Fluctuations and Correlations of Conserved Charges Serving as Signals for QGP Production: An Overview from Polyakov Loop Enhanced Nambu–Jona-Lasinio Model

Abstract

Quark–Gluon plasma driven by the strong force is subject to the conservativeness of the baryon number, net electric charge, strangeness, etc. However, the fluctuations around their mean values at specific temperatures and chemical potentials can provide viable signals for the production of Quark–Gluon plasma. These fluctuations can be captured theoretically as moments of different orders in the expansion of pressure or the thermodynamic potential of the system under concern. Here, we look for possible explanations in the methodologies used for capturing them by using the framework of the Polyakov–Nambu–Jona-Lasinio (PNJL) model under the 2 + 1 flavor consideration with mean-field approximation. The various quantities thus explored can act to signify meaningfully near the phase transitions. Justifications are also made for some of the quantities capable of serving necessarily under experimental scenarios. Additionally, variations in certain quantities are also made for the different collision energies explored in the high-energy experiments. Rectification of the quantitative accuracy, especially in the low-temperature hadronic sector, is of prime concern, and it is also addressed. It was found that most of the observables stay in close proximity with the existing lattice QCD results at the continuum limit, with some artifacts still remaining, especially in the strange sector, which needs further attention.

1. Introduction

At high temperatures and densities, strongly interacting matter is expected to witness a transition from the confined state, the hadronic phase, to a deconfined partonic state, where the restoration of chiral symmetry occurs in conjunction with the deconfinement of quarks [1,2,3,4,5]. In order to understand the fundamental interactions in physics, as well as to have a thorough insight into the early universe and neutron stars, it is very important to understand the phenomena involved. To this end, research on the existence of a possible second-order phase transition is currently ongoing, with studies working toward the conjecture of the possible location of the critical end point (CEP). It is thus important to study and investigate observables, which give a signal of the transition from one phase to the other. Fluctuations in the baryon number, electric charge, and strangeness act as such viable signals. The D-measure, for example, expressed in terms of the fluctuations of the net electric charge, gives one such possible signal [6,7,8]. This quantity takes up distinctly unique values for the hadronic and partonic phases, thus resulting in the excellent signaling of the transition from one phase to the other. In the experimental scenario, the heavy-ion collisions are measurable with an event-by-event (EbE) analysis [9,10], with each event corresponding to a collision.

In this context, various studies are being performed on the theoretical and experimental fronts in search of the critical end point. The framework of the lattice QCD gives a benchmark estimate for the computational baseline [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]; however, QCD-inspired models are proving to be eligible candidates, barring the complexities of the lattice at non-zero chemical potentials. They are found to be highly useful in describing the different attributes of strongly interacting matter at different temperatures and chemical potentials. One such model is the Polyakov–Nambu–Jona-Lasinio or the PNJL model. This model was formed by appending the Polyakov or P-loop to the NJL model [30,31,32]. The NJL, or the Nambu–Jona-Lasinio model, was formed with the global symmetries of the QCD-like chiral symmetry. The multiquark interaction terms give rise to dynamic fermion mass generation, which leads to spontaneous chiral symmetry breaking. However, since the gluon degrees of freedom are integrated at this point, the model fails to incorporate the crucial feature of confinement. This is where the PNJL model, by adding the background gluon field, gives a sense of confinement, thus incorporating the two important transitions of chiral symmetry breaking and its restoration and the confinement–deconfinement transition into a single framework [33,34,35,36,37,38,39]. The various parameters in this model are fixed by adhering to the lattice scenario. The PNJL results showed a satisfactory qualitative agreement with the lattice results. The latter lattice findings were for finite lattice spacings. The lattice QCD data, in the recent past, have been extrapolated to the continuum limit [40], and they have obtained continuum extrapolations for a number of observables. However, the previously obtained PNJL results were rendered incapable of showing the close proximity to the recently obtained lattice continuum data. Henceforth, it became a mandate that the PNJL model parameters were set to show good quantitative agreement with the lattice. Following this, the parameters in the PNJL model were re-set, such that the transition temperature T_c matched with the lattice and also such that the temperature dependence of the pressure was in close conjunction with the lattice. It is also important to check the other derivatives of the pressure and contrast them with the lattice results. For the physical case of two light strange quarks and one heavier strange quark, the crossover temperature was reported by the Hot-QCD and Wupertal–Budapest collaborations [40,41,42,43,44,45] to be in the range of 150–160 MeV. The corresponding readjustment of the Polyakov loop potential brought about a satisfactory overall agreement between the PNJL and lattice results; however, the match was far from perfect. A main region of disagreement was in the low-temperature regime, where the hadronic degrees of freedom played a major role. This was expected, though, since this regime was not yet adequately addressed in the PNJL model. In this context, it needs to be mentioned that the Hadron Resonance Gas (HRG) model [46,47,48,49,50,51,52] has given excellent results in the hadronic sector in central heavy-ion collisions, ranging from the Alternating Gradient Synchrotron (AGS) up to the Relativistic Heavy-Ion Collider (RHIC) energies [53,54]. It is also noteworthy that the susceptibilities of the conserved charges calculated using the background of the lattice QCD have been excellently reproduced by the HRG model for temperatures up to 150 MeV [55]. The regime of high chemical potentials below the critical region can be studied using this model. Henceforth, it is established that the HRG model is quite suitable for describing the hadronic phase of the strong interactions. The HRG model, which is based on the theorem of Dashen, Ma, and Bernstein, illustrates that the dilute system of strongly interacting matter can be described by the gas of free resonances. Though these resonances take care of the long-range attractive part of the hadron interactions, they are incapable of addressing the short-range repulsion, which is also very important for describing the strongly interacting matter. Upon close approach to the critical or crossover regime, the HRG calculations tend toward Hagedorn divergence, which is possibly due to the absence of repulsive interactions.

It is thus of urgent importance to obtain a single model that addresses both the low- and high-temperature domains effectively. This can be achieved by creating an interface between one model, which is appropriate for the partonic phase, and another one for the hadronic phase. This approach is well taken care of by using a switching function to club the HRG and PNJL models. The switching function is crucial for two reasons. Firstly, it is necessary to consider the hadronic contribution at low temperatures, which is missing in the PNJL model. Secondly, at high temperatures, the partonic contribution dominates, and thus hadronic ones need to be suppressed in that regime. On the contrary, at low temperatures, the partonic contribution is already suppressed, thus making the switching function superfluous in this regime. The switching function should be designed to give the correct proportion of the hadronic and partonic contributions. This gives an indeed effective way where by clubbing the two models, good realization of the entire picture is achieved.

An alternative and a more internal way to address the same issue is where the hadronic contributions are added in a simple manner while considering their medium-dependent masses. Using the Polyakov loop, the confinement feature always suppresses the contributions of the constituent quarks at low temperatures and densities. In the former case, the switching function was crucial to cut-off the hadronic contributions at high temperatures and densities. Here, in this case, instead of using an external switching function, the increasing effective masses of hadrons will act as natural in-built switching functions. Thus, the quantitative accuracy is also achieved. This should further aid in obtaining the fluctuations and correlations of the conserved charges in the system with higher accuracy, thus acting as viable signals for QGP production under exotic conditions.

Similar studies have also been performed under the frameworks of various QCD-inspired models like the Linear Sigma Model (LSM) [56,57,58] and the Polyakov Loop Linear Sigma Model (PLSM) [59,60,61,62,63,64,65,66]. These models are equivalent to the NJL or PNJL ones and bear special attention in order to analyze the different features of the matter under concern, including its transition properties, thermodynamics, and observables, to indicate phase structures. Meson masses in the scalar, pseudoscalar, vector, and axial vector channels were also explored with precision. Over the years, following works in similar directions using effective interaction models of quarks embedded in frameworks of the Schwinger–Dyson equation have contributed significantly to the community as well [67,68,69,70,71,72,73] in terms of predicting phase transitions and associated observations. To this end, the PNJL model has also advanced in recent years to encompass the various observations related to the phase transitions and associated symmetry patterns and to study the transport coefficients applicable in the hydrodynamic scenario, thus posing unique ways to portray the phase diagram [74,75,76,77,78,79,80,81,82,83].

2. The Model Frameworks

The Nambu-Jona—Lasinio (NJL) model takes care of the chiral properties but fails to address the confinement–deconfinement transition. To this end, the Polyakov loop-enhanced NJL or the PNJL model explains the deconfinement physics successfully. Studies have been performed using both 2 and 2 + 1 flavors considering up to six- and eight-quark interactions [84,85,86,87]. The thermodynamic potential looks like

$$\begin{array}{cc}\Omega =\hfill & {\mathcal{U}}^{\prime }\left(\Phi \left[A\right],\overline{\Phi }\left[A\right],T\right)+2g_S{\displaystyle \sum _{f=u,d,s}}{\sigma }_f^2-{\displaystyle \frac{g_D}{2}} {\sigma }_u{\sigma }_d{\sigma }_s+3{\displaystyle \frac{g_1}{2}} {\left({\displaystyle \sum _f}{\sigma }_f^2\right)}^2+3g_2{\displaystyle \sum _f}{\sigma }_f^4\hfill \\ & -6{\displaystyle \sum _f}{\int }_0^{\mathsf{\Delta }}{\displaystyle \frac{d^{3}p}{{\left(2\pi \right)}^3}} E_{p_f}\Theta \left(\Lambda -\left|\overrightarrow{p}\right|\right)\hfill \\ & -2{\displaystyle \sum _f}T{\int }_0^{\infty }{\displaystyle \frac{d^{3}p}{{\left(2\pi \right)}^3}} ln\left[1+3\left(\Phi +\overline{\Phi } e^{-{\displaystyle \frac{\left(E_{p_f}-{\mu }_f\right)}{T}}}\right) e^{{\displaystyle \frac{-(E_{p_f}-{\mu }_f)}{T}}}+e^{{\displaystyle \frac{-3(E_{p_f}-{\mu }_f)}{T}}}\right]\hfill \\ & -2{\displaystyle \sum _f}T{\int }_0^{\infty }{\displaystyle \frac{d^{3}p}{{\left(2\pi \right)}^3}} ln\left[1+3\left(\Phi +\overline{\Phi } e^{-{\displaystyle \frac{\left(E_{p_f}+{\mu }_f\right)}{T}}}\right) e^{{\displaystyle \frac{-(E_{p_f}+{\mu }_f)}{T}}}+e^{{\displaystyle \frac{-3(E_{p_f}+{\mu }_f)}{T}}}\right]\hfill \end{array}$$

(1)

${\sigma }_f$ are the condensates corresponding to the up, down, and strange quarks. $g_S$ and $g_D$ are the four and six quark coupling terms, where the latter ensures axial U(1) symmetry breaking. $g_1$ and $g_2$ are the eight-quark coupling terms, which become zero in the absence of the eight quark interactions. The coupling constants are dimensionful, thereby making the theory non-renormalizable. A three-momentum ($\left|\overrightarrow{p}\right|$) cutoff scheme has therefore been introduced and manifested by $\Theta \left(\Lambda -\left|\overrightarrow{p}\right|\right)$. $E_{p_f}=\sqrt{p^2+M_f^2}$ is the quasiparticle energy. The constituent quark masses are given by

$$M_f=m_f-g_S{\sigma }_f+{\displaystyle \frac{g_D}{2}}{\sigma }_{f+1}{\sigma }_{f+2}-2g_1{\sigma }_f\left({\sigma }_u^2+{\sigma }_d^2+{\sigma }_s^2\right)-4g_2{\sigma }_f^3$$

(2)

where ${\sigma }_f$ takes up corresponding values for u, d, and s quarks in a clockwise manner. The integral with finite range gives a zero-point energy, whereas the last two terms indicate the contributions of the constituent quarks in the finite temperature and chemical potential domain. They are nothing but the modified fermion determinants in the presence of the Polyakov loop field $\Phi ={\displaystyle \frac{{Tr}_{c}L}{N_c}}$ and its conjugate $\overline{\Phi }={\displaystyle \frac{{Tr}_c{L}^{†}}{N_c}}$, with L being the Polyakov loop given by

$$L\left(\overrightarrow{x}\right)=\mathcal{P}\exp [i{\int }_0^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$T$}\right.}}d\tau A_4(\overrightarrow{x},\tau )]$$

(3)

$A_4$ is the time component of the background gluonic field. The self-interaction of $\Phi$ and $\overline{\Phi }$ is described by the effective potential ${\mathcal{U}}^{\prime }$:

$${\displaystyle \frac{{\mathcal{U}}^{\prime }[\phi ,\overline{\phi }, T]}{T^4}}={\displaystyle \frac{\mathcal{U}[\phi ,\overline{\phi }, T]}{T^4}}-\kappa \ln \left(J\right[\phi ,\overline{\phi }\left]\right)$$

(4)

Here, $\mathcal{U}$ is one Landau–Ginzburg type potential following the global Z(3) symmetry of the Polyakov loop given by

$${\displaystyle \frac{\mathcal{U}[\phi ,\overline{\phi }, T]}{T^4}}=-{\displaystyle \frac{b_2\left(T\right)}{2}} \overline{\phi }\phi -{\displaystyle \frac{b_3}{6}} \left({\phi }^3+{\overline{\phi }}^3\right)+{\displaystyle \frac{b_4}{4}} {\left(\phi \overline{\phi }\right)}^2$$

(5)

The temperature dependence has been completely absorbed in the coefficient $b_2$ as

$$b_2\left(T\right)=a_0+a_1\left({\displaystyle \frac{T_0}{T}}\right)+a_2{\left({\displaystyle \frac{T_0}{T}}\right)}^3$$

(6)

This truncated series has later been modified to stay in close proximity with the lattice result, the modified form being [88] as follows:

$$b_2\left(T\right)=a_0+a_{1}exp\left(-a_2{\displaystyle \frac{T}{T_0}}\right){\displaystyle \frac{T_0}{T}}$$

(7)

Fixing of the parameters has been detailed in [89], and the crossover temperatures were found to be 166.5 and 162.5 MeV, respectively, for six- and eight-quark frameworks, falling in line satisfactorily with the lattice findings.

The pressure is found from the thermodynamic potential, which is minimized with respect to the corresponding degrees of freedom present in the system, thereby finding the values of the associated fields. The fluctuations of conserved charges and correlations among them are found using the series expansion of the scaled pressure w.r.t the scaled chemical potential as follows:

$$c_n^X\left(T\right)={\displaystyle \frac{1}{n!}} {\displaystyle \frac{{𝜕}^n\left(P/T^4\right)}{𝜕{\left(\frac{{\mu }_X}{T}\right)}^n}}=T^{n-4}{\chi }_n^X(T)$$

(8)

$$c_{m,n}^{X,Y}={\displaystyle \frac{1}{m!n!}}{\displaystyle \frac{{𝜕}^{m+n}\left(P/T^4\right)}{\left(𝜕{\left(\frac{{\mu }_X}{T}\right)}^m\right)\left(𝜕{\left(\frac{{\mu }_Y}{T}\right)}^n\right)}}=T^{m+n-4}{\chi }_{m,n}^{X,Y}(T)$$

(9)

The net charge fluctuation, once obtained from Equation (8), can also be used to define a quantity named D-measure, as elaborated in the following section. This, being a ratio of net charge fluctuations with total charge density, can serve as a viable quantity in the experimental scenario as well, since both of them are experimentally measurable.

On the other side, the HRG model has been quite successful in analyzing the production of hadrons in high-energy, heavy-ion experiments [90,91,92,93,94,95,96,97,98,99,100,101,102]. The observables under concern here are parallelly well-described [103,104,105,106] up to a temperature range of 150 MeV, i.e., the crossover region. However, the absence of repulsive interaction causes Hagedorn divergence there, which is taken care of in the Excluded Volume HRG (EVHRG) model framework [105,107,108,109,110,111,112,113,114,115,116] under acceptance ranges of transverse momentum and pseudo rapidity. It is thus also applied in the systematic hydrodynamic analysis of the system [117,118,119]. In this preliminary study, the goal is, however, to witness the success of incorporating hadronic degrees of freedom to elevate the results of the PNJL model in the low-temperature sector for better agreement with the existing lattice QCD results. The ideal gas of hadrons, following the Particle Data Group [120], has been used, where the grand partition function is

$$\ln Z^{id}={\sum }_i\ln Z_i^{id}$$

(10)

$Z_i^{id}$, the partition function for the ith resonance particle then takes the form

$$\ln Z_i^{id}=\pm {\displaystyle \frac{V{g}_i}{2{\pi }^2}} {\int }_0^{\infty }{p}^{2}dp\ln \left\{1\pm exp\left[-\left(E_i-{\mu }_i\right)/T\right]\right\}$$

(11)

V is the volume of the system, with g_i being the degeneracy factor. $E_i=\sqrt{p^2+m_i^2}$ is the single-particle energy, with m_i being its mass. T is the temperature, and ${\mathit{\mu }}_{\mathit{i}}=B_i{\mu }_B+Q_i{\mu }_Q+S_i{\mu }_S$ is the chemical potential of the ith species, formed using the corresponding chemical potentials of the conserved charges considered for this analysis. ‘+’ and ‘−’ signs refer to the fermions and bosons, respectively. The pressure, and hence other thermodynamic quantities, can be obtained from this partition function using the standard thermodynamic relations.

3. Results

The quantity D-measure was defined as the net charge fluctuation in a form of ratio of two experimentally measurable quantities in order to reduce systematic uncertainties. This is defined as [6] follows:

$$D=4 {\displaystyle \frac{{\mathit{\chi }}_{\mathit{Q}}}{{\mathit{n}}_{\mathit{c}\mathit{h}}/{\mathit{T}}^3}}$$

(12)

Here, ${\chi }_Q$ is the net electric charge fluctuation, and $n_{ch}$ is the total charge density. ${\chi }_Q$ (or ${\chi }_2^Q$) is calculated in a similar way as in Equation (8), i.e., in a dimensionless manner, whereas $n_{ch}$ is obtained using the quark and antiquark distribution functions as they appear in PNJL model. As per the conserved charges considered for this study, viz., the baryon number, net electric charge, and strangeness, the corresponding chemical potentials are used to construct the chemical potentials of quarks of different flavors, ${\mu }_f=B_f{\mu }_B+Q_f{\mu }_Q+S_f{\mu }_S$. While calculating the charge fluctuations, chemical potentials for the other charges are kept constant at zero. Considering a system consisting of pion gas for the hadronic region and calculations from lattice gauge theory for the partonic phase reveals a value of D to be close to 4 and 1, respectively. D has also been measured experimentally by ALICE collaborations in a rapidity window 0.2 < Δη < 1.6 using collision energy, $\sqrt{\mathit{s}}$ = 2.76 TeV in a Pb-Pb run considering different centralities [89,121]. The result shows D to fall consistently with an increase in Δη, obeying results from UrQMD simulation. The results from the STAR collaboration are also available.

Figure 1 shows the variations of ${\chi }_Q$ and $n_{ch}/T^3$ for 2 and 2 + 1 flavor systems with temperature from the PNJL model. These are then employed to obtain $D/D_{free}$, where $D_{free}$ represents the limiting value of D for a free, massless gas of quarks. This quantity is portrayed in Figure 2.

The results are qualitative, but clearly indicate a transition close to T~T_c. The non-monotonic behavior in the low-temperature regime accounts for the absence of explicit hadronic degrees of freedom in the model framework, which was later addressed using two methods; the first one incorporated an appropriate switching function [55] to serve as an adhesive agent between the HRG and PNJL models. The second one [122] was to use the temperature-dependent hadronic masses, thus contributing to the thermodynamic potential accordingly. The quantity D was also computed as a function of collision energy, $\sqrt{\mathit{s}}$, as depicted in the experimental situation, using the freeze-out parametrization [123]. This was found to fall from a higher value before getting saturated, falling in line with the experimental findings, as shown in Figure 3. As in [121], the numerical range of D and its qualitative nature agree with the findings of PNJL outcomes. However, it is to be remembered that the freezeout curve directs to states under complete thermal equilibrium and hence falls solely in the hadronic region. So, a complete match of any observable with experimental findings cannot be expected. Moreover, since considering rapidity range is beyond the scope of the framework of the PNJL model, the dynamics of such a system have not been considered in this case.

The baryon number and strangeness are also conserved charges, and measuring their fluctuations can be of immense importance in this regard. Figure 4 shows the variation of second- and fourth-order susceptibilities of the baryon number, named $c_2^B$ and $c_4^B$, respectively [88]. The quantitative agreement with the lattice QCD data is quite encouraging; however, the mismatch becomes prominent as the order is increased.

Similar quantities for strangeness are shown in Figure 5. The deviation from the existing lattice results is quite noteworthy here and calls for proper modifications with the insertion of hadronic degrees of freedom in the low-temperature regime. The maxima in $c_4^S$ is widely shifted as the melting of strange quark condensate occurs at higher temperatures as compared to the lattice case. This is an artifact in the NJL model parameters, which were fixed entirely at zero temperature and chemical potentials.

The discrepancies were somewhat addressed using a hybrid model by tagging the suitable HRG and PNJL model results using a smooth switching function of the following form [55]:

$$S\left(T\right)={\displaystyle \frac{1}{1+\mathrm{e}\mathrm{x}\mathrm{p} [-\frac{T-T_S}{{\mathsf{\Delta }T}_S\left(T\right)}]}}$$

(13)

which varies continuously from one phase to the other around the crossover region. Hence, this function essentially provides the necessary weightage to the HRG or PNJL model outcome in the corresponding region of temperature. As stated previously, the absence of any single model framework to suitably describe the physics in both the partonic and hadronic sectors necessitates the use of clubbing the two relevant models in an effective manner. Such an idea was successfully implemented by using the HRG and PQCD (perturbative-QCD) results, as in [124], and also merging the HRG and IQ (Interacting Quark) models [125,126]. Considering the hadrons and partons to be states of the same underlying theory of QCD, they can appear simultaneously under some evolved thermodynamic condition of the system. The theory is thus based on a mixture of states in the crossover region, supported by the lattice-QCD results toward the crossover of hadronic dominance to partonic dominance. The coexistence of the hadronic and partonic degrees of freedom near crossover thus makes it possible to explore such frameworks [127,128,129,130,131]. A suitable choice of switching function would then play the role of clustering probability to create a coexisting condition for a simple mixture of the two different sets of degrees of freedom under a specific condition without any overcounting in them. The one used here in Equation (13) is thus used to choose the necessary degrees of freedom suitably in the corresponding window of temperature under concern, with no effect on underlying structures of the PNJL or HRG models and thus keeping all symmetries and their breakings intact. The phase boundary also remains solely dependent on the PNJL model. This helps in fixing the change-over temperature, T_s, close to the crossover temperature. The effects of the derivatives of the switching function while obtaining various thermodynamic quantities may have additional consequences, which, however, have not been considered after detailed investigations [55] by approximating that each observable has an individual dependence on the switching function. The other parameter ${\mathsf{\Delta }T}_S\left(T\right)$ is fixed at par with the width of the crossover region, thus corresponding to a spread in the crossover temperature. The results of pressure from lattice-QCD were used for this purpose. The corresponding parameter set can be found in Table 4 of [55].

On the other hand, as specified, the medium-dependent hadronic masses can also act as built-in switching functions. The low-lying hadrons will be the dominant contributors to the pressure of the system, and hence hadrons of masses up to 1 GeV have been considered. The baryonic masses have been considered to remain fixed as they possess comparatively higher masses than the lowest-lying mesons, which are the predominant partners in this regard. The temperature-dependent mesonic masses can be found by solving the pole condition, as in [122]:

$$1-2G_M{П}_{M }\left(\omega =m_M,\overrightarrow{k}=0\right)=0$$

Here, $G_M$ is the vertex factor for the corresponding flavor combination, and ${П}_{M }$(${\overrightarrow{k}}^2$) is the one-loop polarization function for that mesonic channel, which can be solved using the appropriate quark propagator, as can be found in [122] and references therein. The temperature-dependent masses of mesons which have been considered are shown in Figure 6. These masses have also been studied in the frameworks of the PLSM model by different groups [59,65], and the results show satisfactory equivalence. Variations under the finite temperature and chemical potentials domain have been made in [65]. In the PNJL model, the meson masses are evaluated in the presence of the Polyakov loop fields and quark condensates, which are evaluated at present under mean field approximation, thus helping to solve the pole conditions. The PLSM model, on the other hand, concentrates on the direct attachment of associated terms in the Lagrangian system itself in the scalar, pseudoscalar, vector, and axial vector channels, thereby considering the symmetry effects [60,61,62,63,64,65]. Isospin chemical potentials have been introduced to study the phase diagram in view of the persisting scenario, and thermodynamic quantities were shown to match well with lattice QCD findings, along with the crossover temperature being in close vicinity.

The final contribution of the mesons in the thermodynamic potential thus takes the following form:

$${\delta }^{\prime }{\Omega }_M=-{\nu }_{M}T\int {\displaystyle \frac{d^{3}p}{{\left(2\pi \right)}^3}}\mathrm{l}\mathrm{n} (1-e^{{\displaystyle \frac{{-E}_p}{T}}}),$$

${\nu }_M$ is the corresponding weight factor for the particular meson, and $E_p=\sqrt{{\overrightarrow{p}}^2+m_{pole}^2(T)}$, with $m_{pole}$ being the corresponding mesonic mass. Similarly, the baryonic contribution to the thermodynamic potential is considered as

$${\delta }^{\prime }{\Omega }_B={\nu }_{B}T\int {\displaystyle \frac{d^{3}p}{{\left(2\pi \right)}^3}}\mathrm{l}\mathrm{n} (1+e^{{\displaystyle \frac{{-E}_p}{T}}})$$

where baryonic masses lying within 1 GeV are taken as constants. Thus, finding the full thermodynamic potential and hence the pressure, the susceptibilities can be obtained by suitably Taylor expanding and fitting for the corresponding coefficients as before. The renormalized version of the PNJL model, with medium mass modifications of the hadrons, indeed serves as one potential framework to describe the underlying physics self-consistently. This can be verified from the observed thermodynamic quantities, as portrayed in Figure 7. The previously observed discrepancies in the low-temperature segment are well taken care of now. Figure 7 describes four observables, viz., the pressure (a), energy density (b), speed of sound (c), and specific heat (d) in the four panels, respectively. One additional discrepancy in the hybrid picture of entangling the HRG results with PNJL ones was to incorporate derivatives of the external switching function. Not only would this increase the number of parameters, but it would also have additional effects to deviate the desired scenario from the physical situation. This is also avoided in the new formalism, as can be verified from Figure 7, panels (b), (c), and (d), where derivatives of pressure are involved, yet showing satisfactory agreements with the existing results from lattice calculations.

The inset diagrams picture the zoomed-in region in the low-temperature sector, where quantitative mismatches were profound previously. This essentially boosts beyond and obtains the fluctuations and correlations among the conserved charges present in the system. Figure 8 shows the leading-order baryon number susceptibilities, as found from the hybrid and the mass-modified PNJL model. In both cases, the model shows a satisfactory match for the entire window of temperature with the lattice outcome. The HRG result matches up to the crossover region. For the hybrid model, results from 8-q-type interaction are only shown here. The PNJL outcome itself falls closely to the lattice result, as the baryons are well accounted for by the constituent quarks in this model, as depicted in the distribution functions.

However, significant departure is observed for the electric charge fluctuations. Figure 9 portrays the results from the two different methods along with the PNJL results vis-à-vis lattice data. The mismatch was primarily in the low-temperature domain, where the absence of explicit hadrons creates deviation. The use of the HRG results and, more physically, the use of the temperature-dependent hadron masses, removes the discrepancy to a great extent and provides satisfactory matches. This is quite expected, as the charge sector is dominantly contributed by the light hadrons.

Similar situation occurs in the strange sector. In Figure 10, the HRG results show agreement to the lattice up to a higher temperature than the crossover one. One reason could be that HRG considers only the experimentally found hadrons, whereas the lattice can have contributions from additional strange hadrons predicted by theoretical quark models. In the case of the mass-modified PNJL model, the contributions from the strange sector come from K and η, whose masses are still quite large. Also, the fact remains that constituent strange quark masses do not dilute as fast in the range of temperature where K-mass remains almost constant. The use of any higher-lying strange quark cannot resolve the issue. The artifact can therefore possibly be rectified, as reported in [122], by suitable reparameterization of the NJL part in the strange sector.

Among the leading-order correlations between the conserved charges, $c_{11}^{BQ}$ is shown in Figure 11 for both six- and eight-quark-type interacting hybrid models. The baryon and electric charges remain correlated in the low-temperature regime, as the baryons and anti-baryons have positive and negative electric charges, respectively. Relatively large masses constrain the correlations at a low temperature. They then increase with temperature. But, as in the partonic phase, free quarks with small masses are produced, and the correlation again tends to zero. It therefore acquires a hump-like behavior for a model where transition is incorporated. The mismatch is anticipated to be occurring due to the slow dilution in the constituent masses of the charged strange quarks.

In Figure 12, the correlations between the baryon number and strangeness (BS) for a similar set up are shown, whereas Figure 13 projects the correlations for charge and strangeness (QS). As the contribution of light strange mesons is higher in QS, the correlation is also higher than BS in the intermediate range of temperature. The qualitative behavior is as per expectation from a model with in-between transition. The underestimation of BS and QS correlations of lattice data by the PNJL model is possibly due to the slow melting of the strange quark masses again.

One of the primary causes for the mismatches found comes from the strange sector. This can be verified by examining corresponding quantities at the flavor level. It is also to be kept in mind that the strange particles also contribute to the related observations involving electric charge. Figure 14, displayed below, shows the justification. Some of these have been studied in [88].

The parametrization in the model in its present version has a high current strange quark mass. The constituent mass of the strange quark thereby dilutes quite slowly and, at a high temperature, still acquires a higher mass than expected. This can be verified in Figure 15 below.

Non-monotonicity occurs, where the strange quark mass starts to fall more slowly as compared to the up quark mass. In the higher-temperature region as well, it possesses quite a high value. This suggests necessary rectification in the model parameterization with a lower current mass of the strange quark. The rectified framework, then coupled with the mass-modified version of the PNJL framework, can suitably describe the observables under concern. Moreover, the medium-mass modifications, as described, are performed under the mean-field scenario. Moving to the next level, incorporating the hadronic part of potential into the modifications of the degrees of freedom involved can lead to the closest proximity to the physical picture.

4. Conclusions

Among the various possible signatures of QGP production, studies of fluctuations and correlations among the conserved charges are very effective and efficient. To this end, lattice QCD provides a benchmark theoretically. However, parallel studies of effective models need to be carried out for reasons manifold. The Polyakov-enhanced NJL (PNJL) model, in this regard, is very effective qualitatively. Quantitative accuracy, however, can be achieved once at least low-lying hadrons are incorporated suitably. Two methods to achieve this have been discussed here in parallel. The use of an external switching function behaving smoothly near the transition region can be made to join the HRG and PNJL results simultaneously, giving the necessary weightage corresponding to the applicable temperature domain. The thermodynamic variables and fluctuations of conserved charges now show a proper match near and below the crossover domain. The absence of suitable hadronic degrees of freedom in the low-temperature sector in the PNJL framework is thus solved in this first attempt by considering a mixture of degrees of freedom near the crossover. However, this method has its own difficulties arising from the approximation in its form. The switching function used in this study aims to choose the degrees of freedom adequately in the relevant region of corresponding phases without any overcounting in them. All the observables were considered to depend on the function independently, and hence, the dependence of the derivative terms is excluded after examination. Otherwise, the number of parameters would have been increasing, making the theory more dependent on effective fitting. Again, the applicability in the finite chemical potential domain can be of prime concern. The mismatch in the strange chemical potential, on the other hand, can be attributed to the slow melting of strange condensate in the PNJL model, which demands reparameterization in the sector. This becomes quite evident while studying the observables at the flavor level. The constituent masses of the strange quarks and, in particular, their slow dilution with increasing temperature, cause the problem. To conclude, this simplistic idea has worked quite well to reproduce the physics of strongly interacting matter, at the same time posing a few questions for further improvement toward quantitative accuracy. These issues now motivate building a more self-sustained model by using the medium-dependent masses of the hadrons, which can necessarily suppress the dominance of hadronic degrees of freedom in the higher-temperature region. Thus, the more elementary way, using medium-dependent hadronic masses and thereby including the potential, is introduced. These masses being generated while the system is evolving make the process a fundamental one. The susceptibilities and correlations are thus found to agree with the lattice result much better, as discussed. One major concern of the finite chemical potential in the earlier approach is now automatically addressed, as no additional fitting method is required here. The hadronic masses, being medium-dependent, can extrapolate themselves to choose dominant degrees of freedom that are relevant in the corresponding temperature window. There are, however, still some discrepancies remaining, accounting for some artifacts that need special attention. The masses are extracted by using mean-field values of the degrees of freedom associated. Once the entire thermodynamic potential can be minimized, the theory can somewhat mimic the beyond mean-field scenario, which is planned to be presented in the future.