Energy and system-size dependence of two- and four-particle $v_2$ measurements in heavy-ion collisions at $\sqrt{s_{NN}}=62.4$ and 200 GeV and their implications on flow fluctuations and nonflow

We present STAR measurements of azimuthal anisotropy by means of the two- and four-particle cumulants $v_2$ ($v_2\left\{2\right\}$ and $v_2\left\{4\right\}$) for Au+Au and Cu+Cu collisions at center-of-mass energies $\sqrt{s_{{}_{NN}}}=62.4$ and 200 GeV. The difference between $v_2{\left\{2\right\}}^2$ and $v_2{\left\{4\right\}}^2$ is related to $v_2$ fluctuations (${\sigma }_{v_2}$) and nonflow $\left({\delta }_2\right)$. We present an upper limit to ${\sigma }_{v_2}/v_2$. Following the assumption that eccentricity fluctuations ${\sigma }_{\varepsilon }$ dominate $v_2$ fluctuations $\frac{{\sigma }_{v_2}}{v_2}\approx \frac{{\sigma }_{\varepsilon }}{\varepsilon }$ we deduce the nonflow implied for several models of eccentricity fluctuations that would be required for consistency with $v_2\left\{2\right\}$ and $v_2\left\{4\right\}$. We also present results on the ratio of $v_2$ to eccentricity.

Figure 1

(Left) The two-particle cumulant $v_2{\left\{2\right\}}^2$ for Au+Au collisions at 200 and 62.4 GeV. Results are shown with like-sign combinations (LS) and charge-independent results (CI) for $0.15<p_T<2.0$ GeV/$c$. (Right) The same as the left but for Cu+Cu collisions. The systematic errors are shown as triangles above and below the data points and statistical errors are shown as thick lines with caps at the end. Statistical and systematic errors are very small.

Figure 2

The difference of charge-independent (CI) $v_2\left\{2\right\}$ and like-sign (LS) $v_2\left\{2\right\}$ for Au+Au and Cu+Cu collisions at 200 (top panel) and 62.4 (bottom panel) GeV vs. the log of $\langle d{N}_{\mathrm{ch}}/d\eta \rangle$. The statistical errors are smaller than the marker size and not visible for most of the data.

Figure 3

(Left) The LS and CI four-particle cumulant $v_2{\left\{4\right\}}^4$ for Au+Au collisions at 200 and 62.4 GeV for $0.15<p_T<2.0$ GeV/$c$. The systematic errors are shown as triangles above and below the data points and statistical errors are shown as thick lines with caps at the end. Statistical errors are not visible for most of the points. (Right) The LS and CI four-particle cumulant $v_2{\left\{4\right\}}^4$ for Cu+Cu collisions at 200 and 62.4 GeV for $0.15<p_T<2.0$ GeV/$c$. The most central points (two points for Cu+Cu 62.4 GeV) gives $v_2{\left\{4\right\}}^4<0$ for all the data sets. The negative values are probably due to large fluctuations in agreement with Eq. (1). These may include contributions from impact parameter spread and finite multiplicity bin width.

Figure 4

The difference of charge-independent (CI) $v_2\left\{4\right\}$ and like-sign (LS) $v_2\left\{4\right\}$ for Au+Au collisions at 200 and 62.4 GeV vs. the log of $\langle d{N}_{\mathrm{ch}}/d\eta \rangle$.

Figure 5

(Left) The difference between $v_2{\left\{2\right\}}^2$ and $v_2{\left\{4\right\}}^2$ for 200 GeV Au+Au and Cu+Cu collisions for both LS and CI combinations. (Right) The difference between $v_2{\left\{2\right\}}^2$ and $v_2{\left\{4\right\}}^2$ for 62.4 GeV Au+Au and Cu+Cu collisions for both LS and CI combinations. The statistical and systematic errors are shown as in previous figures.

Figure 6

The upper limit on ${\sigma }_{v_2}/\langle v_2\rangle$ for 200 GeV (left) and 62.4 GeV (right) Au+Au collisions from Eq. (9) compared to ${\sigma }_{\varepsilon }/\varepsilon$ from Eq. (10) for three different models. The upper limit is found using the LS results for $v_2\left\{2\right\}$. Data are from the range $0.15<p_T<2.0$ GeV/$c$. The shaded bands reflect the uncertainties on the models which are dominated by uncertainty on the distribution of nucleons inside the nucleus. The uncertainty is only shown for the MCG-N and fKLN-CGC models. The uncertainty on the MCG-Q model is the same as for the MCG-N model but is not shown for the visual clarity.

Figure 7

The STAR data compared to PHOBOS data [34] on ${\sigma }_{v_2}/\langle v_2\rangle$ with ${\delta }_2$ for $\Delta \eta >2$ taken to be zero (see Fig. 6 from Ref. [34]). The shaded band shows the errors quoted from Ref. [34].

Figure 8

The upper limit on ${\sigma }_{v_2}/\langle v_2\rangle$ for 200 GeV (left) and 62.4 GeV (right) Cu+Cu collisions from Eq. (9) compared to ${\sigma }_{\varepsilon }/\varepsilon$ from Eq. (10) for three different models.

Figure 9

The mean multiplicity scaled nonflow $\left({\delta }_2\right)$ found by assuming Eq. (11) and using the three different models for ${\sigma }_{\varepsilon }^2/{\varepsilon }^2$: MCG-N (left), MCG-Q (middle), and fKLN-CGC (right). The systematic errors are shown as triangles above and below the data points and statistical errors are shown as thick lines with caps. Statistical errors are not visible for most of the points. For clarity, the 200-GeV Cu+Cu data with large point-to-point systematic errors are excluded. For the other data sets the systematic errors are highly correlated but centrality dependent.

Figure 10

(Top panels) The eccentricity scaled $v_2$ for 200- and 62.4-GeV Au+Au and Cu+Cu collisions with eccentricity taken from the MCG-N (left), MCG-Q (middle), or fKLN-CGC (right) model. The statistical and systematic errors are shown as in previous figures. Statistical errors are not visible for most of the points. (Bottom panels) The ratio of the data to a log linear striaght-line fit.

Figure 11

(Top panel) The ratio of the two-particle cumulant $v_2\left\{2\right\}$ for Au+Au collisions at 200 and 62.4 GeV evaluated using the $q$-distribution method and the $Q$-cumulants method. Both results are calculated for combinations of particles independent of their charge (CI). (Bottom panel) The ratio of the $q$-distribution and the $Q$-cumulants method results for $v_2{\left\{2\right\}}^2-v_2{\left\{4\right\}}^2$ (CI) for 200-GeV Au+Au and Cu+Cu collisions. In both panels, systematic errors are shown as triangles above and below the data points and statistical errors are shown as thick lines with caps.

Figure 12

${\varepsilon }_{\mathrm{std}}$ (top) and ${\sigma }_{\varepsilon }$ (bottom) vs. centrality for Au+Au 200 GeV among the Monte Carlo Glauber-nucleon participants, Monte Carlo Glauber-quark constituents, and color glass condensate models. The shaded regions show the systematic errors.

Figure 13

$\varepsilon {\left\{2\right\}}^2$ and $\varepsilon {\left\{4\right\}}^4$ vs. centrality for Monte Carlo Glauber models with nucleon or constituent quark pariticipants and for a color glass condensate model.