ToMCCA-3: A realistic 3-body coalescence model

This paper introduces ToMCCA-3, a realistic three-body coalescence model based on the Wigner function formalism and constrained by modern nuclear interaction data, which successfully predicts light (anti)nuclei yields in proton-proton collisions at 13 TeV and demonstrates their sensitivity to nuclear wave functions when compared with ALICE experimental data.

The Gist

Imagine the universe as a giant, high-speed particle collider, where tiny building blocks of matter (protons and neutrons) smash into each other at incredible speeds. When they collide, they don't just scatter; sometimes, they stick together to form new, heavier "clumps" called light nuclei (like Helium-3 or Tritium).

This paper introduces a new, more realistic way to predict how these clumps form. The authors call their model ToMCCA-3. Here is a breakdown of what they did, using simple analogies:

1. The Problem: The "Clumping" Guesswork

Previously, scientists tried to predict how these particles stick together using a method called "coalescence." Think of this like trying to predict how many people will form a huddle in a crowded room.

• Old Method: They used a simple rule: "If people are close enough in space and moving at similar speeds, they huddle." This worked okay, but it relied on guessing a "magic number" (a parameter) for how close they needed to be. It was like guessing the size of the huddle without knowing how big the people actually are.
• The Issue: This didn't work perfectly for heavier clumps (like 3-body systems: three particles stuck together). The old models were too simple and didn't account for the complex "personality" or internal structure of the particles.

2. The Solution: A "Wigner Function" Map

The authors upgraded their model to use something called a Wigner function.

• The Analogy: Imagine you are trying to predict where a group of three friends will end up after a chaotic dance party.
  • The Old Model just looked at their speed and said, "If they are close, they will dance together."
  • The New Model (ToMCCA-3) looks at a detailed "dance map." It considers not just where they are and how fast they are moving, but also their specific "dance style" (their quantum wave function). It knows exactly how the three particles wiggle and interact with each other before they even try to stick together.

3. The Ingredients: Realistic "Glue"

To make this map accurate, the team used real-world data to describe the "glue" holding these particles together.

• Two-Body Glue: They used a known, highly accurate recipe (the Argonne v18 potential) for how two particles stick.
• Three-Body Glue: They added a special ingredient (the Urbana IX potential) that accounts for how three particles interact all at once. It's like realizing that in a group of three, the third person changes the dynamic between the first two.
• Testing: They tested different "recipes" for the glue. Some were simple (Minnesota potential), and some were complex (Argonne + Urbana). They found that while simple recipes worked okay, the complex ones that included the "three-body glue" gave the most accurate predictions, especially for larger groups.

4. The Experiment: Simulating the Collision

The team used a computer program (an event generator) to simulate billions of proton-proton collisions at the energy levels of the Large Hadron Collider (13 TeV).

• They fed the program the "dance map" (the wave functions) and the "glue recipes."
• They watched to see how many 3-particle clumps (Helium-3, Tritium, and a special "hyper-triton" containing a strange particle called a Lambda) formed.
• The Result: Their predictions matched the real data collected by the ALICE experiment at CERN very well. The model successfully predicted how many of these particles would be created and how fast they would be moving.

5. Key Discoveries

• Size Matters (But Not How You Think): A previous theory suggested that smaller "source" sizes (the area where particles are born) would suppress the formation of larger nuclei. The new model showed this isn't quite right. Instead, the nature of the interaction (the glue) is the most important factor. If the "three-body glue" is attractive, it actually helps form larger nuclei, even in small spaces.
• The Hypertriton: They also modeled a very rare particle called the hypertriton (a proton, neutron, and a Lambda particle). They used a simplified approach where the Lambda particle orbits a stable pair of nucleons (a deuteron). Their predictions for this rare particle are ready for when experimental data becomes available.

Summary

In short, the authors built a high-definition simulation for how three-particle nuclei form in high-energy crashes. By replacing simple guesses with detailed quantum "maps" and realistic "glue" recipes, they created a tool that matches experimental data much better than before. This tool helps scientists understand the fundamental forces that hold matter together and could eventually help us understand how antimatter is formed in the universe.

Technical Summary

Technical Summary: ToMCCA-3: A realistic 3-body coalescence model

Problem Statement
The formation mechanisms of light (anti)nuclei and hypernuclei in high-energy hadronic collisions remain a critical open question in nuclear and particle physics. While models such as statistical hadronization and coalescence have successfully described deuteron ($A=2$) production, discrepancies persist for heavier systems ($A=3$), including $^3$He, $^3$H, and the hypernucleus $^3_\Lambda$H. Traditional coalescence models often rely on free parameters (e.g., coalescence momentum $p_0$) not derived from first principles, while statistical models struggle to account for the specific binding properties and sizes of these states. Furthermore, the production of rare antinuclei is a potential signature for dark matter searches, making accurate theoretical predictions of their production rates in collider environments essential for interpreting cosmic ray data.

Methodology
This work presents an extension of the ToMCCA (Three-body Monte Carlo Coalescence Algorithm) event generator to the $A=3$ case. The framework is based on the Wigner function formalism, which describes the coalescence of nucleons and hyperons into bound states by integrating the product of the nuclear wave function and the phase-space density of the emitting source.

Key methodological components include:

• Formalism: The yield is calculated using a three-particle density matrix formalism, factored into single-particle Wigner functions. The calculation employs Jacobi coordinates to separate center-of-mass motion from internal dynamics.
• Wave Functions: Realistic nuclear wave functions are implemented using the Pair-Correlated Hyperspherical Harmonics (PHH) method.
  • For $^3$He and $^3$H, the model utilizes the Argonne v18 (AV18) two-body potential combined with the Urbana IX (UIX) three-body interaction. The Minnesota potential is also tested as a simplified two-body alternative.
  • For $^3_\Lambda$H, a simplified Congleton approach is adopted, modeling the system as a $\Lambda$ particle orbiting an unperturbed deuteron core (described by AV18) within a $\Lambda$-d potential.
• Source Modeling: The event generator incorporates realistic phase-space correlations extracted from the EPOS 3.6 event generator. This includes two-particle correlations (relative momentum, distance, and transverse mass) and multiplicity-dependent source sizes ($\sigma$) constrained by femtoscopic data from ALICE.
• Implementation: The coalescence probability is pre-calculated on a 4D grid ($|k_1|, |k_2|, \cos\theta_{k12}, \sigma$) and stored as a histogram for efficient evaluation during event generation. The model includes parameterized spectra for protons and $\Lambda$ hyperons dependent on charged-particle multiplicity.

Key Contributions

1. Extension to $A=3$: The paper successfully upgrades ToMCCA from a two-body to a three-body coalescence model, enabling the prediction of $^3$He, $^3$H, and $^3_\Lambda$H production.
2. Realistic Potentials: Unlike previous models using Gaussian approximations, this work integrates realistic nuclear wave functions derived from modern scattering data and three-body forces.
3. Hypernucleus Modeling: The application of the Congleton formalism within a coalescence framework provides a specific prediction for $^3_\Lambda$H production in $pp$ collisions, a system previously lacking detailed coalescence predictions in this context.
4. Source Size Sensitivity: The model explicitly investigates the interplay between the nuclear wave function structure and the size of the particle-emitting source, moving beyond simple momentum-space coalescence criteria.

Results

• $^3$He and $^3$H Spectra: The model reproduces ALICE experimental data for $^3$He transverse momentum ($p_T$) spectra and yield ratios ($^3$He/$p$, $^3$H/$p$) in $pp$ collisions at $\sqrt{s} = 13$ TeV within 2 standard deviations.
• Wave Function Dependence: The results demonstrate that the choice of nuclear wave function significantly impacts the predicted yields. Models using AV18+UIX (including three-body forces) and Minnesota potentials (reproducing binding energies) agree well with data. In contrast, models using only two-body forces (AV18 without UIX) show deviations.
• Yield Ratios: Contrary to predictions based on Gaussian wave functions, the ToMCCA model predicts that the $^3$H/$^3$He yield ratio is largely independent of the charged-particle multiplicity (and thus source size), remaining constant at $\approx 0.996$. This suggests that the underlying nucleon-nucleon interactions, rather than simple geometric size differences, drive the production yields.
• Three-Body Force Effects: The inclusion of the attractive UIX three-body force enhances the yield of $^3$He, particularly at higher multiplicities (larger source sizes), by reducing the effective nuclear size.
• $^3_\Lambda$H Predictions: The model provides the first $p_T$ spectra and yield predictions for $^3_\Lambda$H in $pp$ collisions. The $^3_\Lambda$H/$^3$He ratio is predicted to be almost independent of $p_T$, differing from thermal model predictions which suggest a rising ratio due to radial flow.
• Uncertainties: Global yield uncertainties are estimated at $\pm 17\%$ for $^3$He/$^3$H (dominated by source size and momentum cutoff variations) and $\pm 15.5\%$ for $^3_\Lambda$H.

Significance and Claims
The paper asserts that the ToMCCA-3 model provides a refined, realistic description of light nuclei formation that improves upon traditional coalescence models by incorporating first-principles nuclear interactions. The work highlights that accurate descriptions of nuclear wave functions and three-body forces are crucial for predicting yields, particularly for systems like $^3$He and $^3$H.

The authors claim that their results lay the foundation for extending coalescence studies to a wider range of collision systems and energies. They emphasize the model's utility for:

• Interpreting future high-precision data from LHC Run-3 and Run-4.
• Providing necessary inputs for indirect dark matter searches, where the production of cosmic antinuclei must be distinguished from astrophysical backgrounds.
• Investigating the transition between thermal and coalescence production mechanisms, particularly through the study of hypernuclei like $^3_\Lambda$H.

The paper concludes that while the model successfully reproduces existing ALICE data, further studies with different three-body forces and experimental data from intermediate-mass collision systems (e.g., O–O, Ne–Ne) are required to fully disentangle the effects of source size and nuclear interactions.