$^3$H and $^3$He nuclei production in a combined thermal and coalescence framework for heavy-ion collisions in the few-GeV energy regime

This paper presents a combined thermal and coalescence model for heavy-ion collisions in the few-GeV regime that successfully reproduces proton, pion, and deuteron yields but underpredicts the production of $^3$H and $^3$He nuclei by a factor of two compared to experimental data.

The Gist

Imagine a high-energy particle collision as a massive, chaotic dance party where thousands of tiny particles (protons and neutrons) are created in a flash. For a split second, they are a hot, swirling soup of energy. As the party cools down, these particles try to find partners to form stable groups, like couples or small dance crews.

This paper is about trying to predict how often these particles form specific, larger groups: Tritium (a nucleus with one proton and two neutrons, written as ³H) and Helium-3 (two protons and one neutron, written as ³He).

Here is the breakdown of what the scientists did, using simple analogies:

1. The Two-Step Recipe

The authors combined two different ways of thinking about how these groups form:

• Step 1: The Thermal Model (The "Hot Soup" Phase):
  First, they used a "statistical model." Imagine the collision creates a giant, hot bowl of soup. The particles in this soup are moving randomly. The scientists calculated how many protons and pions (another type of particle) are floating around based on the temperature and pressure of this soup. They already knew this method worked well for predicting how many single particles and pairs (like Deuterium, which is just a proton and a neutron holding hands) were made.

• Step 2: The Coalescence Model (The "Huddle" Phase):
  Next, they asked: "If these particles are close enough, will they stick together to form a trio?" This is called coalescence. Think of it like a game of musical chairs. If three players (nucleons) happen to be standing very close to each other when the music stops (when the system freezes), they grab hands and form a team (a nucleus). The paper uses math to calculate the odds of three specific players being close enough to form a team.

2. The Setup: A Slightly Squashed Ball

The scientists didn't just assume the "soup" was a perfect sphere. They realized the explosion from the collision is more like a slightly squashed ball (a spheroid) that expands outward. They used a more realistic shape for this expansion, which helped them get better numbers for the single particles (protons and pions) before trying to predict the trios.

3. The Prediction vs. Reality

The team ran their calculations to predict how many Tritium and Helium-3 nuclei should be created in gold-gold collisions at a specific energy level (2.4 GeV).

• The Result: Their math predicted that there would be about half as many of these nuclei as what the HADES experiment (a real-world detector) actually observed.

  • For Tritium (³H), they predicted about 3.16, but the experiment found 8.65.
  • For Helium-3 (³He), they predicted about 2.26, but the experiment found 4.55.

• The Good News: Even though they were off by a factor of two, they got the order of magnitude right. In the world of particle physics, predicting that you will get "a few" instead of "zero" or "a million" is a significant success. It proves their combined "Hot Soup + Huddle" idea is on the right track.

4. Why the Discrepancy?

The authors suggest the missing factor of two might come from how they calculated the "formation rate."

• The Analogy: Imagine trying to predict how many people will form a huddle. If you assume everyone stands in a perfect circle, you might get the math wrong. The scientists used a simplified shape for where the particles are standing (a hard sphere). They suspect that if they used a more complex, realistic "wave function" (a better map of exactly where the particles are likely to be), their prediction would get closer to the real numbers.

5. The Shape of the Data

While the total number of nuclei was underestimated, the scientists checked the shape of the data (how the particles are distributed across different speeds and directions).

• They found that their model's shape was slightly too "steep" compared to the experimental data.
• However, if they simply multiplied their prediction by a scaling factor (like turning up the volume), the shape of their curve matched the experimental data very well. This suggests the physics of how they form is correct, even if the exact count needs a slight adjustment.

Summary

The paper is a successful attempt to mix two theories (thermal soup and coalescence huddles) to explain how heavy nuclei form in particle collisions.

• What worked: The model correctly predicted the general size of the effect and the shape of the particle distribution.
• What needs work: The model predicts about half the actual number of nuclei found in experiments. The authors believe this is because their mathematical "map" of where particles are located is a bit too simple, and a more detailed map would fix the count.

They conclude that their framework is a solid foundation for understanding these tiny nuclear teams, even if the final headcount needs a little fine-tuning.

Technical Summary

Technical Summary: 3H and 3He Nuclei Production in a Combined Thermal and Coalescence Framework

Problem Statement
This work addresses the production of light nuclei, specifically tritium ($^3$H) and helium-3 ($^3$He), in heavy-ion collisions within the few-GeV energy regime. While statistical hadronization models successfully describe the spectra of protons and pions, and the coalescence model is often used for light nuclei, these approaches are frequently treated as exclusive alternatives. The authors aim to construct a consistent framework that combines a thermal description of initial particle production with a subsequent coalescence mechanism to predict the yields and spectra of $^3$H and $^3$He in Au-Au collisions at $\sqrt{s_{NN}} = 2.4$ GeV.

Methodology
The authors employ a hybrid approach integrating a statistical hadronization model with a nucleon coalescence framework:

1. Thermal Freeze-out Model:

   • The study utilizes a statistical hadronization model with spheroidal hydrodynamic expansion, which is more realistic than the spherical symmetry assumed in earlier blast-wave models.
   • The freeze-out conditions are constrained by experimental data from the HADES Collaboration (10% centrality). The analysis selects a higher freeze-out temperature ($T = 70.3$ MeV) over a lower alternative ($T = 49.6$ MeV), as the former better reproduces the experimentally measured deuteron yield.
   • The system is characterized by a small radius ($R \approx 6$ fm), baryon chemical potential ($\mu_B = 876$ MeV), and isospin chemical potential ($\mu_{I3} = -21.5$ MeV).
   • The model assumes that all detected protons (both free and those bound in light nuclei) originate from a single thermal system. Consequently, the proton spectra used as input for coalescence are rescaled to account for the fraction of protons found in bound states.

2. Coalescence Framework:

   • The production of $^3$H and $^3$He is calculated using the standard coalescence formalism, where the nucleus momentum distribution is derived from the product of proton and neutron momentum distributions at a fraction of the nucleus momentum.
   • Spatial Distribution: Unlike previous works that often assume a Gaussian spatial profile for the source, this paper adopts a uniform spatial distribution of nucleons within a sphere of radius $R$ at a fixed laboratory time. This choice is consistent with the thermal model assumptions.
   • Wave Functions: The authors use Gaussian approximations for the $^3$H and $^3$He wave functions in Jacobi coordinates.
   • Formation Rate: The nucleus formation rate coefficient ($A_{FR}^X$) is calculated by integrating the product of the spatial distribution function and the squared wave function over the phase space. This calculation yields large numerical values for the coefficients ($A_{FR}^{^3H} \approx 5.51 \times 10^{11}$ MeV$^6$).

3. Numerical Implementation:

   • The Cooper-Frye formula is used to generate invariant momentum spectra for protons and neutrons based on the spheroidal expansion parameters.
   • These nucleon spectra serve as inputs into the coalescence equation (Eq. 6 and Eq. 7) to predict the rapidity and transverse-mass spectra of the light nuclei.

Key Results

• Yields: The model predicts total yields for $^3$H and $^3$He that are approximately a factor of 2 to 2.7 lower than the experimental values reported by the HADES Collaboration. Specifically, the model predicts $N \approx 3.16$ for $^3$H and $N \approx 2.26$ for $^3$He, compared to experimental values of $8.65$ and $4.55$, respectively.
• Spectra Shapes: Despite the normalization discrepancy, the model successfully reproduces the order of magnitude of the multiplicities. The predicted shapes of the rapidity and transverse-mass spectra are compared with preliminary HADES data.
• Scaling Factors: To better match the experimental rapidity spectra shapes, the formation rate coefficients were rescaled by factors ($\alpha$). For $^3$H, a factor of $\alpha \approx 2.4$ provided the best fit to the spectral shape, while for $^3$He, $\alpha \approx 1.9$ was optimal. These factors are slightly different from those required to match the total yields ($\alpha \approx 2.7$ for $^3$H and $\alpha \approx 2.0$ for $^3$He).
• Uncertainties: The authors identify the primary source of uncertainty as the calculation of the formation rate factors, which relies on simplified wave functions and spatial distributions.

Significance and Claims
The paper claims that its primary contribution is demonstrating that thermal production and coalescence mechanisms can coexist and supplement each other in a consistent theoretical scenario for low-energy heavy-ion collisions.

• The authors assert that their approach correctly reproduces the order of magnitude of the light nuclei yields, which is considered a non-trivial success given the complexity of the calculation involving large and small numbers.
• The work extends the successful description of deuteron production (previously established in Ref. [4]) to the next lightest nuclei ($A=3$).
• The authors modestly conclude that while the absolute yields are underestimated, the framework provides a viable baseline for future comparisons with experimental data. They suggest that the discrepancy in yields could be resolved by employing more realistic wave functions for $^3$H and $^3$He in future calculations of the formation rate.
• The paper does not propose new experimental setups but emphasizes that its predictions for spectra can be directly compared with future experimental data to further validate the combined thermal-coalescence picture.