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

Firstly we qualitatively analyze the formation of the dip and peak structures of the kurtosis $\kappa \sigma^2$ 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 $\mu$-dependent PNJL($\mu$ 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 $\kappa \sigma^2$ 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 $\kappa \sigma^2$ 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 $\kappa \sigma^2$ 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.