Non-Gaussian eccentricity fluctuations

We study the fluctuations of the anisotropy of the energy density profile created in a high-energy collision at the LHC. We show that the anisotropy in harmonic $n$ has generic non-Gaussian fluctuations. We argue that these non-Gaussianities have a universal character for small systems such as p+Pb collisions, but not for large systems such as Pb+Pb collisions where they depend on the underlying non-Gaussian statistics of the initial density profile. We generalize expressions for the eccentricity cumulants ${\varepsilon }_2\left\{4\right\}$ and ${\varepsilon }_3\left\{4\right\}$ previously obtained within the independent-source model to a general fluctuating initial density profile.

Figure 1

Closed symbols: values of ${\varepsilon }_2\left\{2\right\}$ and ${\varepsilon }_2\left\{4\right\}$ obtained in a Monte Carlo simulation of the Gaussian independent-source model. Lines: exact result given by Eq. (6). Open squares display, for sake of comparison, the values of ${\varepsilon }_2\left\{2\right\}$ for a uniform average density profile (Sec. 3b). Statistical errors are smaller than symbols.

Figure 2

Same as described in the caption of Fig. 1 for a uniform density distribution in a disk. Symbols are values obtained in a Monte Carlo simulation, and dotted lines are straight lines corresponding to the perturbative formulas, Eqs. (10). Dash-dotted line, labeled “Universal”: value of ${\varepsilon }_2\left\{4\right\}$ derived using Eq. (7), where ${\varepsilon }_2\left\{2\right\}$ in the right-hand side is taken from the Monte Carlo result.

Figure 3

Sketch of the probability of the energy $E$ in a given area. The Gaussian (dashed line) extends into the forbidden region $E<0$ (shaded area). Significant improvement is obtained by skewing the Gaussian (solid line).

Figure 4

${\langle N\rangle }^3{\varepsilon }_2{\left\{4\right\}}^2$ versus $\langle N\rangle$. Closed symbols: fixed $N$, as in Fig. 1. Full line: exact result Eq. (6). Open symbols: Poisson fluctuations. Dotted line: Perturbative result Eq. (29).

Figure 5

Probability distribution of the energy of a single source, corresponding to Eq. (32) with $p=0.892$.

Figure 6

$N_{\mathrm{eff}}{\varepsilon }_2{\left\{2\right\}}^2$ versus $N_{\mathrm{eff}}$. Symbols are results of Monte Carlo calculations, with (closed symbols) and without (open symbols) energy conservation. They are slightly shifted right and left, respectively, so that they do not overlap. The full line is the leading order perturbative result, the dotted and dashed lines are the next-to-leading perturbative results, with [Eq. (35)] and without [Eq. (34)] energy conservation.

Figure 7

$N_{\mathrm{eff}}^3{\varepsilon }_2{\left\{4\right\}}^4$ versus $N_{\mathrm{eff}}$. Symbols are results of Monte Carlo calculations, with (closed symbols) and without (open symbols) energy conservation. As in the previous figure, they are slightly shifted to the left or to the right. Dotted line: perturbative result Eq. (34). Dash-dotted line: Eq. (7), where ${\varepsilon }_2\left\{2\right\}$ in the right-hand side is taken from the Monte Carlo simulation.