Off-diagonal cumulants of net-charge, net-proton and net-kaon multiplicity distributions in Au+Au collisions at $\sqrt{s_{NN}}$ = 7.7-200 GeV from STAR

TL;DR

This paper reports on STAR measurements of second-order off-diagonal cumulants of net-charge, net-proton, and net-kaon multiplicity distributions in Au+Au collisions across a range of energies, providing insights into QCD matter properties.

Contribution

It presents the first experimental measurements of off-diagonal cumulants in heavy-ion collisions and compares them with theoretical models to constrain the QCD phase diagram.

Findings

Measured cumulant ratios show agreement with HRG and UrQMD models.

Results help constrain freeze-out parameters in the QCD phase diagram.

Data across energies reveal trends in conserved charge fluctuations.

Abstract

Fluctuations of conserved quantities such as net-baryon, net-charge, and net-strangeness numbers have generated considerable interest in the study of the thermodynamic properties of the hot and dense QCD matter. Theoretical calculations suggest that the off-diagonal cumulants of conserved charges along with the diagonal cumulants can help better constrain the freeze-out parameters and, therefore, help to map the QCD phase diagram. In this proceeding, we briefly outline the recent STAR measurements on the second-order off-diagonal cumulants of net-charge, net-proton, and net-kaon multiplicity distributions in Au+Au collisions from the RHIC BES-I program in the energy range of $\sqrt{s_{NN}}$ = 7.7-200 GeV. The measured cumulant ratios are compared to the predictions from both thermal (HRG) and non-thermal (UrQMD) models.

Figures

Tables

Table 1. Table 1: Summary of the number of events analyzed.

$\sqrt{s_{\mathrm{NN}}}$ (GeV)	Year	Events ($\times {10}^6$)
7.7	2010	1.5
11.5	2010	2.5
14.5	2014	12.7
19.6	2011	15.6
27	2011	25.2
39	2010	62.3
62.4	2010	31
200	2011	74

Equations

$${\chi }_{B,Q,S}^{m,n,l}=\frac{{\partial }^{m+n+l}(P/T^4)}{{\partial }^m({\mu }_B/T){\partial }^n({\mu }_Q/T){\partial }^l({\mu }_S/T)},$$

$$P=\frac{T}{V}\ln [Z(V,T,{\mu }_B,{\mu }_Q,{\mu }_S)].$$

$${\chi }_{B,Q,S}^{m,n,l}=\frac{1}{V{T}^3}{c}_{B,Q,S}^{m,n,l}.$$

$$c_{\alpha }^2={\sigma }_{\alpha }^2=⟨(\delta N_{\alpha }{)}^2⟩\text{and}{c}_{\alpha ,\beta }^{1,1}={\sigma }_{\alpha ,\beta }^{1,1}=⟨(\delta N_{\alpha })(\delta N_{\beta })⟩,$$

Full text

1 Introduction

The major goal of relativistic heavy-ion collision experiment is to study the formation of a new form matter, called the Quark-Gluon Plasma (QGP). Over the last two decades, a number of possible evidences for the QGP phase has been established experimentally. Currently a large community of physicists are exploring the QCD phase structure and trying to find a possible signature of the QCD critical point. One of the most community used method to explore phase diagram is to study the event-by-event fluctuations of conserved charges, such as net electric charge (Q), net baryon number (B) and net strangeness number (S) in the heavy-ion collisions over a wide range of energy [2, 3]. It is proposed that the higher-order cumulants of the net-multiplicity distributions are related to the higher order thermodynamic susceptibilities of corresponding conserved charges in QCD and expected to diverge near the critical point [2, 4]. Over the past few years, STAR and PHENIX experiments at RHIC published results on the diagonal cumulants ($c_{\alpha}^{n}$) of net-electric charge [5, 6], net-proton ($p$, an experimental proxy for net-baryon) [7, 8] and net-kaon ($k$, an experimental proxy for net-strangeness) [9] multiplicity distributions. Similarly, off-diagonal cumulants ($c_{\alpha,\beta}^{m,n}$) of Q, p and k are related to the mixed susceptibilities that carry the correlation between different conserved charges [1]. Lattice QCD and hadron resonance gas model (HRG) calculations show that normalized baryon-strange correlations, that can be expressed as off-diagonal to diagonal cumulant ($C_{BS}=c_{B,S}^{1,1}/c_{S}^{2}$), are expected to be sensitive to the onset of deconfinement [10]. Another importance of studying off-diagonal cumulants is that it can also be used to constrain the freeze-out parameters in the QCD phase diagram. Different theoretical calculations demonstrate that the $2^{nd}$-order off-diagonal cumulants show a significant sensitivity to the difference between HRG and lattice calculations [11, 12].

In this report, we present the measurements of $2^{nd}$-order diagonal and off-diagonal cumulants of net charge, net proton and net kaon multiplicity distributions in Au+Au collisions ranging in center of mass energy from $\sqrt{{s_{\rm NN}}}$ = 7.7 to 200 GeV, with data taken during the first phase of RHIC Beam Energy Scan (BES-I).

2 Observables

The susceptibilities of the conserved charges of a system in thermal and chemical equilibrium (for a grand-canonical ensemble) can be computed from the partial derivatives of the dimensionless pressure with respect to the chemical potentials:

[TABLE]

where $V$ and $T$ are the system pressure and temperature, respectively, and $m,n,l=1,2,3,...n$ are the order of derivative. $\mu_{Q}$, $\mu_{B}$ and $\mu_{S}$ are the electric charge, baryon and strangeness chemical potentials, respectively. The $P$ is obtained from the logarithm of the QCD partition function:

[TABLE]

These susceptibilities can be related to the cumulants ($c$) of the event-by-event distribution of the associated conserved charges by [3, 13, 14]:

[TABLE]

Due to the limitation in detecting all baryons and strange hadrons experimentally, net proton ($p$) and net kaon ($k$) are considered as proxies for the net baryon and net strangeness, respectively. In this report, we present the measurement of second-order ($m+n+l=2$) diagonal and off-diagonal cumulants of net charge, net proton and net kaon multiplicity distributions, can be expressed as:

[TABLE]

where $\alpha$ and $\beta$ can be $Q$, $p$ or $k$, and $\delta N_{\alpha}=(N_{\alpha^{+}}-N_{\alpha^{-}})-\langle~{}(N_{\alpha^{+}}-N_{\alpha^{-}})~{}\rangle$. Finally, we construct the off-diagonal to diagonal cumulant ratios ($C_{p,k}=c^{1,1}_{p,k}/c^{2}_{k}$, $C_{Q,p}=c^{1,1}_{Q,p}/c^{2}_{p}$ and $C_{Q,k}=c^{1,1}_{Q,k}/c^{2}_{k}$) motivated by Ref. [10], which also cancel the volume dependence.

3 Experimental details

Second-order cumulants of net charge, net proton and net kaon multiplicity distributions for Au+Au collisions at $\sqrt{{s_{\rm NN}}}$ = 7.7, 11.5, 14.5, 19.6, 27, 39, 62.4 and 200 GeV have been studied in the STAR experiment. The minimum-bias (MB) events are analyzed with the requirement that the position of the primary vertex along $z$-direction ($V_{z}$) was reconstructed within $\pm 30$ cm of the center of the STAR detector and within 2 cm on the transverse plane of the beam axis. The number of events analyzed at each energy after applying all event selections criteria is listed in Table 1 .

All charged tracks used in this analysis are required to be within the pseudorapidity range $|\eta|<0.5$, and the transverse momentum range $0.4<p_{\rm{T}}<1.6$ GeV/$c$. To reduce the contamination from the secondary charged particles, only primary tracks are selected within a distance of closest approach (DCA) to the primary vertex of less than 1 cm. The main detectors used in this analysis are the Time Projection Chamber (TPC) and the Time of Flight (TOF). The particle identification (PID) is done with a common acceptance: $0.4<p_{\rm{T}}<1.6$ GeV/$c$ and $|\eta|<0.5$. Within this range, the purities of $K^{\pm}$ and $p(\bar{p})$ identification is estimated to be 98-99%. The collision centrality for this analysis is defined using uncorrected charged particle multiplicity measured within a pseudorapidity range of $0.5<|\eta|<1.0$ in the TPC detector. This way we exclude the particles from the analysis region to determine the centrality in order to suppress autocorrelation effects [15]. We present the results for nine centrality bins, 0-5% (most central), 10-20%, … , 70-80% (most peripheral) as a function of average number of participating nucleons ($\langle N_{part}\rangle$) estimated using a Monte Carlo Glauber model [16]. The cumulants and their ratios were calculated as a function of the reference multiplicity and then averaged over the centrality bins to suppress the volume fluctuations over wide centrality bins [17, 18]. We use embedding Monte Carlo simulation techniques to obtain the efficiencies and an algebra based on binomial detector response to efficiency correction [19, 20, 21]. The statistical uncertainty estimation is based on the numerical error propagation method of multivariate cumulants [22]. The systematic uncertainties are estimated by varying different track quality cuts, tracking efficiency and conditions for particle identification.

4 Results

The centrality dependence of efficiency corrected second-order diagonal cumulants of net proton, net kaon and net charge (top to bottom) for 0-5% most central Au + Au collisions at $\sqrt{{s_{\rm NN}}}$ = 7.7, 11.5, 14.5, 19.6, 27, 39, 62.4 and 200 GeV are shown as a function of $\langle N_{part}\rangle$ in Fig. 1 . We find a linearly increasing trend as expected from a scaling, predicted by the central limit theorem. In a given centrality, the width of net-proton distribution decreases as a function of beam energy in the range $\sqrt{{s_{\rm NN}}}$ = 7.7-39 GeV and then increases at top RHIC energies [8]. This is because of baryon transport that has a strong beam energy dependence.

Figure 2 shows the efficiency corrected second-order off-diagonal cumulants of net-charge, net-proton, net-kaon multiplicity distributions for Au+Au collisions at eight colliding energies. The off-diagonal cumulants between net-charge$-$net-kaon ($\sigma^{1,1}_{Q,k}$) and that of net-charge$-$net-proton ($\sigma^{1,1}_{Q,p}$) increase with centrality. On the contrary, there is a growing anti-correlation behaviour observed between net-proton and net-kaon ($\sigma^{1,1}_{p,k}$) with centrality at $\sqrt{{s_{\rm NN}}}$ $>$ 19.6 GeV.

Figure 3 shows the off-diagonal to diagonal cumulant ratios $C_{p,k}=\sigma^{1,1}_{p,k}/\sigma^{2}_{k}$, $C_{Q,k}=\sigma^{1,1}_{Q,k}/\sigma^{2}_{k}$ and $C_{Q,p}=\sigma^{1,1}_{Q,p}/\sigma^{2}_{p}$ (top to bottom) as a function of beam energy for most central (0-5%) and peripheral (70-80%) collisions. These cumulant ratios are designed to eliminate the effect of system volume. An excess correlation is observed in $C_{Q,p}$ and $C_{Q,k}$ in 0-5% most central in comparison to the peripheral collisions. The values of $C_{Q,p}$ and $C_{Q,k}$ are observed to increase with beam energy, and this increasing trend cannot be explained by the HRG and UrQMD model calculations. It is observed that the normalized $p$-$k$ correlation ($C_{p,k}$) is positive at the lowest BES energy and negative at higher energies. For 0-5% top central bins, $C_{p,k}$ changes sign around 19.6 GeV.

5 Summary

The second-order diagonal and off-diagonal cumulants of net proton, net kaon, and net charge multiplicity distributions in Au+Au collisions from the RHIC BES-I program in the energy range of $\sqrt{s_{NN}}$ = 7.7-200 GeV are presented. Significant excess correlation is observed in $C_{Q,p}$ and $C_{Q,k}$ in central in comparison with peripheral events. Both HRG and UrQMD model underpredict the data and cannot describe the increasing with beam energy trends of $C_{Q,p}$ and $C_{Q,k}$. The value of $C_{p,k}$ in 0-5% central collision is found to be negative at $\sqrt{{s_{\rm NN}}}$ = 200 GeV and positive at $\sqrt{{s_{\rm NN}}}$ = 7.7 GeV. The measurements of the full second-order cumulant matrix elements of net-$p/k/Q$ multiplicity distributions as a function of centrality and beam energy will improve the estimation of freeze-out parameters by theoretical calculations and that help to map the QCD phase diagram. For more details of this analysis we refer the readers to Ref. [1].