ from a Glauber model are indicated by the solid curves for weighted case [Eq. (11)] and dashed lines for the unweighted case.

Figure 8

The ratios of the SP-method correlators to the EP-method correlators, ${〈cos\left(\Sigma \Phi \right)〉}_{\mathrm{w}}/〈cos\left(\Sigma \Phi \right)〉$, for several two-plane correlators, i.e., with $\Sigma \Phi =jk({\Phi }_n-{\Phi }_m)$. The error bars and shaded bands indicate the statistical uncertainties and total systematic uncertainties, respectively.

Figure 9

The ratios of the SP-method correlators to the EP-method correlators, ${\langle cos\left(\Sigma \Phi \right)\rangle }_{\mathrm{w}}/\langle cos\left(\Sigma \Phi \right)\rangle$, for several three-plane correlators, i.e., with $\Sigma \Phi =c_{n}n{\Phi }_n+c_{m}m{\Phi }_m+c_{h}h{\Phi }_h$. The error bars and shaded bands indicate the statistical uncertainties and total systematic uncertainties, respectively.

Figure 10

Comparison of six two-plane correlators, $〈cos\left(\Sigma \Phi \right)〉$, with $\Sigma \Phi =jk({\Phi }_n-{\Phi }_m)$, with results from the AMPT model calculated via the SP method (solid lines) and the EP method (dashed lines) from Ref. [37]. The error bars on the lines represent the statistical uncertainties in the calculation.

Figure 11

Comparison of four three-plane correlators, $〈cos\left(\Sigma \Phi \right)〉$, with $\Sigma \Phi =c_{n}n{\Phi }_n+c_{m}m{\Phi }_m+c_{h}h{\Phi }_h$, with results from the AMPT model calculated via the SP method (solid lines) and the EP method (dashed lines) from Ref. [37]. The error bars on the curves represent the statistical uncertainties in the calculation.

Figure 12

The comparison of the eight two-plane correlators between the calorimeters (default) and ID (cross check) as a function of $\langle N_{\mathrm{part}}\rangle$, both obtained from the EP method. The error bars and the shaded bands indicate the statistical and total systematic uncertainties, respectively.

Figure 13

The comparison of the eight two-plane correlators between the calorimeters (default) and ID (cross check) as a function of $\langle N_{\mathrm{part}}\rangle$, both obtained from the SP method. The error bars and the shaded bands indicate the statistical and total systematic uncertainties, respectively.

Figure 14

The comparison of the six three-plane correlators between the calorimeters (default) and ID (cross check) as a function of $\langle N_{\mathrm{part}}\rangle$, both obtained from the EP method. The error bars and the shaded bands indicate the statistical and total systematic uncertainty, respectively.

Figure 15

The comparison of the six three-plane correlators between the calorimeters (default) and ID (cross check) as a function of $\langle N_{\mathrm{part}}\rangle$, both obtained from the SP method. The error bars and the shaded bands indicate the statistical and total systematic uncertainties, respectively.