Extraction of baryon number susceptibilities at finite density from heavy-ion collisions

Technical Summary: Bayesian Extraction of Baryon Number Susceptibilities from Heavy-Ion Collisions

Problem Statement
Fluctuations of conserved charges, specifically baryon number, serve as sensitive probes of the Quantum Chromodynamics (QCD) phase structure, particularly near a conjectured critical point (CP). While first-principles lattice QCD calculations accurately determine baryon number susceptibilities ($\chi_B^n$) at zero baryon chemical potential ($\mu_B = 0$), their behavior at finite density relies on extrapolations or alternative theoretical approaches. Experimentally, the Relativistic Heavy Ion Collider (RHIC) Beam Energy Scan (BES) program measures event-by-event fluctuations of net-proton numbers in Au-Au collisions to probe this regime. However, a quantitative interpretation of these data is complicated by several non-equilibrium and experimental effects: measured protons carry only a fraction of the total baryon number, experimental acceptance is limited in momentum space, and exact global baryon number conservation non-trivially modifies fluctuations relative to grand-canonical expectations. Previous strategies have typically confronted data with specific model calculations (e.g., Hadron Resonance Gas, hydrodynamics with excluded volume, or transport models), rather than directly inferring the underlying susceptibilities.

Methodology
This work presents, to the authors' knowledge, the first Bayesian extraction of baryon number susceptibilities ($\chi_B^2, \chi_B^3, \chi_B^4$) directly from heavy-ion collision data. The framework constructs a forward map from theoretical susceptibilities to measured proton cumulants through four distinct steps:

1. Hydrodynamic Background: Event-averaged hydrodynamic hypersurfaces are generated using the MUSIC code with 3D initial conditions and particlization at a constant energy density ($\epsilon_{sw} = 0.26$ GeV/fm$^3$).
2. Maximum-Entropy Freeze-out (MaxEnt): Local baryon susceptibilities on the hypersurface are converted into joint baryon-antibaryon cumulants. The framework parametrizes the local susceptibility input as ratios relative to an ideal Hadron Resonance Gas (iHRG) baseline: $\gamma = (\chi_B^2/\bar{\chi}_B^2, \chi_B^3/\chi_B^1, \chi_B^4/\chi_B^2)$. The MaxEnt prescription maps hydrodynamic fields and their fluctuations onto hadron multiplicity irreducible relative cumulants.
3. Acceptance and Filtering: Cooper-Frye acceptance and baryon-to-proton filtering map the joint cumulants onto accepted protons and antiprotons via independent binomial sampling, accounting for the STAR detector's kinematic acceptance.
4. Exact Conservation: The Subensemble Acceptance Method (SAM-3.0) imposes exact global baryon number conservation, converting grand-canonical cumulants into canonical cumulants suitable for comparison with experimental data.

Bayesian inference is performed on STAR data from 0–5% central Au-Au collisions at specific collision energies within the BES-II range of $\sqrt{s_{NN}} = 7.7–27$ GeV (specifically 7.7, 9.2, 11.5, 14.6, 17.3, 19.6, and 27 GeV). Two complementary extractions are conducted using (i) proton factorial cumulant ratios ($\hat{C}_n/\hat{C}_1$) and (ii) net-proton cumulant ratios ($\kappa_n/\kappa_1$). Flat priors are applied to the susceptibility ratios, and a Gaussian likelihood is constructed using statistical and systematic uncertainties.

Key Results

• Second-Order Susceptibility ($\chi_B^2$): The analysis yields tight constraints on the second-order susceptibility normalized by the iHRG value ($\chi_B^2/\bar{\chi}_B^2$), with relative uncertainties of 5–10%. The extracted values exhibit a non-monotonic collision-energy dependence across the measured energies: an enhancement ($\sim 1.06–1.07$) at the lowest energy ($\sqrt{s_{NN}} = 7.7$ GeV), a minimum ($\sim 0.73–0.82$) at intermediate energies ($11.5–19.6$ GeV), and a flat suppression ($\sim 0.8–0.9$) at higher energies.
• Comparison with Lattice QCD: When mapped to the chemical freeze-out line, the extracted $\chi_B^2$ values show quantitative agreement with lattice QCD-based estimates (specifically the 4D-TExS equation of state) for $\mu_B \lesssim 300$ MeV. However, at larger $\mu_B$ (corresponding to $\sqrt{s_{NN}} = 7.7$ GeV), the extracted values deviate upward from the lattice band, showing an enhancement of up to $\sim 50\%$.
• Higher-Order Susceptibilities: The third- and fourth-order susceptibilities ($\chi_B^3, \chi_B^4$) are only weakly constrained due to efficiency losses in the mapping from baryons to protons and distortions from baryon conservation. While the fourth-order ratio shows a local preference for negative values at $\sqrt{s_{NN}} = 19.6$ GeV ($\chi_B^4/\chi_B^2 = -33 \pm 14$), the large uncertainties preclude definitive conclusions about the QCD phase structure from these higher-order moments alone.
• Correlation Dominance: A minimal scenario assuming only irreducible two-baryon correlations (setting $\Delta\chi_B^3 = \Delta\chi_B^4 = 0$) successfully reproduces the observed non-monotonic peak in the proton factorial cumulant ratio $\hat{C}_3/\hat{C}_1$ near $\sqrt{s_{NN}} \approx 11$ GeV. This suggests the observed structure arises from the interplay between the energy dependence of $\chi_B^2$ and exact baryon number conservation, rather than requiring irreducible three- or four-baryon correlations.

Significance and Claims
The paper claims to provide the first data-driven, Bayesian determination of baryon number susceptibilities along the chemical freeze-out line, avoiding assumptions about a specific Equation of State (EoS). The authors assert that their framework successfully isolates the second-order susceptibility, finding it consistent with lattice QCD at lower baryon densities but indicating a potential enhancement of baryon number fluctuations at higher densities ($\mu_B \gtrsim 300$ MeV) relative to non-critical extrapolations.

The study modestly concludes that while the data can be largely described without irreducible multi-baryon correlations beyond the second order, this does not rule out their existence; rather, current measurements are primarily sensitive to the two-baryon contribution. The authors note that the observed enhancement at low collision energies warrants scrutiny with future fixed-target data and that the current extraction is conditional on the fixed hydrodynamic and freeze-out setup, representing a lower-bound estimate for uncertainties in a global analysis. The framework is presented as a general tool for constraining the finite-density QCD EoS and can be extended to include fixed-target energies, volume fluctuation corrections, and other conserved charges once modeling uncertainties are better controlled.