QCD critical end point from a realistic PNJL model
Abstract
With parameters fixed by critical temperature and equation of state at zero baryon chemical potential, a realistic Polyakov–Nambu–JonaLasinio (rPNJL) model predicts a critical end point of chiral phase transition at . The extracted freezeout line from heavy ion collisions is close to the chiral phase transition boundary in the rPNJL model, and the kurtosis of baryon number fluctuations from the rPNJL model along the experimental freezeout line agrees well with the BESI measurement. Our analysis shows that the dip structure of measured is determined by the relationship between the freezeout line and chiral phase transition line at low baryon density region, and the peak structure can be regarded as a clean signature for the existence of CEP.
QCD critical end point from a realistic PNJL model
Kun Xu^{}^{}email: xukun@ihep.ac.cn, Zhibin Li^{}^{}email: lizb@ihep.ac.cn, and Mei Huang^{}^{}email: huangm@ihep.ac.cn
\@textsuperscript1Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
\@textsuperscript2University of Chinese Academy of Sciences, Beijing 100049, China

Abstract.
1 Introduction
Exploring Quantum Chromodynamics (QCD) phase structure and understanding properties of QCD matter under extreme conditions are main topics of heavy ion collisions. It is generally believed that there is a QCD critical end point (CEP) for chiral phase transition at finite baryon density, and searching for the CEP and locating the CEP become one of the central goals of beam energy scan (BES) program at Relativistic Heavy Ion Collisions (RHIC), the Facility for Antiproton and Ion Research (FAIR) in Darmstadt and Nuclotronbased Ion Collider Facility (NICA) in Dubna.
The higher order fluctuations of conserved charges carries the divergency feature of the correlation length at the CEP thus are regarded as good observables of CEP [1]. The measurement on the higher cumulants of conserved charges from BESI at RHIC for Au+Au collisions shows a nonmonotonic energy dependent behavior for the kurtosis of the net proton number distributions in the collision energy from to , corresponding to the baryon chemical potential from to GeV [2, 3, 4]. It is observed that of net proton number distributions starts from about 1 at , decreases to around 0.1 at and rises quickly up to 3.5 at . It calls for theoretical understanding whether this nonmonotonic structure is related to the existence of the QCD CEP.
The QCD CEP has been investigated from lattice theory [5], and the recent lattice result excluded the existence of CEP in the low baryon density region [6]. The QCD CEP has also been widely analyzed through symmetry analysis [7], and by using effective chiral models, e.g., the Nambu–JonaLasinio (NJL) models including the Polyakovloop improved NJL (PNJL) model [8], quarkmeson (QM) models including the Polyakovloop improved QM model [9], the DysonSchwinger equations (DSE) [10], as well as the 5dimension holographic QCD model [11, 12]. The model studies lose the power on predicting the exact location of the CEP, because different models even the same model with different parameter sets give various location of CEP. However, still we can extract some useful information from model studies, e.g., the peak structure of along the freezeout line can be used as a clean signature for the existence of CEP, and the peak location of the measured is close to the real QCD CEP mountain [12, 13].
In this talk, we show the results from a realistic Polyakov–Nambu–JonaLasinio (rPNJL) model [13]. It is interesting to observe that the kurtosis produced from the rPNJL model along the experimental freezeout line agrees with BESI data well!
2 Model setup
The 3flavor rPNJL model takes into account 8quark interaction [14], and its effective potential is given below:
(1)  
Where takes for two light flavors while for strange quark, is quarkantiquark condensate for different flavors, and with the dynamically generated constituent quark mass taking the form of
(2) 
Here if , then and . takes the form of [15]
(3) 
which describes the selfinteraction of the Polyakovloop and , is a dimensionless parameter,
(4) 
with , and
(5) 
5.5  183.468  637.720  2.914  75.968 
(MeV)  

175  6.75  9.8  0.26  0.805  7.555  0.1 
The parameters of the NJL part shown in Table 1 are fixed by vacuum properties, and the parameters of Polyakov loop part listed in Table 2 are fixed by global fitting of the pressure density at zero chemical potential. With these parameters, the critical temperature and equation state at zero chemical potential in the rPNJL model agree well with lattice data [14].
3 Numerical results
From the pressure , the minus thermodynamical potential, one can obtain the netbaryon number fluctuations [4]
(6) 
which gives the cumulants of baryon number distributions . By introducing the variance and kurtosis , we can write the ratio of fourth and second order cumulants of netbaryon number fluctuations as
(7) 
The value of of baryon number fluctuation is 1 in the hadron resonance gas (HRG) phase and it takes the value of in the ideal free quark gas (FQG) limit at very high temperature [16].
With the parameters fixed in Table 1 and 2, we can obtain of baryon number fluctuation in the rPNJL model as shown in Fig.1 and compare with lattice data in Ref.[16]. The HRG limit and FQG limit are also shown in Fig.1. It is noticed that above the chiral phase transition , the result of agrees well with lattice result, and below , of baryon number fluctuation is only about half of the HRG limit. However, in the regular NJL model, the magnitude of of baryon number fluctuation is much smaller than the the lattice result [13]. This indicates that gluodynamics contribution plays dominant role in the baryon number fluctuations.
The chiral phase boundary in the rPNJL model is shown in Fig. 2, with the CEP located at . We also show the freezeout temperatures and baryon number chemical potentials extracted from experiment in Fig. 2, where the freezeout temperatures and baryon number chemical potentials extracted from BESI at RHIC [17] are shown in dots, and the freezeout temperatures and baryon number chemical potentials extracted from lower energy heavyion collisions [18] are shown in triangles. We use two fitted freezeout lines
(8) 
which are shown in dashed and dasheddotted lines, respectively. The second freezeout line is taken from Ref.[4]. Both fitted experimental freezeout lines are very close to the chiral phase boundary from the rPNJL model. The only difference is that at low baryon density region, the fitted experimental freezeout line starts from above the chiral phase boundary, and starts from below the phase boundary. The 3D plot of the kurtosis as functions of the temperature and chemical potential calculated from the rPNJL model is shown in the right of Fig. 2, from which it can be clearly seen that the first freezeout line starts from the back ridge of the phase boundary and starts from the front ridge of the phase boundary.
We show the kurtosis of baryon number fluctuations calculated from the rPNJL model along the two fitted freezeout lines from heavy ion collisions in Fig. 3 and compare with BESI measurement of (shown in dots with error bars) [2, 3, 4]. The left figure is as a function of the baryon number chemical potential and the right figure is as a function of the collision energy, where the relation between the chemical potential and the collision energy [18] has been used. As a reference, the calculated from a realistic regular NJL model along the freezeout line [19] is also shown as long dashed line. Because there is no gluon contribution in the regular NJL model, the value of kurtosis in general is much smaller than that in the rPNJL model.
It is observed that the kurtosis produced from the realistic PNJL model along the freezeout line develops a dip structure around () and a peak structure at around (, while along the freezeout line , only develops a peak structure at around ( and no dip structure is developed. As we have discussed that the only difference between the two freezeout lines is at the low baryon density region: crosses the chiral phase boundary from above, i.e. starts from the back ridge of the phase boundary, from the 3D plot in Fig. 2, it is natural to see how the dip structure is formed. For the case along the freezeout line , which always lies below the phase boundary, there is no chance to form the dip structure. The peak structure along the freezeout line is solely determined by the existence of the CEP. From this analysis, we can see that the dip structure of measured is determined by the relationship between the freezeout line and chiral phase transition line at low baryon density region, and the peak structure can be regarded as a clean signature for the existence of CEP.
Another unexpected result is that the kurtosis from the equilibrium result in the rPNJL model along the experimental freezeout line agrees with BESI measurement very well. This is a surprising result, because in general, one should take into account the nonequilibrium evolution of the system in heavy ion collisions. However, as shown in Ref.[20] that the kurtosis of baryon number fluctuation even changes sign and becomes negative. It is worthy of mentioning that the measurement of is always positive, therefore it deserves further understanding on how the "little bang" after collision evolves with time. As shown in [21], one possible scenario is that the system after collision reaches thermalization very quickly in quite high temperature and then evolves in equilibrium state. From our analysis, to form the dip structure, the freezeout temperature at low baryon density region should be higher than the phase transition temperature! If this scenario is correct, the equilibrium result from a realistic PNJL model might work.
4 Summary
In summary, we have investigated the kurtosis of the baryon number fluctuations in a reparameterized realistic PNJL model, in which the critical temperature, equation of state and baryon number fluctuations are in good agreement with lattice data at zero chemical potential. This rPNJL model predicts a CEP located at . The extracted experimental freezeout line from heavy ion collisions is very close to the chiral phase transition boundary in the rPNJL model. It is surprised to see that the kurtosis of baryon number fluctuations from the rPNJL model along the experimental freezeout line agrees well with the BESI measurement. We also analyzed the formation of the dip structure and peak structure of measured along the freezeout line. Our analysis shows that the dip structure of is determined by the relationship between the freezeout line and chiral phase transition line at low baryon density region, and the peak structure is solely determined by the divergency at CEP, and the peak structure can be regarded as a clean signature for the existence of CEP. It is worthy of mentioning that at low baryon density region, the extracted freezeout temperatures from BESI measurement at RHIC [17] are indeed higher than the critical temperatures, this supports our analysis on the formation of dip structure of along the freezeout line.
Our result shows that the equilibrium result can explain the BESI data on baryon number fluctuations, this may indicate that the system after collision reaches thermalization quickly and evolve in equilibrium before the freezeout and phase transition, which is in agreement with the analysis in [21]. For example, in the collision energy of , the system reaches thermalization at around the temperature of , which is much higher than the freezeout temperature and the phase transition temperature around .
Acknowledgments
This work was supported in part by the NSFC under Grant Nos. 11725523, 11735007, and 11261130311 (CRC 110 by DFG and NSFC).
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