Azimuthal Harmonics in Small and Large Collision Systems at RHIC Top Energies

The first ($v_1^{\mathrm{fluc}}$), second ($v_2$), and third ($v_3$) harmonic coefficients of the azimuthal particle distribution at midrapidity are extracted for charged hadrons and studied as a function of transverse momentum ($p_T$) and mean charged particle multiplicity density $⟨N_{\mathrm{ch}}⟩$ in $U+U$ ($\sqrt{s_{NN}}=193\mathrm{GeV}$), $\mathrm{Au}+\mathrm{Au}$, $\mathrm{Cu}+\text{Au}$, $\mathrm{Cu}+\mathrm{Cu}$, $d+\mathrm{Au}$, and $p+\mathrm{Au}$ collisions at $\sqrt{s_{NN}}=200\mathrm{GeV}$ with the STAR detector. For the same $⟨N_{\mathrm{ch}}⟩$, the $v_1^{\mathrm{fluc}}$ and $v_3$ coefficients are observed to be independent of the collision system, while $v_2$ exhibits such a scaling only when normalized by the initial-state eccentricity (${ϵ}_2$). The data also show that $\ln (v_2/{ϵ}_2)$ scales linearly with $⟨N_{\mathrm{ch}}{⟩}^{-1/3}$. These measurements provide insight into initial-geometry fluctuations and the role of viscous hydrodynamic attenuation on $v_n$ from small to large collision systems.

Figure 1

(a)–(f) Two-particle azimuthal correlation functions and (g) four-particle cumulants for $p_T$-integrated track pairs ($-1\lesssim \eta \lesssim 1$). Results are shown for (a) $U+U$ collisions ($\sqrt{s_{NN}}=193\mathrm{GeV}$) and (b) $\mathrm{Au}+\mathrm{Au}$, (c) $\mathrm{Cu}+\mathrm{Au}$, (d) $\mathrm{Cu}+\mathrm{Cu}$, (e) $d+\mathrm{Au}$, (f) and $p+\mathrm{Au}$ collisions ($\sqrt{s_{NN}}=200\mathrm{GeV}$) for $⟨N_{\mathrm{ch}}⟩=21\pm 3$. The solid curves show the result of a Fourier fit to the data. (g) The second-order cumulant $c_2\{4\}$ vs $⟨N_{\mathrm{ch}}⟩$, obtained with the three subevents method for the same datasets.

Figure 2

$v_1^{\mathrm{fluc}}$ (a–c), $v_2$ (g–i), $v_3$ (d–f) and $v_2/{ϵ}_2$ (j–l) vs $p_T$ for several $⟨N_{\mathrm{ch}}⟩$ selections. Results are compared for $U+U$, $\mathrm{Au}+\mathrm{Au}$, $\mathrm{Cu}+\text{Au}$, and $\mathrm{Cu}+\mathrm{Cu}$ for $⟨N_{\mathrm{ch}}⟩=140$, and $⟨N_{\mathrm{ch}}⟩=70$ and for $U+U$, $\mathrm{Au}+\mathrm{Au}$, $\mathrm{Cu}+\text{Au}$, $\mathrm{Cu}+\mathrm{Cu}$, $d+\mathrm{Au}$, and $p+\mathrm{Au}$ for $⟨N_{\mathrm{ch}}⟩=21\pm 3$. For the latter, the $p+\mathrm{Au}$ and $d+\mathrm{Au}$ data points are shifted by 0.1 and $-0.1\mathrm{GeV}/c$, respectively, to aid clarity.

Figure 3

Comparison of the $⟨N_{\mathrm{ch}}⟩$ dependence of (a) $v_1^{\mathrm{fluc}}$, (b) $v_3$, (c) and $v_2$ for all collision systems for the $p_T$ selections indicated. The dashed curve in (c) represents a hydrodynamic model calculation [66] for $\mathrm{Au}+\mathrm{Au}$ collisions. The $⟨N_{\mathrm{ch}}⟩$ values for $p+\mathrm{Au}$ and $d+\mathrm{Au}$ correspond to $\sim 0\%–20\%$ central collisions. The inset in (a) compares the extracted values of $K$ vs $⟨N_{\mathrm{ch}}{⟩}^{-1}$ for each system; the dashed line is drawn to guide the eye.

Figure 4

$v_2/{ϵ}_2$ vs $⟨N_{\mathrm{ch}}{⟩}^{-1/3}$ for $U+U$, $\mathrm{Au}+\mathrm{Au}$, $\mathrm{Cu}+\text{Au}$, $\mathrm{Cu}+\mathrm{Cu}$, $d+\mathrm{Au}$, and $p+\mathrm{Au}$ collisions as indicated. The open boxes indicate systematic uncertainties. The $v_2$ data are the same as in Fig. 3. The dotted line represents an exponential fit to the data with Eq. (4). (Inset) The respective ratios of the slopes extracted for each system relative to the slope extracted from a fit to the combined data sets ($⟨\text{Slope}⟩=8.2\times {10}^{-1}\pm 0.02$).