Probing the QCD Critical End Point with Finite-Size Scaling of Net-Baryon Cumulant Ratios

Finite-size scaling (FSS) is applied to net-baryon cumulant ratios $C_2/C_1$, $C_3/C_2$, $C_4/C_2$, $C_3/C_1$, and $C_4/C_1$ measured in Au+Au collisions over the Beam Energy Scan Phase~I range $\sqrt{s_{NN}}=7.7$–$200$~GeV to constrain the location and universality class of the QCD critical end point (CEP). Although finite-size and finite-time effects suppress non-monotonic signatures in unscaled data, the FSS analysis reveals a collapse of measurements from different beam energies and centralities onto universal scaling functions. All cumulant ratios collapse under a single, common set of critical exponents and exhibit divergence patterns characteristic o 3D Ising critical behavior. The scaling results indicate a CEP at $\sqrt{s}{\rm CEP}\approx33.0$~GeV, corresponding to $μ{B,\rm CEP}\approx130$~MeV and $T_{\rm CEP}\approx158.5$~MeV. These findings demonstrate that finite-size scaling provides a robust, model-independent framework for accessing critical behavior in finite, dynamically evolving systems, where non-equilibrium baryon-number transport can enhance the experimental visibility of susceptibility-driven fluctuations without modifying the underlying universality class.

💡 Research Summary

This paper presents a comprehensive analysis aimed at locating and characterizing the QCD Critical End Point (CEP) using experimental data from relativistic heavy-ion collisions. The core challenge in this search is the finite size and short lifetime of the fireball created in collisions, which suppresses the divergent correlation length characteristic of equilibrium critical phenomena, thereby masking potential non-monotonic signatures. To overcome this, the study employs the powerful framework of Finite-Size Scaling (FSS).

The analysis uses net-baryon cumulant ratios—specifically C2/C1, C3/C2, C4/C2, C3/C1, and C4/C1—measured in Au+Au collisions across the full Beam Energy Scan Phase I range at RHIC (√s_NN = 7.7 to 200 GeV) and for various centralities. A characteristic system size (L), proportional to the transverse extent of the collision overlap region, is calculated for each energy and centrality using Monte Carlo Glauber simulations. The fundamental premise of FSS is that near a critical point, these cumulant ratios should follow scaling relations governed by the universality class of the phase transition. The paper assumes the expected 3D Ising universality class, using its critical exponents (ν, γ, β, Δ).

Since the theoretical reduced temperature (t) and field (h) variables are not directly accessible, the analysis constructs two complementary experimental scaling variables. The first is a “field-driven” variable, h_√s, built using the inverse baryon chemical potential (1/μ_B), which is linked to √s via established chemical freeze-out parameterizations. This variable is sensitive to fluctuations driven by the baryon chemical potential. The second is a “density-driven” variable, t_√s, constructed directly from the beam energy (√s - √s_CEP), reflecting changes in baryon density.

The raw, unscaled cumulant ratios show no pronounced non-monotonic behavior as a function of beam energy, consistent with the expected suppression from finite-size and finite-time effects. However, when the data are rescaled according to the FSS formulas and plotted against the scaled variables h_√s * L^(Δ/ν) or t_√s * L^(1/ν), a remarkable collapse is observed. Measurements from different beam energies and centralities (i.e., different system sizes L) fall onto single, universal scaling curves for each cumulant ratio. This collapse only occurs successfully for a specific set of parameters: the 3D Ising critical exponents and a CEP located at a beam energy of √s_CEP ≈ 33.0 GeV. Using the freeze-out mapping, this corresponds to a baryon chemical potential μ_B,CEP ≈ 130 MeV and a temperature T_CEP ≈ 158.5 MeV. The fact that both the field-driven and density-driven scaling trajectories collapse with this same parameter set provides a stringent internal consistency check.

The paper also addresses the role of non-equilibrium baryon number transport mechanisms, such as baryon junctions, which can enhance cumulant ratios at lower beam energies. Crucially, the FSS framework provides a means to distinguish such non-critical contributions from genuine critical behavior. Non-critical effects are not required to exhibit the universal scaling with system size that is the hallmark of critical phenomena. The successful collapse observed across multiple independent observables strongly indicates the presence of underlying 3D Ising critical dynamics.

In conclusion, this work demonstrates that Finite-Size Scaling provides a robust, model-independent methodology for probing critical behavior in the finite, dynamically evolving systems created in heavy-ion collisions. By turning system-size limitations into an analytical tool, FSS simultaneously constrains both the location and the universality class of the QCD Critical End Point, offering compelling evidence for its existence at √s_CEP ≈ 33 GeV.