Limitation of multiparticle correlations for studying the event-by-event distribution of harmonic flow in heavy-ion collisions

The sensitivity of flow harmonics from cumulants to the event-by-event flow distribution $p\left(v_n\right)$ is investigated using a simple central moment expansion approach. For a narrow distribution whose width is much smaller than the mean ${\sigma }_n\ll \langle v_n\rangle$, the differences between the first three higher-order cumulant estimates $v_n\left\{4\right\},v_n\left\{6\right\}$, and $v_n\left\{8\right\}$ are not very sensitive to the shape of $p\left(v_n\right)$. For a broad distribution ${\sigma }_n\gtrsim 〈v_n〉$, the higher-order cumulant estimates differ from each other but may change sign and become ill defined. This sign change arises from the choice of $p\left(v_n\right)$, without the need to invoke nonflow effects. Direct extraction of $p\left(v_n\right)$ via a data-driven unfolding method used by the ATLAS experiment is a more preferred approach for flow distribution measurement.

Figure 1

Cumulants for second-order azimuthal harmonics as a function of number of charged tracks in $p$+Pb collisions: (a) $c_2\left\{4\right\}$ from Ref. [22], (b) flow harmonics $v_2\left\{2k\right\}$ for $k=1$ to 4 from Ref. [23].

Figure 2

The two flow probability distributions represented by Eqs. (18) and (19). Parameter $a$ controls the width of these distributions.

Figure 3

(Top panels) The flow harmonics from cumulants $v_n\left\{2k\right\}$ and (bottom panels) ratios between neighboring cumulant estimates $v_n\left\{2k\right\}/v_n\{2k-2\}$ as a function of $a$ for two probability distributions represented by Eq. (18) (left panel) and Eq. (19) (right panel).

Figure 4

The flow harmonics from cumulants $v_n\left\{2k\right\}$ as a function of $k$ for four values of $a$ for the two probability distributions represented by Eq. (18) (top panel) and Eq. (19) (bottom panel).

Figure 5

(a) The event-by-event distributions of $v_n^{\prime }{\left\{2k\right\}}^2$ defined by Eq. (28) and $q_n^2$ from the toy simulation. (b) The event-by-event differences between these variables.