⚛️ nuclear theory

Unveiling QCD Criticality with Cross-Rapidity Net-Baryon Cumulants

This paper proposes using cross-rapidity net-baryon cumulants to distinguish QCD critical fluctuations from global baryon-number conservation effects by exploiting their opposite correlation signs, thereby offering a refined strategy for identifying the critical endpoint in beam-energy-scan experiments.

Original authors: Jianing Li, Shuzhe Shi, Lipei Du

Published 2026-09-03

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CC BY 4.0

Original authors: Jianing Li, Shuzhe Shi, Lipei Du

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The universe as we know it is built from a vast zoo of particles, but at the very beginning, moments after the Big Bang, everything was a seething, super-hot soup of quarks and gluons. This primordial state, known as the quark-gluon plasma, existed before protons and neutrons had formed. As the universe expanded and cooled, this soup underwent a dramatic transformation, condensing into the stable matter that makes up stars, planets, and people today. Physicists have long suspected that this transition was not a simple, smooth cooling process, but rather a complex phase change, similar to how water turns into ice. At a specific point of extreme density and temperature, this change might have involved a critical endpoint, a unique spot on the map of the universe's history where the rules of matter shift in a way that creates wild, unpredictable fluctuations. Finding this spot is one of the most important goals in modern nuclear physics, as it would reveal the fundamental laws governing how matter behaves under the most extreme conditions imaginable.

To hunt for this elusive critical endpoint, scientists smash heavy atomic nuclei together at nearly the speed of light, recreating the tiny, fleeting conditions of the early universe. By varying the energy of these collisions, they can scan different regions of temperature and density, looking for signs of the critical point. The primary tool for this search is the measurement of fluctuations in the number of protons and neutrons produced in each crash. If the collision passes near the critical endpoint, the theory predicts that these numbers should wiggle in a very specific, non-random way. However, a major obstacle has always stood in the way: the law of conservation. Just as money cannot be created or destroyed in a closed system, the total number of protons and neutrons in a collision is fixed. This rule forces a trade-off; if a detector sees an extra proton in one region, it must be missing a proton somewhere else. This "conservation effect" creates a background noise that mimics or hides the very signals scientists are trying to find, making it incredibly difficult to tell if a fluctuation is due to the critical point or simply the result of this accounting rule.

A new study by Jianing Li, Shuzhe Shi, and Lipei Du proposes a clever way to cut through this confusion by looking at how fluctuations in one part of the collision relate to fluctuations in a completely different part. Instead of counting particles in a single, central zone, the researchers suggest measuring the net number of protons and neutrons in two separate, non-overlapping regions of the collision debris. They found that the two main forces at play—the conservation law and the critical point—leave opposite fingerprints on the relationship between these two regions. The conservation law acts like a strict budget; if one region has a surplus, the other must have a deficit, creating a negative connection between them. In contrast, a critical fluctuation acts like a shared wave that ripples through the entire system, causing both regions to rise or fall together, creating a positive connection.

The researchers developed a method to mathematically separate these two opposing effects. By calculating the correlation between the two distant regions and then subtracting the predictable negative effect caused by the conservation law, they can isolate the remaining signal. What is left is a clean measure of the critical fluctuations. Their simulations show that this "subtracted" signal grows stronger and behaves in a distinct, non-smooth pattern as the collision energy passes near the hypothetical critical endpoint. This pattern is directly linked to the size of the critical fluctuations, which grow larger as the system approaches the critical point. The study also reveals that the size of the detector windows matters; using narrower windows preserves the clarity of the signal, while wider windows blur the details, much like looking at a distant object through a foggy lens.

This approach offers a powerful new strategy for experiments currently underway and planned for the future, such as those at the Relativistic Heavy Ion Collider. By focusing on how different parts of the collision debris talk to each other, rather than just looking at the total count in one spot, scientists can distinguish the subtle whispers of the critical point from the loud noise of conservation. The authors emphasize that while their work is currently a theoretical proof of concept based on simulations, it provides a clear roadmap for experimentalists. They suggest that by measuring these cross-region correlations across a range of energies and window sizes, researchers can not only detect the presence of a critical point but also map out its properties with much greater precision than ever before. This method turns a long-standing problem—the difficulty of separating conservation from criticality—into a diagnostic tool, using the very structure of the collision to reveal the hidden architecture of the universe's phase transitions.

Technical Summary: Unveiling QCD Criticality with Cross-Rapidity Net-Baryon Cumulants

Problem Statement
The search for the QCD critical endpoint (CEP) relies heavily on measuring event-by-event fluctuations of conserved charges, such as net-proton cumulants, as a function of collision energy ( $\sqrt{s_{NN}}$ ). However, interpreting the beam-energy dependence of these measurements is complicated by global baryon-number conservation. As collision energy decreases, the experimental acceptance window covers an increasingly large fraction of the total net-baryon number in the system. Global conservation imposes a constraint that suppresses fluctuations within the measured acceptance and generates negative correlations with the unobserved part of the system. This conservation-induced background is non-trivial and changes across the beam-energy scan, potentially mimicking or obscuring the nonmonotonic signatures expected from critical dynamics near the CEP. Current analyses struggle to disentangle these changing noncritical contributions from a genuine critical signal.

Methodology
The authors propose a strategy utilizing cross-rapidity net-baryon cumulants, specifically the covariance ( $\kappa_{1,1}$ ) of net-baryon yields measured in two non-overlapping rapidity windows ( $A$ and $B$ ). The theoretical framework employs a canonical independent-source model coupled to a fluctuating critical mode:

Nested Statistical Averages: The system is treated as a closed system with fixed total net-baryon number $B$ . The calculation involves two nested averages: an inner canonical average over baryon degrees of freedom for a fixed critical field configuration ( $\sigma$ ), and an outer average over the event-by-event fluctuations of the critical field $\sigma$ .
Exact Decomposition: Using the law of total covariance, the measured cross-rapidity covariance is decomposed into two distinct terms:
- A conservation-induced term ( $\langle \kappa^{ce}_{1,1}[\sigma] \rangle_\sigma$ ): Arising from the fixed total baryon number constraint at a fixed field configuration. This term represents the redistribution of a fixed charge among disjoint acceptances.
- A critical-field term ( $C^\sigma_{1,1}$ ): Arising from the event-by-event fluctuations of the mean yields driven by the long-wavelength fluctuations of the critical order parameter $\sigma$ .
Sign Distinction: The authors demonstrate that these two mechanisms produce qualitatively opposite signs. Global conservation generates a negative correlation (anticorrelation) between the windows. Conversely, a common long-wavelength critical fluctuation modulates the local baryon density coherently, leading to a positive correlation between the windows.
Conservation-Subtracted Correlator: Based on this sign contrast, the authors define a yield-scaled, conservation-subtracted correlator:
$S_{1,1} \equiv \frac{1}{\bar{B}_1 \bar{B}_2} \left( \kappa_{1,1} - \kappa^{ce}_{1,1}[0] \right)$
where $\kappa^{ce}_{1,1}[0]$ is the leading-order conservation contribution estimated from the mean yields in the full phase space and the acceptance windows. This subtraction aims to isolate the critical contribution.

Key Results

Additive Structure: Within the canonical independent-source framework, the conservation background and the critical signal enter as separate additive terms at leading order. This allows the leading conservation term to be estimated and subtracted.
Analytic Formulation: The authors derive an analytic expression for $S_{1,1}$ assuming a cylindrical freeze-out hypersurface and narrow adjacent rapidity windows. The result shows that $S_{1,1}$ is proportional to the square of the equilibrium correlation length ( $\xi_{eq}^2$ ) modified by geometric and thermal factors involving modified Struve functions.
Equilibrium Response: In a proof-of-principle calculation mapping the 3D Ising model to the QCD phase diagram, the subtracted correlator $S_{1,1}$ exhibits a nonmonotonic enhancement along a freeze-out trajectory passing near a hypothetical CEP. This enhancement closely tracks the behavior of the input equilibrium correlation length $\xi_{eq}$ .
Rapidity Window Dependence: The study finds that narrower rapidity windows preserve a larger normalized critical signal. Wider windows introduce thermal smearing, where particles in a momentum-rapidity bin originate from a broader range of source rapidities, diluting the overlap of correlated source regions and reducing the scaled response.

Significance and Claims
The paper claims that cross-rapidity cumulants provide a rapidity-differential strategy to reduce the leading conservation background and sharpen fluctuation-based searches for QCD criticality.

Diagnostic Power: Unlike single-window cumulants, the cross-rapidity observable retains information on how fluctuations are organized across rapidity. The distinct signs of conservation (negative) and criticality (positive) offer a mechanism to separate the two.
Experimental Utility: The proposed observable is not presented as a standalone critical signature but as a diagnostic tool. By measuring the correlator as a function of beam energy, window width, and window separation, experiments can determine not only the presence of an excess correlation but also its rapidity extent. This pattern could help distinguish long-wavelength critical modes from other noncritical effects like baryon stopping, volume fluctuations, and hadronic transport.
Future Directions: The authors state that quantitative interpretation requires realistic noncritical baselines, detector effects, and a controlled mapping from conserved net baryons to measured net protons. They propose that the next step involves embedding this observable in dynamical multistage simulations with nonequilibrium critical evolution to optimize rapidity acceptances for future beam-energy scan programs at facilities such as RHIC, NA61/SHINE, CBM, HIAF, and NICA.

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