Investigation of fully heavy tetraquark within chiral quark model

Using the Chiral quark model and the real-scaling method, this study finds no bound states for fully charmed ($cc\bar{c}\bar{c}$) or fully bottomed ($bb\bar{b}\bar{b}$) tetraquarks but predicts specific resonance states that could correspond to the observed $X(6900)$ and $X(7200)$ in the charmed sector and a new resonance in the bottomed sector, respectively.

The Gist

Imagine the universe is built out of tiny, invisible Lego bricks called quarks. Usually, these bricks snap together in pairs (like a proton and an antiproton) or triplets (like a proton or neutron). But physicists have been wondering: "What if we tried to build a structure with four bricks?"

This paper is a theoretical investigation into a very specific, heavy-duty version of these four-brick structures, called fully heavy tetraquarks. Instead of using light bricks, the authors tried to build them using only the heaviest bricks available: charm quarks and bottom quarks.

Here is a simple breakdown of what they did and what they found, using some everyday analogies.

The Setup: Two Ways to Build the House

The researchers wanted to see if these four heavy bricks could stick together to form a stable "house" (a bound state) or if they would just wobble and fall apart (resonance).

They considered two different blueprints for how the bricks could be arranged:

1. The "Molecule" Blueprint: Two pairs of bricks holding hands (Quark-Antiquark + Quark-Antiquark). Think of this like two couples dancing together.
2. The "Clump" Blueprint: Two bricks of the same type huddled together, and two of the opposite type huddled together (Quark-Quark + Antiquark-Antiquark). Think of this like two teams of friends huddled up.

They ran their calculations using a set of rules called the Chiral Quark Model, which is like a sophisticated physics simulation game that predicts how these particles interact.

The Results: No Stable Houses, But Some "Bouncy" Resonances

1. The Search for a Stable House (Bound States)
First, they asked: "Can these four heavy bricks lock together so tightly that they form a permanent, stable object?"

• The Answer: No.
• The Analogy: Imagine trying to stack four heavy, slippery bowling balls on top of each other. No matter how you arrange them, they just slide apart. The math showed that for both the charm-quark version and the bottom-quark version, there is no way to build a stable, permanent house. They are too heavy and repel each other too much to stay locked in place.

2. The Search for "Bouncy" Resonances
Since a stable house wasn't possible, they asked a second question: "Can they form a temporary, wobbly structure that exists for a split second before falling apart?" In physics, this is called a resonance.

• The Analogy: Think of a trampoline. If you jump on it, you go up and come down. You aren't "stuck" to the trampoline, but you interact with it for a moment. A resonance is like a particle that "jumps" into existence, hangs out for a tiny fraction of a second, and then decays.

To find these, the authors used a special trick called the Real-Scaling Method.

• The Analogy: Imagine you are trying to find a hidden island in a foggy ocean. If you just look at the water, you might see waves that look like islands but aren't (false alarms). The "Real-Scaling Method" is like slowly changing the tide. A real island (a genuine resonance) stays put and looks different as the tide changes, while a fake wave (a false signal) just washes away. This method helped them separate the real temporary structures from the noise.

What They Found

The Charm Quark System (The "Heavy" Version)
They found two "bouncy" structures that could explain some mysterious signals scientists have already seen in experiments:

• Structure A: A resonance with a mass of about 7,002 MeV.
  • The Match: This looks very much like a particle recently discovered by the LHCb experiment called X(6900).
• Structure B: A resonance with a mass of about 7,227 MeV.
  • The Match: This looks like another structure hinted at in experiments, called X(7200).

The authors suggest that these two "bouncy" structures are likely the physical explanations for the X(6900) and X(7200) that experimentalists are seeing.

The Bottom Quark System (The "Super-Heavy" Version)
They did the same test with the even heavier bottom quarks.

• The Result: They found one "bouncy" structure at a mass of about 19,743 MeV.
• The Suggestion: Since we haven't seen this one in experiments yet, the authors are telling experimentalists: "Go look for this specific signal in the data from particle colliders, specifically by looking at the collision products of two Upsilon (Υ) particles."

The Bottom Line

In simple terms, this paper says:

1. You can't build a permanent, stable house out of four heavy quarks; they are too unstable.
2. However, they can form temporary, "bouncy" structures that last for a split second.
3. Two of these temporary structures likely explain the mysterious X(6900) and X(7200) particles we've already seen.
4. There is likely a third, super-heavy temporary structure waiting to be discovered in the bottom-quark world, and the authors have given experimentalists a specific target to hunt for.

The paper is essentially a theoretical map telling experimental physicists exactly where to look in the data to confirm these exotic, four-quark "ghosts."

Technical Summary

Technical Summary: Investigation of Fully Heavy Tetraquark within Chiral Quark Model

Problem Statement
Following the experimental discovery of the $X(6900)$ and subsequent structures like $X(7200)$ in the $J/\psi J/\psi$ invariant mass spectrum by the LHCb, CMS, and ATLAS collaborations, there is a pressing need to theoretically characterize fully heavy tetraquark states ($cc\bar{c}\bar{c}$ and $bb\bar{b}\bar{b}$). While previous theoretical works have explored these systems, recent experimental data suggests that the quantum numbers of fully charmed tetraquarks may favor $J^{PC} = 2^{++}$. Furthermore, the existence and properties of fully bottomed tetraquarks remain inconclusive, with conflicting experimental reports and a lack of definitive theoretical consensus on their bound or resonant nature. This study aims to systematically investigate the $cc\bar{c}\bar{c}$ and $bb\bar{b}\bar{b}$ systems with $J^{PC} = 2^{++}$ to identify candidates for the observed experimental structures and to clarify the status of the fully bottomed sector.

Methodology
The authors employ the Chiral Quark Model (ChQM) to investigate the four-quark systems. The Hamiltonian includes kinetic energy, confinement interactions (central and spin-orbit), and one-gluon-exchange (OGE) interactions (central, spin-orbit, and tensor). Notably, Goldstone and $\sigma$ meson exchange terms are omitted as the systems consist entirely of heavy quarks.

The wave functions are constructed considering two structural configurations:

1. Meson-meson structure: $Q\bar{Q} - Q\bar{Q}$
2. Diquark-antidiquark structure: $QQ - \bar{Q}\bar{Q}$

The total wave function incorporates orbital, flavor, spin, and color degrees of freedom. The spatial part is expanded using the Gaussian Expansion Method (GEM). The spin and orbital angular momenta are coupled to form the total $J^{PC} = 2^{++}$ state, considering six specific combinations of angular momentum couplings in the S-wave ($L_3=0$). The color space includes four structures: color-singlet-singlet ($1 \otimes 1$), color-octet-octet ($8 \otimes 8$), color-triplet-antitriplet ($\bar{3} \otimes 3$), and color-sextet-antisextet ($6 \otimes \bar{6}$).

To distinguish between bound states, genuine resonances, and scattering states, the authors utilize the Real-Scaling Method (also known as the stabilization method). This technique involves scaling the finite volume of the system; genuine resonances exhibit an "avoid-crossing" configuration in the energy spectrum as the scaling factor increases, whereas scattering states decay into threshold channels. The decay widths of identified resonances are calculated using a formula derived from the energy difference between the resonance and scattering states at the crossing point.

The study considers two scenarios for channel coupling:

• Scenario 1: Coupling of three S-wave channels ($J/\psi J/\psi$, $[J/\psi]_8[J/\psi]_8$, and $[cc]_{\bar{3}}[\bar{c}\bar{c}]_3$ for the charmed system; analogous for bottomed).
• Scenario 2: Inclusion of four additional channels composed of excited mesons ($\chi_{Q0}\chi_{Q2}$, $\chi_{Q1}\chi_{Q1}$, $\chi_{Q1}\chi_{Q2}$, $\chi_{Q2}\chi_{Q2}$), totaling seven channels.

Key Results

• Bound State Analysis:

  • For both $cc\bar{c}\bar{c}$ and $bb\bar{b}\bar{b}$ systems with $J^{PC} = 2^{++}$, the bound-state calculations reveal no bound states below the lowest physical thresholds. The energies of all considered channels remain above the respective thresholds, even when channel coupling is included.

• Resonance States in $cc\bar{c}\bar{c}$:

  • Using the real-scaling method, two genuine resonance states are identified in the $cc\bar{c}\bar{c}$ system.
  • $R(7002)$: A resonance with a mass of approximately 7002 MeV and a decay width of 54 MeV. This state persists even when channels with excited mesons are included (shifting slightly to 7023 MeV in the 7-channel calculation). The authors identify this as a candidate for the experimentally observed $X(6900)$.
  • $R(7227)$: A resonance with a mass of approximately 7227 MeV and a decay width of 66 MeV. This state is also robust against the inclusion of excited meson channels. The authors propose this as a candidate for the $X(7200)$.
  • The analysis confirms that these are genuine resonances, as their color structure proportions are high (90-92% for the 3-channel case, and significant for the 7-channel case), distinguishing them from false resonances formed by scattering states.

• Resonance States in $bb\bar{b}\bar{b}$:

  • For the fully bottomed system, only one genuine resonance state is found, regardless of whether excited meson channels are included.
  • $R(19743)$: A resonance with a mass of approximately 19743 MeV and a decay width of 67 MeV (calculated as 68 MeV in the 3-channel case and 64 MeV in the 7-channel case).
  • Other avoid-crossing structures observed (e.g., around 19645 MeV or 19745 MeV in specific configurations) were determined to be false resonances dominated by scattering channel components.

Significance and Claims
The paper claims to provide a systematic theoretical investigation of fully heavy tetraquarks with $J^{PC} = 2^{++}$ within the Chiral Quark Model framework. The primary significance lies in:

1. Interpretation of Experimental Data: The calculated resonance $R(7002)$ and $R(7227)$ offer theoretical support for the $X(6900)$ and $X(7200)$ structures observed by LHCb, CMS, and ATLAS, consistent with the $2^{++}$ quantum number assignment suggested by recent experimental trends.
2. Prediction for Future Searches: The study predicts a specific resonance state in the fully bottomed sector ($bb\bar{b}\bar{b}$) at approximately 19743 MeV with a width of ~67 MeV. The authors suggest that future experiments should search for this state in the invariant mass spectrum of $\Upsilon\Upsilon$ or $\Upsilon\Upsilon(2S)$.
3. Methodological Validation: The work demonstrates the utility of the real-scaling method in distinguishing genuine resonances from scattering states in complex multi-channel systems, confirming that the inclusion of excited meson channels does not eliminate the identified resonances but may slightly shift their energies.

The authors conclude that while no bound states exist for these $2^{++}$ systems, the existence of narrow resonances provides a viable explanation for the experimental observations in the charmed sector and offers a concrete target for future experimental verification in the bottomed sector.