Fully heavy tetraquark states with a diquark-antidiquark configuration

This paper systematically investigates the mass spectra and decay channels of fully heavy tetraquark states using a diquark-antidiquark model, concluding that while $X(6600)$, $X(6900)$, and $X(7200)$ are unlikely to be $1S$-wave fully charmed tetraquarks,$X(6200)$is a strong candidate for the$2^{++}$ state and predicting several narrow states for future experimental observation.

The Gist

Imagine the universe is built from tiny, invisible Lego bricks called quarks. Usually, these bricks snap together in very specific, stable patterns: two bricks make a "meson" (like a proton's cousin), and three bricks make a "baryon" (like a proton or neutron).

But physicists have recently started finding strange, "exotic" structures made of four bricks stuck together. These are called tetraquarks.

This paper is a theoretical investigation into the heaviest, most extreme version of these four-brick structures: Fully Heavy Tetraquarks. These are made entirely of the universe's heaviest bricks: either four "charm" quarks or four "bottom" quarks.

Here is a breakdown of what the authors did, using simple analogies:

1. The Setup: A Heavy Metal Dance Floor

Usually, when quarks interact, they can exchange light particles (like pions) to stick together, kind of like dancers holding hands. But in a "fully heavy" tetraquark, the bricks are so massive and heavy that they can't do the light dance. They are stuck in a tight, compact cluster, interacting only through the "glue" of the strong force (gluons).

The authors treated these four quarks as two pairs: a diquark (two quarks holding hands) and an antidiquark (two anti-quarks holding hands). They are like two heavy couples dancing in a very small room, spinning and orbiting each other.

2. The Goal: Predicting the "Heights" and "Moves"

The researchers wanted to answer two main questions:

• How heavy are these dancers? (Calculating their mass).
• How do they fall apart? (Calculating how they decay into lighter particles).

To do this, they built a mathematical model based on how quarks spin and orbit. Think of it like a complex dance choreography where the spin of the dancers and their orbital speed determine the total energy (mass) of the group.

3. The Big Discovery: The "X" Mystery

In recent years, giant particle colliders (like the LHC) have spotted some mysterious bumps in their data, labeled X(6600), X(6900), and X(7200). Scientists were guessing: "Are these the four-charm-quark tetraquarks we've been looking for?"

The Paper's Verdict:

• No, not those ones. The authors calculated that the simplest, most stable version of these four-charm-quark groups (called the "1S-wave") should be much lighter than 6600 or 6900.
• Yes, maybe this one. They found a candidate at 6260 MeV (roughly 6.26 GeV) with specific spin properties. This matches a signal called X(6200) seen in recent data. They suggest X(6200) is the real "four-charm" tetraquark, while the heavier X(6600/6900/7200) might be something else entirely (perhaps excited states or different combinations).

4. The "Bottom" Heavyweights

They also looked at tetraquarks made of four "bottom" quarks.

• The Prediction: They calculated these should exist around 19,000 MeV.
• The Reality Check: Experiments have seen hints of something around 18,400 MeV, but the authors' calculations suggest the stable versions are slightly heavier. They conclude that while these particles likely exist, we haven't clearly seen them yet, and the signals we have might be something else.

5. The "Mixed" Couples

The paper also looked at tetraquarks made of a mix: two charm quarks and two bottom quarks (or one of each).

• Because these mix different types of quarks, the rules of symmetry are looser. This allows for a much richer "menu" of possible states.
• The authors identified several of these mixed states that are narrow (meaning they don't fall apart quickly). In the world of particle physics, "narrow" is good—it means the particle lives long enough to be clearly spotted by detectors. They predict these could be found in future experiments.

Summary: What Does This Mean for You?

Think of this paper as a map for treasure hunters.

• The "treasure" is the fully heavy tetraquark, a new form of matter.
• The "map" says: "Don't dig at the spots marked X(6600) or X(6900); you won't find the simple four-charm treasure there."
• Instead, "Dig at X(6200); that's where the simple four-charm treasure is likely buried."
• Also, "Keep an eye out for the mixed charm-bottom treasures; they might be hiding in plain sight because they are very stable."

The Takeaway:
This research helps us understand the fundamental rules of how matter holds together. By proving that some experimental signals are not what we thought they were, and pointing toward new candidates, the authors are helping physicists refine their search for the building blocks of the universe. It's a reminder that in the subatomic world, just because something looks heavy and exotic, it doesn't always mean it's the specific thing we were looking for.

Technical Summary

Here is a detailed technical summary of the paper "Fully heavy tetraquark states with a diquark-antidiquark configuration" by Xi Xia and Tao Guo.

1. Problem Statement

The paper addresses the theoretical characterization of fully heavy tetraquark states (systems composed of four heavy quarks: $cc\bar{c}\bar{c}$, $bb\bar{b}\bar{b}$, and mixed flavors like $cb\bar{c}\bar{b}$).

• Context: Recent experimental observations by LHCb, CMS, and ATLAS have reported structures in the di-$J/\psi$ and di-$\Upsilon$ invariant mass spectra, notably $X(6200)$, $X(6600)$, $X(6900)$, and $X(7200)$.
• The Challenge: There is significant theoretical disagreement regarding the nature of these states. Some models interpret them as $1S$-wave states, while others suggest$2S$or$P$-wave configurations. Furthermore, the existence of bound states below the double-meson thresholds remains a subject of debate, with lattice QCD and various quark models yielding contradictory results.
• Goal: To systematically investigate the mass spectra and strong decay channels of fully heavy tetraquarks using a specific diquark-antidiquark model based on one-gluon exchange, aiming to clarify the quantum numbers and identities of the observed experimental structures.

2. Methodology

The authors employ a non-relativistic effective Hamiltonian approach within the diquark-antidiquark configuration $[QQ][\bar{Q}\bar{Q}]$.

• Effective Hamiltonian:
  The system is modeled using an improved non-relativistic Hamiltonian (Eq. 1) consisting of:
  $$H = \sum m_i + H_{SS} + H_{SL} + H_L$$

  • $H_{SS}$: Spin-spin interaction between quark pairs.
  • $H_{SL}$: Spin-orbit coupling.
  • $H_L$: Pure orbital angular momentum contribution.
  • The interaction is dominated by one-gluon exchange, as light-meson exchange is forbidden in fully heavy systems.

• Wave Functions and Symmetry:

  • Basis wave functions are constructed considering flavor, color, and spin.
  • The Pauli exclusion principle is strictly applied: for identical flavor diquarks (e.g., $cc$), the spin and color symmetries must be opposite.
  • The color structure is treated as a superposition of $\bar{3}-3$ (antitriplet-triplet) and $6-\bar{6}$ (sextet-antisextet) configurations.

• Parameter Determination:

  • Constituent quark masses ($m_c = 1556$ MeV, $m_b = 4755$ MeV) and coupling parameters ($\kappa_{ij}$) are fitted to experimental data of conventional mesons and baryons.
  • For unobserved states, parameters are cross-referenced with Lattice QCD calculations.
  • Spin-orbit and orbital coefficients ($A$ and $B$) are fitted using charmonium and bottomonium spectra ($J/\psi, \chi_c, \Upsilon, \chi_b$).

• Decay Calculations:

  • Strong decay widths are calculated using the quark rearrangement mechanism, where the tetraquark decays into two mesons.
  • Partial decay widths are derived using phase space integration (Eq. 11).
  • Conservation laws (angular momentum, parity, and charge conjugation $C$) are strictly enforced to determine allowed decay channels.

3. Key Contributions

1. Systematic Spectroscopy: The paper provides a comprehensive mass spectrum for $S$-wave ($L=0$) and $P$-wave ($L=1$) fully heavy tetraquarks across three systems: $[cc][\bar{c}\bar{c}]$, $[bb][\bar{b}\bar{b}]$, and mixed flavors ($[cb][\bar{c}\bar{b}]$, $[cc][\bar{c}\bar{b}]$, etc.).
2. Resolution of Experimental Ambiguities: The authors explicitly test the hypothesis that the recently observed $X(6600)$, $X(6900)$, and $X(7200)$ are $1S$-wave fully charmed tetraquarks.
3. Candidate Identification: The study proposes a specific candidate for the $X(6200)$ resonance and identifies several narrow states in the $P$-wave and mixed-flavor sectors that are promising targets for future experiments.

4. Key Results

A. Fully Charmed Tetraquarks ($cc\bar{c}\bar{c}$)

• $1S$-Wave States:

  • The calculated masses for $1S$-wave states are:
    • $0^{++}$: 6098 MeV and 6314 MeV.
    • $1^{+-}$: 6204 MeV.
    • $2^{++}$: 6261 MeV.
  • Conclusion on $X(6600/6900/7200)$: The results do not support interpreting $X(6600)$, $X(6900)$, or $X(7200)$ as $1S$-wave states, as their predicted masses do not align with the experimental values.
  • Conclusion on $X(6200)$: The state with $J^{PC} = 2^{++}$ at 6260.89 MeV is identified as a strong candidate for the experimental $X(6200)$. It decays primarily to $J/\psi J/\psi$ with a narrow width, consistent with recent LHCb observations.

• $P$-Wave States:

  • Several $P$-wave states are predicted above the double-meson thresholds (e.g., $\eta_c \chi_{c0}$, $J/\psi \chi_{c1}$).
  • The $1^{--}$state at 6740 MeV is noted as potentially observable in the$J/\psi \chi_{c1}$and$J/\psi \chi_{c2}$ channels.

B. Fully Bottom Tetraquarks ($bb\bar{b}\bar{b}$)

• All calculated $S$-wave states lie above the $\Upsilon \Upsilon$ threshold (lowest mass $\approx 18949$ MeV).
• The results suggest no bound states below the non-interacting thresholds, consistent with some lattice QCD findings.
• The predicted masses are higher than the excess observed by CMS/ANDY ($\sim 18.1-18.4$ GeV), implying the experimental signal might not be a simple $1S$-wave tetraquark or requires further investigation.

C. Mixed Flavor Systems ($cb\bar{c}\bar{b}$, etc.)

• $[cb][\bar{c}\bar{b}]$ System: Due to the lack of identical quarks, the Pauli principle imposes fewer restrictions, leading to a richer spectrum.
  • Narrow Resonances: Several states are predicted with narrow decay widths, making them excellent candidates for future observation.
  • Specific Candidates:
    • A $1^{++}$state at **12665.6 MeV** is predicted to be narrow and observable in the$B_c^* \bar{B}_c^*$ channel.
    • A $1^{--}$state at **13067.9 MeV** is very close to the$B_c^* \bar{B}_{c1}$ threshold, suggesting it could appear as a narrow resonance.
    • A $1^{-+}$state at **12993.1 MeV** is predicted to form a peak in the$h_c \Upsilon$ invariant mass spectrum.

5. Significance

• Theoretical Clarification: The work provides a rigorous constraint on the interpretation of LHCb/CMS/ATLAS data, effectively ruling out the $1S$-wave assignment for the higher-mass structures ($X(6600/6900/7200)$) within this specific diquark model.
• Experimental Guidance: By predicting specific masses, quantum numbers, and dominant decay channels (e.g., $J/\psi J/\psi$ for $X(6200)$, $B_c^* \bar{B}_c^*$ for mixed states), the paper offers concrete targets for future experimental searches at the LHC and other high-energy facilities.
• Model Validation: The successful identification of the $X(6200)$ as a $2^{++}$ state reinforces the validity of the diquark-antidiquark model with one-gluon exchange for describing compact fully heavy hadrons.

In summary, this paper refines our understanding of the fully heavy tetraquark spectrum, successfully linking a theoretical $2^{++}$state to the experimental$X(6200)$ while predicting a host of new, narrow resonances in the mixed-flavor sector that await experimental confirmation.