Exploring the QCD phase diagram through correlations and fluctuations

TL;DR

The paper addresses locating the QCD critical point and mapping the phase diagram of strongly interacting matter by leveraging fluctuations of conserved charges in heavy-ion collisions. It integrates theoretical predictions from lattice QCD, functional methods, and holography with experimental fluctuation measurements (notably cumulants of net-proton and net-charge) and emphasizes non-critical baselines to isolate potential critical signals. Theoretical efforts consistently place the critical point in a narrow region near $T_C \sim 100-120\,\mathrm{MeV}$ and $\mu_C \sim 550-650\,\mathrm{MeV}$, while high-energy data mostly align with non-critical expectations but low-energy BES data show intriguing non-monotonic behavior that may signal critical dynamics or volume effects. Robust CP searches thus require careful handling of global conservation, acceptance, and volume fluctuations, with upcoming CBM measurements at FAIR poised to test the predicted region.

Abstract

The exploration of the Quantum Chromodynamics (QCD) phase diagram is a central goal of relativistic heavy-ion collision experiments. This review focuses on the role of fluctuations and correlations as sensitive probes of the phase structure. We discuss theoretical advancements and experimental methodologies employed to map the QCD phase diagram, highlighting constraints derived from both lattice QCD calculations and existing experimental data. Key observables such as cumulants and factorial cumulants of conserved charges (e.g., net-proton, net-charge) are explored as promising signatures of phase transitions and the QCD critical point. We discuss how these quantities are measured experimentally and compared with theoretical predictions, addressing challenges and best practices for meaningful comparisons. Special attention is given to predictions and current experimental results at high baryon density, including recent findings from the STAR collaboration at RHIC. Finally, we identify open issues and future directions for fluctuation and correlation studies at lower collision energies, relevant for future measurements, for example by the CBM experiment.

Figures (7)

• Figure 1: Based on Shah:2024img. A compilation of predictions for the location of the QCD critical point on the $T$-$\mu_B$ phase diagram of QCD. The black point with a red covariance ellipse shows the estimate from Ref. Shah:2024img, based on the extrapolation of constant entropy density contours from $\mu_B = 0$. The stars depict estimates from other approaches, functional methods (fRG Fu:2019hdw, DSE-fRG Gao:2020fbl, DSE Gunkel:2021oya) and holography (BHE Hippert:2023bel, VQCD Ecker:2025vnb). The orange line represents the chemical freeze-out estimate from Ref. Lysenko:2024hqp, with points on the line corresponding to various collision energies (in terms of $\sqrt{s}_{\rm NN}$ in GeV). Both the chemical freeze-out line and all CP estimates [except DSE ($\mu_S = 0$) and VQCD ($\beta$-equilibrium)] correspond to $\mu_S = \mu_B / 3$ ($n_S \neq 0$) conditions. The dashed green line depicts the chiral crossover line from Borsanyi:2020fev.
• Figure 2: Cumulant ratio $\kappa_{4}/\kappa_{2}$ (upper panel) and $\kappa_{6}/\kappa_{2}$ (lower panel) as a function of the acceptance window in rapidity, $\Delta Y$, for a system created in heavy-ion collisions at the LHC (left panel) and RHIC-BES collider energies (right panel). Left panel: The horizontal gray lines represent the result from lattice QCD calculations for the net baryons Borsanyi:2018grbBazavov:2017dus. The black dashed lines show the effect of global charge conservation while the red lines also include thermal smearing. The blue points are the results for the net-proton cumulant ratio, again with charge conservation and thermal smearing included. The blue diamonds are the results for net-proton cumulants using the method of Kitazawa:2011whKitazawa:2012at. For details see Vovchenko:2020kwg, where this figure is adapted from. Right panel: Hydro-EV model calculations from Vovchenko:2021kxx depicting cumulant ratios of (i) net baryons in the grand canonical ensemble without momentum cuts (dash-dotted black line), (ii) the same but for net protons (dashed blue line), (iii) net protons with momentum cuts (dashed magenta line), and (iv) net protons with momentum cuts and baryon number conservation effects included (solid red line).
• Figure 3: Corrected scaled variance $\tilde{\omega}_{y}$ of particle number in rapidity acceptance as a function of the fixed acceptance fraction $\alpha_y$, which is the ratio of accepted to total number of particles. Calculations are performed for a system of $N = 400$ particles at $T = 1.06 T_c$ and $n = 0.95 n_c$ Different bands correspond to different magnitudes of the collective flow corresponding to the collision energies in a Bjorken picture. The limiting cases of coordinate, red band, labeled $\tilde{\omega}_{\rm coord}$, and rapidity acceptance, black line, labeled $\tilde{\omega}_y (y_{\rm cm} = 0)$, in the absence of collective expansion are also shown. For details see Kuznietsov:2024xyn where this figure is adapted from.
• Figure 4: Cumulants (top row) and factorial cumulants (bottom row) obtained by the STAR collaboration from the second phase of the RHIC beam energy scan STAR:2025zdq. Note that, contrary to common practice, STAR uses $C_{n}$ to denote cumulants and $\kappa_{n}$ to denote factorial cumulants. Also shown are the baselines of Vovchenko:2021kxx (blue dashed line), Braun-Munzinger:2020jbk (dotted black line) as well as UrQMD calculations by STAR (brown band). Also shown are lattice QCD results for net baryons uncorrected for global baryon number conservation Bazavov:2020bjn. Figure adapted from STAR:2025zdq
• Figure 5: Energy dependence of cumulants (left panel) and factorial cumulants (right panel) obtained from UrQMD simulations Zhang:2025ale for different rapidity acceptance windows and for fixed impact parameter (blue crosses) and centrality selection a la STAR (red square and black filled circle). Figure adapted from Zhang:2025ale.