⚛️ nuclear theory

Hadronization into Nuclei: Does Size Matter?

This paper demonstrates that incorporating nuclear size corrections into the statistical hadronization model fails to consistently describe (hyper)nucleus production data across various collision systems, whereas the uncorrected model supports the interpretation that these nuclei form from compact multi-quark configurations that expand after hadronization.

Original authors: Anton Andronic, Peter Braun-Munzinger, Hjalmar Brunßen, Johanna Stachel

Published 2026-10-02

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CC BY 4.0

Original authors: Anton Andronic, Peter Braun-Munzinger, Hjalmar Brunßen, Johanna Stachel

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

In the most violent collisions imaginable, where atomic nuclei smash together at nearly the speed of light, a tiny, seething drop of matter is born. Scientists call this a fireball, a state of existence so hot and dense that protons and neutrons melt into a soup of their constituent parts, quarks and gluons. As this fireball expands and cools, it undergoes a phase transition, much like steam condensing into water, and the quarks recombine to form the familiar particles of our universe. Among the most surprising products of this process are not just individual particles, but entire atomic nuclei, including some that are so fragile they barely hold together. The question that has long puzzled physicists is how these delicate structures manage to form in such a chaotic environment. Do they assemble piece by piece from free-floating particles as the fireball dies down, or do they emerge as compact, pre-formed objects that simply expand as the heat fades?

A team of researchers set out to test a specific idea about this formation process, focusing on the physical size of these newly born nuclei. In the standard view of how these particles are created, scientists often treat nuclei as if they were single, point-like dots with no internal space. However, some nuclei, particularly a type called the hypertriton, are incredibly large and loosely bound. The hypertriton is so spread out that its size is comparable to the fireball itself. If these nuclei were truly formed as large, diffuse clouds of particles at the moment of creation, a significant portion of them would be too big to fit inside the fireball. Logic suggests that if a nucleus is larger than the container holding it, it cannot exist there; the parts of the nucleus would spill out, and the object would be destroyed. Therefore, if the standard model of formation were correct and these large nuclei were created as big clouds, their numbers should drop dramatically in smaller collisions where the fireball is tiny.

The researchers investigated this possibility by taking a well-established model of particle production and adding a new rule that accounts for the actual size of the nuclei. They calculated how much the production of light nuclei and the hypertriton should be suppressed if the fireball was too small to contain their full, spread-out shapes. They then compared these predictions against real data collected from particle collisions of varying sizes, ranging from massive lead-ion crashes down to smaller proton collisions. The results were clear and decisive: the data did not show the massive drop in numbers that the size-based calculation predicted. In the smallest collisions, where the fireball is tiny, the number of nuclei produced remained far higher than the "size correction" model allowed. In fact, the model that ignored the size of the nuclei entirely, treating them as if they were compact points, matched the experimental data perfectly across all collision sizes.

This finding rules out the idea that these nuclei are born as large, diffuse clouds that must fit entirely within the fireball at the moment of their creation. Instead, the evidence points to a different story. The researchers conclude that these nuclei must form as tight, compact bundles of quarks, small enough to fit easily inside the fireball. It is only after they have been created and the fireball has cooled that they expand into the large, fragile shapes we observe. The time it takes for a nucleus like the deuteron to expand to its final size is far longer than the lifetime of the fireball itself. This means the final, spacious shape of the nucleus is not a condition for its birth, but a result of its growth after the chaotic event has ended. The data suggests that nature first builds a small, sturdy core and only later allows it to stretch out into the vast, delicate structure we detect, resolving the paradox of how such large objects can survive in such small spaces.

Technical Summary: Hadronization into Nuclei: Does Size Matter?

Problem Statement
The production mechanism of light nuclei and hypernuclei in high-energy heavy-ion collisions remains a subject of active debate. While the Statistical Hadronization Model (SHM) successfully describes yields of hadrons and (hyper)nuclei across nine orders of magnitude in abundance, the physical interpretation of how loosely bound states (e.g., deuteron, hypertriton) form is contested. A specific paradox arises when applying the SHM to large, loosely bound systems like the hypertriton ( $^3_\Lambda$ H), whose root-mean-square (rms) separation (~11 fm) is comparable to the fireball radius in central Pb–Pb collisions. If these nuclei are treated as having their final-state spatial wave functions at the moment of chemical freeze-out, a significant portion of their wave function would extend outside the fireball volume, implying a suppression of their yield. This paper investigates whether a "size correction" based on these final-state wave functions is necessary to describe experimental data from the ALICE experiment.

Methodology
The authors extend the Statistical Hadronization Model by incorporating a size correction factor ( $S_i$ ) that accounts for the spatial extent of nuclear wave functions relative to the fireball volume ( $V_{\Delta y=1}$ ).

Size Correction Formulation: For a nucleus to be formed, all constituents must lie within the fireball volume. The correction factor $S_i$ $S_{i}$ is calculated as the probability of finding all constituents within $V$ $V$ .
- For two-body systems (deuteron, hypertriton treated as $\Lambda$ -d), $S_i$ is derived by integrating the squared spatial wave function $|\psi(x_1 - x_2)|^2$ over the volume.
- For $A \ge 3$ nuclei ( $^3$ He, $^3$ H, $^3_\Lambda$ H), the calculation utilizes normalized one-particle density distributions ( $\rho_p, \rho_n$ ) and specific wave function factorizations (e.g., separating the $\Lambda$ -d distance from the core nucleon distance).
Wave Function Inputs:
- Deuteron: Based on the Argonne v18 nucleon-nucleon potential (S-wave and D-wave components), yielding an rms matter radius of 1.94 fm.
- Hypertriton: Modeled as a $\Lambda$ loosely bound to a deuteron core using chiral effective field theory (EFT) radial correlation densities. The $\Lambda$ -d separation has an rms of 11.1 fm, highlighting a halo structure.
- $^3$ He and $^3$ H: Calculated using one-body densities from chiral EFT.
Fireball Geometry: The fireball is modeled as a cylinder. The longitudinal length is derived from the freeze-out proper time ( $\tau_f \approx 10.4$ fm/c) via Bjorken flow, and the transverse radius is fixed by the midrapidity volume ( $V \approx 4175$ fm $^3$ ) determined in central Pb–Pb collisions.
Ensemble Treatment: The study compares three scenarios:
1. Grand-canonical SHM.
2. Canonical SHM (accounting for exact conservation of quantum numbers like baryon number and strangeness in a correlation volume $V_c$ ).
3. Canonical SHM with the additional size correction ( $S_i < 1$ ).

Key Results
The study compares the model predictions against ALICE measurements for deuterons, $^3$ He, $^3$ H, and $^3_\Lambda$ H across pp, p–Pb, and Pb–Pb collisions at various energies ( $\sqrt{s_{NN}} = 2.76 - 13$ TeV).

Deuterons: The canonical SHM (without size correction) reproduces the experimental rapidity densities over a dynamic range of three orders of magnitude in charged-particle multiplicity ( $dN_{ch}/d\eta$ ). The inclusion of the size correction introduces a suppression of nearly one order of magnitude for small systems (pp, p–Pb), causing the theoretical predictions to fall significantly below the data.
Mass-3 Nuclei ( $^3$ He, $^3$ H): Similar to deuterons, the experimental data for $^3$ He and $^3$ H follow the trend of the canonical SHM with only canonical suppression. The size-corrected calculations predict yields that are too low, particularly at small multiplicities.
Hypertriton ( $^3_\Lambda$ H): Due to its large spatial extent, the hypertriton is the most sensitive probe for size effects. While the absolute yield in central Pb–Pb collisions at 5.02 TeV is somewhat lower than the standard SHM prediction (and closer to the size-corrected value), the multiplicity dependence tells a different story. The experimental yield drop as system size decreases is significantly less steep than predicted by the size-corrected model. The size-corrected SHM underpredicts the yields in minimum-bias pp collisions by more than an order of magnitude.
Consistency: Across all studied species, the SHM without a nuclear-size correction provides a substantially better description of the data than the version including the size correction.

Significance and Conclusions
The paper concludes that the hypothesis of nuclei forming with their final-state spatial wave functions at chemical freeze-out is inconsistent with experimental observations. The failure of the size-corrected model suggests that the large spatial extent of the final-state wave function is not present at the moment of hadronization.

Instead, the results support the interpretation that nuclei and hypernuclei are formed from initially compact, multi-quark configurations (with small virtuality) that possess the correct quantum numbers of the final state. The expansion to the final, large spatial wave function occurs after hadronization, during the time evolution of the system, well beyond the chemical freeze-out time. This is consistent with the characteristic formation time scales (e.g., $\hbar/E_B \approx 100$ fm/c for the deuteron) which exceed the decoupling time of the nuclear fireball. Consequently, the statistical hadronization model remains a robust framework for describing (hyper)nuclei production without requiring ad-hoc size corrections based on asymptotic wave functions.

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