A Fourier-Cumulant Analysis for Multiharmonic Flow Fluctuation

The Fourier analysis of the final particle distribution followed by cumulant study of the Fourier coefficient event-by-event fluctuation is one of the main approaches for testing the collective evolution in the heavy-ion collision. Using a multidimensional generating function, we propose a method to extract any possible cumulant of multiharmonic flow fluctuations and classify them in terms of the order of cumulants and harmonics involved in them. In particular, we show that there are 33 distinct cumulants with orders $2,3,4,5$ and harmonics $2,3,4,5$. We compute the normalized version of these cumulants from hydrodynamic simulation for Pb--Pb collisions based on {\tt T$_\text{R}$ENTo}+{\tt VISH2+1}+{\tt UrQMD}. We compare the simulation with those normalized cumulants that the LHC has measured and predict the unmeasured ones. Comparing the initial and final state fluctuation normalized cumulants, we compute the linear and nonlinear hydrodynamic response couplings. We finally introduce the genuine three-particle correlation function containing information of all third-order cumulants.