The Critical Fluctuations Of High-order Cumulants From Spin Models And The Statistical Fluctuations Of High-order Cumulants Of Conserved Charges In Heavy-ion Collisions

Quantum Chromodynamics (QCD)-the gauge theory of describing strong interaction, predict that the quarks confined in hadrons will be deconfined at high enough energies, and formed quark gluon plasma. At low temperature and big baryon chemical potential, the phase transition is of first order from hadronic matter to quark gluon plasma. The endpoint of the first order phase transition line is the critical point of QCD. At high temperature and small baryon chemical potential, hadrons transform to quark gluon plasma through a crossover. One of the main goals of current experiments at international large hadron colliders is to locate the QCD critical point. Because the correlation length diverges at the critical point, and the high-order cumulants of conserved charges are more sensitive to the correlation length, what’s more, they may have non-monotonic behavior and sign changes, so the high-order cumulants are suggested to be used searching for the critical point.Due to at large baryon chemical potential and near the chiral limit, the thermodynamics is not accessible from the first principle Lattice QCD calculations, many phenomenological models and effective theories have been used to help study these thermodynamics. In this paper, based on the universality of the critical behavior of high-order cumulants, we used the three dimensional Ising model, O(4) and O(2) spin models, and three dimensional three-state Potts model to calculate the high-order cumulants of magnetization and energy, which can belong to the same universality classes with QCD transitions. We found the high-order cumulants indeed have non-monotonic behavior and sign change in the vicinity of the critical point.In the three dimensional Ising universality class, when the critical point is approached from the crossover side, the fourth and sixth order cumulants of magnetization present negative valley. And the valley of the sixth order cumulant is more obvious than that of the fourth order. These results are consistent with that from the linear parametric model of the Ising universality class in the thermodynamic limit. Since the volume of the system formed in relativistic heavy-ion collisions is finite, we study the finite-size behavior of high-order cumulants in the Ising model. Based on the fact that the temperature where the sign change occur in the fourth and sixth order cumulants increases as the increase of system size, we infer that the pseudo-critical temperature will also increase as the increase of system size, and be closer and closer to the critical point. This trend is consistent with the prediction of two-flavor quark model and the linear sigma model.From the high-order cumulants of energy in the three dimensional O(4) and O(2) spin model, we found in the chiral phase transition of QCD at vanishing baryon chemical potential, the fourth order cumulant of baryon number has a peak. While from the sixth order cumulant, the sign change occurs. These results are consistent with that from the lattice QCD calculation and effective model simulations, which both keep the chiral O(4) symmetry.The three dimensional three-state Potts model is a simple effective model of finite-temperature pure gauge. We study the behavior of generalized susceptibilities of magnetization in the area of first order phase transition, critical point and crossover. When crossing the phase boundary at fixed external fields, the second to sixth order susceptibilities all have non-monotonic behavior or sign change. And their behavior is similar in three kinds of phase transitions. Non-monotonic behavior or sign change is not only related to the critical point, but also can occur in the first order phase transition and crossover. This behavior cannot distinguish different orders of phase transition effectively. We further study the finite-size scaling behavior of the second and fourth order susceptibilities. Their scaling exponents are different in different orders of phase transitions, which can distinguish different kinds of phase transitions.We also calculate the second and fourth order susceptibilities on the phase boundary and along the line away from the phase boundary by some temperature. We found that with the increase of the external field, the second order susceptibility decreases monotonically from the first order phase transition side to crossover side. While the fourth order susceptibility is negative on the phase boundary and increases monotonically. When away from the phase boundary some extent, it changes to positive at the first order phase transition side. While away from the phase boundary too far, it becomes positive in all sides, and decreases monotonically.The behavior of the fourth order susceptibility is related to the distance to the phase boundary. So we give the density plot of its sign on the temperature-external field plane. We found that the values of the fourth order susceptibility are all negative near the phase boundary. But the negative area is narrower in the first order phase transition side than in crossover side. That’s why when it’s away from the phase boundary to some extent, the value of fourth order susceptibility will change to positive in the first order phase transition side first.The number of produced particles is finite in the relativistic heavy-ion collisions, so the influence of statistical fluctuations on the high-order cumulants cannot be neglected. Supposing each kind of particles follows the Poisson or binomial distribution, we deduce the high-order cumulants of statistical fluctuations of the net-baryon number, net-electric charges and net-strangeness. The results based on the Poisson distribution are consistent with the baseline of fluctuations of conserved charges from the hadron resonance gas model. Using the mean and variance of the proton and anti-proton from the experiments, we calculate the statistical part of high-order cumulants of net-proton. Comparing the statistical fluctuations to the data, we found the high-order cumulants of net-proton calculated by experiment is dominated by statistical fluctuations.In order to see the difference between experimental results with statistical fluctuations, we suggest measuring the dynamical cumulants of conserved charges. That’s the high-order cumulants calculated by experimental data minus the corresponding part caused by Poisson statistical fluctuations. Using the two different versions of a multi-particle transport model (default version and string melting version) and ultra-relativistic quantum molecular dynamics model (UrQMD), we calculate the dynamical kurtosis of net-proton and total-proton at seven RHIC energies and nine centralities. In contrast to the sign change of dynamical kurtosis of net-proton observed in data, they are all positive in these models, and have no sign change. But the dynamical kurtosis of total-proton is the same. That’s to say no matter in the traditional models or experimental data, there is no sign change.The traditional particle production mechanisms implemented in the three transport models cannot repeat the observed sign change behavior of the dynamical kurtosis of net protons at experiments. The inconsistency indicates that there should be extra correlations at experiments which have not been taken into account in the three transport models. Whether the correlations are related to the critical phenomena or not needs further study.