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Topological analysis of scale-invariant spatial fluctuations in ultrarelativistic heavy-ion collisions

This paper presents a novel two-stage topological machine learning framework that combines Topological Data Analysis with deep learning to successfully isolate weak critical fluctuation signals from overwhelming non-critical backgrounds in ultrarelativistic heavy-ion collisions, thereby restoring power-law scaling and enabling the reliable identification of the QCD critical point.

Original authors: Salman Khurshid Malik, Ramni Gupta, Fakhar Ul Haider, Balwan Singh

Published 2026-08-11

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CC BY 4.0

Original authors: Salman Khurshid Malik, Ramni Gupta, Fakhar Ul Haider, Balwan Singh

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 Cosmic Soup and the Search for a Hidden Recipe

Imagine the universe just a fraction of a second after the Big Bang. It wasn't the cold, empty space we see today, but a super-hot, super-dense soup where the basic building blocks of matter—quarks and gluons—were swimming freely, unbound by the usual rules. This state of matter is called the Quark-Gluon Plasma (QGP). Scientists today try to recreate this ancient soup by smashing heavy atomic nuclei together at nearly the speed of light in massive machines called particle colliders. When these collisions happen, the QGP forms for a tiny instant before cooling down and turning back into the ordinary particles (like protons and neutrons) that make up our world.

The big question is: how does this transformation happen? Does it happen smoothly, like ice melting into water, or is there a specific "tipping point" or critical moment where the rules change dramatically? This is known as the QCD critical point. To find it, scientists look for tiny, chaotic ripples in the way particles fly out after a collision. If the universe hit that critical tipping point, these ripples would look like a specific, self-similar pattern (like a fractal snowflake) across different sizes. However, finding these patterns is incredibly hard because the signal is like a whisper in a hurricane; the "whisper" of the critical point is drowned out by the "roar" of billions of ordinary particles produced in the crash.

Hunting for a Ghost in the Machine

This paper is a story about how a team of physicists decided to use a new kind of detective work to find that whisper. They didn't just listen harder; they built a machine that could "see" the shape of the chaos.

The Problem: The Needle in a Haystack
Imagine you are trying to find a few specific, glowing marbles hidden inside a giant bucket filled with millions of dull, grey pebbles. If you just dump the whole bucket onto a table and count the marbles, you might miss them because the grey pebbles are so overwhelming. In the world of heavy-ion collisions, the "grey pebbles" are the normal particles produced by the collision (simulated by a model called EPOS), and the "glowing marbles" are the rare particles that might be showing signs of the critical point (simulated by a model called CMC).

The researchers found that even if they injected a "signal" of critical particles into their simulations, it was so weak—constituting only about 1% to 5% of the total particles—that traditional methods of counting and measuring failed. The signal was completely washed out by the background noise. It was like trying to hear a single violin in a stadium full of cheering fans.

The Solution: Topological Machine Learning
To solve this, the authors used a clever two-step strategy involving Topological Data Analysis (TDA) and Machine Learning. Think of TDA as a way to look at the "shape" of the data rather than just the numbers. Instead of looking at individual particles, they treated the collision as a cloud of points and asked: "What does the shape of this cloud look like?"

Step One: The Shape Shifter (Event Classification)
First, they turned the particle tracks into a 2D map (using coordinates called $\eta$ and $\varphi$ ). They then used a mathematical tool called Delaunay triangulation to connect the dots, creating a web of triangles. As they zoomed in and out on this web, they tracked how the "holes" and "loops" in the shape appeared and disappeared. This created a unique "fingerprint" called a Betti curve.

They fed these fingerprints into two different types of AI "brains": a TopoPointNet (a type of deep learning network) and a Boosted Decision Tree (BDT). These AIs learned to spot the subtle difference between the messy, random shape of the background noise and the tight, clustered shape of the critical signal. The result? The AIs became incredibly good at spotting the signal. When the signal made up 5% of the mix, the AI could identify the right events with 99% accuracy, effectively filtering out the "grey pebbles" to find the bucket containing the "glowing marbles."
Step Two: The Particle Filter (Density Filter)
However, the researchers realized that just picking the right bucket wasn't enough. Even inside the "signal" bucket, most of the marbles were still the dull grey pebbles. The critical particles were still a tiny minority. So, they added a second step: a particle-level density filter.

They looked at the distance between every single particle. The critical particles were packed tightly together (like a dense cluster), while the background particles were spread out. By keeping only the particles that were very close to their neighbors (within a distance of 0.02 radians), they stripped away the diffuse background. It was like using a sieve to remove all the loose sand, leaving only the tightly packed gems.

The Discovery: Restoring the Pattern
When they applied this two-stage filter to their simulated data, something magical happened. The chaotic, flat lines that traditional methods produced suddenly transformed into a clear, straight line on a graph. This line showed a power-law scaling, which is the mathematical signature of a critical point.

Without the filter, the "intermittency index" (a number that measures how strong the pattern is) was close to zero, meaning no pattern was found.
With the filter, the index recovered to $\phi_2 \approx 0.73$ , which is very close to the theoretical prediction for a critical system ( $\approx 0.75$ in their specific simulation setup).

What This Means
The paper demonstrates that by combining topological shape analysis with machine learning, it is possible to extract a critical signal that is 50 times weaker than the background noise. The authors explicitly state that traditional methods are insufficient for this task because they get lost in the noise. Their new method, however, successfully "restores" the signal, proving that the critical geometry was there all along, just hidden.

It is important to note that this is a simulation. The researchers used computer models (EPOS and CMC) to generate the data, not real collision data from a particle accelerator yet. While the results are highly promising and suggest this method could work in real life, the authors caution that real-world detectors have their own quirks (like tracks merging together) that need to be tested before this technique can be used to find the actual QCD critical point in nature.

In short, this paper shows that if you want to find a ghost in a crowded room, you shouldn't just listen for a whisper; you should build a machine that recognizes the specific shape of the ghost's shadow.

Technical Summary: Topological Analysis of Scale-Invariant Spatial Fluctuations in Heavy-Ion Collisions

Problem Statement
The identification of the Quark-Gluon Plasma (QGP) to hadronic matter phase transition and the QCD critical point in heavy-ion collisions relies on detecting scale-invariant spatial fluctuations (intermittency) in final-state particles. While the Critical Monte Carlo (CMC) model has successfully simulated these fluctuations in the two-dimensional transverse momentum ( $p_x$ - $p_y$ ) space, experimental signals are extremely weak, constituting only a few percent of the total event sample. These critical signals are severely diluted by the overwhelming non-critical thermal background (e.g., from EPOS simulations), rendering traditional Normalized Factorial Moment (NFM) analyses insufficient for reliable signal extraction. Furthermore, previous intermittency studies have largely been confined to momentum space, leaving a gap in the application of CMC-based critical fluctuation models to the experimentally accessible angular ( $\eta, \phi$ ) phase space used by detectors like ALICE at the LHC.

Methodology
The authors propose a novel two-stage topological machine learning framework designed to extract weak critical signals from dominant backgrounds in the ( $\eta, \phi$ ) phase space.

Event Generation and Signal Embedding:
- Background: $6 \times 10^5$ minimum bias Pb-Pb collision events at $\sqrt{s_{NN}} = 5.02$ TeV were generated using the EPOS hybrid event generator.
- Signal: The CMC model, utilizing a Lévy random walk with an exponent $\mu = 1/6$ (corresponding to the 3D Ising universality class), was adapted to generate critical clusters directly in the ( $\eta, \phi$ ) acceptance window.
- Embedding: Critical signal tracks were embedded into EPOS events at fractions ( $\lambda$ ) of 1%, 3%, and 5%. The original angular coordinates of selected EPOS tracks were replaced by CMC coordinates while preserving transverse momentum ( $p_T$ ).
Topological Data Analysis (TDA) Pipeline:
- Delaunay Filtration: Particle events were treated as 2D point clouds. A Delaunay triangulation was constructed with a "phantom-point" scheme to enforce periodic boundary conditions on the azimuthal angle $\phi$ .
- Sub-level Set Filtration: A continuous geometric filtration was performed based on the nearest-neighbor distance ( $d_{NN}$ ) of particles. As the scale parameter $\epsilon$ increased, simplicial complexes (vertices, edges, faces) were activated.
- Betti Curves: The evolution of topological invariants—specifically the 0th Betti number ( $\beta_0$ , connected components) and 1st Betti number ( $\beta_1$ , loops)—was tracked to generate Betti curves.
- Bias Correction: To remove multiplicity bias and collective flow effects, the raw Betti curves were normalized by event multiplicity and corrected via azimuthal randomization. This produced $\Delta$ -Betti curves ( $\beta^\Delta_k = \beta_k - \beta^{rand}_k$ ), isolating dynamical spatial correlations.
Machine Learning Classification:
- The 600-dimensional feature vectors (concatenated $\Delta\beta_0$ and $\Delta\beta_1$ curves) were fed into two distinct classifiers: a 1D Convolutional Neural Network (TopoPointNet) and a Gradient Boosted Decision Tree (XGBoost/BDT).
- Two-Stage Filtering:
  - Stage 1 (Event-Level): Events were classified as signal or background. Only events with a signal probability $S \ge 0.90$ were retained.
  - Stage 2 (Particle-Level): Within selected events, a geometric density filter was applied ( $d_{NN} \le \epsilon_{cut}$ ) to strip the diffuse thermal background and isolate the densely packed critical clusters.

Key Results

Signal Dilution: Standard NFM analysis on mixed samples (EPOS + 5% CMC) failed to recover the power-law scaling, yielding an intermittency index $\phi_2 \approx 0$ (background-like) rather than the critical value.
Classification Performance: Both TopoPointNet and BDT achieved high discrimination power. At a 5% signal injection fraction, the Area Under the Curve (AUC) reached $\approx 0.99$ for both models. The models demonstrated robustness across different resolutions ( $N_\epsilon = 300$ was optimal).
Feature Attribution: SHAP and gain-based importance analyses revealed that the models relied on specific topological scales:
- $\epsilon \approx 0.001$ rad: The onset of connectivity in the densest Lévy clusters ( $\Delta\beta_0$ ).
- $\epsilon \approx 0.02\text{--}0.05$ rad: The formation of closed loops within clusters ( $\Delta\beta_1$ ).
- $\epsilon \approx 0.08$ rad: The coalescence of separate clusters.
Recovery of Intermittency: After applying the two-stage topological filter, the NFM analysis on the remaining tracks successfully restored the power-law scaling. The recovered intermittency index for the 5% diluted mixture was $\phi_2 \approx 0.73$ , closely matching the pure CMC reference value of $\phi_2 \approx 0.75$ (shifted from the theoretical $2/3$ due to the $(\eta, \phi)$ geometry and finite acceptance).

Significance and Claims
The paper claims to present the first application of the CMC model combined with topological machine learning in the ( $\eta, \phi$ ) phase space. The authors assert that this work:

Establishes a robust, data-driven methodology for probing the QCD critical point where traditional factorial moment analysis fails due to signal dilution.
Demonstrates that topological features (Betti curves) contain sufficient geometric information to distinguish critical fluctuations from thermal backgrounds even at low signal fractions (1–5%).
Validates that a two-stage pipeline (event classification followed by particle-level density filtering) is necessary to recover the underlying critical scaling laws.
Provides a baseline for future experimental searches at LHC energies (specifically ALICE), noting that while the current study is at the generator level, the framework is designed to leverage the high-statistics capabilities of Run 3 data.

The authors remain modest regarding immediate experimental application, acknowledging that the current analysis does not simulate detector effects (e.g., track merging/splitting) and that the CMC model represents a static fractal geometry rather than a dynamic phase evolution. Future work is proposed to incorporate dynamic models (like the SCR model) and realistic detector simulations.

The Cosmic Soup and the Search for a Hidden Recipe

Hunting for a Ghost in the Machine

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