Critical net-proton number fluctuations with hydrodynamics

We compute the net-proton number fluctuations and their ratios $C_2/C_1$, $C_3/C_2$ and $C_4/C_2$ on the hydrodynamic freeze-out hypersurface of particlization at nine collision energies, $\sqrt{s_{\mathrm{NN}}}=7.7-200$ GeV, based on the fluctuations obtained from the functional renormalization group (fRG) approach, where both the regular and the critical fluctuations arising from the critical end point (CEP) are included. The transverse momentum and rapidity acceptance windows as same as the experimental measurements, the isospin randomization for the proton number fluctuations, and the global baryon conservation effect are implemented in the calculations. The results are also compared with the baseline results without critical fluctuations. It is found that for the low-order cumulants, e.g., $C_2/C_1$ the difference between the critical and non-critical results is small, while the difference increases with the increasing order of cumulants in the region of low collision energy. A non-monotonic dependence on the collision energy is observed in $C_4/C_2$ with critical fluctuations, which is absent in the results without critical fluctuations.