Measurement of Fifth- and Sixth-Order Fluctuations of (Net-)proton Number in Au+Au Collisions from Phase II of the Beam Energy Scan Program at RHIC

TL;DR

This study reports high-statistics measurements of the fifth- and sixth-order fluctuations of net-proton numbers in Au+Au collisions across $\sqrt{s_{NN}} = 7.7$–$27$ GeV from the STAR BES-II program. Using efficiency- and volume-fluctuation-corrected factorial cumulants $\kappa_4$, $\kappa_5$, $\kappa_6$ and cumulant ratios $C_5/C_1$, $C_6/C_2$, the authors test predictions from lattice QCD, FRG, and HRG-CE, as well as noncritical hadronic models like UrQMD. The results show no sign-alternating behavior in $\kappa_5$ or $\kappa_6$, and $C_5/C_1$, $C_6/C_2$ fluctuate around zero within uncertainties, consistent with a smooth crossover and with noncritical baselines; $C_4/C_2$ at higher energies aligns with QCD-based predictions, while UrQMD describes lower-energy trends. These measurements place meaningful constraints on baryon-number fluctuations and the QCD phase structure in the explored $\mu_B$ range, informing the search for the critical point and first-order transition regions.

Abstract

We report high-statistics measurements of fifth- and sixth-order factorial cumulants and cumulant ratios of (net-)proton multiplicity distributions in Au+Au collisions at $\sqrt{s_{NN}} = 7.7$--27 GeV, using data from the STAR experiment collected during the Beam Energy Scan Phase~II at RHIC. Protons and antiprotons are identified at midrapidity ($|y| < 0.5$) with transverse momentum $0.4 < p_T < 2.0$ GeV/$c$. The proton factorial cumulants $κ_4$, $κ_5$, and $κ_6$ increase with order but exhibit no sign alternation within current uncertainties, offering no evidence for a two-component structure in the proton multiplicity distribution, as might be expected near a first-order phase transition. The cumulant ratios $C_{5}/C_{1}$ and $C_{6}/C_{2}$ fluctuate around zero in collisions at 0--40\% centrality. The results are consistent with both the negative predictions from lattice QCD (LQCD) and the positive trends obtained from the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) model. At $\sqrt{s_{NN}} \gtrsim 27$ GeV, the $C_4/C_2$ and $C_5/C_1$ results are compatible with predictions from lattice QCD, functional renormalization group (FRG), and hadron resonance gas (HRG) models, while UrQMD describes the data better at lower energies. These measurements place constraints on baryon number fluctuations and offer valuable insights into the QCD phase structure.

Figures (5)

• Figure 1: (a) Proton acceptance in rapidity ($y$) vs. transverse momentum ($p_{\mathrm T}$) for Au+Au collisions at $\sqrt{s_{NN}} = 19.6$ GeV from BES-II. The blue box indicates the kinematic window used in this analysis. The TPC is used for PID in the range $0.4 < p_T < 0.8$ GeV/$c$, while both the TPC and TOF are used for $0.8 < p_T < 2.0$ GeV/$c$. The numbers within each ($y$, $p_T$) bin represent the purity of proton identification. (b) Event-by-event proton multiplicity distributions for 0--40% centrality of Au+Au collisions at BES-II energies ($\sqrt{s_{NN}} = 7.7$--27 GeV).
• Figure 2: Proton factorial cumulants ($\kappa_5$, $\kappa_6$) and net-proton cumulant ratios ($C_{5}/C_{1}$, $C_{6}/C_{2}$) in Au+Au collisions at $\sqrt{s_{NN}} = 7.7$--27 GeV as a function of collision centrality, quantified by the average number of participant nucleons $\langle N_{\mathrm{part}} \rangle$. BES-II results are compared with those from BES-I where applicable. Also shown is a comparison between BES-II measurements using two different centrality estimators: RefMult3X (red) and RefMult3 (blue); see text for details. The RefMult3X data points (red) are slightly shifted along the positive x axis for better visualization. Error bars and bands represent statistical and systematic uncertainties, respectively. For clarity in presentation, BES-I data points for 0-40% centrality (both factorial cumulants and cumulant ratios) are scaled down by a factor of 10 at each beam energy. Similarly, for the 0-40% centrality class in BES-II, the factorial cumulant data ($\kappa_5$, $\kappa_6$) from RefMult3 and RefMult3X are scaled down by a factor of 10 uniformly across all energies. No scaling is applied to the BES-II net-proton cumulant ratio data ($C_{5}/C_{1}$, $C_{6}/C_{2}$).
• Figure 3: Proton factorial cumulants ($\kappa_4$, $\kappa_5$, $\kappa_6$) in Au+Au collisions at $\sqrt{s_{NN}} = 7.7$--27 GeV (BES-II) and $\sqrt{s_{NN}} = 39$--200 GeV (BES-I) STAR:2022vloSTAR:2025zdq, shown as a function of collision energy. Results at $\sqrt{s_{NN}} = 3$ GeV from the fixed-target (FXT) program STAR:2022vlo are also shown. Error bars and shaded bands represent statistical and systematic uncertainties, respectively. Model calculations from UrQMD and the two-component model are shown for comparison.
• Figure 4: Net-proton cumulant ratios ($C_4/C_2$, $C_5/C_1$, $C_6/C_2$) in Au+Au collisions at $\sqrt{s_{NN}} = 7.7$--27 GeV (BES-II) and $\sqrt{s_{NN}} = 39$--200 GeV (BES-I) STAR:2020tgaSTAR:2022vloSTAR:2025zdq, shown as a function of collision energy. Results at $\sqrt{s_{NN}} = 3$ GeV from the fixed-target (FXT) program STAR:2022vlo are also shown. Error bars and shaded bands indicate statistical and systematic uncertainties, respectively. Model calculations from lattice QCD, FRG, HRG (canonical ensemble), and UrQMD are included for comparison. At 3 GeV, cumulant ratios (filled red circles), $C_{4}/C_{2}$, $C_{5}/C_{1}$, and $C_{6}/C_{2}$, are scaled down by a factor of 4 to visually accommodate other data points.
• Figure 5: Cumulant ratios $C_3/C_1$ (stars), $C_4/C_2$ (crosses), $C_5/C_1$ (diamonds), and $C_6/C_2$ (circles) for net-proton distributions in 0--40% centrality Au+Au collisions at $\sqrt{s_{NN}} = 7.7$--27 GeV (BES-II). The shaded bands represent predictions from FRG and UrQMD models. At 7.7 and 9.2 GeV, the $C_6/C_2$ points (open red circles) are scaled down by a factor of 2 for clarity of presentation.