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

Roy A. Lacey

Abstract

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.

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

Abstract

Finite-size scaling (FSS) is applied to net-baryon cumulant ratios , , , , and measured in Au+Au collisions over the Beam Energy Scan Phase~I range --~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 ~GeV, corresponding to ~MeV and ~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.

Paper Structure

This paper contains 3 equations, 4 figures.

Figures (4)

  • Figure 1: (Color online) Relationship between $1/\mu_B$ and the beam energy $\sqrt{s}$. The values of $\mu_B$ are obtained from empirical chemical freeze-out parametrizations Cleymans:2005xvAndronic:2009gj. Over the Beam Energy Scan range, $1/\mu_B$ exhibits an approximately linear dependence on $\sqrt{s}$, motivating its use as a practical proxy for field-driven scaling analyses.
  • Figure 2: (Color online) Illustration of the finite-size scaling procedure. Panel (a) shows the beam-energy dependence of the unscaled cumulant ratio $C_2(\mathrm{cent})/C_1(\mathrm{cent})$ for Au+Au collisions across the indicated centrality intervals. Panels (b) and (c) show the corresponding field-driven and density-driven finite-size scaling functions. The collapse of data from different system sizes in the scaled representations reveals the upward divergence expected for $C_2/C_1$, consistent with a compressibility-related observable in the vicinity of the CEP.
  • Figure 3: (Color online) Same as Fig. 2, but for the cumulant ratio $C_3(\mathrm{cent})/C_2(\mathrm{cent})$. Panel (a) shows the unscaled beam-energy dependence across centralities, while panels (b) and (c) display the corresponding field-driven and density-driven scaling functions. The scaled data exhibit the downward divergence expected for $C_3/C_2$, consistent with the emergence of skewness near the CEP.
  • Figure 4: (Color online) Field-driven finite-size scaling functions for the cumulant ratios $C_4(\mathrm{cent})/C_2(\mathrm{cent})$ (a), $C_3(\mathrm{cent})/C_1(\mathrm{cent})$ (b), and $C_4(\mathrm{cent})/C_1(\mathrm{cent})$ (c). The scaled data exhibit distinct divergence patterns—downward for $C_4/C_2$ and $C_3/C_1$, and upward for $C_4/C_1$—consistent with expectations for 3D Ising-like critical behavior in the vicinity of the CEP.