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In the context of the ongoing search for the QCD critical point at theRelativistic Heavy-Ion Collider, we study the equation of state near thecritical point in the temperature and baryon chemical potential plane. We usethe parametric representation introduced in earlier literature, which maps theuniversal 3D Ising equation of state onto the QCD phase diagram using severalnon-universal parameters. We focus on the quartic cumulant of the baryonnumber, or baryon number susceptibility~$\chi_4^B$, which can be accessedexperimentally via net-proton fluctuation kurtosis measurements. It wasoriginally predicted, through universality arguments based on the {\em leading}singular contribution, that $\chi_4^B$ and net-proton kurtosis should show aspecific non-monotonic behavior due to the critical point. In particular, whenfollowing the freeze-out curve on the phase diagram by decreasing beam energy,the kurtosis is expected to dip, and then peak, when the beam energy scanpasses close to the critical point. We study the effects of the non-universaland thus far unknown parameters of the Ising-to-QCD mapping on the behaviorof~$\chi_4^B$. We find that, while the peak remains a solid feature, thepresence of the critical point does not necessarily cause a dip in $\chi_4^B$on the freezeout line {\em below} the transition temperature. The criticalpoint contribution to the dip appears only for a narrow set of mappingparameters, when subleading singular terms are sufficiently suppressed.

Quartic cumulant of baryon number in the presence of QCD critical point