Net-Proton Fluctuations at FAIR Energies Using PHQMD Model

Abstract

One of the main goals of the Compressed Baryonic Matter (CBM) experiment at the Facility for Antiproton and Ion Research (FAIR) is to investigate the properties of strongly interacting matter under high baryon densities and explore the QCD phase diagram. Fluctuations of conserved quantities like baryon number, electric charge, and strangeness are key probes for phase transitions and critical behavior, as are connected to thermodynamic susceptibilities predicted by lattice QCD calculations. In this paper, we report on up-to-the-fourth-order cumulants of (net-)proton number distributions in gold–gold ion collisions at the nucleon–nucleon center of mass energies $\sqrt{s_{NN}}$ = 3.5–19.6 GeV using the Parton–Hadron-Quantum-Molecular Dynamics (PHQMD) model. Protons and anti-protons are selected at midrapidity ($\left|y\right|$ < 0.5) within a transverse momentum range 0.4 $<p_T<$ 2.0 GeV/c of STAR experiment and 1.08 $<y<$ 2.08 and 0.4 $<p_T<$ 2.0 GeV/c of CBM acceptances. The results obtained from the PHQMD model are compared with the existing experimental data to undersatand potential signatures of critical behavior and to probe the vicinity of the critical end point in the CBM energy range. The results obtained here with the PHQMD calculations for $\kappa {\sigma }^2$ (the distribution kurtosis times variance squared) are consistent with the overall trend of the measurement results for the most central (0–5% centrality) collisions, although the calculations somewhat overestimate the experimental values.

1. Introduction

One of the unsolved questions in heavy-ion collisions is to understand how quark–gluon plasma (QGP) evolves with decreasing beam energy and to determine the energy threshold below which QGP formation no longer occurs. The main goal of the experiments with relativistic heavy-ion collisions is to explore the phase structure of quantum chromodynamics (QCD). Lattice QCD calculations predict a smooth crossover from hadronic matter to a deconfined state of quarks and gluons at high temperatures and vanishing net-baryon chemical potential ${\mu }_B=0$ [1]. However, at higher values of ${\mu }_B$, theoretical QCD-based models suggest the existence of a first-order phase transition boundary [2,3], which in turn implies the existence of the QCD critical point at the boundary [4,5]. Meantime, due to the sign problem in lattice QCD at finite baryon density, the location of the critical point remains quite uncertain [4].

Fluctuations of different conserved quantities, such as net-baryon, net-charge, and net-strangeness number, are analyzed on an event-by-event basis. Study of these fluctuations allows probing the system’s correlation length $\xi$, the latter characterizing an extent of spatial and temporal correlations. In the vicinity of the QCD critical point, the correlation length is expected to diverge in an infinite system. As a result, fluctuation measurements provide a sensitive tool for exploring critical phenomena and searching for the QCD critical point [6,7,8,9,10].

The CBM experiment at FAIR, Darmstadt, Germany, is considered to provide a unique opportunity to investigate the model-predicted first-order phase transition from hadronic matter to QGP with high precision, enabled by the experiment’s exceptionally high collision rate of up to 10 MHz, significantly exceeding those of earlier experiments [11].

In this study, we present the first measurements of the moments of net-proton (a proxy for net-baryon) distributions in Au+Au collisions at the nucleon–nucleon center-of-mass energy $\sqrt{s_{NN}}$ spanning 3.5 to 19.6 GeV, using the PHQMD model within the Solenoidal Tracker at Relativistic Heavy Ion Collider (STAR) acceptance, namely, the midrapidty $\left|y\right|$ < 0.5 and the transverse momenta 0.4 $<p_T<$ 2.0 GeV/c and the CBM detector acceptance of 1.08 $<y<$ 2.08 and 0.4 $<p_T<$ 2.0 GeV/c, with c the speed of light. These results serve as baseline calculations for proton cumulants and correlation functions from the PHQMD model, contributing to the ongoing search for the QCD critical point in heavy-ion collisions. At comparably low collision energies, the net-proton number is primarily dominated by protons, making it a sensitive observable in this regime.

The paper is organized as follows. Section 2 introduces the observables of fluctuation study, the relation between the cumulants. Section 3 discusses the PHQMD model and the data samples generated with it. Section 4 presents the results, where we show the energy and centrality dependences of net-proton number cumulants in Au+Au collisions at $\sqrt{s_{NN}}$ spanning 3.5 to 19.6 GeV generated using the PHQMD model. The paper ends with a summary (Section 5) and conclusions (Section 6).

2. Cumulants

To quantify intricacies of a distribution, fluctuations are typically characterized using correlation functions—moments of distributions or cumulants. The cumulants of a distribution up to the fourth order are defined as follows:

$$\begin{array}{ccc}\hfill C_1& =& \langle N\rangle ,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill C_2& =& \langle \delta N^2\rangle ,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill C_3& =& \langle \delta N^3\rangle ,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill C_4& =& \langle \delta N^4\rangle -3{\langle \delta N^2\rangle }^2,\hfill \end{array}$$

(4)

where N is the net-particle number per event, $\langle N\rangle$ is the average of N across all the events, and $\delta N=N-\langle N\rangle$. The first- and second-order moments represent the mean and variance squared, whereas the third- and fourth-order moments reflect the skewness and kurtosis,

$$\begin{array}{ccc}\hfill M& =& \langle N\rangle ,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill {\sigma }^2& =& \langle \delta N^2\rangle ,\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill S& =& \langle \delta N^3\rangle /{\left(\langle \delta N^2\rangle \right)}^{3/2},\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill \kappa & =& \langle \delta N^4\rangle /{\left(\langle \delta N^2\rangle \right)}^2-3,\hfill \end{array}$$

(8)

respectively. The products of moments $\kappa {\sigma }^2$ and $S\sigma$ can be expressed in terms of cumulant ratios:

$$\kappa {\sigma }^2=C_4/C_2$$

(9)

and

$$S\sigma =C_3/C_2.$$

(10)

The moment ${\sigma }^2$, skewness S, and kurtosis $\kappa$ have been found to be proportional to the powers of the correlation length $\xi$, specifically ${\xi }^2$, ${\xi }^{4.5}$, and ${\xi }^7$ [7], respectively. The $n\mathrm{th}$-order susceptibilities ${\chi }^{\left(n\right)}$ are related to the cumulant $C_n$ as ${\chi }^{\left(n\right)}=C_n/V{T}^3$ [12], where V and T represent, respectively, the volume and the temperature of the system. The event-by-event fluctuation of the net-proton number (the difference between the number of protons and antiprotons observed) has been proposed as a key observable for signaling the QCD critical end point [13]. In the energy range $\sqrt{s_{NN}}$ = 3.5–4.9 GeV of the CBM experiment, the expectation is that the QCD critical point may be discovered.

3. PHQMD: Model and Data Samples

3.1. The Model

The PHQMD model has been developed to explore the complex dynamics of heavy-ion collisions and the formation of clusters and hypernuclei over a broad spectrum of beam energies, from several hundred AMeV up to ultra-relativistic scales. The model is built upon the Quantum Molecular Dynamics (QMD) model and the Parton–Hadron-String Dynamics (PHSD). The PHQMD model describes the particle production from SIS up to the LHC energies [14] based on the following.

• QMD approach. In PHQMD, baryons are treated as Gaussian wave packets and propagated using the QMD model, which includes density-dependent two-body potential interactions. This approach enables the preservation of full n-body phase-space correlations among the baryons, contrasting with traditional mean-field models that average out those correlations [15,16].
• PHSD approach. PHQMD considers the collision integral from PHSD to simulate the full evolution of the system, including hadronic collisions, QGP formation, partonic scatterings, hadronization, propagation of mesons and final-state hadronic interactions. The partonic phase is treated using the Kadanoff–Baym equations [17,18]. The model also incorporates in-medium effects such as collisional broadening and modifications of spectral functions for vector mesons (e.g., $\rho$, $\omega$, $\varphi$) and strange mesons (K, $\overline{K}$, $K^{*}$, ${\overline{K}}^{*}$), enabling a realistic description of hadronic dynamics in dense and hot nuclear matter.
• Dynamic cluster formation. PHQMD takes a dynamic approach to cluster formation with QMD. Instead of applying a static coalescence criterion at a specific time, it allows clusters to emerge through ongoing potential interactions throughout the system’s evolution. Cluster recognition is carried out at selected times using the Simulated Annealing Clusterization Algorithm (SACA) [19] or the Minimum Spanning Tree (MST) method [14]. Before the SACA or MST algorithms are applied, baryon resonances are decayed “virtually” and the decay nucleons are considered for cluster formation (whereas the baryon resonances continue to propagate within the PHQMD framework until they naturally decay) [20].
• QGP phase identification: In PHQMD, the regions where the local energy density exceeds 0.5 GeV/fm³ are considered to be in the QGP phase, as hadrons are expected to dissolve into their constituent quarks and gluons. In the QGP phase, the quarks, antiquarks, and gluons are scattered and dynamically propagated within a self-generated scalar mean-field potential. As the system expands and the local energy density decreases to the critical value, the partons undergo hadronization into color-neutral off-shell hadrons—mesons and baryons. This process is described by covariant transition rates that conserve the energy, momentum, and quantum numbers per event.

3.2. Generated Data Samples

In the analysis performed, we use the MST algorithm in the PHQMD model for clusterization. In this approach, cluster formation is determined entirely based on coordinate-space information. Two nucleons are considered part of the same cluster if their spatial separation is less than a threshold distance $r_0=4\mathrm{fm}$. Nucleons connected by this spatial criterion collectively form a cluster. All clusters are initially counted without distinguishing their physical relevance. For this study, from the clusters identified by the MST method, only combinations of baryons that form physical nuclei are selected.

In the current study, we use the PHQMD QGP-on and QGP-off data sets. The QGP-on state refers to the system being in the QGP phase, characterized by a local energy density exceeding 0.5 GeV/fm³. If this condition is not met, the system is considered to be in the QGP-off state.

Two million minimum-bias Au+Au collision events were generated at $\sqrt{s_{NN}}$ = 3.5, 4.0, 4.9, 7.7, and 19.6 GeV for STAR acceptance and at $\sqrt{s_{NN}}$ = 3.5, 4.0, and 4.9 GeV for CBM acceptance.

Although PHQMD offers the flexibility to explore by changing the input parameters, resource and storage constraints prevented the generation of two million minimum-bias events for each QGP-on and QGP-off scenario involving physical clustering. To study how the centrality-wise $S\sigma$, $\kappa {\sigma }^2$ differs for the QGP-on and QGP-off cases, we compare two million minimum-bias events for the $\sqrt{s_{NN}}$ = 7.7 GeV. Our analysis, based on varying model parameters, focuses on results obtained using a reduced maximum-impact parameter of $b_{\max }=3.2$ fm ($b_{\max }$ obtained using the Glauber model [21] as $3.2$ fm), corresponding to the 0–5% of most central collision events. This data set allows us to investigate and compare the effects of physical clustering and QGP dynamics under the top-central collision condition.

In the model calculations, the volume fluctuations are not applied, whereas this analysis technique has been used in the real data analysis of STAR. Instead of applying volume fluctuation, we performed the analysis considering smaller centrality bins. We determined centrality using the multiplicity distribution of all particles, whereas the STAR experiment employed a different method for centrality determination in BES-II [22]. In addition, the Delta theorem method [23,24] has been used to estimate the statistical error in cumulants. This paper focuses on the results from the 0–5% central (most central) Au+Au collision events and centrality-dependent results.

4. Results

Figure 1 shows the net-proton multiplicity distributions for 0–5% collision centrality. With increasing energy, proton as well as net-proton numbers decrease because the net-baryon number tends to zero in the higher-energy range.

Figure 2 and Figure 3 show the centrality dependence of net-proton cumulants $C_n$ (1)–(4), generated by the PHQMD model for the STAR and CBM acceptances, respectively. In Figure 2, the cumulants tend to deviate towards the central collisions, while for the peripheral collisions (higher centrality), the values are close to zero across all energies. The trend observed in the net-proton $C_1$ (mean) (5)) reflects an increase in particle production, favoring matter over antimatter, as collisions go from peripheral to central. Similarly, the $C_2$ (variance) (6)) rises due to the same effect. As both proton and antiproton yields increase with the increasing centrality, the variance of their distributions also becomes larger. It is necessary to note that higher-order cumulants are associated with larger statistical uncertainties.

Figure 4 and Figure 5 show $C_1$–$C_4$ similar to those in Figure 2 and Figure 3, but now for 0–5%-centrality events and as a function of $\sqrt{s_{NN}}$. $C_1$, $C_2$, and $C_3$ cumulants decrease with an increase of $\sqrt{s_{NN}}$, while $C_4$ cumulants demonstrate opposite trend is seen, with larger statistical error for lower energies.

The collision energy dependence of the cumulant ratios $S\sigma$ (10) (Figure 6a) and $\kappa {\sigma }^2$ (9) (Figure 6b) for net-proton distributions are presented in the energy range $\sqrt{s_{NN}}$ = 3.5–19.6 GeV considering the STAR acceptance. To eliminate volume dependence, the ratios of cumulants of different orders are utilized. The results are shown for the most central (0–5%) collisions. The STAR data [22] are compared to calculations of PHQMD without a critical point. Figure 7 shows the $S\sigma$ (10) (Figure 7a) and $\kappa {\sigma }^2$ (9) (Figure 7b) of net-proton distribution for top-central PHQMD data for the CBM acceptance. $\kappa {\sigma }^2$ values are negative at lower energies but appear to approach zero or be somewhat positive at higher energies. This might suggest a non-Gaussian distribution, possibly reflecting critical dynamics or non-Poissonian fluctuations.

Figure 8 shows the centrality dependence of cumulant ratios of net-protons in the PHQMD data for Au+Au collisions at $\sqrt{s_{NN}}$ = 3.5–19.6 GeV for STAR acceptance. In the case of CBM acceptance, the centrality dependence of $S\sigma$ and $\kappa {\sigma }^2$ are shown in Figure 9a and Figure 9b, respectively. Overall, $S\sigma$ decreases from central to peripheral collisions and there are more pronounced negative $\kappa {\sigma }^2$ values at lower energies and in central events. For the energy range $\sqrt{s_{NN}}=3.5$–4.9 GeV, the values of $\kappa {\sigma }^2$ (Figure 8b) initially decrease with centrality and then exhibit a rise in the peripheral region. In contrast, for higher beam energies, such as 7.7 and 19.6 GeV, $\kappa {\sigma }^2$ decreases with centrality and remains approximately constant towards the peripheral region.

We present a comparison of the centrality-dependent PHQMD results for QGP-on and QGP-off scenarios, showing $S\sigma$ and $\kappa {\sigma }^2$ of the net-proton multiplicity distribution in Au+Au collisions at $\sqrt{s_{NN}}=7.7$ GeV, within the STAR acceptance, in Figure 10. In Figure 10a, both the blue QGP-on and QGP-off points show a decreasing trend with centrality, indicating that the values of $S\sigma$ are more pronounced in central collisions. The difference between QGP-on and QGP-off is quite small but visible for central collisions, where QGP effects are expected to be stronger due to higher energy density.

For the top-central collisions (0–5%), there is a noticeable difference between the QGP-on and QGP-off scenarios, which one can see in Figure 10b. QGP-on sample shows a significantly larger value of possibly reflecting enhanced critical fluctuations or non-Gaussian tails. For mid-central to peripheral collisions, both QGP-on and QGP-off values remain almost constant and quite close to unity. Considerably large errors at top-centrality reflect statistical uncertainty due to less number of events.

Figure 11 shows the collision energy dependence of net-proton fluctuation observables $S\sigma$ (10) and $\kappa {\sigma }^2$ (9) obtained from PHQMD simulations for Au+Au collisions under STAR acceptance ($\left|y\right|<0.5$, $0.4<p_T<2.0$ GeV/c) with $b_{\max }=3.2$ fm. The results compare the scenarios with and without a QGP phase (QGP-on and QGP-off). A noticeable difference is observed in $S\sigma$ at lower energies ($\sqrt{s_{NN}}<10$ GeV), where the QGP-on case shows suppressed values, suggesting a damping of fluctuations due to QGP formation. $\kappa {\sigma }^2$ exhibits non-monotonic behavior in both cases, with a somewhat stronger suppression in the QGP-on scenario.

5. Discussion

One has to note that in peripheral collisions, the system formed may not reach the necessary temperature and density to undergo a phase transition or approach the QCD critical point. Therefore, the calculations performed here mainly focused on the most central collisions. The STAR Collaboration [22] observed a non-monotonic energy dependence of net-proton kurtosis ($\kappa {\sigma }^2$) in central Au+Au collisions what suggests possible signatures of the QCD critical point. In contrast, peripheral collisions showed a monotonic trend consistent with non-critical models. Within uncertainties, the values of $\kappa {\sigma }^2$ in the analysis performed here are consistent with the STAR experimental data for central (0–5%) Au+Au collision energies for 7.7 and 14.5 GeV. The fluctuations are found to decrease with increasing centrality, showing some abrupt behavior in the 5–10% centrality interval in Figure 8a and Figure 9a. For $\kappa {\sigma }^2$, similar irregularities are observed in Figure 8b and Figure 9b, particularly within the 0–30% centrality interval. The observed dip can be influenced either by limited event statistics or by dynamical cluster formation effects. Using the generated data set, we have investigated the differences between the QGP-on and QGP-off scenarios as implemented in the PHQMD model. A visible distinction is observed in the $\kappa {\sigma }^2$ values for central (0–5%) Au+Au collisions at $\sqrt{s_{NN}}=7.7$ GeV within the STAR acceptance in Figure 10b, highlighting the sensitivity of higher-order fluctuations to the presence of a QGP phase. The results shown in Figure 11 considering $b_{\max }=3.2$ fm indicate that the presence of a QGP phase can significantly influence net-proton fluctuation observables, particularly at lower collision energies, where a suppression in $S\sigma$ and a stronger non-monotonic trend in $\kappa {\sigma }^2$ are observed in the QGP-on scenario.

The exploration of the energy region ($\sqrt{s_{NN}}$ = 3.0–4.9 GeV) with precision measurements is currently underway using STAR’s FXT program. It has to be noted that the PHQMD model does not explicitly incorporate the critical point in its current implementation. While non-monotonic behavior in observables such as $\kappa {\sigma }^2$ is often regarded as a potential signal for the existence of the QCD critical point, its presence alone does not provide conclusive evidence.

Previous studies employing conventional transport models such as UrQMD [25] and AMPT [26] have demonstrated that hadronic scatterings can significantly influence the net-proton distributions in heavy-ion collisions. In the present PHQMD framework, such rescattering effects have not yet been taken into account; however, their inclusion constitutes a promising direction for future investigations.

Furthermore, UrQMD does not account for realistic cluster formation, which plays a significant role in the evolution and final-state composition of the system. In contrast, PHQMD includes a dynamic cluster formation mechanism, allowing for the generation of light nuclei and fragments based on nucleon correlations in phase space. This aspect provides a more realistic scenario for studying fluctuations of conserved quantities, particularly in the high-baryon-density regime relevant to FAIR energies. As a result, PHQMD offers a more comprehensive framework for investigating fluctuation observables in heavy-ion collisions. The upcoming CBM experiment is designed to explore this region and is expected to provide a large enough amount of data. A significantly larger statistics data set is required to investigate the QCD critical point.

6. Conclusions

By maintaining n-body correlations and dynamically modeling interactions from early to late stages, PHQMD enhances the capabilities of earlier models and offers a comprehensive picture of both hadronic and partonic matter, as well as cluster and hypernuclei formation in heavy-ion collisions.

Fluctuations of conserved quantities—such as baryon number, electric charge, and strangeness—are considered sensitive probes in heavy-ion collisions for investigating the QCD phase transition and identifying the critical point. The STAR experiment has reported measurements of the moments of net-proton distributions in Au+Au collisions over a collision energy range of $\sqrt{s_{NN}}$ = 7.7–200 GeV [22]. The PHQMD calculations for $\kappa {\sigma }^2$ are consistent with the trend of STAR data for the most central 0–5% of Au+Au collisions within current uncertainties, whereas the same for $S\sigma$ somewhat overestimate the data.

QCD-based model calculations suggest that net-proton fluctuations are particularly sensitive observables. In this study, the cumulant ratios of net-proton distributions from PHQMD simulations display a non-monotonic trend as a function of collision energy within the CBM energy range, albeit within the current statistical uncertainties. This analysis offers new insights into the behavior of net-proton fluctuations in the CBM energy domain and contributes to establishing a baseline for interpreting net-proton and net-baryon fluctuations. Such a baseline is essential for future high-statistics measurements within the CBM experiment, aimed at probing possible critical phenomena near the QCD critical end point.