Impact of the underlying O(4) criticality on net-proton number fluctuations.

We discuss the beam-energy dependence of ratios of the first four cumulants of the net-proton number, calculated using a phenomenologically motivated model in which critical mode fluctuations couple to protons and anti-protons. The model takes into account an impact of the underlying $O(4)$ criticality on the QCD phase diagram in the presence of a $Z(2)$ critical point. This allows us to capture qualitatively both the monotonic behavior of the lowest-order ratio and the non-monotonic behavior of higher-order ratios which are seen in the experimental data from the STAR Collaboration. We also discuss the dependence of our results on the coupling strength between the critical mode and the protons as well as the location of the critical point.