Kurtosis of elliptic flow fluctuations

Elliptic flow ($v_2$) in ultrarelativistic nucleus-nucleus collisions fluctuates event to event, both in magnitude and in orientation with respect to the reaction plane. Even though the reaction plane is not known event to event in experiment, we show that the statistical properties of $v_2$ fluctuations in the reaction plane can be precisely extracted from experimental data. Previous studies have shown how to measure the mean, variance and skewness using the first three cumulants $v_2\left\{2\right\}$, $v_2\left\{4\right\}$, and $v_2\left\{6\right\}$. We complement these studies by providing a formula for the kurtosis, which requires an accurate determination of the next cumulant $v_2\left\{8\right\}$. Using existing data, we show that the kurtosis is positive for most centralities, in contrast with the kurtosis of triangular flow fluctuations, which is negative. We argue that these features are robust predictions of fluid-dynamical models.

Figure 1

Kurtosis of eccentricity fluctuations along the reaction plane in the $\mathrm{T}R\mathrm{ENTo}$ model for $\text{Pb}+\text{Pb}$ collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV, as a function of centrality percentile. Full line shows Eq. (3). Dotted line shows Eq. (15). In these equations, we replace the elliptic flow with the initial eccentricity.

Figure 2

Contour plots of (a) the standardized kurtosis ${\gamma }_2$, defined in Eq. (3), and (b) the estimated kurtosis ${\gamma }_2^{\mathrm{expt}}$, defined by Eq. (15), computed by using the elliptic power distribution. The quantities ${\varepsilon }_0$ and $N$ are the eccentricity of the distribution, and the number of sources, respectively, as in Eq. (5). Symbols correspond to elliptic power fits to the $\mathrm{T}R\mathrm{ENTo}$ calculations (see Fig. 1) in a few centrality windows.

Figure 3

Kurtosis of $v_2$ fluctuations in $\text{Pb}+\text{Pb}$ collisions at $\sqrt{s_{{}_{\text{NN}}}}=5.02$ TeV estimated using Eq. (15) and CMS data [19].