# Event-by-event multi-harmonic correlations of different flow amplitudes in Pb-Pb collisions at $\sqrt{s_{_{\rm NN}}}=2.76$ TeV

The genuine event-by-event correlations between three flow amplitudes are measured for the first time in Pb-Pb collisions at $\sqrt{s_{\rm {NN}}} = 2.76$ TeV by the ALICE Collaboration at the Large Hadron Collider. The results are obtained with recently developed observables, the higher order Symmetric Cumulants (SC), in the midrapidity region $|\eta| <~ 0.8$ and the transverse momentum range $0.2 <~ p_{\mathrm T} <~ 5.0$ GeV/$c$. These higher order observables show the same robustness against systematic biases arising from nonflow effects as the two-harmonic SC. The new results cannot be interpreted in terms of lower order flow measurements, since they are dominated by different patterns of event-by-event flow fluctuations. The results are compared with expectations from initial state models such as T$_{\rm{R}}$ENTo and next-to-leading order perturbative-QCD+saturation model of initial conditions, followed by iEBE-VISHNU and EKRT viscous hydrodynamic calculations. Model comparisons provide an indication of the development of genuine correlations between the elliptic $v_2$, the triangular $v_3$ and the quadrangular $v_4$ flow amplitudes during the collective evolution of the medium. The comparison with the predictions for the correlations between $v_2$, $v_3$ and the pentagonal flow magnitude $v_5$ illustrate the need for further tuning of model parameterizations. Therefore, these results can provide new and independent constraints for the initial conditions and system properties of nuclear matter created in heavy-ion collisions, complementary to previous flow measurements.