Examination of the lattice QCD-motivated strong attractive $ΩN$ potentials in the $Ω^- n p$ system

Using Faddeev equations in configuration space, this study demonstrates that the large binding energy of the $\Omega^- np$ system arises from the short-range behavior of strong attractive $\Omega N$ potentials, while revealing that the Coulomb interaction has only a marginal perturbative effect on the system's spatial configuration and binding energy.

The Gist

Imagine the universe as a giant, complex LEGO set. Most of the time, we see two main types of blocks: mesons (made of two smaller pieces stuck together) and baryons (made of three pieces). The most famous baryons are protons and neutrons, which build the atoms in everything around us.

But physicists have long wondered: What happens if we try to build something with more pieces? Specifically, what if we mix a very rare, heavy, and strange block called the Omega baryon ($\Omega^-$) with the standard proton and neutron blocks?

This paper is like a detailed blueprint for a theoretical experiment to see if these three blocks can snap together to form a stable, tiny "super-nucleus."

The Cast of Characters

• The Proton ($p$) and Neutron ($n$): The standard building blocks of our world. They are like two siblings who are very similar but not identical.
• The Omega Baryon ($\Omega^-$): A rare, heavy cousin made entirely of "strange" quarks. It's like a heavy, exotic brick that doesn't usually hang out with the standard ones.
• The Strong Force: The "glue" that holds these blocks together. In normal atoms, this glue has a tricky rule: it pushes blocks apart if they get too close (like a repulsive core), preventing them from crushing each other.

The Big Question: Can They Stick?

The scientists wanted to know: If you take one Omega, one proton, and one neutron, will they stick together to form a new, exotic object (called a tribaryon or $\Omega^-np$)?

To answer this, they used two different "instruction manuals" (potentials) for how the Omega and the nucleons interact:

1. The Lattice QCD Manual: This is like a super-computer simulation of the fundamental laws of physics, calculating exactly how these particles behave based on the math of quarks.
2. The Meson Exchange Manual: This is a more traditional approach, imagining the particles as throwing tiny messenger particles (mesons) back and forth to create the glue.

The Two Ways to Look at the System

The paper explores two ways to model this trio:

1. The "Identical Twins" Model (AAC): This treats the proton and neutron as if they were identical twins. This is a common shortcut in physics, but it ignores a tiny detail: the proton has a positive electric charge, while the neutron is neutral.
2. The "Three Distinct Individuals" Model (ABC): This is the paper's main focus. It treats the Omega, proton, and neutron as three completely different people. Because the proton is charged, it feels a tiny electric tug (Coulomb force) from the negatively charged Omega. This breaks the symmetry, making the proton and neutron act differently.

The Surprising Results

Here is what the authors found, using some creative analogies:

1. The "Weak Link" becomes a "Super Glue"
Individually, the Omega and a single nucleon (proton or neutron) only stick together very weakly. It's like two people holding hands loosely; they might let go easily.

• The Analogy: Imagine two people holding a single rope. They are barely connected.
• The Twist: When you add a third person to the group, the whole system suddenly becomes incredibly tight. The paper found that the three-body system is bound (stuck together) with an energy ten times stronger than the two-body pair.
• Why? In normal atoms, the "glue" (strong force) has a repulsive core that stops things from getting too close. But the Omega is so different from the proton and neutron that this "repulsive core" doesn't exist. The Omega and nucleons can get very close, and the glue becomes incredibly strong at short distances. It's like the three people hugging so tightly they become a single solid unit.

2. The Electric Tug is a Minor Disturbance
The authors checked if the electric charge of the proton (which pulls it toward the Omega) would ruin the structure.

• The Analogy: Imagine the three people hugging in a perfect triangle. The proton is wearing a magnet that pulls it slightly closer to the Omega.
• The Result: The magnet does pull the proton a little closer, making the triangle slightly lopsided (breaking the perfect symmetry). However, because the "hug" (the strong nuclear force) is so incredibly strong, the electric magnet barely changes the overall shape or stability of the group. The strong force dominates the electric force.

3. The Shape of the New Nucleus
The resulting object is very compact. The particles are packed much closer together than in a normal nucleus. The paper suggests this is because the Omega is heavy (it moves slowly, like a heavy anchor) and the glue is so strong at short range that it pulls everything into a tight ball.

What This Means (According to the Paper)

• Existence: The math strongly suggests that this exotic "Omega-Proton-Neutron" nucleus does exist as a bound state. It is a real, physical possibility, not just a fantasy.
• Stability Warning: While the math says they stick together, the paper notes a catch. The Omega particle is naturally unstable and decays (breaks apart) very quickly (in about 0.1 nanoseconds). So, even if this super-nucleus forms, it won't last long enough to build a permanent structure. It's a fleeting moment of stability.
• Future Potential: The authors speculate that if we could somehow stabilize these particles or add more of them, we might create "strange matter" with much higher binding energy than normal atoms. This is relevant to understanding the cores of neutron stars, where such exotic matter might exist under extreme pressure.

Summary

In simple terms, this paper is a theoretical investigation that says: "If you mix a rare Omega particle with a proton and a neutron, they will snap together into a very tight, very stable (but short-lived) cluster."

The key discovery is that the usual rules preventing particles from crushing each other don't apply here, allowing for a much stronger bond than we see in normal matter. The electric charge of the proton makes a tiny difference in the shape, but the powerful "strong force" glue is the real boss of the system.

Technical Summary

Technical Summary: Examination of the Lattice QCD-Motivated Strong Attractive $\Omega N$ Potentials in the $\Omega^-np$ System

Problem Statement
The existence and properties of multiquark systems, specifically dibaryons and tribaryons containing strange quarks, remain a significant area of investigation in hadron physics. While the $\Omega N$ dibaryon (strangeness $S=-3$) has been predicted to be bound in various quark models and lattice QCD calculations, the formation of the $\Omega NN$ tribaryon system (the $\Omega^-np$ system) requires a rigorous three-body treatment. Previous studies have largely employed the "AAC model," treating the two nucleons as identical particles and neglecting the Coulomb interaction between the negatively charged $\Omega^-$ baryon and the proton. This approximation ignores the symmetry breaking introduced by the Coulomb force, which distinguishes the $\Omega^-p$ pair from the $\Omega^-n$ pair. Furthermore, there is a need to reconcile results obtained using different $\Omega N$ interaction potentials: those derived from lattice QCD (HAL QCD Collaboration) and those based on meson-exchange models. A central question addressed is whether the strong binding observed in the $\Omega NN$ system arises from the specific short-range behavior of the $\Omega N$ potential, particularly given that the two-body $\Omega N$ system is only weakly bound.

Methodology
The authors employ the Faddeev equations in configuration space to solve the three-body problem for the $\Omega^-np$ system. The study utilizes two distinct formalisms:

1. The AAC Model: Treats the system as having two identical nucleons, neglecting the Coulomb interaction or using isospin formalism where protons and neutrons are indistinguishable.
2. The ABC Model: Treats the system as three non-identical particles ($\Omega^-$, $n$, $p$), rigorously incorporating the attractive Coulomb force between the $\Omega^-$ and the proton. This breaks the isosceles triangle symmetry of the spatial configuration.

Two primary $\Omega N$ interaction potentials are tested:

• $V_{L\Omega N}$: A local potential derived from HAL QCD lattice QCD simulations (Ref. [41]), fitted with Gaussian and Yukawa-squared forms to reproduce scattering phase shifts and binding energies.
• $V_{Yu\Omega N}$: A local meson-exchange potential (Ref. [40]) based on a baryon-baryon interaction model, incorporating $\eta$ meson exchange and correlated two-meson exchanges, with short-range contact interactions.

For the nucleon-nucleon ($NN$) interaction, the Malfliet-Tjon (MT-I-III) and Afnan-Tang (ATS3) potentials are used. The calculations are performed in the $S$-wave, spin-2 channel ($^5S_2$), corresponding to the maximal spin state $(I)J^P = (0)5/2^+$. The authors calculate ground state energies, scattering lengths, effective ranges, and root-mean-square (rms) distances. The numerical approach involves both direct finite-difference methods and cluster reduction techniques (expanding wave functions on eigenfunction bases).

Key Results

• Binding Energies: Both the HAL QCD and meson-exchange potentials predict a bound $\Omega NN$ system.
  • Using the HAL QCD potential ($V_{L\Omega N}$), the binding energy is approximately 19.6 MeV (without Coulomb) and 20.5 MeV (with Coulomb).
  • Using the meson-exchange potential ($V_{Yu\Omega N}$), the binding energy is approximately 16.8 MeV (without Coulomb) and 18.1 MeV (with Coulomb).
  • These values represent a binding energy roughly ten times larger than the weakly bound $\Omega N$ pair (1–2 MeV), contrasting with the nucleon-nucleon system where the triton binding energy is only ~3.8 times the deuteron binding energy.
• Impact of Coulomb Interaction: In the ABC model, the Coulomb force increases the binding energy by approximately 0.9 MeV for the HAL QCD potential and 1.3 MeV for the meson-exchange potential. The Coulomb interaction acts as a marginal perturbation, shifting the three-body binding energy primarily by the Coulomb energy of the two-body $\Omega^-p$ subsystem.
• Spatial Configuration: The strong $\Omega N$ attraction dominates the system's structure. While the Coulomb force breaks the isosceles triangle symmetry (making the $\Omega^-p$ distance slightly smaller than $\Omega^-n$), the deviation is small. The system remains compact, with rms distances between particles around 1.5–2.0 fm. The inclusion of the $NN$ interaction further increases compactness, masking the symmetry violation caused by the Coulomb force.
• Potential Sensitivity: Low-energy characteristics (scattering length, effective range, binding energy) are highly sensitive to the form of the $\Omega N$ potential. The HAL QCD potential is more short-ranged and deeper at short distances compared to the medium-range meson-exchange potential, leading to significant differences in calculated observables.
• Mass Polarization: The study quantifies the mass polarization term (MPT), finding it to be a small contribution (~0.4 MeV) that depends weakly on the specific potential parameters but is sensitive to the mass ratios of the particles.

Significance and Claims
The paper claims to demonstrate that the large binding energy of the $\Omega^-np$ system arises primarily from the short-range behavior of the $\Omega N$ potentials, specifically the absence of a repulsive core (unlike the $NN$ force) and the presence of a strong attractive core. This allows the three-body system to achieve a deeply bound state despite the weak binding of the constituent two-body pairs.

The authors assert that the ABC model, which rigorously includes the Coulomb interaction, yields low-energy characteristics that differ slightly from previous AAC model calculations but qualitatively agree with them. The study confirms the existence of a bound or quasi-bound exotic $\Omega NN$ system (a "strange nucleus" or $\Omega d$) across different theoretical approaches.

The paper concludes that while the $\Omega d$ bound state is theoretically possible, its physical stability is questionable due to the short lifetime of the $\Omega$ baryon (~0.1 ns). However, the existence of such bound systems provides a theoretical guideline for future experimental searches and contributes to the understanding of strange matter, potentially relevant to the interior of neutron stars where high densities might stabilize systems with multiple strange quarks. The work highlights the necessity of using rigorous three-body formalisms (Faddeev equations) and accurate interaction potentials to predict the properties of exotic multiquark systems.