Fixed point behavior of cumulants in the three-dimensional Ising universality class

High-order cumulants and factorial cumulants of conserved charges are suggested for the study of the critical dynamics in heavy-ion collision experiments. In this paper, using the parametric representation of the threedimensional Ising model which is believed to belong to the same universality class as quantum chromo-dynamics,the temperature dependence of the second-to fourth-order(factorial) cumulants of the order parameter is studied. It is found that the values of the normalized cumulants are independent of the external magnetic field at the critical temperature, which results in a fixed point in the temperature dependence of the normalized cumulants. In finite-size systems simulated using the Monte Carlo method, this fixed point behavior still exists at temperatures near the critical. This fixed point behavior has also appeared in the temperature dependence of normalized factorial cumulants from at least the fourth order. With a mapping from the Ising model to QCD, the fixed point behavior is also found in the energy dependence of the normalized cumulants(or fourth-order factorial cumulants) along different freezeout curves.