Three-body resonances of $ααM$ clusters ($M=ϕ$, $J/ψ$, $η_c$) in $^{8}_{M}{\mathrm{Be}}$ nuclei

This study investigates the structural effects of $\phi$, $J/\psi$, and $\eta_c$ mesons on $^8$Be nuclei using HAL QCD potentials and the Gaussian expansion method, revealing that the $\phi$ meson strongly binds the system into stable states while $J/\psi$ and $\eta_c$ interactions are weaker and form only shallow bound states, with specific predictions for new $\alpha$-charmonium bound states and the non-Borromean nature of $^{8}_{J/\psi}\text{Be}$.

The Gist

Imagine the nucleus of an atom not as a solid ball of mush, but as a tiny, dancing group of smaller balls called alpha particles. In a specific type of atom called Beryllium-8 ($^8$Be), you have exactly two of these alpha particles. They are like two friends holding hands, but they are very unstable; they are constantly trying to let go and fly apart. They are in a "resonant" state, meaning they are vibrating on the edge of breaking up.

This paper asks a fascinating question: What happens if we introduce a third character to this dance?

The researchers imagine adding a "guest" particle—a specific type of subatomic particle called a meson—to this two-alpha dance floor. They tested three different types of guests:

1. The $\phi$ (phi) meson (a heavy particle containing strange quarks).
2. The $J/\psi$ (J/psi) meson (a heavy particle containing charm quarks).
3. The $\eta_c$ (eta-c) meson (another charm particle).

Here is what they found, using simple analogies:

1. The "Super Glue" Guest ($\phi$ meson)

When the $\phi$ meson joins the party, it acts like super-strong glue.

• The Effect: In the normal world, the two alpha particles in Beryllium-8 are shaky and unstable. But when the $\phi$ meson arrives, it grabs onto both of them so tightly that it forces them to stay together. It doesn't just stabilize them; it pulls them closer together, shrinking the distance between the two alpha particles.
• The Result: The unstable, vibrating states of the original atom become stable, solid "bound" states. The paper calls this a "glue-like" effect because the particle acts as a binding agent that holds the whole nucleus together more tightly than before.

2. The "Weak Handshake" Guests ($J/\psi$ and $\eta_c$ mesons)

When the $J/\psi$ or $\eta_c$ mesons join the party, the effect is much different.

• The Effect: These particles are like people who only offer a very weak handshake. They don't have the same "glue" power as the $\phi$ meson. In fact, they are so weak that they can only hold onto the most stable version of the atom (the ground state). They cannot stabilize the shaky, excited versions.
• The Result: Instead of pulling the alpha particles closer, these guests actually push them slightly further apart. The nucleus expands a tiny bit. They form very shallow, fragile bonds, but they don't transform the unstable atom into a stable one in the same dramatic way the $\phi$ meson does.

3. The "Borromean" Mystery

The paper also solves a little puzzle about a specific type of connection called a Borromean state.

• The Analogy: Imagine three rings linked together. If you remove any one ring, the other two fall apart. That is a Borromean state.
• The Discovery: Previous studies suggested that the atom with the $J/\psi$ meson ($^8_{J/\psi}$Be) might be Borromean (meaning the two alpha particles and the meson stick together, but the alpha and meson alone would not stick).
• The Correction: This paper found that the alpha particle and the $J/\psi$ meson can actually stick together on their own, even if just barely. Therefore, the whole system is not a Borromean state. It's more like a standard family where the parents can hold hands even without the child.

4. The Sensitivity of the Dance

The researchers also discovered that the stability of these atoms is incredibly sensitive to the "size" of the alpha particles.

• The Analogy: Imagine the alpha particles are balloons. If the balloons are slightly bigger or smaller, the "glue" effect changes.
• The Finding: For the $\phi$ meson, if the alpha particles are a certain size, the atom is stable. But if the alpha particles are just a tiny bit larger, that same stable atom suddenly becomes unstable and starts to vibrate (resonate) again. This shows that the physics of these tiny worlds is extremely delicate and precise.

Summary

In short, this paper uses advanced computer simulations to predict how adding different heavy particles to a tiny nucleus changes its behavior.

• The $\phi$ meson is a super-glue that stabilizes the nucleus and squeezes it tight.
• The $J/\psi$ and $\eta_c$ mesons are weak connectors that barely hold on and actually let the nucleus expand slightly.
• The study corrects a previous misunderstanding, showing that the $J/\psi$ system is not a "Borromean" mystery but a more standard, albeit weak, bond.

These findings provide a roadmap for future experiments, telling scientists exactly what kind of signals to look for when they try to create these exotic, heavy-atom versions of Beryllium in particle accelerators.

Technical Summary

Technical Summary: Three-body resonances of $\alpha\alpha M$ clusters ($M=\phi, J/\psi, \eta_c$) in $^8_M$Be nuclei

Problem Statement
This study investigates the structure of exotic nuclei formed by embedding heavy mesons ($\phi$, $J/\psi$, and $\eta_c$) into the $^8$Be nucleus, modeled as an $\alpha + \alpha + M$ three-body system. While meson-nucleus interactions have been studied theoretically and experimentally, the low-energy interactions of the $\phi$ meson with nucleons remain poorly understood due to the non-perturbative nature of the strong interaction. Furthermore, previous literature has largely focused on bound states, leaving the resonant states of these systems and the specific dynamic effects imparted by different mesons unexplored. A key objective is to determine whether these mesons exhibit a "glue-like" effect—contracting the nuclear core and binding excited states—similar to the $\Lambda$ hyperon, or if they behave differently due to OZI rule suppression (in the case of charmonia).

Methodology
The authors employ a cluster model where $^8$Be is treated as two $\alpha$ clusters and the exotic nucleus as an $\alpha\alpha M$ three-cluster system. The study utilizes the following computational framework:

• Hamiltonian Construction:
  • The $\alpha$-$\alpha$ interaction is described using a phenomenological potential model (double Gaussian nuclear part plus Coulomb part), with parameters selected to accurately reproduce the $^8$Be ground state and scattering phase shifts.
  • The nucleon-meson ($N$-$M$) interactions are derived from recent HAL QCD potentials obtained via (2+1)-flavor lattice QCD simulations. These potentials are fitted to Gaussian forms (and a Woods-Saxon form for specific channels) based on the HAL QCD data.
  • The effective $\alpha$-$M$ interaction is constructed by folding the $N$-$M$ potentials into the nucleon density distribution of the $\alpha$ particle (using a Gaussian density function). The study explicitly tests the sensitivity of results to the $\alpha$-particle root-mean-square (rms) matter radius ($R_\alpha$), utilizing values of 1.84, 1.70, and 1.56 fm.
• Numerical Solution:
  • The Schrödinger equation is solved using the Gaussian Expansion Method (GEM), a high-precision variational technique. The wave functions are expanded using a set of Gaussian basis functions in three Jacobian coordinates to ensure rapid convergence and accurate description of both short-range correlations and asymptotic behavior.
  • Resonant states are identified using the Complex Scaling Method (CSM). By applying a complex rotation to the coordinates, resonant states appear as $\theta$-independent complex eigenvalues ($E = E_r - i\Gamma/2$), distinct from the rotating continuum and stable bound states.

Key Contributions and Results

1. The $\phi$ Meson as a "Glue": The $\phi$ meson exhibits a strong attractive interaction, significantly stronger than that of charmonia. It binds the $0^+_1$, $2^+_1$, and $4^+_1$ resonant states of $^8$Be into stable bound states within $^8_\phi$Be. This interaction causes a significant contraction of the $\alpha$-$\alpha$ distance ($R_{\alpha\alpha}$), confirming a strong "glue-like" effect analogous to the $\Lambda$ hyperon but more pronounced.
2. Weak Charmonium Interactions: In contrast, the interactions of $J/\psi$ and $\eta_c$ with the nucleus are weaker. These mesons form only shallow bound states with the $0^+_1$ ground state of $^8$Be. Unlike the $\phi$ meson, the addition of $J/\psi$ or $\eta_c$ to the excited $2^+_1$ and $4^+_1$ states does not bind them; they remain resonant. Furthermore, for these excited states, the $\alpha$-$\alpha$ separation actually increases compared to pure $^8$Be, indicating that the meson does not compress the core in these configurations.
3. Re-evaluation of Borromean Nature: The study predicts weakly bound $\alpha$-$J/\psi$ states in the $4S_{3/2}$ and $2S_{1/2}$ channels. This finding contradicts prior literature (specifically Ref. [3]) which suggested $^8_{J/\psi}$Be might be a Borromean nucleus (bound only as a three-body system with unbound two-body subsystems). The authors conclude that $^8_{J/\psi}$Be is likely not Borromean because the $\alpha$-$J/\psi$ subsystem itself is bound. Conversely, $^8_{\eta_c}$Be remains a candidate for a Borromean nucleus as no two-body bound states were found for its subsystems.
4. Sensitivity to Nuclear Radius: The results highlight a strong sensitivity to the $\alpha$-particle radius ($R_\alpha$). For instance, the $^8_\phi$Be($4^+_1$) state transitions from a bound state to a resonant state depending on whether $R_\alpha$ is set to 1.84 fm or 1.70 fm, underscoring the subtle dynamics involved.

Significance and Claims
The paper claims to provide the first systematic theoretical comparison of how different vector mesons ($\phi$, $J/\psi$, $\eta_c$) modify nuclear clustering in the $^8$Be system using modern HAL QCD potentials. The work offers critical predictions for future experimental searches, suggesting that:

• The $\phi$ meson induces deep binding and core contraction, making $^8_\phi$Be a prime candidate for observation.
• The nature of $^8_{J/\psi}$Be is distinct from previous assumptions, specifically regarding the existence of two-body $\alpha$-$J/\psi$ bound states.
• Experimental validation of these spectra would serve as a crucial test for the HAL QCD potentials and our understanding of non-perturbative QCD dynamics in nuclear environments.

The authors note that while experimental challenges exist (such as large momentum transfer for charm production), recent upgrades at facilities like JLab and proposed experiments at J-PARC (e.g., E29) could potentially access these predicted exotic hadron-nucleus systems.