Impact of limited statistics on the measured hyperorder cumulants of net-proton distributions in heavy-ion collisions

Hyperorder cumulants $C_5/C_1$ and $C_6/C_2$ of net-baryon distributions are anticipated to offer crucial insights into the phase transition from quark-gluon plasma to hadronic matter in heavy-ion collisions. However, the accuracy of $C_5$ and $C_6$ is highly contingent on the fine shape of the distribution's tail, the detectable range of which could be essentially truncated by low statistics. In this paper, we use the fast Skellam-based simulations, as well as the ultrarelativistic quantum molecular dynamics model, to assess the impact of limited statistics on the measurements of $C_5/C_1$ and $C_6/C_2$ of net-proton distributions at lower energies available at the BNL Relativistic Heavy Ion Collider. Both ratios decrease from the unity baseline as we reduce statistics and could even turn negative without a pertinent physics mechanism. By incorporating statistics akin to experimental data, we can replicate the net-proton $C_5/C_1$ and $C_6/C_2$ values comparable to the corresponding measurements for $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{NN}}=7.7$, 11.5, and 14.5 GeV. Our findings underscore a caveat to the interpretation of the observed beam energy dependence of hyperorder cumulants.

Figure 1

Probability of the net-proton number ($\mathrm{\Delta }{N}_p$) based on the Skellam distribution in Eq. (7).

Figure 2

(a) $C_5/C_1$ and (b) $C_6/C_2$ calculated with artificial truncation on the distribution in Fig. 1, excluding events with $\mathrm{\Delta }{N}_p$ exceeding the upper bound $\mathrm{\Delta }{N}_p^{\mathrm{up}}$. The red dashed lines at unity serve as the expected baselines.

Figure 3

Skellam-based Monte Carlo simulations of $C_6/C_2$ (a) before and (b) after randomly rejecting one event with $\mathrm{\Delta }{N}_p>58$ from each subsample of ${10}^6$ events.

Figure 4

UrQMD calculations of net-proton $C_5/C_1$ (upper panels) and $C_6/C_2$ (lower panels) in 0–40% $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{NN}}=11.5$ GeV. The entire sample of $9.6\times {10}^7$ events is divided into subsamples of $5\times {10}^6$ (left panels) or ${10}^6$ (right panels) events. The red dashed lines represent the ensemble averages over all subsamples.

Figure 5

Skellam-based simulations of net-proton $C_6/C_2$ for 0–40% $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{NN}}=11.5$ GeV using subsamples of (a) $5\times {10}^6$ or (b) ${10}^6$ events. The RefMult3 distribution and $\langle N_p\rangle$ and $\langle N_{\overline{p}}\rangle$ for each RefMult3 are taken from the UrQMD model. The red dashed lines represent the ensemble averages over 1000 subsamples. The blue shaded bands denote the UrQMD calculations of $\langle C_6/C_2\rangle \pm 2\sigma$ extracted from Figs. 4 and 4.

Figure 6

Statistics dependence of net-proton $\langle C_5/C_1\rangle$ (upper panels) and $\langle C_6/C_2\rangle$ (lower panels) in 0–40% $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{NN}}=7.7$, 11.5, and 14.5 GeV from Skellam-based Monte Carlo simulations. $N_{\mathrm{Evt}}$ denotes the number of events in each subsample. $\langle C_5/C_1\rangle$ and $\langle C_6/C_2\rangle$ are computed by averaging over 1000 subsamples. The dashed lines at unity serve as the expected baselines.

Figure 7

Skellam-based simulations of the beam energy dependence of net-proton (a) $\langle C_5/C_1\rangle$ and (b) $\langle C_6/C_2\rangle$ in 0–40% $\mathrm{Au}+\mathrm{Au}$ collisions. For comparison, we also show the STAR measurements [12, 13] that have the same statistics as the simulations at the overlapping energies. The UrQMD calculations for $\sqrt{s_{NN}}=7.7$ and 11.5 GeV are slightly offset horizontally to improve clarity. The dashed lines at unity serve as the statistical baselines.