⚛️ phenomenology

Short-range baryon-baryon potentials in constituent quark model revisited

Revisiting short-range baryon-baryon potentials within the constituent quark model, this study predicts that specific flavor antidecuplet states and the $N\Omega$ system possess sufficient attraction to form dibaryon bound states as hadronic molecules, while also comparing these findings with lattice QCD simulations.

Original authors: Takayasu Sekihara, Taishi Hashiguchi

Published 2026-07-22

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CC BY 4.0

Original authors: Takayasu Sekihara, Taishi Hashiguchi

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

Imagine the universe is built from a cosmic LEGO set, but instead of plastic bricks, the smallest pieces are tiny, invisible particles called quarks. These quarks are the fundamental building blocks of matter, snapping together in groups of three to form larger structures called baryons. The most famous baryons are protons and neutrons, the heavyweights that make up the nucleus of every atom in your body. But just like LEGO bricks, these quarks have rules about how they can stack up. They have "flavors" (like up, down, and strange) and "spins" (a kind of internal twirl), and they are governed by a mysterious force called the strong interaction, which acts like a super-strong rubber band holding them together.

Now, here is the big mystery: when two of these baryon-LEGO structures get close to each other, what happens? Do they bounce off like magnets with the same pole facing each other, or do they stick together to form a brand-new, double-sized super-structure? Scientists have long known that protons and neutrons can stick together to form atomic nuclei, but they are still hunting for other, stranger combinations that might exist for a split second before flying apart. Finding these "dibaryons" is like discovering a new species of animal in a deep jungle; it tells us exactly how the rules of the universe work at the smallest scales. If we can figure out which combinations stick and which don't, we unlock the secret code of how matter is held together.

In this study, two researchers from Kyoto Prefectural University decided to play a high-stakes game of "what if" using a computer simulation based on a model called the "constituent quark model." Instead of trying to catch these fleeting particles in a real lab, they built a digital playground where they could calculate exactly how different pairs of baryons would interact. They focused on the "short-range" part of the interaction, which is what happens when the two baryons get very close—closer than the width of a single proton. They used a mathematical technique called the "Gaussian expansion method" to map out the shape of the baryons and the "resonating group method" to see how two baryons dance together. Think of it as creating a 3D map of the invisible forces that push or pull these particles.

The researchers found some exciting results. They discovered that certain pairs of baryons, specifically those involving heavy, spinning particles called the $\Delta$ (Delta) and $\Omega$ (Omega), seem to have a strong enough attraction to stick together and form a bound state. These aren't just random clumps; they belong to a special family called the "flavor antidecuplet" with a total spin of $J=3$ . The study suggests that systems like $\Delta\Delta$ (two Deltas), $\Delta\Sigma^*$ , and $\Delta\Omega$ could form stable "molecules" made of six quarks. One of these, the $\Delta\Delta(3,0)$ system, is particularly interesting because it might be the same thing as a particle called $d^*(2380)$ , which has already been spotted in experiments. The researchers calculated that these bound states would have a binding energy of about 10 MeV, which is the energy holding them together.

However, the story isn't simple. The paper also rules out the idea that these particles are just compact, tight balls of six quarks squeezed into a tiny space. Instead, the simulations suggest these bound states are "hadronic molecules," meaning the two baryons are more like two friends holding hands rather than two people fused into a single body. The distance between them is larger than a typical particle, stretching out to about 1.7 to 5.2 femtometers (a femtometer is one-quadrillionth of a meter). This is a crucial distinction: it means the particles keep their own identities while sticking together.

The study also looked at a famous pair: the Nucleon (a proton or neutron) and the $\Omega$ (Omega) baryon. In a simple, single-channel view, these two shouldn't interact at all because the rules of quark shuffling would force them to change into something else immediately. But when the researchers let them interact with other possible "channels" (other ways the quarks could rearrange), a strong attraction appeared. This suggests that the Nucleon-Omega pair could form a bound state with a binding energy of 2.1 MeV, or even 10.3 MeV if you add in the effects of meson exchanges (another type of particle interaction).

It is important to note that these findings come from computer simulations, not direct measurements of these specific new states (except for the $d^*(2380)$ which was already seen). The authors are careful to say their model "suggests" these bound states exist and "indicates" they are molecules. They compared their results with data from lattice QCD (a different, very powerful type of computer simulation) and found that while both methods agree the forces are attractive, the details of the force at the very center differ. Their model shows a tiny bit of repulsion right at the start, while the other simulations show none. This difference is a puzzle that scientists are still trying to solve.

Ultimately, this paper provides a roadmap. It tells us which combinations of baryons are the most likely to stick together and form these exotic six-quark molecules. By identifying the "flavor antidecuplet" members as the strongest candidates, the study gives experimentalists a clear target list for future searches in particle accelerators. It's a bit like a treasure map that says, "Don't look here, look there," helping the scientific community hunt down the hidden secrets of how the universe's building blocks hold hands.

Technical Summary: Short-range baryon-baryon potentials in constituent quark model revisited

Problem Statement
Understanding baryon-baryon interactions is crucial for elucidating quark dynamics within hadrons. While phenomenological nucleon-nucleon (NN) potentials successfully describe low-energy scattering and the deuteron, they rely on meson exchanges for medium- and long-range forces, leaving the short-range repulsive core ( $r < 1$ fm) to be explained by quark degrees of freedom. Specifically, the Pauli exclusion principle among valence quarks and spin-spin interactions are known to govern the short-range repulsion in the NN system. However, a systematic understanding of these mechanisms across the full flavor SU(3) sector—encompassing interactions between octet ( $N, \Lambda, \Sigma, \Xi$ ) and decuplet ( $\Delta, \Sigma^*, \Xi^*, \Omega$ ) baryons—remains incomplete. Recent experimental data (e.g., $\Sigma^- p$ scattering, $\Xi$ hypernuclei) and lattice QCD simulations (e.g., HAL QCD method) have highlighted the need for precise theoretical calculations of short-range potentials to identify potential dibaryon bound states and understand the underlying quark dynamics.

Methodology
The authors employ a constituent quark model to systematically calculate short-range baryon-baryon potentials. The methodology proceeds in two main stages:

Single-Baryon Wave Functions: The internal structure of ground-state baryons (octet and decuplet) is modeled as a three-quark system. The authors solve the three-quark Schrödinger equation using the Gaussian expansion method. The inter-quark potential includes color Coulomb, linear confining, and color magnetic forces. The coupling constant for the color Coulomb force is treated as dependent on the reduced mass of the quark pair, a feature suggested by lattice QCD. Model parameters are fixed by reproducing the experimental masses of ground-state baryons.
Two-Baryon Potentials: To derive the interaction between two baryons, the authors utilize the Resonating Group Method (RGM) within a six-quark framework. They construct the two-baryon state using creation and annihilation operators for quarks, accounting for the antisymmetrization required by the Pauli principle. The RGM equation yields a non-local interaction kernel and a normalization kernel. To facilitate comparison with lattice QCD results and practical applications, the authors extract equivalent local potentials ( $V_{eq}(r)$ ) that reproduce the relative wave functions obtained from the RGM calculation. This extraction involves reducing the wave function to eliminate unphysical forbidden states (zero eigenstates of the normalization kernel) and solving for a local potential that generates the same reduced wave function at the ground-state energy.

The study systematically covers all S-wave two-baryon channels derived from ground-state baryons, analyzing both single-channel cases and coupled-channel effects where inelastic transitions (e.g., $\Delta \Xi^* \leftrightarrow \Sigma^* \Sigma^*$ ) are significant.

Key Contributions and Results

Systematic Potential Calculation: The paper provides a comprehensive set of equivalent local potentials for S-wave two-baryon systems across the flavor SU(3) sector, including channels involving unstable decuplet baryons.
Flavor Antidecuplet Bound States ( $J=3$ ): The most significant finding is the identification of attractive potentials sufficient to generate dibaryon bound states in the flavor antidecuplet sector with total spin $J=3$ $J = 3$ . Specifically, the $\Delta\Delta(3,0)$ $ΔΔ (3, 0)$ , $\Delta\Sigma^*(3,1/2)$ $Δ Σ^{*} (3, 1/2)$ , $\Delta\Xi^*-\Sigma^*\Sigma^*(3,1)$ $Δ Ξ^{*} - Σ^{*} Σ^{*} (3, 1)$ , and $\Delta\Omega-\Sigma^*\Xi^*(3,3/2)$ $ΔΩ - Σ^{*} Ξ^{*} (3, 3/2)$ systems exhibit strong attraction.
- The $\Delta\Delta(3,0)$ system is found to have a binding energy of approximately 13.1 MeV, which the authors interpret as the hadronic molecule $d^*(2380)$ , recently confirmed experimentally.
- Coupled-channel calculations confirm that the $\Delta\Xi^*-\Sigma^*\Sigma^*$ and $\Delta\Omega-\Sigma^*\Xi^*$ systems also form bound states (with binding energies $\sim 11$ MeV) due to the mixing of channels within the antidecuplet.
- Analysis of the potential components reveals that the color Coulomb plus linear confining force provides the primary attraction, which is enhanced by the color magnetic force. The strength of this attraction correlates with the spin-flavor $[33]$ component ( $N_{33}$ ), where larger $N_{33}$ values (indicating less Pauli repulsion) yield stronger attraction.
Nature of Bound States: The calculated root-mean-square radii of these bound states significantly exceed the typical hadron size ( $\sim 1$ fm), leading the authors to conclude that these states are hadronic molecules rather than compact hexaquarks.
$N\Omega$ Interaction: In the single-channel limit, the $N\Omega(2,1/2)$ interaction is absent due to inevitable transitions to inelastic channels. However, when coupled-channel effects ( $N\Omega-\Lambda\Xi^*-\Sigma^*\Xi-\Sigma\Xi^*$ ) are included, a strong attraction emerges, generating a bound state with a binding energy of 2.1 MeV. The inclusion of meson exchange contributions (specifically $\eta$ , correlated two-meson, and $K$ meson exchanges) further deepens this binding to 10.3 MeV with a decay width of 4.6 MeV, consistent with recent lattice QCD predictions.
Resonance States: The study identifies a resonance state in the $\Delta\Sigma(2,1/2)$ channel (with eigenenergy $E = 40.6 - 38.0i$ MeV) when decay channels are treated perturbatively.
Comparison with Lattice QCD: The authors compare their equivalent local potentials with those from HAL QCD simulations for $\Delta\Delta(3,0)$ , $\Omega\Omega(0,0)$ , and $N\Omega(2,1/2)$ . While both approaches predict attraction, discrepancies exist at short distances ( $r \lesssim 0.4$ fm). The constituent quark model predicts a weak repulsive core at the origin (arising from color Coulomb and linear forces), whereas the HAL QCD potentials show no such core. The authors suggest this difference may be resolved by including meson exchange contributions or investigating quark mass dependence.

Significance
The paper claims that its systematic calculation of short-range baryon-baryon potentials provides critical clues for understanding the mechanism of attractive and repulsive forces in hadron physics. By demonstrating that the flavor antidecuplet states with $J=3$ are natural candidates for dibaryon bound states (hadronic molecules), the work offers a theoretical framework to interpret experimental findings like the $d^*(2380)$ . Furthermore, the extraction of equivalent local potentials allows for direct comparison with lattice QCD results and provides a tool for other research groups to investigate two-baryon systems. The study emphasizes that both the Pauli exclusion principle for quarks and color magnetic interactions are essential for generating the observed bound states, reinforcing the role of quark degrees of freedom in short-range nuclear forces. The calculated potential strengths ( $\beta$ ) serve as a guideline for future searches for dibaryon states in lattice QCD simulations, relativistic ion collisions, and scattering experiments.

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