Standardized cumulants of flow harmonic fluctuations

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 (v2 and v3) distribution. For the second harmonic, it turns out that finding two dimensional cumulants in terms of 2q-particle correlation functions c2{2q} is limited to the skewness. We also show that c3{2}, c3{4}, and c3{6}, are related to the second, fourth, and sixth standardized cumulants of the v3 distribution, respectively. The cumulant cn{2q} can be also written in terms of vn{2q}. Specifically, −(v3{4}/v3{2}) 4 turns out to be the kurtosis of the v3 eventby-event fluctuation distribution. We introduce a new parametrization for the distribution p(v3) with v3{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 v3 distribution. II. ELLIPTIC-POWER DISTRIBUTIONIn this section, we will review a simple and interesting model of the heavy-ion collision initial state, introduced in References [12,17,18]. Consider N independent point-like sources distributed in a two-dimensional (2D) plane with a 2D Gaussian probability distribution. This distribution function could have different widths along two directions. We can imagine the sources as the loca-arXiv:1704.06295v2 [nucl-th]

Primordial non-Gaussianity in heavy-ion collisions

“…Our aim in this paper is to derive expressions relating them. We show that the correlation between the magnitudes of v 2 and v 3 , as measured by the symmetric cumulant SC(3, 2) [11][12][13], and the kurtosis of the distribution of v 3 [14] as measured by v 3 {4} [15], have a common origin. Their negative values can be ascribed to the positive skewness of the distribution of energy in a given transverse area.…”

“…The numerator is a sum of 3-point averages, which are evaluated using Eq. (14). It is directly proportional to the 3-point function of the density field C 3 , as the contribution (21) to the kurtosises.…”

“…For short-range correlations, one can make the approximation that the weighting functions f, g, h, k in Eqs. (13), (14) and (16) are evaluated at the same point [20], and integrate C n over relative positions [32]. Thus δf δg is just a weighted integral of f (s)g(s), with a positive weight, and so on.…”

“…where we use the subscript S because this contribution involves three-point averages, which are proportional to the skewness of the density field C 3 , according to Eq. (14). The only contribution to c n {4} S is from the second term in the right-hand side of Eq.…”

“…While this quantity is typically negative, its negative sign cannot be attributed to the positive skewness of the underlying density field, unlike the negative signs of c 3 {4} and SC(3, 2). Figure 3 displays three measurements of non-Gaussianity of ε n fluctuations: the scaled skewness of elliptic flow fluctuations, the kurtosis of triangular flow [14], and the mixed kurtosis. Based on the perturbative expansion, one typically expects these quantities to be much smaller than unity, and roughly of the same order or magnitude.…”