Model Study of Eigen-Microstate Signatures of Criticality in Relativistic Heavy-Ion Collisions

TL;DR

The paper addresses locating the QCD critical point in relativistic heavy-ion collisions amidst sizeable non-critical backgrounds. It introduces the Eigen-Microstate Approach (EMA), which diagonalizes the event-event correlation matrix to yield eigen-microstates and weights, with the leading weight $w_1=\lambda_1$ acting as an order-parameter-like indicator. Through analyses of non-critical baselines (UrQMD and stochastic models) and hybrid UrQMD+CMC samples, the study shows that non-critical dynamics do not produce cluster-like EM patterns, while embedding critical fluctuations leads to robust cluster structures and a growing $w_1$, accompanied by fixed-point-like convergence of eigenvalue ratios such as $w_3/w_1$ and scale-invariant EM patterns. These findings demonstrate that EMA provides a robust, background-filtering method for critical-point searches in RHIC BES II and future facilities, with guidance on event-level versus particle-level signatures and finite-size scaling expectations that enhance practical applicability.

Abstract

We present a comprehensive model study of the eigen-microstate approach (EMA) for identifying critical fluctuations in relativistic heavy-ion collisions. Using UrQMD and two stochastic baseline models, we demonstrate that EMA is insensitive to conventional short-range correlations and effectively filters out non-critical backgrounds. Critical fluctuations embedded via event-level or particle-level replacement with CMC events generate characteristic cluster-like eigen-microstate patterns and enhanced leading eigenvalues, with event-level criticality producing stronger responses. The eigen microstates exhibit the same pattern across different scales, demonstrating that the fractal nature of critical fluctuations is captured by the eigen microstates. Finite-size scaling of eigenvalue ratios exhibits fixed-point behavior, confirming the largest eigenvalue as an effective order-parameter-like quantity. These results demonstrate that EMA offers a robust and background-independent method for critical-point searches in the RHIC Beam Energy Scan and future heavy-ion experiments.

Figures (6)

• Figure 1: (Color online) First three EMs (EM$_1$, EM$_2$, EM$_3$) for UrQMD and for the two stochastic non-critical ensembles: Stoch. II and Stoch. I with and without kinetic constraints.
• Figure 2: (Color online) Cumulative weight $C(m)$ for the UrQMD ensemble, the stochastic model II and I with and without kinetic constraints.
• Figure 3: (Color online) First three EMs for hybrid UrQMD+CMC ensembles constructed with event-level replacement (left columns) and particle-level replacement (right columns) for several critical fractions $\alpha_{\rm e}$ and $\alpha_{\rm p}$.
• Figure 4: (Color online) Top three eigenvalues $w_1$, $w_2$, $w_3$ as functions of the critical fractions $\alpha_{\rm e}$ (event-level replacement) and $\alpha_{\rm p}$ (particle-level replacement), together with the corresponding cumulative weight $C(m)$.
• Figure 5: (Color online) First three EMs at fixed $\alpha_{\rm e} = 4\%$ for three scales $L=10$, $40$, and $100$.