Factorial cumulants of proton multiplicity near a critical point using maximum entropy freeze-out prescription

TL;DR

The paper develops a first equilibrium-based baseline for proton multiplicity fluctuations near a conjectured QCD critical point by mapping the 3D Ising universality to a parametrized QCD equation of state and applying a maximum-entropy freeze-out prescription. It identifies how factorial cumulants, when normalized, isolate critical fluctuations and depend on non-universal mapping parameters through clear scaling relations: the peak location scales with $\bar{\rho}=\rho w^{1-1/(\beta\delta)}$ and the peak height with $w^{-1-1/\delta}$, while higher cumulants obey $\Delta T_f^{1+1/\delta-k}$. The results, including the impact of freeze-out offsets $\Delta T_f$ and the role of direct versus decay protons, provide a controlled equilibrium baseline for interpreting heavy-ion data and guiding future out-of-equilibrium analyses and Bayesian constraints on the critical-point location.

Abstract

We present the first application of the maximum-entropy freeze-out prescription to calculate factorial cumulants of proton multiplicities near the conjectured QCD critical point in thermal equilibrium. We map the Gibbs free energy of the 3D Ising model to a parameterized class of possible EoS near QCD critical point. This equilibrium baseline highlights how factorial cumulants isolate critical fluctuations by subtracting trivial self-correlations, setting the stage for future out-of-equilibrium analyses. We identify the key non-universal aspects of the mapping to the Ising model that strongly control the characteristic properties, such as magnitude and location of the peaks of the factorial cumulants along the freeze-out curve.

Figures (4)

• Figure 1: The second factorial cumulant of the proton multiplicity distribution, $\hat{\Delta}\omega_{2p}$, along the three freezeout curves from Fig. \ref{['fig-0']} characterized by Eq. \ref{['eq:FreezeoutCurve']} with $\Delta T_f=4$, 6 and 9 MeV (blue dashed, red dashed and black dotted respectively). The panels show $\hat{\Delta}\omega_{2p}$ for various values of the nonuniversal mapping parameters $w$ and $\rho$ (specified on top of the figure), with $\mu_c=600 \, \text{MeV}$, $T_c=90$ MeV and $\alpha_2=0^{\circ}$. The peak height decreases as $w$ increases, consistent with the scaling $w^{-6/5}$, while the location of the peak is controlled by the quantity $\bar{\rho}=\rho w^{2/5}$.
• Figure 2: The third factorial cumulant of the proton multiplicity distribution, $\hat{\Delta}\omega_{3p}$, along the three freezeout curves from Fig. \ref{['fig-0']} characterized by Eq. \ref{['eq:FreezeoutCurve']} with $\Delta T_f=4$, 6 and 9 MeV (blue dashed, red dashed and black dotted respectively).
• Figure 3: The second, third and fourth factorial cumulants of the proton multiplicity distribution, along the freezeout curve of Eq. (\ref{['eq:FreezeoutCurve']}) with $\Delta T_f =6$ MeV. The red dashed curves show $\hat{\Delta}\omega_{4p}$, which includes contributions of direct and child protons, whereas the blue solid curves include only the direct protons, as in Figs. \ref{['fig-1']} and \ref{['fig-3']}.
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