Importance of the Volume Fluctuation Correction on Higher Order Cumulants

TL;DR

This paper investigates the significance of correcting for volume fluctuations in higher-order cumulant measurements, comparing the effectiveness of the Volume Fluctuation Correction (VFC) with traditional methods using toy models and UrQMD simulations.

Contribution

It evaluates the validity of the VFC method under different models, highlighting its limitations beyond the independent particle production assumption.

Findings

VFC performs well in toy models assuming IPP.

VFC does not work effectively in UrQMD simulations.

Residual volume fluctuation effects may persist after traditional corrections.

Abstract

Initial volume fluctuation (VF) caused by participant fluctuation would be the background which should be subtracted from experimentally measured higher-order cumulants. STAR experiment has been applied Centrality Bin Width Correction (CBWC) to suppress VF. However, CBWC is not perfect method, there might be thus residual fractions of VF backgrounds. Recently, Volume Fluctuation Correction (VFC) was developed under the assumption of the independent particle production (IPP) model. In this talk, the importance of subtracting VF and validity of the VFC are studied by using simple toy models assuming IPP as well as UrQMD model. The results showed that VFC worked well in toy model but did not work well in UrQMD which imply that IPP model could be broken in UrQMD.

Figures

Equations

$$C_n=\sum _r{w}_r{C}_{(n,r)},w_r=\frac{N_r}{{\sum }_r{N}_r},$$

$${\kappa }_1(\Delta N)$$

$${\kappa }_2(\Delta N)$$

$${\kappa }_3(\Delta N)$$

$${\kappa }_4(\Delta N)$$

Full text

Quark Gluon Plasma, Event-by-event fluctuation, volume fluctuation, net-charge

1 Introduction

In higher order event-by-event fluctuation analysis, initial volume fluctuation (VF) is one of the experimental backgrounds which should be taken into account. In order to remove VF, STAR experiment has been applying Centrality Bin Width Correction (CBWC) [1] . In CBWC, cumulants for each centrality bin are calculated by taking weighted average for each multiplicity bin as follows:

[TABLE]

where $N_{r}$ and $C_{(n,r)}$ are number of events and $n^{th}$-order cumulants in $r^{th}$ multiplicity bins respectively. Recently, a new correction method called Volume Fluctuation Correction (VFC) [2] is proposed. Up to the fourth-order cumulnats can be written as

[TABLE]

where $\kappa_{n}{(\Delta{N})}$ and $\kappa_{n}{(\Delta{n})}$ are the measured cumulants and cumulants of net-quantities produced by each ”source” which is assumed to be the number of participant ($N_{W}$) respectively.

2 Analysis method

In order to estimate the VFC corrections, we need to determine the cumulants of $N_{W}$ distribution in the Eq. (3)-(5). Since we cannot directly measure the $N_{W}$ in experiment, one simple way is to use the Glauber model. In Glauber model, the final state multiplicity, which is used for centrality determination, is produced from each source independently. This model is called Independent Particle Production (IPP) model because particles are produced from each source independently. The number of source ($N_{source}$) is defined by two-component model as $N_{source}=(1-x)\frac{N_{W}}{2}+xN_{coll}$, where $N_{coll}$ represents the number of collisions. The number of produced particle from each source is fluctuating under the Negative Binomial Distribution (NBD), and the parameters of Glauber and NBD are the same as net-charge analysis from STAR in Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV [3]. The top left panel of Fig. 1 shows the correlation between multiplicity and $N_{W}$, where red line represent the 10 % step centralities divided by multiplicity. The top right panel shows the $N_{W}$ distributions for each centrality, and second to fourth-order cumulants as a function of $N_{W}$ are shown in the bottom panels.

Next, particles of interested, whose event-by-event distributions are analyzed, are generated from each participant nucleons independently (IPP) based on two Poisson distributions. Cumulants of net-particle distribution should thus include VF defined by the Glauber model, which can subtracted by using Eq. (2)-(5). The parameters of the Poisson distributions are determined that number of positively and negatively charged particles describe the real experiment respectively. On the other hand, in UrQMD simulation, $N_{W}$ can be obtained directly. Cumulants are also measured by using UrQMD approach in addition to toy model approach.

3 Results

3.1 Toy model approach

Fig. 2 shows the second to fourth-order cumulants of net-charge distribution as a function of $\langle{N_{W}}\rangle$ by using toy model for 10% centrality step. For red points, $N_{W}$ is fixed at the value of the averaged number of participant nucleons ($\langle{N_{W}}\rangle$) in each centrality bin, they thus do not include VF. Blue symbols include the fluctuation of the $N_{W}$ in each centrality. Red and blue dotted line show the Poisson baseline and the expectation line of $N_{W}$ fluctuation which is estimated from Eq. (2)-(5) respectively. $N_{W}$ fixed results (red) are consistent within Poisson baseline, and $N_{W}$ fluctuation results (blue) are also consistent with the baseline in all cases. $K_{2}(N_{+}-N_{-})$ which corresponds to $K_{2}(\Delta{n})$ in Eq. (3) is not affected by VF. This is because small $\Delta{n}$ leads to small VF according to Eq. (3). For $K_{3}$ and $K_{4}$, $N_{W}$ fluctuation results are larger than $N_{W}$ fixed results which means that $N_{W}$ fluctuation results are enhanced by VF.

Then, we tried both CBWC and VFC to subtract VF from $N_{W}$ fluc results. VFC results (green) are consistent with $N_{W}$ fixed results (red) which means that VFC works well in this model. On the other hand, CBWC results are smaller than $N_{W}$ fluctuation results but larger than $N_{W}$ fixed results. This results mean that CBWC can reduce VF but can not completely eliminate the VF. Therefore, in toy model case, CBWC is not enough and VFC works well.

Fig. 3 shows the $\kappa\sigma^{2}$ of net-charge distribution as a function of $\langle{N_{W}}\rangle$ for 10%, 5% and 2.5% centrality step. In 10% centrality step, CBWC results contain larger VF compared to the results with 5% and 2.5% step centrality. However, the differences between CBWC and $N_{W}$ fluctuation results become smaller in 5% centrality step and consistent in 2.5% step. This results imply that 2.5% centrality step can reduce VF as well as CBWC. However, there remain VF in both CBWC and $N_{W}$ fluctuation results in any case.

3.2 UrQMD approach

Fig. 4 shows the second to fourth-order cumulants of net-charge distribution as a function of $\langle{N_{W}}\rangle$ by using UrQMD model for 10% centrality step. Red open star symbols ”CBWC-$N_{W}$” mean that CBWC is applied for each $N_{W}$ bin. Standard CBWC is applied for each multiplicity bin which is represented by blue open star symbol. CBWC-$N_{W}$ results are considered as ”no-VF” results which correspond to the red round symbol in the toy model case. Blue symbols contain VF without any corrections, and VFC results are shown in green markers. As discussed in previous section, $K_{2}$ is not affected by VF due to the small value of $\Delta{n}$. However, trends at $K_{3}$ and $K_{4}$ are not consistent with toy model case. For example, CBWC results are smaller than CBWC-$N_{W}$ results, and VFC results are smaller than both of them. VFC seems over correction and does not work well. One of the reason could be that IPP is broken in UrQMD model.

4 Conclusions

Importance of the volume fluctuation correction on higher cumulants are presented by using toy model assuming IPP and UrQMD simulation. From these studies, 2.5% centrality division can reduce VF as well as CBWC but 5% and 10% centrality divisions include the effect from VF. In toy model, even though CBWC has applied, effect from VF can not be removed completely and VFC works well. However, VFC does not work well in UrQMD model, which could be because IPP model is broken in UrQMD. Therefore, we have to consider these effect if VFC is applied to experimental data, and further studies are needed in order to fully understand how to correctly subtract the VF from the measured cumulants.