Introduction

Mapping the phase diagram of quantum chromodynamics (QCD)is a primary objective in nuclear physics, which involves chiral and deconfinement phase transitions related to the transformation of quark-gluon plasma to hadronic matter [1]. The calculations from lattice QCD and hadron resonance gas (HRG) model indicate that a smooth crossover tranformation occurs at high temperatures and small chemical potentials [2-8]. Furthermore, studies on effective quark models [9-23], the Dyson-Schwinger equation approach [24-29], the functional renormalization group theory [30-32] and machine learning [33] suggest that a first-order chiral phase transition occurs at large chemical potentials.

The fluctuations and correlations of conserved charges (baryon number B, electric charge Q, and strangeness S) are sensitive observables for studying the phase transitions of strongly interacting matter [34, 35]. The net proton (proxy for net baryon) cumulants measured in the beam energy scan (BES) program at the Relativistic Heavy Ion Collider (RHIC) [36-42] has inspired extensive studies on QCD phase transition, particularly the QCD critical endpoint (CEP). More impressively, the distributions of the net proton number at the center-of-mass energy $\sqrt{s_{\text{NN}}}=3\text{ GeV}$ and 2.4 GeV are different from those at 7.7 GeV and above because the fluctuation distributions of the net proton number are primarily dominated by the interaction among hadrons [40].

The experimental results at 3 GeV and below necessitate studies on the effect of hadronic interactions on the fluctuations of conserved charges at lower energies [43-46]. The nuclear liquid-gas phase transition (LGPT) may be involved at lower collision energies [47-63].

A van der Waals model was used to study the high-order distributions of the net baryon number in both pure and mixed phases of the LGPT[64-66]. The second-order susceptibility of the net baryon number for positive- and negative-parity nucleons was examined near the chiral and nuclear liquid-gas phase transitions using a double-parity model, in which both the chiral phase transition and nuclear LGPT are effectively included [45]. The net baryon kurtosis and skewness were considered in the nonlinear Walecka model to analyze the experimental signals at lower collision energies [55, 56]. The hyperskewness and hyperkurtosis of the net baryon number were recently calculated to explore the relationship between nuclear LGPT and experimental observables [67].

Because the interactions among hadrons dominate the density fluctuations in lower-energy regimes (below 3 GeV), the BES program at collision energies lower than 7.7 GeV is expected to provide detailed information on the phase structure of strongly interacting matter. Additionally, relevant experiments have been planned at the High Intensity Heavy-ion Accelerator Facility (HIAF). Meanwhile, the HADES collaboration at the GSI Helmholtzzentrum für Schwerionenforschung planned to measure the higher-order net proton and net charge fluctuations in the central Au + Au reactions at collision energies ranging from 0.2 A GeV to 1.0 A GeV to probe the LGPT region [68]. These experiments are significant for investigating nuclear liquid-gas and chiral phase transitions through density fluctuations.

In addition to the fluctuations in conserved charges, the correlations between different conserved charges provide important information for exploring phase transitions. The correlations of conserved charges or off-diagonal susceptibilities have been calculated to study the chiral and deconfinement phase transitions at high temperatures in lattice QCD and some effective quark models (e.g., [69-75]). However, correlations between the net baryon number and electric charge in nuclear matter and their relationship with nuclear LGPT, which are useful for diagnosing the phase diagram of strongly interacting matter at low temperatures, have not yet been explored. In this study, we explored the correlations between the net baryon number and electric charge up to the sixth order in nuclear matter using the nonlinear Walecka model. The characteristic behaviors of correlations evoked by the nucleon-nucleon interaction, both near and far away from the nuclear LGPT, were obtained. These results are expected to aid future analyses of chiral phase transitions, nuclear LGPT, and related experimental signals.

The remainder of this paper is organized as follows. In Sect. 2, we introduce formulas to describe the correlations between conserved charges and the nonlinear Walecka model. In Sect. 3, we illustrate the numerical results for the correlation between the net baryon number and electric charge. Finally, a summary is presented in Sect. 4.

Theoretical descriptions

The fluctuations and correlations of conserved charges are related to the equation of state of a thermodynamic system. In the grand canonical ensemble of strongly interacting matter, the pressure is the logarithm of the partition function [76]: $P=\frac{T}{V}\ln Z\left(V\mathrm{,}T\mathrm{,}{\mu }_{\text{B}}\mathrm{,}{\mu }_{\text{Q}}\mathrm{,}{\mu }_{\text{S}}\right)\mathrm{,}$ (1) where ${\mu }_{\text{B}}\mathrm{,}{\mu }_{\text{Q}}\mathrm{,}{\mu }_{\text{S}}$ are the chemical potentials of the conserved charges, that is, the baryon number, electric charge, and strangeness in a strong interaction, respectively. The generalized susceptibilities are derived by taking the partial derivatives of the pressure with respect to the corresponding chemical potentials [39] ${\chi }_{ijk}^{\text{BQS}}=\frac{{\partial }^{i+j+k}\left[P/T^4\right]}{\partial {\left({\mu }_{\text{B}}/T\right)}^i\partial {\left({\mu }_{\text{Q}}/T\right)}^j\partial {\left({\mu }_{\text{S}}/T\right)}^k}\mathrm{.}$ (2) The cumulants of the multiplicity distributions of the conserved charges are usually measured experimentally. These are related to the generalized susceptibilities by $C_{ijk}^{\text{BQS}}=\frac{{\partial }^{i+j+k}\ln \left[Z\left(V\mathrm{,}T\mathrm{,}{\mu }_{\text{B}}\mathrm{,}{\mu }_{\text{Q}}\mathrm{,}{\mu }_{\text{S}}\right)\right]}{\partial {\left({\mu }_{\text{B}}/T\right)}^i\partial {\left({\mu }_{\text{Q}}/T\right)}^j\partial {\left({\mu }_{\text{S}}/T\right)}^k}=V{T}^3{\chi }_{ijk}^{\text{BQS}}\mathrm{.}$ (3) To eliminate the volume dependence in heavy-ion collision experiments, observables are usually constructed using the ratios of cumulants and then compared with the theoretical calculations of the generalized susceptibilities with $\frac{C_{ijk}^{\text{BQS}}}{C_{\text{lmn}}^{\text{BQS}}}=\frac{{\chi }_{ijk}^{\text{BQS}}}{{\chi }_{\text{lmn}}^{\text{BQS}}}\mathrm{.}$ (4) In this study, the nonlinear Walecka model was used to calculate the correlations between the net baryon number and electric charge in nuclear matter at low temperatures. This model describes the properties of finite nuclei and the equation of state of nuclear matter. The approximate equivalence of this model with the HRG model at low temperatures and densities is also indicated in Ref. [77]. This model was recently used to explore fluctuations in the net baryon number in nuclear matter, for example, kurtosis and skewness [55, 56] and hyperskewness and hyperkurtosis [67].

The Lagrangian density for the nucleon–meson system in the nonlinear Walecka model [54, 78] is $\begin{array}{c}\mathcal{L}={\displaystyle \sum _N}{\overline{\psi }}_N\left[i{\gamma }_{\mu }{\partial }^{\mu }-\left(m_N-g_{\sigma }\sigma \right)-g_{\omega }{\gamma }_{\mu }{\omega }^{\mu }-g_{\rho }{\gamma }_{\mu }\tau \cdot {\rho }^{\mu }\right]{\psi }_N\\ +\frac{1}{2}\left({\partial }_{\mu }\sigma {\partial }^{\mu }\sigma -m_{\sigma }^2{\sigma }^2\right)-\frac{1}{3}b{m}_N{\left(g_{\sigma }\sigma \right)}^3-\frac{1}{4}c{\left(g_{\sigma }\sigma \right)}^4\\ +\frac{1}{2}{m}_{\omega }^2{\omega }_{\mu }{\omega }^{\mu }-\frac{1}{4}{\omega }_{\mu \nu }{\omega }^{\mu \nu }\\ +\frac{1}{2}{m}_{\rho }^2{\rho }_{\mu }\cdot {\rho }^{\mu }-\frac{1}{4}{\rho }_{\mu \nu }\cdot {\rho }^{\mu \nu }\mathrm{,}\end{array}$ (5) where ${\omega }_{\mu \nu }={\partial }_{\mu }{\omega }_{\nu }-{\partial }_{\nu }{\omega }_{\mu }$, ρμν=∂μρν-∂νρμ and mN is the nucleon mass in vacuum. The interactions between nucleons are mediated by $\sigma \mathrm{,}\omega \mathrm{,}\rho$ mesons..

The thermodynamic potential can be derived in the mean-field approximation as $\begin{array}{c}\text{Ω}=-{\beta }^{-1}{\displaystyle \sum _{N}2}{\displaystyle \int }\frac{d^{3}k}{{\left(2\pi \right)}^3}[\ln \left(1+e^{-\beta \left(E_N^{*}\left(k\right)-{\mu }_N^{*}\text{}\right)}\right)\\ +\ln \left(1+e^{-\beta \left(E_N^{*}\left(k\right)+{\mu }_N^{*}\right)}\right)]+\frac{1}{2}{m}_{\sigma }^2{\sigma }^2+\frac{1}{3}b{m}_N{\left(g_{\sigma }\sigma \right)}^3\\ +\frac{1}{4}c{\left(g_{\sigma }\sigma \right)}^4-\frac{1}{2}{m}_{\omega }^2{\omega }^2-\frac{1}{2}{m}_{\rho }^2{\rho }_3^2\mathrm{,}\end{array}$ (6) where $\beta =1/T$, $E_{\text{N}}^{*}=\sqrt{k^2+m_{\text{N}}^{*2}}$, and ρ₃ denotes the third component of the ρ meson field. The effective nucleon mass $m_{\text{N}}^{*}=m_{\text{N}}-g_{\sigma }\sigma$ and effective chemical potential ${\mu }_{\text{N}}^{*}$ is defined as ${\mu }_{\text{N}}^{*}={\mu }_{\text{N}}-g_{\omega }{\omega }_{}-{\tau }_{\text{3N}}{g}_{\rho }{\rho }_3$ (${\tau }_{\text{3N}}=1/2$ for proton, -1/2 for neutron).

Minimizing the thermodynamical potential $\frac{\partial \text{Ω}}{\partial \sigma }=\frac{\partial \text{Ω}}{\partial \omega }=\frac{\partial \text{Ω}}{\partial {\rho }_3}=\mathrm{0,}$ (7) the meson field equations can be derived as: $g_{\sigma }\sigma ={\left(\frac{g_{\sigma }}{m_{\sigma }}\right)}^2\left[{\rho }_{\text{p}}^s+{\rho }_{\text{n}}^s-b{m}_{\text{N}}{\left(g_{\sigma }\sigma \right)}^2-c{\left(g_{\sigma }\sigma \right)}^3\right]\mathrm{,}$ (8) $g_{\omega }\omega ={\left(\frac{g_{\omega }}{m_{\omega }}\right)}^2\left({\rho }_{\text{p}}+{\rho }_{\text{n}}\right)\mathrm{,}$ (9) $g_{\rho }{\rho }_3=\frac{1}{2}{\left(\frac{g_{\rho }}{m_{\rho }}\right)}^2\left({\rho }_{\text{p}}-{\rho }_{\text{n}}\right)\mathrm{.}$ (10) In Eqs.(8)–(10), the nucleon number density ${\rho }_i=2{\displaystyle \int }\frac{{\text{d}}^{3}k}{{\left(2\pi \right)}^3}\left[f\left(E_i^{*}-{\mu }_i^{*}\right)-\overline{f}\left(E_i^{*}+{\mu }_i^{*}\right)\right]\mathrm{,}$ (11) and scalar density ${\rho }_i^{\text{s}}=2{\displaystyle \int }\frac{{\text{d}}^{3}k}{{\left(2\pi \right)}^3}\frac{m_i^{*}}{E_i^{*}}\left[f\left(E_i^{*}-{\mu }_i^{*}\right)+\overline{f}\left(E_i^{*}+{\mu }_i^{*}\right)\right]\mathrm{,}$ (12) where $f\left(E_i^{*}-{\mu }_i^{*}\right)$ and $\overline{f}\left(E_i^{*}+{\mu }_i^{*}\right)$ are the fermion and antifermion distribution functions, respectively, with $f\left(E_i^{*}-{\mu }_i^{*}\right)=\frac{1}{1+\exp \left\{\left[E_i^{*}-{\mu }_i^{*}\right]/T\right\}}\mathrm{,}$ (13) and $f\left(E_i^{*}+{\mu }_i^{*}\right)=\frac{1}{1+\exp \left\{\left[E_i^{*}+{\mu }_i^{*}\right]/T\right\}}\mathrm{.}$ (14) The mesonfield equations can be solved for a given temperature and chemical potential (or baryon number density). The model parameters, $g_{\sigma }\mathrm{,}{g}_{\omega }\mathrm{,}{g}_{\rho }\mathrm{,}b$ and c, are listed in Table 1. They were fitted with the compression modulus $K=240\text{ MeV}$, symmetric energy $a_{\text{sym}}=31.3\text{ MeV}$, effective nucleon mass $m_{\text{N}}^{*}=m_{\text{N}}-g_{\sigma }\sigma =0.75m_{\text{N}}$, and binding energy $B/A=-16.0\text{ MeV}$ at nuclear saturation density with ${\rho }_0=0.16{\text{ fm}}^{-3}$.

Results and discussion

In this section, we present the numerical results for the correlation between the net baryon number and electric charge in the nonlinear Walecka model. To simulate the physical conditions in the BES program at RHIC STAR, the isospin asymmetric nuclear matter was considered in the calculation with the constraint ρ_Q/ρ_B=0.4. In the present Walecka model, strange baryons were not included; thus, the strangeness condition ρ_S=0 was automatically satisfied. ρ_Q/ρ_B = 0.4 might deviate slightly owing to isospin dynamics. The influence of different isospin asymmetries on the fluctuations and correlations of conserved charges will be explored in detail in a separate study.

The correlations between the baryon number and electric charge are related to the baryon (μ_B) and isospin μ_Q (${\mu }_{\text{Q}}={\mu }_{\text{p}}-{\mu }_{\text{n}}$) chemical potentials. In Fig. 1, we demonstrate the value of μ_Q as a function of the temperature and baryon chemical potential by first plotting the contour map of μ_Q in the $T-{\mu }_{\text{B}}$ plane derived under the constraint of ρ_Q/ρ_B=0.4. Additionally, the corresponding liquid-gas phase transition line with a CEP located at T=13 MeV and μ_B=919 MeV is plotted in this figure. To compare with the chiral crossover phase transition of quarks, the dashed “Line A” in Fig. 1 was derived under the condition that the value of $\partial \sigma /\partial {\mu }_{\text{B}}$ was maximum for each given temperature. To a certain degree, this line is analogous to a chiral crossover transformation, although it is not a true phase transition in nuclear matter. It indicates the location at which the dynamic nucleon mass changes most rapidly with an increase in the chemical potential. The reason for plotting “Line A” is to emphasize that both the σ field in nuclear matter and the quark condensate in quark matter are associated with the dynamic mass of fermions, and therefore, the rapid change of mass might have a universal effect on the fluctuation distributions of conserved charges. As indicated in our previous studies [54, 55, 67], the location of line A helps us understand the behavior of the interaction measurement (trace anomaly) and the fluctuations of conserved charges near the phase transition [55, 67].

Fig. 1Full-screen View

(Color online) Contour of μ_Q in the $T-{\mu }_{\text{B}}$ plane derived in the nonlinear Walecka model with the constraint of ρ_Q/ρ_B=0.4. The solid line is the liquid-gas transition line with a CEP located at T=13 MeV and μ_B=919 MeV. The blue line is the chemical freeze-out line fitted in Ref. [79]. The dash-dotted line corresponds to the temperature and chemical potential for ρ_B=0.1ρ_0. “Line A” is derived using the maximum value of $\partial \sigma /\partial {\mu }_{\text{B}}$ for each given temperature

Additionally, “Line A” can be defined by the maximum point of $\partial \omega /\partial {\mu }_{\text{B}}$ or $\partial n_{\text{B}}/\partial {\mu }_{\text{B}}$, because the density can be considered as the order parameter for liquid-gas phase transition. Under this definition, the results obtained using the quark model do not correspond to a chiral crossover phase transition. However, this was not the purpose of the present study. Our aim was to identify common properties related to the dynamical fermion mass near the critical region of a first-order phase transition. However, the calculation indicates that the curves (“Line A”) under the two definitions coincide near the critical region and gradually deviate at higher temperatures away from the critical region.

For convenience, in the subsequent discussion of the experimental observables, we include a plot of the chemical freeze-out line fitted to the experimental data at high energies in Fig. 1 [79], which can be described by $T\left({\mu }_{\text{B}}\right)=a-b{\mu }_{\text{B}}^2-c{\mu }_{\text{B}}^4\mathrm{,}$ (15) where $a=0.166\text{ GeV}\mathrm{,}b=0.139{\text{ GeV}}^{-1}$ and $c=0.053{\text{GeV}}^{-3}$.

It should be noted that the trajectories of the present relativistic heavy-ion collisions do not pass through T_C of nuclear LGPT. It is still not known how far the realistic chemical freeze-out line is from the critical region at the present time. However, similar to the chiral phase transition of quarks, the existence of nuclear LGPT affects the fluctuation and correlation of the net baryon and electric charge numbers in the region not adjacent to the critical endpoint in intermediate-energy heavy-ion collision experiments. The numerical results for the parameterized chemical freeze-out line in this study can be used as a reference. The realistic chemical freeze-out conditions at intermediate and low energies will be extracted in future heavy-ion collision experiments. The contribution from LGPT needs to be considered when analyzing the experimental data.

Figure 1 shows that the value of $\left|{\mu }_{\text{Q}}\right|$ is less than 40 MeV in the red area. In this region, the baryon number density is very small, which is approximately indicated by the temperature and chemical potential curves for ρ_B=0.1ρ_0(dash-dotted line). The value of $\left|{\mu }_{\text{Q}}\right|$ increases with the baryon density(corresponding to a larger chemical potential). This trend in $\left|{\mu }_{\text{Q}}\right|$ is illustrated in Fig. 1. Along the chemical freeze-out line(solid blue line), changes in μ_Q during freeze-out with decreasing temperature or collision energy are observed.

Figure 2 shows the second-order correlation between the baryon number and electric charge, ${\chi }_{11}^{\text{BQ}}/{\chi }_2^{\text{Q}}$, as a function of the baryon chemical potential for T=75, 50, 25, MeV. To derive a physical quantity comparable with future experimental data, the correlated susceptibility was divided by ${\chi }_2^{\text{Q}}$ to eliminates volume dependence. For each temperature, the rescaled second-order correlation ${\chi }_{11}^{\text{BQ}}/{\chi }_2^{\text{Q}}$ shown in Fig. 2 displays nonmonotonic behavior with a peak at a certain chemical potential. The values of these peaks increase with decreasing temperature, which indicates that the correlation between the baryon number and electric charge is enhanced near the phase transition region. The solid dots in Fig. 2 show the values for the chemical freeze-out described by Eq. (15), which illustrates that the value of ${\chi }_{11}^{\text{BQ}}/{\chi }_2^{\text{Q}}$ increases along the freeze-out line when moving from the high-temperature region to the critical region.

Fig. 2Full-screen View

(Color online) Second order correlation between the baryon number and electric charge as a function of the chemical potential for different temperatures. The solid dots indicate the values on the chemical freeze-out line given in Fig. 1

Figure 3 shows the third-order correlations ${\chi }_{12}^{\text{BQ}}/{\chi }_2^{\text{Q}}$ and ${\chi }_{21}^{\text{BQ}}/{\chi }_2^{\text{Q}}$ as functions of the chemical potential at several temperatures. Compared with ${\chi }_{12}^{\text{BQ}}/{\chi }_2^{\text{Q}}$, ${\chi }_{21}^{\text{BQ}}/{\chi }_2^{\text{Q}}$ exhibits relatively larger fluctuations at the same temperature. The solid dots on the chemical freeze-out line show the same trend. This means that the measurement of ${\chi }_{21}^{\text{BQ}}/{\chi }_2^{\text{Q}}$ is more sensitive than ${\chi }_{12}^{\text{BQ}}/{\chi }_2^{\text{Q}}$ in heavy-ion collision experiments. Figure 3 also indicates that the correlations between the baryon number and electric charge intensify with decreasing temperatures. Evident oscillations of ${\chi }_{12}^{\text{BQ}}/{\chi }_2^{\text{Q}}$ and ${\chi }_{21}^{\text{BQ}}/{\chi }_2^{\text{Q}}$ appear for T=25 MeV, accompanied by alternating positives and negatives. As the temperature decreases, divergent behavior occurs at the CEP of LGPT. These features can be used to determine the signal for the phase transition in the experiments.

Fig. 3Full-screen View

(Color online) Third order correlations between the baryon number and electric charge as functions of the chemical potential at different temperatures. The solid dots indicate the values on the chemical freeze-out line plotted in Fig. 1

In Fig. 4, we plot the fourth-order correlations between the baryon number and electric charge: ${\chi }_{13}^{\text{BQ}}/{\chi }_2^{\text{Q}}$, ${\chi }_{22}^{\text{BQ}}/{\chi }_2^{\text{Q}}$, and ${\chi }_{31}^{\text{BQ}}/{\chi }_2^{\text{Q}}$. Compared to the second- and third-order correlations, Figs. Fig. 2, Fig. 3, 4 shows that the fourth-order correlations rescaled by ${\chi }_2^Q$ are weaker at higher temperatures such as T=75 MeV. However, the correlations are much stronger at T=25 MeV, near the critical region of LGPT. Correspondingly, with an increase in the chemical potential at lower temperatures, a bimodal structure evidently exists for all three correlations. In addition, the maximum values of ${\chi }_{13}^{\text{BQ}}/{\chi }_2^{\text{Q}}$, ${\chi }_{22}^{\text{BQ}}/{\chi }_2^{\text{Q}}$ and ${\chi }_{31}^{\text{BQ}}/{\chi }_2^{\text{Q}}$ increase sequentially. Furthermore, the solid dots demonstrate that the value of each correlation during freeze-out increases with the decreasing temperatures. Moreover, ${\chi }_{13}^{\text{BQ}}/{\chi }_2^{\text{Q}}<{\chi }_{22}^{\text{BQ}}/{\chi }_2^{\text{Q}}<{\chi }_{31}^{\text{BQ}}/{\chi }_2^{\text{Q}}$ during chemical freeze-out at each temperature, which implies that ${\chi }_{31}^{\text{BQ}}/{\chi }_2^{\text{Q}}$ is the most sensitive among the three fourth-order correlations.

Fig. 4Full-screen View

(Color online) Fourth order correlations between the baryon number and electric charge as functions of the chemical potential for different temperatures. The solid dots indicate the values on the chemical freeze-out line given in Fig. 1

Figure 5 presents the fifth-order correlations between the baryon number and electric charge, ${\chi }_{14}^{\text{BQ}}/{\chi }_2^{\text{Q}}$, ${\chi }_{23}^{\text{BQ}}/{\chi }_2^{\text{Q}}$, ${\chi }_{32}^{\text{BQ}}/{\chi }_2^{\text{Q}}$, and ${\chi }_{41}^{\text{BQ}}/{\chi }_2^{\text{Q}}$ for T=75, 50, 25, MeV. At T=75 MeV, the values of the four rescaled correlations are small; however, they become significant at T=25 MeV. Combined with the phase diagrams shown in Fig. 1, it is seen that the high-order correlated fluctuations strengthen as they approach the liquid-gas transition. Similar to the fourth-order correlations, the rescaled fifth correlations fulfill the relationships $\left|{\chi }_{14}^{\text{BQ}}/{\chi }_2^{\text{Q}}\right|<\left|{\chi }_{23}^{\text{BQ}}/{\chi }_2^{\text{Q}}\right|<\left|{\chi }_{32}^{\text{BQ}}/{\chi }_2^{\text{Q}}\right|<\left|{\chi }_{41}^{\text{BQ}}/{\chi }_2^{\text{Q}}\right|$ at chemical freeze-out. Moreover, all the four fifth-order correlation fluctuations are negative at chemical freeze-out for T=75 and 50 MeV; however, they are positive at T=25 MeV, near the liquid-gas transition. This is a prominent feature for exploring the interactions and phase transitions of nuclear matter.

Fig. 5Full-screen View

(Color online) Fifth order correlations between the baryon number and electric charge as functions of the chemical potential for different temperatures. The solid dots indicate the values on the chemical freeze-out line given in Fig. 1

Figure 6 shows the sixth order correlations of the baryon number and electric charge, that is, ${\chi }_{15}^{\text{BQ}}/{\chi }_2^{\text{Q}}$, ${\chi }_{24}^{\text{BQ}}/{\chi }_2^{\text{Q}}$, ${\chi }_{33}^{\text{BQ}}/{\chi }_2^{\text{Q}}$, ${\chi }_{42}^{\text{BQ}}/{\chi }_2^{\text{Q}}$, and ${\chi }_{51}^{\text{BQ}}/{\chi }_2^{\text{Q}}$. Each sixth-order correlation exhibits a double peak and double valley structure, although one of the two peaks is not prominent. The oscillating behavior intensifies when moving towards the phase transition region from high to low temperatures. Similarly, the intensity of the oscillations increases from ${\chi }_{15}^{\text{BQ}}/{\chi }_2^{\text{Q}}$, ${\chi }_{24}^{\text{BQ}}/{\chi }_2^{\text{Q}}$, ${\chi }_{33}^{\text{BQ}}/{\chi }_2^{\text{Q}}$, ${\chi }_{42}^{\text{BQ}}/{\chi }_2^{\text{Q}}$ to ${\chi }_{51}^{\text{BQ}}/{\chi }_2^{\text{Q}}$.

Fig. 6Full-screen View

(Color online) Sixth order correlations between the baryon number and electric charge as functions of the chemical potential for different temperatures. The solid dots demonstrate the values on the chemical freeze-out line given in Fig. 1

For a given order of correlations, the numerical results shown in Fig. 2, 3, 4, 5, and 6 indicates that the signals become stronger when taking the higher-order partial derivatives of the baryon chemical potential. Additionally, we examined the pure baryon number fluctuation and found that its highest sensitivity was of the same order as the LGPT critical endpoint, possibly because the baryon number fluctuation includes both proton and neutron contributions. However, the electric charge fluctuation involves the isospin density ${\rho }_{\text{N}}-{\rho }_{\text{P}}$. The baryon number density is always larger than the isospin density, which is associated with stronger fluctuations when there are more derivatives with respect to the baryon chemical potential than that with to electric chemical potential for a given order of correlations.

In addition, a comparison of the results shown in Fig. 2, 3, 4, 5, and 6 shows that the rescaled higher-order correlations fluctuate more strongly near the phase transition region, whereas the lower-order correlations at high temperatures are larger than most of the higher-order correlations away from the phase transition region. A similar phenomenon occurs in the correlations of conserved charges in quark matter [74]. According to the fluctuations of net baryon number [55, 67] and the correlations between net baryon number and electric charge in this study, the fluctuations and correlations of conserved charges have similar organizational structures for nuclear and quark matter. This is primarily attributed to the fact that the two phase transitions belong to the same universal class, and both describe the interaction of matter with temperature- and chemical-potential-dependent fermion masses.

Because the QCD phase transition and nuclear LGPT possibly occur sequentially from high to low temperatures (even if LGPT is not triggered), the energy-dependent behaviors of the fluctuations and correlations can be referenced to determine the phase transition signals of the strongly interacting matter. Although the latest reported BES II high-precision data at 7.7-3.9 GeV do not display a drastic change in the net baryon number kurtosis, stronger fluctuation signals may appear in heavy-ion experiments with collision energies lower than 7.7 GeV. Furthermore, in the hadronic interaction dominant evolution with collision energies lower than the threshold of the generation of QGP, the nuclear interaction and phase structure of LGPT will dominate over the behavior of fluctuations and correlations of conserved charges. The nature of the changes in fluctuations and correlations with decreasing collision energy during experiments requires investigation.

Summary

Fluctuations and correlations between conserved charges are sensitive probes for investigating the phase structure of strongly interacting matter. In this study, we used the non-linear Walecka model to calculate the correlations between the net baryon number and electric charge up to the sixth order, which originated from hadronic interactions in nuclear matter and explored their relationship with the nuclear liquid-gas phase transition.

The calculation indicated that the correlations between the net baryon number and electric charge gradually became stronger from the high-temperature region to the critical region of the nuclear LGPT. In particular, the correlations were significant at the location where the σ field or nucleon mass changed rapidly near the critical region. Similar behavior was observed for the chiral crossover phase transition of quark matter, primarily because of the similar dynamic mass evolution and same universal class of the chiral phase transition of quark matter and the liquid-gas phase transition of nuclear matter.

Compared to the lower-order correlations, the higher-order correlations fluctuated more strongly near the phase transition region, whereas the rescaled lower-order correlations were relatively stronger than most of the higher-order correlations away from the phase transition region at high temperatures. At the chemical freeze-out for each temperature, the calculation indicated that ${\chi }_{13}^{\text{BQ}}/{\chi }_2^{\text{Q}}<{\chi }_{22}^{\text{BQ}}/{\chi }_2^{\text{Q}}<{\chi }_{31}^{\text{BQ}}/{\chi }_2^{\text{Q}}$ for the fourth order correlation, $|{\chi }_{14}^{\text{BQ}}/{\chi }_2^{\text{Q}}|<|{\chi }_{23}^{\text{BQ}}/{\chi }_2^{\text{Q}}|<|{\chi }_{32}^{\text{BQ}}/{\chi }_2^{\text{Q}}|<|{\chi }_{41}^{\text{BQ}}/{\chi }_2^{\text{Q}}|$ for the fifth order correlations, and $|{\chi }_{15}^{\text{BQ}}/{\chi }_2^{\text{Q}}|<|{\chi }_{24}^{\text{BQ}}/{\chi }_2^{\text{Q}}|<|{\chi }_{33}^{\text{BQ}}/{\chi }_2^{\text{Q}}|<|{\chi }_{42}^{\text{BQ}}/{\chi }_2^{\text{Q}}|<|{\chi }_{51}^{\text{BQ}}/{\chi }_2^{\text{Q}}|$ for the sixth order correlations. In particular, the values of the fifth- and sixth-order correlations changed from negative to positive when approaching the critical region of LGPT from the high-temperature side along the extrapolated chemical freeze-out line. With the availability of more precise data from experiments below 7.7 GeV in the future, realistic chemical freeze-out conditions can be fitted, and the results obtained in this study can be used to analyze the signals of QCD phase transitions and the influence of the nuclear liquid-gas phase transition.