Symmetric cumulants and event-plane correlations in Pb + Pb collisions

The ALICE Collaboration has recently measured the correlations between amplitudes of anisotropic flow in different Fourier harmonics, referred to as symmetric cumulants. We derive approximate relations between symmetric cumulants involving $v_4$ and $v_5$ and the event-plane correlations measured by the ATLAS Collaboration. The validity of these relations is tested using event-by-event hydrodynamic calculations. The corresponding results are in better agreement with the ALICE data than existing hydrodynamic predictions. We make quantitative predictions for three symmetric cumulants which have not yet been measured.

Figure 1

Schematic picture of the relation between the event-plane angle ${\mathrm{\Phi }}_{24}$ in Eq. (4) and the decomposition equation, Eq. (3). The legs of the triangle correspond to the rms values of the linear and nonlinear parts, and the hypothenuse is the rms $v_4$. A similar figure can be drawn for $V_5$.

Figure 2

Test of Eqs. (8) using hydrodynamic calculations. Symbols correspond to the left-hand sides of Eqs. (8); dark-shaded bands correspond to the right-hand sides. Light-shaded bands correspond to Eqs. (9) and (12). Errors are statistical and estimated via jackknife resampling.

Figure 3

Open symbols: ALICE data for $sc(4,2)$ [17]. Closed symbols: Value obtained using the right-hand side of Eq. (8) using the ATLAS data for the moments of $v_2$ [46] and the event-plane correlation [16].

Figure 4

Predictions using the right-hand sides of Eqs. (9) and (12), using the ATLAS data for the moments of $v_2$ and $v_3$ [46] and the event-plane correlations [16], and using the ALICE data for $sc(3,2)$ [17].