Revealing long-range multiparticle collectivity in small collision systems via subevent cumulants

Multiparticle azimuthal cumulants, often used to study collective flow in high-energy heavy-ion collisions, have recently been applied in small collision systems such as $pp$ and $p+A$ to extract the second-order azimuthal harmonic flow $v_2$. Recent observation of four-, six-, and eight-particle cumulants with “correct sign” $c_2\left\{4\right\}<0$, $c_2\left\{6\right\}>0$, $c_2\left\{8\right\}<0$ and approximate equality of the inferred single-particle harmonic flow, $v_2\left\{4\right\}\approx v_2\left\{6\right\}\approx v_2\left\{8\right\}$, have been used as strong evidence for a collective emission of all the soft particles produced in the collisions. We show that these relations in principle could be violated due to the non-Gaussianity in the event-by-event fluctuation of flow and/or nonflow. Furthermore, we show, using $pp$ events generated with the pythia model, that $c_2\left\{2k\right\}$ obtained with the standard cumulant method are dominated by nonflow from dijets. An alternative cumulant method based on two or more $\eta$-separated subevents is proposed to suppress the dijet contribution. The new method is shown to be able to recover a flow signal as low as 4% imposed on the pythia events, independently of how the event activity class is defined. Therefore the subevent cumulant method offers a more robust way of studying collectivity based on the existence of long-range azimuthal correlations between multiple distinct $\eta$ ranges. The prospect of using the subevent cumulants to study collective flow in $A+A$ collisions, in particular its longitudinal dynamics, is discussed.

Figure 1

The $\eta$ ranges for the subevents in the three-subevent method (a) and the two-subevent method (b).

Figure 2

The $c_2\left\{4\right\}$ (left panel) and $c_3\left\{4\right\}$ (right panel) calculated for particles in $0.3<p_{\mathrm{T}}<3$ GeV with the standard cumulant method. The event averaging is performed for $N_{\mathrm{ch}}^{\mathrm{sel}}$ calculated for various $p_{\mathrm{T}}$ selections as indicated in the figure, which is then mapped to $〈N_{\mathrm{ch}}〉$, the average number of charged particles with $p_{\mathrm{T}}>0.4$ GeV.

Figure 3

The $c_2\left\{4\right\}$ (left panel) and $c_3\left\{4\right\}$ (right panel) calculated for particles in $0.5<p_{\mathrm{T}}<5$ GeV with the standard cumulant method. The event averaging is performed for $N_{\mathrm{ch}}^{\mathrm{sel}}$ calculated for various $p_{\mathrm{T}}$ selections as indicated in the figure, which is then mapped to $〈N_{\mathrm{ch}}〉$, the average number of charged particles with $p_{\mathrm{T}}>0.4$ GeV.

Figure 4

The $c_2\left\{4\right\}$ (left panels) and $c_3\left\{4\right\}$ (right panels) calculated for particles in $0.3<p_{\mathrm{T}}<3$ GeV (top panels) or $0.5<p_{\mathrm{T}}<5$ GeV (bottom panels) with the two-subevent cumulant method. The event averaging is performed for $N_{\mathrm{ch}}^{\mathrm{sel}}$ calculated for various $p_{\mathrm{T}}$ selections as indicated in the figure, which is then mapped to $〈N_{\mathrm{ch}}〉$, the average number of charged particles with $p_{\mathrm{T}}>0.4$ GeV.

Figure 5

The $c_2\left\{4\right\}$ (left panels) and $c_3\left\{4\right\}$ (right panels) calculated for particles in $0.3<p_{\mathrm{T}}<3$ GeV (top panels) or $0.5<p_{\mathrm{T}}<5$ GeV (bottom panels) with the three-subevent cumulant method. The event averaging is performed for $N_{\mathrm{ch}}^{\mathrm{sel}}$ calculated for various $p_{\mathrm{T}}$ selections as indicated in the figure, which is then mapped to $〈N_{\mathrm{ch}}〉$, the average number of charged particles with $p_{\mathrm{T}}>0.4$ GeV.

Figure 6

The $c_2\left\{4\right\}$ calculated for particles in $0.3<p_{\mathrm{T}}<3$ GeV (left panel) or $0.5<p_{\mathrm{T}}<5$ GeV (right panel) compared between the three cumulant methods. The event averaging is performed for $N_{\mathrm{ch}}^{\mathrm{sel}}$ calculated for same $p_{\mathrm{T}}$ range, which is then mapped to $〈N_{\mathrm{ch}}〉$, the average number of charged particles with $p_{\mathrm{T}}>0.4$ GeV.

Figure 7

The $c_2\left\{4\right\}$ calculated for particles in $0.3<p_{\mathrm{T}}<3$ GeV compared among the three cumulant methods with 4% (left panel) or 6% (right panel) $v_2$ imposed. The event averaging is based on $N_{\mathrm{ch}}^{\mathrm{sel}}$ calculated for the same $p_{\mathrm{T}}$ range, which is then mapped to $〈N_{\mathrm{ch}}〉$, the average number of charged particles with $p_{\mathrm{T}}>0.4$ GeV.

Figure 8

The $c_2\left\{4\right\}$ calculated in different bin widths of $N_{\mathrm{ch}}(0.3<p_{\mathrm{T}}<3\text{GeV})$ and then averaged over the $40\le N_{\mathrm{ch}}(0.3<p_{\mathrm{T}}<3\text{GeV})<80$ interval. The difference from the unit-bin-width case is plotted as a function of bin width.

Figure 9

The $c_2\left\{4\right\}$ (left panels) and $c_3\left\{4\right\}$ (right panels) calculated for particles in $0.3<p_{\mathrm{T}}<3$ GeV (top panels) or $0.5<p_{\mathrm{T}}<5$ GeV (bottom panels) with the second type of two-subevent cumulant method [Eq. (27)]. The event averaging is performed for $N_{\mathrm{ch}}^{\mathrm{sel}}$ calculated for various $p_{\mathrm{T}}$ selections as indicated in the figure, which is then mapped to $〈N_{\mathrm{ch}}〉$, the average number of charged particles with $p_{\mathrm{T}}>0.4$ GeV.

Figure 10

The $c_2\left\{4\right\}$ (left panels) and $c_3\left\{4\right\}$ (right panels) calculated for particles in $0.3<p_{\mathrm{T}}<3$ GeV (top panels) or $0.5<p_{\mathrm{T}}<5$ GeV (bottom panels) with the second type of three-subevent cumulant method [Eq. (33)]. The event averaging is performed for $N_{\mathrm{ch}}^{\mathrm{sel}}$ calculated for various $p_{\mathrm{T}}$ selections as indicated in the figure, which is then mapped to $〈N_{\mathrm{ch}}〉$, the average number of charged particles with $p_{\mathrm{T}}>0.4$ GeV.