Eccentricity distributions in nucleus-nucleus collisions

We propose a new parametrization of the distribution of the initial eccentricity in a nucleus-nucleus collision at a fixed centrality, which we name the elliptic power distribution. It is a two-parameter distribution, where one of the parameters corresponds to the intrinsic eccentricity, while the other parameter controls the magnitude of eccentricity fluctuations. Unlike the previously used Bessel-Gaussian distribution, which becomes worse for more peripheral collisions, the new elliptic power distribution fits several Monte Carlo models of the initial state for all centralities.

Figure 1

(a) Two-dimensional plot of the distribution of $({\varepsilon }_x,{\varepsilon }_y)$, with $n=2$, in a Monte Carlo Glauber [31] simulation of Pb + Pb collisions at 2.76 TeV per nucleon pair, in the 75–80 % centrality range. $4\times {10}^4$ events are generated in this centrality. (b) Fit using the elliptic power distribution Eq. (3) with ${\varepsilon }_0=0.61$ and $\alpha =3.3$. (c) Fit using an isotropic two-dimensional Gaussian [26], corresponding to Eq. (5) with ${\varepsilon }_0=0.61$ and ${\sigma }_x={\sigma }_y=0.10$. The $z$ axis of the right panel has been reduced by 0.6 to match the others. All fit parameters are obtained by standard ${\chi }^2$ fits to the distribution of ${\varepsilon }_2$, see Fig. 2.

Figure 2

Distribution of ${\varepsilon }_n$ in 75–80 % central Pb-Pb collisions. (a) ${\varepsilon }_2$, (b) ${\varepsilon }_3$, (c) ${\varepsilon }_4$. Histograms are Monte Carlo Glauber simulations [same as in Fig. 1]. Dashed lines are Bessel-Gaussian fits using Eq. (11). Full lines are elliptic power fits using Eq. (8) for ${\varepsilon }_2$ [parameters as in Fig. 1] and ${\varepsilon }_4$ ($\alpha =3.2$ and ${\varepsilon }_0=0.22$). The dash-dotted line for ${\varepsilon }_3$ is a power fit using Eq. (10) ($\alpha =3.6$).

Figure 3

Histograms of the distribution of ${\varepsilon }_2$ in Pb + Pb collisions at 2.76 TeV using the PHOBOS Monte Carlo Glauber [31] (a)–(d) and the IP-Glasma [37, 41] (e)–(h) for four centrality bins (with decreasing centrality or increasing centrality percentile from left to right). Solid curves are fits using the elliptic power distribution, Eq. (8), dashed curves are fits using the Bessel-Gaussian distribution, Eq. (11). Each bin contains $\sim 4\times {10}^4$ events for the Glauber simulation, and $\sim$2000 events for the IP-Glasma, which explains the larger statistical fluctuations for the bottom row even though the bins are twice as wide. The area under the curves is $0.5$ for the 5–10 % centrality bin and 1 for the other bins.

Figure 4

(a) Power parameter $\alpha$ from elliptic power fits to the eccentricity distribution and power fits to the triangularity distribution [cf. Fig. 2], as a function of centrality for the Monte Carlo Glauber model [31] and the IP-Glasma model [37]. (b) Symbols correspond to the value of the ellipticity parameter ${\varepsilon }_0$. Lines correspond to the mean eccentricity in the reaction plane ${\varepsilon }_{\mathrm{RP}}$ for the Glauber model, either calculated directly (full line), or estimated using the elliptic power distribution Eq. (A2) (dotted line).

Figure 5

Relative fluctuations, as defined by Eq. (12), as a function of collision centrality. Lines: Monte Carlo Glauber results for ${\varepsilon }_2$ (full) and ${\varepsilon }_3$ (dash-dotted). Symbols are results using the elliptic power distribution Eq. (8) for ${\varepsilon }_2$ (circles), and the power distribution Eq. (10) for ${\varepsilon }_3$ (squares), with parameters fitted to the histograms. The horizontal dashed line indicates the value $\sqrt{4/\pi -1}$ corresponding to pure Gaussian fluctuations [Eq. (11) with ${\varepsilon }_0$].

Figure 6

(a) Cumulant ratio ${\varepsilon }_2\left\{4\right\}/{\varepsilon }_2\left\{2\right\}$. (b) Cumulant ratios ${\varepsilon }_2\left\{6\right\}/{\varepsilon }_2\left\{4\right\}$ and ${\varepsilon }_2\left\{8\right\}/{\varepsilon }_2\left\{6\right\}$. Lines are Monte Carlo Glauber results. Symbols are results using the elliptic power distribution Eq. (A5) with parameters fitted to the distribution of ${\varepsilon }_2$.