Criticality of the net-baryon number probability distribution at finite density

We compute the probability distribution $P(N)$ of the net-baryon number at finite temperature and quark-chemical potential, $\mu$, at a physical value of the pion mass in the quark-meson model within the functional renormalization group scheme. For $\mu/T<1$, the model exhibits the chiral crossover transition which belongs to the universality class of the $O(4)$ spin system in three dimensions. We explore the influence of the chiral crossover transition on the properties of the net baryon number probability distribution, $P(N)$. By considering ratios of $P(N)$ to the Skellam function, with the same mean and variance, we unravel the characteristic features of the distribution that are related to $O(4)$ criticality at the chiral crossover transition. We explore the corresponding ratios for data obtained at RHIC by the STAR Collaboration and discuss their implications. We also examine $O(4)$ criticality in the context of binomial and negative-binomial distributions for the net proton number.