Effects of volume corrections and resonance decays on cumulants of net-charge distributions in a Monte Carlo hadron resonance gas model

The effects of volume corrections and resonance decays (the resulting correlations between positive charges and negative charges) on cumulants of net-proton distributions and net-charge distributions are investigated by using a Monte Carlo hadron resonance gas ({\tt MCHRG}) model. The required volume distributions are generated by a Monte Carlo Glauber ({\tt MC-Glb}) model. Except the variances of net-charge distributions, the {\tt MCHRG} model with more realistic simulations of volume corrections, resonance decays and acceptance cuts can reasonably explain the data of cumulants of net-proton distributions and net-charge distributions reported by the STAR collaboration. The {\tt MCHRG} calculations indicate that both the volume corrections and resonance decays make the cumulant products of net-charge distributions deviate from the Skellam expectations: the deviations of $S\sigma$ and $\kappa\sigma^{2}$ are dominated by the former effect while the deviations of $\omega$ are dominated by the latter one.

For the data reported by the STAR collaboration [8], on the other hand, the volume corrections play significant role on the cumulants (cumulant products) of net-charge distributions [19,20]. Therefore, besides the information of chemical Email address: haojiexu@pku.edu.cn (Hao-jie Xu) potential and temperature investigated in previous studies, the volume information is also important for theroetical baseline studies [20]. Meanwhile, with the volume information, one can study the centrality dependence of multiplicity distributions in relativistic heavy ion collisions.
In this work, with the volume information generated by a Monte Carlo Glauber (MC-Glb) model, I propose a Monte Carlo hadron resonance gas (MCHRG) model to study the effect of volume corrections and resonance decays on the cumulants of netproton distributions and net-charge distributions in a transparent way. Instead of primordial particles used in previous studies, the acceptance cuts can be applied to the decay products in the MCHRG simulations. Based on these advantages, the MCHRG model can be used to give more realistic baselines for the cumulants of multiplicity distributions than previous HRG models.

Monte Carlo hadron resonance gas model
In the MCHRG model, the multiplicities of different particle species in each event are randomly generated by Poisson distributions, and the Poisson parameters are calculated by [31] where g i is degeneracy factor, m i is particle mass, T ch is chemical freeze-out temperature. The chemical potential µ i = B i µ B + S i µ S + Q i µ Q with the baryon number B i , strangeness number S i , charge number Q i and the corresponding chemical potentials µ B , µ S , µ Q . K is modified Bessel function. Note that I have neglected the effects of quantum statistics on multiplicity fluctuations.
To study the effect of resonance decays, I use 319 primordial particle species as inputs and 26 stable particle species after performing Monte Carlo resonance decays, as the particle species used in [29]. The resonance decay channels are taken from [32] and the contributions from weak decays are not taken into account in the present study.
For more realistic simulations of acceptance cuts, the transverse momentum (p T ) spectra of primordial particles are simulated by the blast-wave model [33] dN where m T = p 2 T + m 2 , T kin is kinetic freeze-out temperature, ρ = tanh −1 (β) and β = β S x with β S being the velocity at volume boundary. And the rapidity (y) distribution is modeled by [34] with L = ln( √ s NN /m p ) and m p is the mass of proton.
To study the centrality dependence of multiplicity distributions, as well as the volume corrections on high order cumulants of multiplicity distribution in MCHRG model, a Monte Carlo Glauber (MC-Glb) model [35] is employed to generate the volume information in each event (thermal system), where n part is the number of participant nucleons and n coll is the number of binary nucleon-nucleon collisions.
With the events generated from the MCHRG model, the first four cumulants of multiplicity distributions are calculated by [7,8,10] where ∆N = N − N with N being the multiplicity of fluctuation measures, and ... denotes the event average. Here M, σ 2 , S and κ are the mean value, variance, skewness and kurtosis of the probability distribution. To reduce the centrality bin width effect [36], the cumulants of multiplicity distributions are calculated in each reference multiplicity bin -the finest centrality bin in heavy ion experiments. The statistical errors are estimated by the Delta theorem [37,38]. In this work, as an illustrate, I focus on the STAR measure- . For more precise constraint, the parameters x and h can be determined by the distribution of reference multiplicity if the data becomes available [19,20]. are used to include the effect of flow and more realistic simulation of acceptance cuts. The chemical freeze-out parameters T ch = 151 MeV and µ B = 98.5 MeV are determined by the multiplicity ratios p/π and p/p, which are consist with the parameters obtained in Ref. [24]. Since I do not focus on the details of strangeness fluctuations, the chemical freeze-out parameter µ S is just obtained from and e S = 0.161 [23]. The chemical freeze-out parameter µ Q is determined by the multiplicity ratio of positive charges and negative charges. The mean multiplicity of net-charges is the remainder of two large values, i.e. mean multiplicity of positive and negative charges, which are two orders of magnitude larger than the remainder. To give a good description of centrality dependence of mean multiplicity of net-charges, therefore, a centrality-dependent µ Q is parameterized as with µ Q0 = −4.7 MeV and a Q = 1.5 MeV. Besides µ Q , in general, all the chemical and kinetic freeze-out parameters are centrality-dependent for more precise constraint. For simplicity in this work, all of them except µ Q are set to constants to roughly reproduce the mean multiplicity of positive charges, negative charges, net-charges [8], protons, anti-protons and netprotons [10] with some specific acceptance cuts. Note that the main conclusions obtained in the present study are almost independent of the selection of collision energy and model parameters, and the impact of Glauber parameters on cumulant calculations has been investigated in my previous study [20]. I apply the same acceptance cuts as used in experiment [8,10]. More specifically, |η| < 0.5, 0.2 < p T < 2.0 GeV for fluctuation measures (π ± , K ± and p/p after removing protons and anti-protons with p T < 0.4 GeV) and 1.0 > |η| > 0.5, 0.2 < p T < 2.0 GeV for reference particles (total charged hadrons) in the net-charge case; |y| < 0.5, 0.4 < p T < 0.8 GeV for fluctuation measures (p/p) and |η| < 1.0, 0.2 < p T < 2.0 GeV for reference particles (π ± and K ± ) in the net-proton case. Here η is pseudo-rapidity.

Results and Discussions
the other one is the independent product (IP) baseline, which is obtained by Here c + n and c − n are the cumulants of proton (positive charge) distributions and anti-proton (negative charge) distributions in the net-proton (net-charge) case calculated by the MCHRG model. According to the Central Limit Theorem (CLT), the cumulants c n ∝ n part , σ ∝ n part , S ∝ 1/ n part and κ ∝ 1/ n part . The CLT baselines are shown in Fig. 1 with red-solid curves, which follow the general trend of the MCHRG baselines and experimental data.
The MCHRG baselines, Skellam baselines and IP baselines are almost the same for the first four cumulants of net-proton distributions shown in Fig. 1 (a-d). The reasons are twofold: First, in the absence of critical fluctuations, the volume corrections on cumulants of net-proton distributions at √ s NN = 39GeV can be neglected. Second, the resonance decay processes make no contributions to the correlation between protons and antiprotons [29]. Therefore, for the study of net-proton distributions, the MCHRG model and the HRG model established in Ref. [29] are almost the same. Note that the MCHRG model give more realistic simulations of acceptance cuts for the fluctuation measures, but it is more computational expansive due to the high statistics in Monte Carlo simulations. In general the MCHRG model can reasonably reproduce the data of net-proton distributions, but, for more precise predictions, more non-critical effects need to be investigated. The situation is very different for the net-charge fluctuations shown in Fig. 1(e-h). The volume corrections play a significant role for the cumulants of net-charge distributions reported by the STAR collaboration. As I have explained in Ref. [19,20], such difference comes from the different magnitude of reduced cumulants [20,39] b ≃ M/(k +1) (M is the mean multiplicity of fluctuation measures and k is the corresponding reference multiplicity) in the net-charge case and net-proton case, corresponding to the data reported by the STAR collaboration [8,10]: the reduced cumulants of positive and negative charges are of the order O(1), while the reduced cumulants of protons and antiprotons are much smaller than 1. The MCHRG baselines support my conclusions about volume corrections on multiplicity distributions given in Ref. [19,20] with a simple statistical model. Meanwhile, the resonance decays and resulting correlations between positive charges and negative charges are included in the MCHRG model. Therefore, Monte Carlo simulations are very appropriate for the study of net-charge fluctuations than a simple statistical model.
The large gaps between the Skellam baselines and IP baselines indicate that the effects of volume corrections and resonance decays on cumulants of positive charge distributions and negative charge distributions are important. Analogously, the deviations of the MCHRG baselines from Skellam baselines indicate that the effect of volume corrections and resonance decays on skewness and kurtosis of net charge distributions are important, see Fig. 1(g,h). However, I will show that the effect of volume corrections on the variances of net-charge distributions can be neglected, which make the MCHRG baselines close to the Skellam baselines, see Fig 1(f). The deviations of the MCHRG baselines from the IP baselines indicate strong correlations between positive charges and negative charges. With the volume corrections, resonance decays, and more realistic simulations of acceptance cuts, the MCHRG model can reasonably reproduce the skewness and kurtosis of net-charge distributions reported by the STAR collaboration [8], but shown obvious deviations for the variances. The results indicate that the correlations between positive and negative charges from other sources are required to quantitatively reproduce the data of variances of net-charge distributions, which are beyond the scope of the MCHRG model proposed in this work. The skewness and kurtosis of net-charge distributions seem insensitive to these correlations and its deviations from Skellam baselines are dominated by the effect of volume corrections.  To explore in depth the effects of volume corrections and resonance decays on the cumulants of net-charge distributions individually, I further calculate the cumulant products ω = c 2 /c 1 , S σ = c 3 /c 2 , κσ 2 = c 4 /c 2 (9) in three different cases: (1) MCHRG simulations with MC-Glb volume distributions discussed before. (2) Similar to case (1) but for the primordial particles before resonance decays, The results are shown in Fig. 2. The volume corrections on ω of net-charge distributions can be neglected [40], though the volume corrections on ω of positive charges and negative charges are important [19,20]. This is because the mean multiplicity of net-charge distribution is much smaller than the reference multiplicity, though the magnitudes of mean multiplicity of positive and negative charges are the same order of reference multiplicity. The volume corrections play significant role for S σ and κσ 2 , which make their values deviate far away from the Skellam predictions. The resonance decays make ω of netcharge distributions smaller than the Skellam predictions, but it make S σ and κσ 2 larger than the Skellam predictions. From the MCHRG calculations, I find that the deviations of ω from Skellam distributions are mainly due to the effect of resonance decays, while the deviations of S σ and κσ 2 are mainly due to effect of volume corrections.
For the effect of resonance decays, the multi-charged hadrons play special role in study of net-charge distribution [41]. To identify the effect of multi-charged hadrons on net-charge distributions through the resonance decay process, I then calculate the cumulant products of net-charge distributions in case (3) without the decay channels of multi-charged hadrons. The results are shown in red-dotted curves of Fig. 2. The resonance decays of multi-charged hadron enhance the fluctuations of netcharges, as it have been investigated in Ref. [41]. Comparison the results with (black-dashed lines) and without (red-dotted lines) multi-charged hadrons in Fig. 2, I find that the effect of multi-charged hadrons can be neglected for ω and κσ 2 of netcharge distributions, while they make substantial contributions to S σ of net-charge distributions.
The deviations of ω, S σ and κσ 2 from data are mainly due to the fact that the MCHRG model fail to quantitatively reproduce the variances of net-charge distributions(see Fig. 1(f)). The results imply that, for more realistic baselines predictions of the first four cumulants of net-charge distributions, some other effects are expected to make relevant contributions to the variances, without significant affecting its skewness and kurtosis.

Conclusions
I investigated the cumulants of net charge distributions and net-proton distributions within a Monte Carlo hadron resonance gas (MCHRG) model. To study the centrality dependence of multiplicity distributions, as well as the effect of volume corrections on its cumulants, the volume distributions are generated by a Monte Carlo Glauber (MC-Glb) model. For the net-proton distributions, even with more realistic simulations of acceptance cuts and volume corrections, the MCHRG calculations are consist with the semi-analytical calculations given in Ref. [29]. However, both the effect of volume corrections, resonance decays, as well as the resulting correlations between positive charges and negative charges, that are important for the cumulants of netcharge distributions. With these effect, the MCHRG calculations provide more realistic baseline predictions for the cumulants of net-charge distributions than previous HRG studies.
Except the variances of net-charge distributions, the MCHRG model can reasonably explain the cumulants of net-proton distributions and net-charge distributions reported by the STAR collaboration. To explore in depth the effect of volume corrections and resonance decays on the cumulants of net-charge distributions individually, I also calculated the cumulant products of net-charge distributions with primordial particles, as well as the cumulant products of net-charge distributions in a fixed volume with and without multi-charged hadrons. The deviations of ω of net-charge distributions from Skellam expectations are mainly due to the effect of resonance decays, while the deviations of S σ and κσ 2 are mainly due to the effect of volume corrections.
Note that, for more realistic baseline predictions in the future, more additional effects beyond the non-interacting MCHRG model used in this work are in order, e.g. the correlations between positive charges (protons) and negative charges (antiprotons) from the charge (baryon) conservation laws [18], the dynamic evolution in hadronic phase [42], etc. The results in the present study imply that these effects are expected to make substantial contributions to the variances of net-charge distributions, without significant affecting its skewness and kurtosis.