Importance of correlations and fluctuations on the initial source eccentricity in high-energy nucleus-nucleus collisions

In relativistic heavy-ion collisions, anisotropic collective flow is driven, event by event, by the initial eccentricity of the matter created in the nuclear overlap zone. Interpretation of the anisotropic flow data thus requires a detailed understanding of the effective initial source eccentricity of the event sample. In this paper, we investigate various ways of defining this effective eccentricity using the Monte Carlo Glauber (MCG) approach. In particular, we examine the participant eccentricity, which quantifies the eccentricity of the initial source shape by the major axes of the ellipse formed by the interaction points of the participating nucleons. We show that reasonable variation of the density parameters in the Glauber calculation, as well as variations in how matter production is modeled, do not significantly modify the already established behavior of the participant eccentricity as a function of collision centrality. Focusing on event-by-event fluctuations and correlations of the distributions of participating nucleons, we demonstrate that, depending on the achieved event-plane resolution, fluctuations in the elliptic flow magnitude $v_2$ lead to most measurements being sensitive to the root-mean-square rather than the mean of the $v_2$ distribution. Neglecting correlations among participants, we derive analytical expressions for the participant eccentricity cumulants as a function of the number of participating nucleons, $N_{\mathrm{part}}$, keeping nonnegligible contributions up to $\mathcal{O}(1/N_{\mathrm{part}}^3)$. We find that the derived expressions yield the same results as obtained from mixed-event MCG calculations which remove the correlations stemming from the nuclear collision process. Most importantly, we conclude from the comparison with MCG calculations that the fourth-order participant eccentricity cumulant does not approach the spatial anisotropy obtained assuming a smooth nuclear matter distribution. In particular, for the Cu+Cu system, these quantities deviate from each other by almost a factor of 2 over a wide range in centrality. This deviation reflects the essential role of participant spatial correlations in the interaction of two nuclei.

Figure 1

Schematic illustration of a nucleus-nucleus collision depicted in the transverse plane. The principal axes ($x^{\text{'}}$ and $y^{\text{'}}$) of the area formed by the participants are tilted with respect to the reaction plane given by the axes ($x$ and $y$) of the transverse plane.

Figure 2

Participant (top) and reaction-plane (bottom) eccentricity in Au+Au (left) and Cu+Cu (right) collisions as a function of $N_{\mathrm{part}}$ for variations of the Glauber parameters as given in Table . The inset shows the ratio with respect to the baseline calculation in each case.

Figure 3

Comparison of participant vs binary weighted participant eccentricity, as well as the $x=0.13$ mixture, as a function of $N_{\mathrm{part}}$ in Au+Au (upper set of curves) and Cu+Cu (lower set of curves) collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV. The inset shows the ratio with respect to the participant-weighted eccentricity.

Figure 4

Comparison of participant eccentricity as a function of $N_{\mathrm{part}}$ in Au+Au (upper set of curves) and Cu+Cu (lower set of curves) collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV for pointlike, hard-sphere, and Gaussian matter distributions. The inset shows the ratio with respect to the pointlike participant eccentricity.

Figure 5

Dependence of α, Eq. (6), on the event-plane resolution $R$ for $P(v_2)$ uniform with various combinations of $v_{2,\min }$ and $v_{2,\max }$, number of signal $N$ and background $f_{\mathrm{bkg}}N$ particles in the MC calculation (see text for more details). The shaded band covers the parameter errors obtained from a polynomial fit to the data.

Figure 6

Participant-eccentricity-scaled elliptic flow vs transverse charged particle area density at midrapidity for Cu+Cu and Au+Au collisions at $\sqrt{s_{\mathrm{NN}}}=62.4$ and $\sqrt{s_{\mathrm{NN}}}=200$ GeV. The horizontal and vertical error bars originate from the combined statistical and systematic errors of the data (90% C.L.). The shaded boxes are the result of the systematic errors assigned to ${\epsilon }_{\mathrm{part}}\{2\}$ and $S$ by the variation of the MCG parameters (90% C.L.).

Figure 7

Schematic of densities for the different approaches: (a) optical limit, (b) full MCG with correlated participants originating from each of the two nuclei in one MCG event, and (c) mixed-event MCG with uncorrelated participants where every participant originates from an individual nucleon-nucleon collision obtained in a different MCG event.

Figure 8

Participant eccentricity and cumulants, ${\epsilon }_{\mathrm{part}},{\epsilon }_{\mathrm{part}}\{2\}$, and ${\epsilon }_{\mathrm{part}}\{4\}$, as a function of $N_{\mathrm{part}}$ in Au+Au (left) and Cu+Cu (right) collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV for the full (upper set of curves) and mixed-event (lower set of curves) MCG calculations. The inset shows the ratio of mixed-event to the full MCG results.

Figure 9

Comparison of ${\epsilon }_{\mathrm{part}}\{4\}$ from the full MCG (upper set of curves), the mixed-event MCG (lower set of curves) with the semianalytical approach, Eq. (B37), and B&O's approximation [Eq. (13) from Ref. [23]], as well as with the standard eccentricity as a function of $N_{\mathrm{part}}$ in Au+Au (left) and Cu+Cu (right) collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV. The inset shows the ratio with respect to ${\epsilon }_{\mathrm{part}}\{4\}$ from the full MCG calculation.

Figure 10

Participant eccentricity and cumulants as well as reaction-plane and standard eccentricities as functions of $N_{\mathrm{part}}$ in Au+Au (left) and Cu+Cu (right) collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV obtained from full MCG calculations.