Extracting freeze-out conditions in beam energy scan via functional QCD

Abstract

Fluctuations of conserved charges provide a link between high quality theoretical results and precision measurements of heavy ion collisions. We compare results for ratios of the lowest order baryon number susceptibilities from functional QCD approaches to proton number cumulants extracted from experiments. We find that they meet at a specific temperature and chemical potential for each collision energy. This is indicative of the respective freeze-out point. From this self-consistent determination of the freeze-out parameters we extract a prediction for the kurtosis on the freeze-out line. We find quantitative agreement with experimental data where available, despite comparing apples (baryons) with oranges (protons). At a collision energy around 5 GeV, our kurtosis exhibits a peak structure indicative of the critical end point of QCD.

Figures (7)

• Figure 1: Freeze-out points extracted from \ref{['eq:freezeout-extract-C1C2', 'eq:freezeout-extract-C3C1']} in the $T$-$\mu_B$ plane, using the cumulant data $C_{1,2,3}$ from BES-I STAR:2021iop and BES-II STAR:2025zdq, with the collision energies labeled as the black numbers in GeV unit. Also shown are the locations of the CEP extracted in Fu:2019hdwGao:2020fblGunkel:2021oya and the corresponding result Lu:2025cls obtained with the scheme used in this work. The prediction on the freeze-out condition towards lower $\sqrt{s_{\small \textrm{NN}}}$ or higher $\mu_B$ is shown as the blue band, which is determined by fitting our freeze-out points with \ref{['eq:muBfsNNfit', 'eq:TfmuBffit']}. We also compare the freeze-out points with the chiral phase transition line, and the freeze-out points extracted from particle yields Andronic:2017pugSagun:2017eye whose collision energies are indicated by the grey numbers in GeV unit. The peak position of the kurtosis along the freeze-out lines within the band is indicated by the blue-dashed curve, with its peak width indicated by the purple region, in match with \ref{['fig:muBf-sqrtsNN']} and \ref{['fig:kurtosis-along-extracted']}.
• Figure 2: Freeze-out chemical potential $\mu_{B,f}^{}$ as a function of collision energy $\sqrt{s_{\small \textrm{NN}}}$. Our result for the $\mu_{B,f}^{}$-$\sqrt{s_{\small \textrm{NN}}}$ relation is displayed as the blue band, which is compared with the freeze-out points predicted from statistical models Andronic:2017pugSagun:2017eye. The collision energies in STAR BES-I and BES-II are marked in black, and those collision energies covered in other experiments are marked in gray, all in GeV units.
• Figure 3: Collision energy ($\sqrt{s_{\small \textrm{NN}}}$) dependence of the net-baryon number kurtosis along the fitted freeze-out band in \ref{['fig:freezeout-T-muB']}. The error estimate of the kurtosis is marked by the blue band, which corresponds to its variation along all possible freeze-out lines within the freeze-out band in \ref{['fig:freezeout-T-muB']}.
• Figure 4: Net-baryon number susceptibility ratios $\chi_1^B/\chi_2^B$ and $\chi_3^B/\chi_1^B$, compared with the experimental data of net-proton cumulant ratios $C_1/C_2$ and $C_3/C_1$ from BES-I $\sqrt{s_{\small \textrm{NN}}}$ = 200 to 7.7 GeV, whose mean values are denoted by the dashed horizontal lines with the collision energies in GeV unit marked by the numbers alongside. The extracted freeze-out temperature $T_f$ and $\mu_{B,f}$ are denoted by stars, with the error bars denoted by the shaded rectangles. The vertical height of the rectangle corresponds to the experimental error at each $\sqrt{s_{\small \textrm{NN}}}$.
• Figure 5: Net-baryon number susceptibility ratios $\chi_1^B/\chi_2^B$ and $\chi_3^B/\chi_1^B$, compared with the experimental data of net-proton cumulant ratios $C_1/C_2$ and $C_3/C_1$ from BES-I $\sqrt{s_{\small \textrm{NN}}}$ = 27 to 7.7 GeV. All symbols have the same meaning as those illustrated in \ref{['fig:cumulant-ratio-compare-bes1']}.