Sexaquarks and $H$ dibaryons in the $uuddss$ system: a comparison within a constituent quark model

Using a constituent quark model and diffusion Monte Carlo method, this study compares fully antisymmetric sexaquarks and baryon-baryon-like $H$ dibaryons in the $uuddss$ system, finding that only the latter configuration yields loosely bound states near the two-baryon threshold while all other structures remain compact.

The Gist

Imagine the universe is built from tiny, invisible Lego bricks called quarks. Usually, these bricks snap together in small, stable groups: three bricks make a proton or a neutron (which we call baryons). Two bricks make a meson.

But what happens if you try to snap six specific bricks together at once? Specifically, two "up" bricks, two "down" bricks, and two "strange" bricks?

Physicists have been arguing about this for decades. They call this six-brick cluster the H dibaryon. The big question is: Is it a single, tight, compact ball of six bricks (a "sexaquark"), or is it just two separate three-brick balls (like two protons) that happen to be holding hands loosely?

This paper is like a high-tech simulation lab where the authors built a digital model to see which of these two shapes actually works.

The Two Competing Shapes

The researchers tested two different ways these six bricks could arrange themselves:

1. The "Super-Compact" Ball (The Sexaquark):
   Imagine taking six Lego bricks and gluing them so tightly together that they become one single, dense marble. In this scenario, you can't tell one brick from another; they are all mixed up perfectly. The rules of quantum physics (the "Pauli Exclusion Principle") demand that if they are all mixed up, the whole ball must be perfectly symmetrical and antisymmetric (a fancy way of saying the arrangement must be perfectly balanced so the bricks don't crash into each other).

2. The "Hand-Holding" Pair (The H Dibaryon):
   Imagine taking two separate groups of three bricks (like two tiny triangles) and just letting them float close to each other. They might touch, but they keep their own identities. In this model, the bricks inside the first triangle are mixed up, and the bricks inside the second triangle are mixed up, but the two triangles don't have to be perfectly symmetrical with each other. This is like two people holding hands; they are a pair, but they are still two distinct people.

The Experiment: A Digital "Diffusion"

To figure out which shape is real, the authors used a super-computer method called Diffusion Monte Carlo (DMC).

Think of this like a massive, digital game of "musical chairs" played in a foggy room.

• The "chairs" are all the possible positions the six quarks could be in.
• The "fog" represents the uncertainty of quantum mechanics.
• The computer sends out thousands of "walkers" (digital ghosts) that bounce around, trying to find the most comfortable, lowest-energy spot for the six bricks to sit.

They ran this simulation twice: once forcing the bricks into the "Super-Compact" ball shape, and once allowing them to form the "Hand-Holding" pair shape.

The Results: The Verdict

Here is what they found, translated into plain English:

1. The "Super-Compact" Ball is too heavy.
When they forced the six bricks into a tight, single ball, the resulting object was very heavy. In physics, heavy usually means "unstable." It was so heavy that it would immediately fall apart into two separate three-brick groups. It's like trying to glue six heavy magnets together; the magnetic repulsion is too strong, and they pop apart.

2. The "Hand-Holding" Pair is lighter, but still not a winner.
When they let the bricks form two separate groups (the H dibaryon style), the object was lighter. However, it was still too heavy.

• The Threshold: There is a "price tag" for stability. If the six-brick object weighs less than the cost of two separate three-brick objects, it's stable. If it weighs more, it falls apart.
• The Outcome: Even the "Hand-Holding" pair weighed about 20 to 40 MeV (a tiny amount in human terms, but huge in particle physics) more than two separate groups.

3. The "Loose" Structure:
For the few cases where the "Hand-Holding" pair was the lightest option, the simulation showed something interesting: the two groups of three bricks were far apart. They were separated by a distance of about 2.5 femtometers (roughly the width of a small atomic nucleus).

• Analogy: It's not a tight hug; it's more like two people standing on opposite sides of a room, barely holding hands. They aren't really a single unit; they are just two separate people who happen to be in the same room.

The Big Conclusion

The paper concludes that, according to their model:

• There is no "Super-Compact" Sexaquark. The universe doesn't seem to like six quarks glued into a single tight marble.
• The H Dibaryon is likely not a stable, bound particle. It doesn't stick together tightly enough to exist as a permanent new particle. Instead, it looks more like two separate particles (like a Lambda and a Lambda) that are just floating near each other, ready to drift apart.

Why Does This Matter?

For decades, scientists hoped the H dibaryon might be a "dark matter" candidate or a new form of stable matter. This paper suggests that if you look for a tight, six-quark ball, you won't find it. The universe prefers to keep its six-quark systems as two separate three-quark teams, not as a single super-team.

In short: Six quarks don't want to be one big ball; they'd rather be two small teams standing a little apart.

Technical Summary

1. Problem Statement

The existence and nature of the H dibaryon (a six-quark state with quark content $uuddss$, isospin $I=0$, and spin-parity $J^P=0^+$) remains a long-standing open problem in hadron physics.

• Historical Context: Originally proposed by Jaffe (1977) using the MIT bag model as a deeply bound state ($\sim 2150$ MeV), well below the $\Lambda\Lambda$ threshold.
• Current Status: Experimental searches (e.g., NAGARA event, double-$\Lambda$ hypernuclei) and recent lattice QCD studies suggest that if the H dibaryon exists, it is either weakly bound or unbound (lying near or above the $\Lambda\Lambda$ threshold).
• Theoretical Gap: While the H dibaryon is often modeled as a loosely bound system of two baryons (e.g., $\Lambda\Lambda$, $N\Xi$, $\Sigma\Sigma$), an alternative compact structure known as the sexaquark has been proposed. In this model, all six quarks are treated as indistinguishable within a fully antisymmetric wavefunction.
• Objective: This study aims to rigorously compare the sexaquark (compact, fully antisymmetric) and the H dibaryon (baryon-baryon-like, cluster-antisymmetric) configurations within the same theoretical framework to determine their masses, stability, and spatial structures.

2. Methodology

The authors employ a Constituent Quark Model solved via the Diffusion Monte Carlo (DMC) method.

• Hamiltonian: The system is governed by a non-relativistic Schrödinger equation using the AL1 empirical potential (from Ref. [23]). The potential includes:
  • One-Gluon Exchange (OGE) with color-spin interaction.
  • Color Confinement (linear potential).
  • Center-of-mass kinetic energy subtraction.
• Wavefunction Construction: The total wavefunction is $\Psi = \Psi_{\text{radial}} \times \Psi_{\text{scf}}$ (spin-color-flavor).
  • Radial Part: A totally symmetric function dependent on inter-particle distances ($\prod e^{-a_{ij}r_{ij}}$), satisfying cusp conditions.
  • Internal Part ($\Psi_{\text{scf}}$): Constructed from eigenfunctions of spin ($S$), flavor ($F^2$), and color ($C^2$) operators, restricted to Isospin $I=0$.
• Two Distinct Configurations: The core innovation lies in how antisymmetry is imposed on the same set of quarks:
  1. Sexaquark: All six quarks are treated as indistinguishable. The wavefunction is fully antisymmetric under the exchange of any two quarks ($N_p = 720$ permutations). This represents a compact, single-cluster object.
  2. H Dibaryon: The system is partitioned into two clusters of three quarks (mimicking baryons). Antisymmetry is imposed only within each cluster, while quarks between clusters are treated as distinguishable ($N_p = 36$ permutations). This allows for "hidden color" components and baryon-baryon correlations.
• Calculation: The DMC algorithm uses importance sampling with trial wavefunctions to find the ground state energy for various quantum numbers ($S=0, 1, 2, 3$ and various flavor representations).

3. Key Contributions

• Direct Comparison: The paper provides the first direct, apples-to-apples comparison of the sexaquark and H dibaryon configurations using the same Hamiltonian and DMC solver, isolating the effect of antisymmetry constraints.
• Flavor Symmetry Analysis: The study explores both the $SU(3)_f$ symmetric limit (where $u, d, s$ are equivalent) and a symmetry-broken scenario (where $s$ quarks are distinguished by mass/position) to test the robustness of binding.
• Structural Characterization: Beyond mass, the authors calculate inter-quark distances to determine if the system forms a compact object or separates into two distinct baryon-like clusters.

4. Key Results

A. Masses and Stability

• No Bound States: For all calculated quantum numbers ($S$ and $F^2$), neither the sexaquark nor the H dibaryon forms a bound state. All calculated masses lie above the lowest two-baryon thresholds (e.g., $N\Xi$ at $\sim 2410$ MeV, $\Lambda\Lambda$ at $\sim 2424$ MeV).
• H Dibaryon vs. Sexaquark: When both configurations are allowed by symmetry, the H dibaryon configuration consistently yields lower masses than the compact sexaquark.
  • Example: For $S=0, F^2=0$, the H dibaryon mass is $2656 \pm 5$ MeV, while the sexaquark is $2659 \pm 6$ MeV.
  • The energy difference is attributed to the larger number of available channels in the H dibaryon configuration due to relaxed symmetry constraints.
• Proximity to Threshold: The lowest mass states (e.g., $S=0, F^2=3, 8$) are only $\sim 20\text{--}40$ MeV above the threshold. These are interpreted as weakly unbound or resonant states, not deeply bound particles.
• Symmetry Breaking: Breaking $SU(3)_f$ symmetry (treating $s$ quarks distinctly) does not improve binding; in fact, the states move further away from the threshold compared to the symmetric case.

B. Structural Analysis

• Compact vs. Separated: The spatial structure correlates with the mass:
  • Compact States: Configurations with higher masses (further from threshold) exhibit small inter-quark distances ($\sim 0.9\text{--}1.0$ fm) across all pairs, indicating a compact, single-cluster object.
  • Near-Threshold States: The states closest to the two-baryon threshold (e.g., $S=0, F^2=3$) show a distinct baryon-baryon structure.
    • The distance between quarks within a cluster is $\sim 0.9$ fm (similar to a standard baryon).
    • The distance between the two clusters is large ($\sim 2.5$ fm).
• Conclusion on Structure: The system naturally evolves toward a two-baryon-like configuration (two loosely bound clusters) rather than a compact sexaquark. The DMC method successfully "breaks" the system into subunits when energetically favorable, even with a symmetric radial trial function.

5. Significance and Conclusions

• Refutation of Deep Binding: Within the AL1 constituent quark model, the H dibaryon is not a deeply bound state. The results align with experimental constraints (NAGARA) that rule out deep binding.
• Nature of the State: The study suggests that if a $uuddss$ state exists near the threshold, it is likely a molecular-like state (two loosely bound baryons) rather than a compact sexaquark. The compact sexaquark configuration is energetically disfavored.
• Model Robustness: The finding that symmetry breaking does not induce binding suggests that the lack of a bound state is a robust feature of this specific quark model framework.
• Implications: The results challenge the "sexaquark" dark matter hypothesis (which relies on a stable, compact $uuddss$ state) within this specific theoretical framework, suggesting that such a state would likely decay rapidly into two baryons.

In summary, the paper concludes that the $uuddss$ system does not favor a compact, fully antisymmetric sexaquark configuration. Instead, the system prefers a baryon-baryon-like structure that remains unbound (or at best, very weakly resonant) relative to the two-baryon threshold.