# Standardized Cumulants of Flow Harmonic Fluctuations

## TL;DR

This paper models flow harmonic fluctuations in heavy ion collisions using standardized cumulants, introduces a new parametrization for $v_3$ distribution, and compares theoretical predictions with experimental data.

## Contribution

It provides a novel parametrization of the $v_3$ distribution based on standardized cumulants and relates cumulants to multi-particle correlations, improving the modeling of flow fluctuations.

## Key findings

- Standardized cumulants effectively characterize flow harmonic fluctuations.
- The new $v_3$ distribution parametrization fits experimental data better than Gaussian.
- Comparison shows consistency between simulation and experimental kurtosis values.

## Abstract

The distribution of flow harmonics in heavy ion experiment can be characterized by standardized cumulants. We first model the ellipticity and power parameters of the elliptic-power distribution by employing MC-Glauber model. Then we use the elliptic-power distribution together with the hydrodynamic linear response approximation to study the two dimensional standardized cumulants of elliptic and triangular flow ($v_2$ and $v_3$) distribution. For the second harmonic, it turns out that finding two dimensional cumulants in terms of $2q$-particle correlation functions $c_2\{2q\}$ is limited to the skewness. We also show that $c_3\{2\}$, $c_3\{4\}$, and $c_3\{6\}$, are related to the second, fourth, and sixth standardized cumulants of the $v_3$ distribution, respectively. The cumulant $c_{n}\{2q\}$ can be also written in terms of $v_n\{2q\}$. Specifically, $-(v_3\{4\}/v_3\{2\})^4$ turns out to be the kurtosis of the $v_3$ event-by-event fluctuation distribution. We introduce a new parametrization for the distribution $p(v_3)$ with $v_3\{2\}$, kurtosis and sixth-order standardized cumulant being its free parameters. Compared to the Gaussian distribution, it indicates a more accurate fit with experimental results. Finally, we compare the kurtosis obtained from simulation with that of extracted from experimental data for the $v_3$ distribution.