QCD phase diagram at finite density

• : crossover established
• Finite : extrapolation

Contours of constant entropy density

• Contours cross: multi-valued → first-order transition
• Spinodal region between the touching contours
• Onset of crossing = critical point:

Lattice input and effective-theory tests

• ✅ Effective theories (holographic BH model, fRG–DSE): extrapolated CP is within 10% of the true one [Shah et al., PRD 114, 034501 (2026)]
• ⚠️ Ideal HRG: spurious CP at (80, 830) MeV from the extrapolation [Marczenko et al., PRD 112, 034019 (2025)]
• ⚠️ The critical structure is conditional on the contour expansion (truncation uncertainty not included)

A four-dimensional QCD EoS from entropy contours

• diverges
• changes sign
• : peak and dip

Candidate critical region

Equilibrium expectations along the freeze-out line

• Strong non-monotonic below GeV, where freeze-out passes closest to the CP
• Similar non-monotonic behavior in functional QCD: fRG [Fu et al., PRD 111, L031502 (2025), Zhao et al., arXiv:2607.26912] and DSE [Lu, Gao, Liu, Pawlowski, PRD 113, 054019 (2026)]

What RHIC measures: net-proton cumulants

Factorial cumulants show more structure

Theory vs experiment

hydro background

event selection

baryons → protons

track reconstruction

acceptance

efficiency correction

conservation

purity

CBWC

MEME: the complete forward framework

Proton cumulants are baryon susceptibilities!

Protons, acceptance, conservation. Try MEME.

Maximum entropy freeze-out

Baryon-to-proton mapping and kinematic acceptance

• MaxEnt: → baryons and antibaryons
• Baryons → protons: isospin randomization and decays, proton fraction from the ideal HRG [Kitazawa, Asakawa, PRC 85, 021901 (2012)]
• Acceptance: probability that a particle from cell lands in , GeV/, from the Cooper–Frye formula (thermal emission from the moving cell) [VV, Koch, Shen, PRC 105, 014904 (2022)]

SAM-3.0: exact global baryon conservation

• In this pipeline: global conservation only, without , or local conservation
• Large-system (saddle-point) result, checked against exact sampling

From the forward model to Bayesian inference

One energy: what does the posterior constrain?

Main result: second-order baryon susceptibility

• Tight constraints: 3–9% on (68% intervals, for fixed hydro and freeze-out setup). The proton and net-proton fits agree at every energy, the latter retains more statistical precision
• Energy dependence: below the HRG reference from 11.5 to 200 GeV, then a pronounced rise toward 7.7 GeV, with central values slightly above it ()

Interpreting the second-order response

• 11.5–200 GeV: suppression consistent with repulsive baryon interactions, as in Hydro EV [VV, Koch, Shen, PRC 105, 014904 (2022)]
• 7.7 GeV: a pronounced rise, slightly above the reference. Onset of attraction at large ? Possible connection to criticality remains to be established. [see also Friman, Redlich, Rustamov, JHEP 07, 020 (2026)]

Absolute vs lattice-based expectations

• Absolute scale depends on the hadron list
• Fluctuations frozen at chemical freeze-out
• 4D-TExS: truncation uncertainty not included
• No volume, initial-state or stopping fluctuations

Higher orders

• Large uncertainties at higher orders: baryons → protons and acceptance dilute higher-order signals, similar to an efficiency loss. Exact baryon conservation adds correlations and mixes the orders.
• Largest deviation: at 19.6 GeV (net-), a local preference for negative values
• Both the Skellam baseline and a critical-point-like non-monotonic remain possible within the uncertainties

Nonmonotonic from two-baryon correlations

• Grand canonical:
• Exact conservation alone (Skellam): monotonic
• Adding the energy dependence of : the peak near 11 GeV

Scope of the present inference

• Error bars: conditional on fixed hydro and freeze-out, without model systematics, so likely smaller than in a future global analysis
• Checked: GeV/fm leaves the net-proton extraction stable. Other model uncertainties remain to be explored.

Back to entropy contour susceptibilities

• : agreement at GeV. At 7.7 and 9.2 GeV the extracted values lie above the EoS.
• : consistent within the large uncertainties. : extraction errors too large to compare.
• Deviations at lowest energies: truncation of the EoS, or dynamics missing in the extraction (memory, stopping)?
• Next target: the non-monotonic below 7.7 GeV

Extending the calculation to lower collision energies

• New hydro runs down to GeV (●)
• Particlization at lower energy density for lower energies: 0.35 (7.7–14.5 GeV), 0.25 (6.2), 0.22 (5.2), 0.18 GeV/fm (3.9, 4.5)
  • so that the particlization points (●) lie close to the chemical freeze-out curve (◇)
• Energy labels shifted: each hydro point sits near the freeze-out point of the next higher energy (e.g. hydro at 14.5 GeV near freeze-out at 19.6 GeV)

Preliminary: factorial cumulants from a forward pipeline

• inputs: ideal HRG and constant entropy contours (with CP)
• : the curves separate below 20 GeV. The entropy-contour EoS rises and peaks at 9.2 GeV, and ideal HRG stays monotonic.
• Model uncertainties still to be added

Mean- correlations: a complementary probe

• Mean- fluctuations may reflect event-by-event temperature fluctuations [Stodolsky, PRL 75, 1044 (1995)]
• STAR 0–5%: non-monotonic (), minimum near 7.7 GeV

• DynFlu (dynamic fluidization): a sizable effect [Góes-Hirayama et al., PRC 113, 044906 (2026)]
• Our calculations show it can constrain the dynamics at lower energies, which may also help for cumulants

Conclusions

• The entropy-contour construction now gives a full 4D QCD EoS with a critical surface, predictions under heavy-ion charge constraints
• MEME pipeline connects susceptibilities to measured proton fluctuations. The RHIC Bayesian analysis constrains the second-order baryon response most strongly: suppressed over most of the collider range, then a pronounced rise to a slight excess at 7.7 GeV.
• Together with baryon conservation, this energy dependence of also describes the non-monotonic . Higher-order susceptibility constraints remain weak.

• Pihan, Poberezhniuk, Kuznietsov, VV, arXiv:2608.10238 (Bayesian extraction)
• Poberezhniuk, Kuznietsov, Pihan, VV, JHEP 10 (2026) 080 (SAM-3.0)
• Gyure, Shah, Formaggio, Di Clemente, Hippert, Ratti, VV, arXiv:2610.20806 (4D EoS)

Heavy-ion slice: and