# The kurtosis of net baryon number fluctuations from a realistic Polyakov--Nambu--Jona-Lasinio model along the experimental freeze-out line

## Abstract

Firstly we qualitatively analyze the formation of the dip and peak structures of the kurtosis of net baryon number fluctuation along imagined freeze-out lines and discuss the signature of the existence of the QCD critical end point (CEP) in the Nambu–Jona-Lasinio (NJL) model, Polyakov-NJL (PNJL) model as well as -dependent PNJL( PNJL) model with different parameter sets, and then we apply a realistic PNJL model with parameters fixed by lattice data at zero chemical potential, and quantitatively investigate its along the real freeze-out line extracted from experiments. The important contribution from gluodynamics to the baryon number fluctuations is discussed. The peak structure of along the freeze-out line is solely determined by the existence of the CEP mountain and can be used as a clean signature for the existence of CEP. The formation of the dip structure is sensitive to the relation between the freeze-out line and the phase boundary, and the freeze-out line starts from the back-ridge of the phase boundary is required. To our surprise, the kurtosis produced from the realistic PNJL model along the experimental freeze-out line agrees with BES-I data well, which indicates that equilibrium result can explain the experimental data. It is worth to point out that the extracted freeze-out temperatures from beam energy scan measurement are indeed higher than the critical temperatures at small chemical potentials, which supports our qualitative analysis.

###### pacs:

12.38.Mh,25.75.Nq,11.10.Wx,11.25.Tq^{1}

^{2}

^{3}

^{4}

## I Introduction

The phase transition and phase structure of Quantum Chromodynamics (QCD) under extreme conditions is the main topic of relativistic heavy ion collisions, and it is also highly related to the evolution of the early universe and the equation of state inside the compact stars. Lattice QCD calculation shows that at small baryon density, the QCD phase transitions including the chiral phase transition as well as deconfinement phase transition are of crossover at finite temperature Fodor:2001au (); Ding:2015ona (); Schmidt:2017bjt (). From symmetry analysis and effective chiral models, it is generally believed that at high baryon density the chiral phase transition is of first order and there exists a QCD critical end point (CEP) for chiral phase transition in the temperature and baryon chemical potential plane Pisarski:1983ms (); Hatta:2002sj (); Schwarz:1999dj (); Zhuang:2000ub (); Chen:2014ufa (); Chen:2015dra (); Fan:2016ovc (); Fan:2017kym (); Fu:2010ay (); Bowman:2008kc (); Mao:2009aq (); Schaefer:2011ex (); Schaefer:2012gy (); Qin:2010nq (); Luecker:2013oda (); Fu:2016tey (); Stephanov:2008qz (); Stephanov:2011pb (); Asakawa:2009aj (); Athanasiou:2010kw (). The QCD CEP have been widely analyzed in different models, e.g., Nambu–Jona-Lasinio (NJL) model, the Polyakov-loop improved NJL (PNJL) model, linear sigma model, quark-meson (QM) model, the Polyakov-loop improved QM model, the Dyson-Schwinger equations (DSE), and the holographic QCD model Schwarz:1999dj (); Zhuang:2000ub (); Chen:2014ufa (); Chen:2015dra (); Fan:2016ovc (); Fan:2017kym (); Fu:2010ay (); Bowman:2008kc (); Mao:2009aq (); Schaefer:2011ex (); Schaefer:2012gy (); Qin:2010nq (); Luecker:2013oda (); Fu:2016tey (); Critelli:2017oub (). However, different models even the same model with different parameter sets give various location of CEPVovchenko:2017gkg (). Therefore, to search for the existence of the CEP and further to locate the CEP is one of the most central goals at Relativistic Heavy Ion Collisions (RHIC) as well as for the future accelerator facilities at Facility for Antiproton and Ion Research (FAIR) in Darmstadt and Nuclotron-based Ion Collider Facility (NICA) in Dubna.

The cumulants of conserved quantities up to fourth order of net-proton, net-charge and net-kaon multiplicity distributions have been measured in the first phase of beam energy scan program (BES-I) at RHIC for Au+Au collisions at and , and the results are summarized in Adamczyk:2013dal (); Aggarwal:2010wy (); Luo:2017faz (). A non-monotonic energy dependent behavior for the kurtosis of the net proton number distributions has been observed in the most central Au+Au collisions: firstly decreases from around 1 at the colliding energy to 0.1 at and then rises quickly up to around 3.5 at . From the BES-I observed oscillation behavior of kurtosis, we need to know what information we can extract about the CEP and where is the CEP located. Before the second phase of beam energy scan (BES-II) at RHIC performed in 2019-2020, it is very urgent for both experimentalists and theorists to extract a clean signature to identify the existence of the QCD CEP and further to locate the CEP. On the one hand, we should try to extract useful information about QCD phase transitions from the measurement along the freeze out line, and to find the evidence of the QCD CEP. On the other hand, we should explore carefully how QCD phase transitions will shed light on properties of the cumulants of conserved quantities at freeze-out.

By using the Ising model, it was pointed out in Stephanov:2011pb () that the quartic cumulant (or kurtosis) of the order parameter is universally negative when the critical point is approaching to the crossover side of the phase separation line. From the results in Stephanov:2011pb (), the oscillation behavior of the kurtosis along the freeze out line has been regarded as a typical signature of the existence of the CEP. Many interests have focused on the sign changing of various cumulants around the CEP, and the sign changing for higher order susceptibilities has been recently discussed in Fan:2017kym (). It is noticed that the sign changing for the 6th and 8th order susceptibilities starts at the baryon chemical potential quite far away from the CEP, it may indicate that the sign changing of cumulans of conserved quantities is not directly related to the CEP. In the holographic QCD model Li:2017ple (), we explained that the sign changing of the baryon number susceptibilities along the freeze-out line is not necessarily related to the CEP, but the peaked baryon number susceptibilities along the freeze-out line is solely determined by the CEP thus can be used as an evident signature for the existence of the CEP, and the peak position is close to the location of the CEP in the QCD phase diagram. Furthermore, it is found that at zero chemical potential, the magnitude of around the phase transition line in the gluodynamics dominant holographic QCD model is around , which is in agreement with lattice result, and is much larger than that in the Ising model (close to zero), and also much larger than that in the NJL model (around 0.1). Therefore, it is natural to speculate that the gluodynamics contribution is dominant to the baryon number susceptibilities, which is quite surprising!

In order to check the gluodynamics contribution to the baryon number susceptibilities, in this work, we will try to explore the structure of the kurtosis of the net proton number distribution along the freeze out line in the Nambu–Jona-Lasinio (NJL) model, the Polyakov-loop improved NJL (PNJL) model as well as -dependent Polyakov-loop potential improved NJL (PNJL) model. Also in order to investigate the relation between the CEP and the structure of the along the freeze out line, we will need to shift the location of the CEP by introducing the interaction in the vector channel. This paper is organized as following: After Introduction, in Sec. II, we give a brief introduction to the two-flavor NJL model, the Polyakov-loop improved NJL (PNJL) model as well as Polyakov-loop improved NJL (PNJL) model, and qualitatively analyze the formation of the dip and peak structures of the kurtosis of net baryon number fluctuation along imagined freeze-out lines and discuss the signature of the existence of the QCD critical end point (CEP). Then in Sec. III we apply a realistic PNJL model with parameters fixed by lattice data at zero chemical potential, and quantitatively investigate its along the real freeze-out line extracted from experiments. Finally, the discussion and conclusion part is given in Sec. IV.

## Ii Qualitative analysis of baryon number fluctuations in the NJL, PNJL and PNJL Models

We will investigate the gluodynamics contribution to the kurtosis of the net baryon number distribution , therefore we will compare in the framework of NJL model, the Polyakov-loop improved NJL (PNJL) model as well as NJL with -dependent Polyakov-loop potential (PNJL) model, and analyze the structure of along the imagined freeze-out lines and its relation with the existence and location of the QCD CEP.

### ii.1 The NJL, PNJL and PNJL models with vector interaction

In order to shift the location of the CEP, we introduce the two-flavor NJL model with the vector interaction, and the Lagrangian is given by Nambu:1961tp (); Nambu:1961fr (); Klevansky:1992qe ()

(1) |

Where is the doublet of the two light quark flavors and with the current mass , and the isospin Pauli matrix. and are the coupling constants in the (psudo)scalar channel and the vector channel, respectively. By introducing the auxiliary fields for scalars and vectors, and in the vacuum we take the mean-field approximation with

(2) |

the quark condensate and the net quark number density of flavor respectively. Then the thermodynamical potential of this NJL model takes the following form:

(3) |

with the number of colors. The quark quasiparticle energies and constituent quark masses for flavors are given by

(4) |

and the effective chemical potentials are shifted by

(5) |

In order to solve the minimum of the thermal potential , we have the following gap equations

(6) |

and

(7) |

For numerical calculations, we fix in all the models. In the regular NJL model, we choose the parameters as in Ref.Dutra:2013lya (). To shift the location of the CEP in the NJL model we choose different coupling constant in the vector channel: , and Fan:2017kym (), which is indicated as NJL-1, NJL-2, NJL-3 respectively, and three sets of parameters are shown in Table. 1. Correspondingly, the locations of CEP in the NJL-1 and NJL-2 are , , respectively, there is no CEP in the NJL-3.

NJL-1 | 651 | 5.04 | -0.5 | 76.8 | 796.7 |
---|---|---|---|---|---|

NJL-2 | 651 | 5.04 | 0 | 34.6 | 1005.2 |

NJL-3 | 651 | 5.04 | 0.67 | No | No |

In the 2-flavor PNJL and PNJL models, we need to add the Polyakov-loop effective potential with the following ansatz Fukushima:2008wg (); Weiss:1980rj (); Ratti:2006wg (); Shao:2017yzv ()

(8) |

where and . The critical temperature of the confinement-deconfinement phase transition in pure gluon system is , and we rescale to because of the presence of fermion fields. The thermodynamical potential of this PNJL model is given by

(9) | |||||

with the same definitions of , and as in the NJL model. The gap equations are determined by

(10) |

and

(11) |

In the PNJL model, we fix , and choose three different parameter sets for numerical calculations for PNJL-1, PNJL-2 and PNJL-3 as shown in Table.2. For the PNJL-1, at zero chemical potential , the critical temperature for the chiral phase transition is , and for deconfinement phase transition is , and the CEP is located at . For the PNJL-2, at , the critical temperature for the chiral phase transition is , and for deconfinement phase transition is , and the CEP is located at . For the PNJL-3, at , the critical temperature for the chiral phase transition is , and for deconfinement phase transition is , and there is no CEP in the PNJL-3.

PNJL-1 | 648 | 4.98 | -0.35 | 3.51 | -2.47 | 14.5 | -1.95 | 149.1 | 203.4 |
---|---|---|---|---|---|---|---|---|---|

PNJL-2 | 651 | 5.04 | 0 | 3.51 | -2.47 | 15.2 | -1.75 | 103.2 | 971.5 |

PNJL-3 | 651 | 5.04 | 0.67 | 3.51 | -2.47 | 15.2 | -1.75 | No | No |

By considering the back-reaction of quark to the glue sector, we take the so called PNJL model with the -dependent Schaefer:2007pw (); Xin:2014dia (); Shao:2016fsh ()

(12) |

where and . The thermodynamical potential of PNJL model has the same form of PNJL model, and the gap equations take the following form:

(13) |

and

(14) |

In the PNJL model, we choose two different parameter sets as shown in Table 3. With this parameter set, for the PNJL-1, at , the critical temperature for the chiral phase transition is , and for deconfinement phase transition is , and the CEP is located at . For PNJL-2, at , the critical temperature for the chiral phase transition is , and for deconfinement phase transition is , and the CEP is located at .

PNJL-1 | 651 | 5.04 | 1770 | 0.304 | 0.2 | 3.51 | -2.47 | 15.2 | -1.75 | 82.0 | 983.5 |
---|---|---|---|---|---|---|---|---|---|---|---|

PNJL-2 | 648 | 4.98 | 1870 | 0.304 | 0.2 | 3.51 | -2.47 | 14.5 | -1.95 | 63.9 | 967.8 |

### ii.2 The gluodynamics contribution to

BES-I measures the net-proton number susceptibilities, which can be approximately regarded as net-baryon number fluctuations and are defined as the derivative of the dimensionless pressure with respected to the reduced chemical potential Luo:2017faz ()

(15) |

with the pressure which is just the minus thermodynamical potential. The cumulants of baryon number distributions are given by

(16) |

By introducing the variance , kurtosis , one can have the following relation between observable quantities and theoretical calculations:

(17) |

which relates with the ratio of fourth and second order cumulants of net-baryon number fluctuations.

In Fig.1, we show as a function of in three parameter sets of NJL model, and three parameter sets of PNJL model and two parameter sets of PNJL model at zero chemical potential, and compare with lattice results in Bazavov:2017dus (), with the critical temperature for chiral phase transition at zero chemical potential . The critical temperature of lattice data at is and we choose . The value of of net baryon number fluctuations is unity in the limit of hadron resonance gas (HRG) and, in the ideal free quark gas (FQG) limit at infinite temperature it takes the value of , which are also shown in Fig.1. It is worthy of mentioning that from the lattice result shown in Ref.Bazavov:2017dus (), that at the critical temperature , the magnitude of is around 0.8, which is smaller than its value of 1 at the temperature of , at which quark matter transfers to hadron gas. Here describes a physical “confinement-deconfinement” transition, which is different from the describing the confinement-deconfinement phase transition by using the order parameter of Polyakov loop.

In the NJL model, the magnitude of at is around in the NJL-1, NJL-2 and NJL-3, which is much smaller than the lattice result. In the whole temperature region , the magnitude of is quite small comparing with the lattice results.

When the gluodynamics contribution is taken into account in the PNJL model, the critical temperatures for the chiral phase transition and deconfinement phase transition are separated, and the critical temperature for the deconfinement phase transition is lower than that of the chiral phase transition at zero chemical potential. It is observed that the magnitude of shows a peak at the critical temperature for deconfinement phase transition , and in the hadron gas phase with , the magnitude of decreases with the decreasing temperature, and smaller than the HRG limit. With the parameter sets used in the PNJL-1, the peak value of can reach 1 at , and is around at , which is in agreement with the lattice result. However, with a negative coupling constant in the vector channel , the magnitude of at high temperature is obviously not correct, it does not decrease to the free gas limit but increases rapidly when and rise up again to 1 at around . However, in this work, we only need the information of the model around phase transition and its location of CEP, we can still use the parameter set in PNJL-1.

For the PNJL model, the magnitude of also shows a peak at the critical temperature of deconfinement phase transition . However, the peak value is around 2 and much larger than the lattice result. In the region of , the magnitude of does not decreases to the free quark gas limit but increases with the temperature and is larger than lattice result. This indicate that the dependent Polyakov-loop potential is not properly added, which can be improved in the future.

From the magnitude of in the NJL model, PNJL model and PNJL model, we can see that the dominant contribution to comes from gluodynamics at zero chemical potential. In the future for model construction, as a function of the temperature can be used to constrain models.

### ii.3 The kurtosis of baryon number fluctuation in the NJL model

In Fig. 2, we show the 3D plot for the kurtosis of baryon number fluctuation as a function of the temperature and baryon chemical potential in the NJL model. In Fig.3 we show the chiral phase transition line and 2D plot for as a function of the baryon chemical potential along different freeze-out lines for NJL-1,NJL-2 and NJL-3, respectively.

By changing the coupling constant in the vector channel, the location of the CEP will shift. The CEP in the NJL-1 and NJL-2 is located at , , respectively, and it is observed that develops a high CEP mountain in the NJL-1 and NJL-2. There is no CEP in the NJL-3, therefore, no CEP mountain of develops in the NJL-3. From the 3D plot in Fig. 2, one can observe an obvious chiral phase boundary in the crossover side, and the magnitude of its ridge decreases with the increase of the baryon chemical potential. Here we define the ridge as along the phase boundary, the back/front ridge as the higher/lower temperature side comparing with the phase boundary, respectively. If there is a CEP located on the phase diagram, a CEP mountain rises up around the CEP, and one can observe a negative region of around the CEP above the chiral phase boundary extended from the crossover side.

Because the measurement of heavy-ion collision is along the freeze-out line, we choose three different freeze-out lines in the NJL model: 1) Starting from the back ridge of the chiral phase boundary, goes through the negative region and then cross the phase boundary; 2) Starting from the front ridge of the chiral phase boundary and then cross the foot of the CEP mountain; 3) Far away from the phase boundary. These three free-out lines are indicated by long dashed, dashed-dotted and dashed lines, respectively.

It is observed that when the freeze-out line crosses the CEP mountain or crosses the foot of the CEP mountain, there will be a peak showing up for along the freeze-out line, and the location of the peak is close to the location of the CEP mountain. In the case of no CEP, it is found that keeps flat in almost the whole region and does not show any structure. The BES-I measurement is also shown in the 2D plot of as a function of the baryon chemical potential for comparing. In the NJL-1 and NJL-2 with the existence of CEP in the phase diagram, it is found that for the first case freeze-out line, because the freeze-out line starts from the back ridge of the chiral phase boundary and goes through the negative region of , then crosses the CEP mountain, therefore, decreases from around 0.1 and then down to negative value then rises up quickly and shows up a high peak around the critical chemical potential, which shows a dip and then a peak structure of along the freeze-out line. For the second case of freeze-out line, because the freeze-out line starts from the front ridge of the chiral phase boundary, and then crosses the foot of the CEP mountain, we can only see a peak structure of along the freeze-out line at high chemical potential. For the third case of freeze-out, if the freeze-out line is far away from the phase boundary, one can only observe a weak peak of along the freeze-out line.

It is found that the magnitude of in the NJL model at small baryon chemical potential region is small comparing with experiment measurement.

### ii.4 The kurtosis of the baryon number fluctuation in the PNJL model

In Fig. 4, we show the 3D plot for the kurtosis of baryon number fluctuation as a function of the temperature and baryon chemical potential in the PNJL model. In Fig.5 we show the phase transition lines and 2D plot for as a function of the baryon chemical potential along different freeze-out lines for PNJL-1,PNJL-2 and PNJL-3, respectively.

Different from the NJL model, in the PNJL model, when the gluodynamics is taken into account, there will be two phase transitions: one for the chiral phase transition and another for the deconfinement phase transition, and the deconfinement phase transition line lays below the chiral phase transition line at small chemical potential region. The two separate phase transition lines can be obviously seen from the phase diagram in Fig.5. In the 3D plot Fig.4 one can observe two separate phase boundaries at small baryon chemical potentials, and a valley forms in between the two phase boundaries. It is noticed that the magnitude of baryon number fluctuation is quite small along the chiral phase boundary, but around 1 along the deconfinement phase boundary.

By changing the coupling constant in the vector channel and parameter sets, the CEP of the chiral phase transition in the PNJL model can shift. The CEPs are located at and in the PNJL-1 and PNJL-2 models, respectively. Even though the two critical baryon chemical potentials in the PNJL-1 and PNJL-2 are almost the same, the critical temperature in the PNJL-1 model is higher than that in the PNJL-2. For the parameters used in the PNJL-3, there is no CEP shows up in the phase diagram. From the 3D plot in Fig. 4, one can observe two obvious phase boundaries for the chiral and deconfinement phase transitions in the crossover side, and the magnitude of the deconfinement ridge is much higher than the chiral ridge. If a CEP for the chiral phase transition exists in the PNJL model, the structure of the CEP mountain for the chiral phase transition looks as the same as that in the NJL model, and one can observe a negative region of around the CEP above the chiral phase boundary extended from the crossover side. The only difference is that the deconfinement phase boundary extends to the CEP mountain and merges with the chiral phase boundary.

The structure of along the freeze-out line in the PNJL model is more complicated due to the two separated phase boundaries. Comparing with the NJL model, the magnitude of at small baryon chemical potentials in the PNJL model is in agreement with experiment measurement due to the contribution from gluodynamics. We also choose three different freeze-out lines: 1) Starting from the back ridge of the chiral phase boundary, goes through the negative region and then crosses the foot of the CEP mountain; 2) Starting from the back ridge of the deconfinement phase boundary, and then crosses the foot of the CEP mountain; 3) Starting from the front ridge of the deconfinement phase boundary and keeps far away from the CEP mountain. These three different freeze-out lines are indicated by long dashed, dashed-dotted and dashed lines, respectively. Here the back/ front ridge also means the higher/lower temperature side comparing with the phase boundary, respectively.

The same as in the NJL model, the peak structure of along the freeze-out line in the PNJL model is solely related to the CEP mountain, when the freeze-out line crosses the CEP mountain or crosses the foot of the CEP mountain, there will be a peak showing up for , and the location of the peak is related to the location of the CEP mountain. If there is no CEP, keeps flat in almost the whole chemical potential region. In the PNJL-1 and PNJL-2 models with the existence of CEP in the phase diagram, it is found that for both the first and second cases of the freeze-out lines, one can observe the dip-peak structure for along the freeze-out lines. The difference lies in that for the first freeze-out line case, goes to negative at the dip, and for the second freeze-out line case, is positive at the dip. This is because that for the first case, the freeze-out line starts from the back ridge of the chiral phase boundary and goes through the negative region of , then crosses the foot of the CEP mountain, therefore, decreases from around 0.1 and then down to negative value then rises up quickly and shows up a high peak around the critical chemical potential, thus shows a dip and then a peak structure of along the freeze-out line. For the second case, the freeze-out line starts from the back ridge of the deconfinement phase boundary, and has no chance to go through the negative region, then crosses the foot of the CEP mountain, though we can observe a dip structure but the magnitude of at the dip is positive. For the third case when the freeze-out line is far away from the phase boundary, along the freeze-out line is almost flat.

It is noticed that in the PNJL-1 model, along the second cases of freeze-out line shows a dip structure and a peak structure, in the PNJL-3 model, there is no CEP in the phase diagram, and no special structure of along the freeze-out lines is observed.

### ii.5 The kurtosis of the baryon number fluctuation in the PNJL model:

In Fig. 6, we show the 3D plot for the kurtosis of baryon number fluctuation as a function of the temperature and baryon chemical potential in the PNJL model. In Fig.7 we show the phase transition lines and 2D plot for as a function of the baryon chemical potential along different freeze-out lines for PNJL-1 and PNJL-2 models, respectively.

Same as that in the PNJL model, in the PNJL model, there also exist two separate phase transitions for the chiral phase transition and deconfinement phase transition, and the deconfinement phase transition line also lays below the chiral phase transition line at small chemical potential region. From the 3D plot Fig.6 one can observe two separate phase boundaries at small baryon chemical potentials. The height of the ridge along the deconfinement phase boundary is around 2 at small chemical potentials in the PNJL model, which is higher than that in the PNJL model.

Similar to that in the PNJL model, we choose three different freeze-out lines: 1) Starting from the back ridge of the chiral phase boundary, goes through the negative region and then crosses the foot of the CEP mountain; 2) Starting from the back ridge of the deconfinement phase boundary, and then crosses the foot of the CEP mountain; 3) Starting from the front ridge of the deconfinement phase boundary and keeps far away from the CEP mountain. These three different freeze-out lines are indicated by long dashed, dashed-dotted and dashed lines, respectively. The structure of along the freeze-out line in PNJL-1 and PNJL-2 can show the dip and peak structure for both the first and second cases of the freeze-out lines, goes to negative at the dip in the first case, and keeps positive at the dip for the second freeze-out line case.

It is worthy of mentioning that for the PNJL-1 with the imagined freeze-out line starting from the back ridge of the deconfinement phase boundary and then crossing the foot of CEP mountain, the behavior of along the freeze-out line agree with the measurement of BES-I at RHIC. It is shown that indeed the extracted freeze-out temperature at BES-I Das:2014qca () starts from the temperature above the critical temperature, and crosses the effective “deconfinement” transition line at about , where a peak is predicted to show up around.

## Iii The kurtosis of the baryon number fluctuation in a realistic PNJL model

In last section, we have investigated gluodynamics contribution to the baryon number fluctuations, and analyzed the formation of the dip and peak structures of the kurtosis along the imagined freeze-out lines. In this section, we will investigate the kurtosis along the experimental freeze-out line in a realistic 3-flavor PNJL model which takes into account 8-quark interaction Bhattacharyya:2016jsn (). The effective potential is given below:

(18) | |||||

where corresponds to quark condensates and takes for two light flavors while for strange quark. with the dynamically generated constituent quark mass:

(19) |

If , then and , and so on in a clockwise manner.

describes the contribution from self interaction of and and it reads Ghosh:2007wy ():

(20) |

where

(21) |

and

(22) |

correspond to the effective potential of the Polyakov loop and the Jacobian of the transformation from the Polyakov loop to its trace, respectively. Besides, is a dimensionless parameter. is a temperature dependent coefficient which is chosen to have the form of

(23) |

Follow Bhattacharyya:2016jsn (), the parameters of the NJL part are fixed by vacuum properties and the parameters of Polyakov loop part are fixed by global fitting of the pressure density at zero chemical potential, and the details are listed in Table 4 and Table 5, respectively.

5.5 | 183.468 | 637.720 | 2.914 | 75.968 |

(MeV) | ||||||
---|---|---|---|---|---|---|

175 | 6.75 | -9.8 | 0.26 | 0.805 | 7.555 | 0.1 |

With these parameters, as shown in Bhattacharyya:2016jsn (), at zero chemical potential , the equation of state, baryon number fluctuations above the critical temperature are in good agreement with Lattice data. The kurtosis of baryon number fluctuation as a function of the temperature at zero baryon number density is shown in Fig. 8. It is noticed that for temperatures higher than the critical temperature , i.e., , the value of agrees well with lattice data. The phase transition line is shown in Fig. 9, and the CEP is located at . In Fig. 9, the freeze-out temperatures and baryon number chemical potentials extracted from BES-I at RHIC Das:2014qca () are shown in dots, and the freeze-out temperatures and baryon number chemical potentials for lower energy heavy-ion collisions summarized in Begun:2016pdy () are shown in triangles, and the two fitted freeze-out lines described by and as used in Luo:2017faz () are shown in long dashed and dashed-dotted lines, respectively. The first freeze-out line starts from the back ridge of the phase boundary and starts from the front ridge of the phase boundary which can be seen explicitly from the 3D plot in Fig. 9.

The kurtosis from the realistic PNJL model along the two freeze-out lines fitted from experimental data are shown in Fig. 10, the left figure is shown as a function of the baryon number chemical potential and the right figure is shown as a function of the collision energy, where we have used the relation between the chemical potential and the collision energy Begun:2016pdy () :

(24) |

We can see that the kurtosis from the realistic PNJL model along the freeze-out line , which starts from the back ridge of the phase boundary, develops a dip structure around (), and a peak structure at around (. To our surprise, the kurtosis from the realistic PNJL model along this experimental freeze-out line agree with BES-I result very well, except in the small chemical potential region. This is because in this realistic PNJL model, even though the Polyakov loop potential takes into account part of the confinement effect, however the real confinement from quark-gluon degrees of freedom to hadrons is still missing, thus the kurtosis below the critical temperature at zero chemical potential is lower than 1. This part can be improved in the future when we take into account hadron gas contribution. Along the second freeze-out line , which starts from the front ridge of the phase boundary, the kurtosis only develops a peak structure at around ( and no dip structure is developed. This supports our qualitative analysis in Sec. II. Furthermore, we also show the kurtosis of the baryon number fluctuations in a reparameterized NJL model Fan:2016ovc () with long dashed line in Fig. 10. Because of the lack of gluon contribution in the NJL model, the value of kurtosis in general is smaller than that in the PNJL model.

## Iv Conclusion and discussion

In this work, firstly we qualitatively investigate the kurtosis of net baryon number fluctuation and analyze the formation of its dip and peak structures along the imagined freeze-out lines in the NJL model, PNJL model as well as PNJL model with different parameter sets, and then we apply a realistic PNJL model and quantitatively investigate its along the real freeze-out line extracted from experiments.

Through qualitative analysis, we find that: 1) At zero chemical potential, the magnitude of is rather small in the NJL model comparing with lattice result, and it can reach around 1 in the PNJL model and 2 in the PNJL model around the critical temperature. This indicates that gluodynamics plays important role in the baryon number fluctuation; 2) The peak structure of along the freeze-out line is solely determined by the existence of the CEP mountain. When the freeze-out line crosses the CEP mountain or crosses the foot of the CEP mountain, there will be a peak showing up for along the freeze-out line, and the location of the peak is close to the location of the CEP mountain. The higher the peak, the closer the peak location to the CEP. If the height of the peak is higher than 1, it is for sure that the freeze-out line crosses the foot of the CEP mountain, and if the height of the peak is higher than 15, then the freeze-out line should crosses the CEP mountain, and the peak location is almost the location of the CEP. In the case of no CEP, it is found that keeps flat almost in the whole chemical potential region and does not show any structure; 3) The formation of the dip structure is more complicated: Firstly, it requires the existence of the CEP in the QCD phase diagram; Secondly, it is sensitive to the relation between the freeze-out line and the phase boundary. The dip structure can be formed if the freeze-out line starts from the temperature above the critical temperature at and crosses the phase boundary from above when the CEP exists in the phase diagram. Because the CEP is for chiral phase transition and there is a negative region of from the crossover side, if the freeze-out line starts from the back ridge of the chiral phase boundary and if it goes through the negative region of , and then crosses the CEP mountain or the foot of the CEP mountain, in this case one can observe a dip structure and a peak structure, and the magnitude of will go to negative at the dip. If the freeze-out line starts from the back ridge of the deconfinement phase boundary, and then crosses the foot of the CEP mountain, we can also see a dip and peak structure of along the freeze-out line, but the magnitude of will not go to negative at the dip. Therefore we can read the information on how does the freeze-out line crosses the chiral phase boundary from the negative/positive value at the dip of measured along the freeze-out line.

Quantitatively, we use a reparameterized realistic PNJL model, with its critical temperature, equation of state and baryon number fluctuations in good agreement lattice data at zero chemical potential. To our surprise, the kurtosis produced from the realistic PNJL model along the experimental freeze-out line agrees with BES-I data well. This may indicate that the equilibrium result can explain the BES-I data on baryon number fluctuations. Indeed, from the analysis in BraunMunzinger:2003zd (), after collision, the system reaches thermalization quickly in quite high temperature and then evolves in equilibrium state, e.g. in the collision energy of , the system reaches thermalization at around , which is much higher than the freeze-out temperature as well as the phase transition temperature. It is worth to point out that the extracted freeze-out temperatures from beam energy scan measurement are indeed higher than the critical temperatures at small chemical potentials, which supports our qualitative analysis on the formation of dip structure of along the freeze-out line.

At last, we should mention that in this work, even though our quantitative result from static thermodynamics can describe BES-I data well, the non-thermal effect, or memory effect Memory (), and finite size effect deserves further studies to locate the CEP.

###### Acknowledgements.

We thank valuable discussions with H.T.Ding, W.J.Fu, X.F.Luo and G.Y.Shao. M.H. acknowledges Aerospace Central Hospital where most part of the manuscript is written. This work is supported by the NSFC under Grant No. 11725523, 11735007 and 11261130311(CRC 110 by DFG and NSFC).### Footnotes

- thanks: lizb@mail.ihep.ac.cn, co-first author
- thanks: xukun@mail.ihep.ac.cn, co-first author
- thanks: wangxinyang@mail.ihep.ac.cn
- thanks: huangm@mail.ihep.ac.cn, correspondence author

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