Fully Heavy Tetraquark States

• Fully heavy tetraquarks are exotic hadrons composed exclusively of heavy quarks (charm and bottom), providing a unique laboratory to probe QCD dynamics.
• They are studied using nonrelativistic models, QCD sum rules, and lattice techniques to resolve complex color configurations and predict mass spectra.
• Predicted mass ranges indicate these states form resonances above quarkonium thresholds with widths of 30–80 MeV, underscoring the interplay of color–electric and chromomagnetic forces.

A fully heavy tetraquark state is a color-singlet hadron composed exclusively of heavy quarks—such as charm ($c$) and/or bottom ($b$)—without valence light quarks present. The canonical configurations include $QQ\bar Q'\bar Q'$ ($Q, Q' = c, b$), encompassing systems like $cc\bar c\bar c$, $bb\bar b\bar b$, $bc\bar b\bar c$, and similar variants. These states represent an extreme in multiquark QCD spectroscopy, providing a unique laboratory for studying the interplay between color, spin, and heavy-quark symmetries, and for testing the predictive power of constituent quark models, potential models, QCD sum rules, and lattice results. The properties, stability, and decay patterns of fully heavy tetraquarks are central to understanding exotic QCD bound states beyond the standard quark–antiquark and three-quark pictures.

1. Theoretical Frameworks and Structural Classification

The majority of theoretical investigations adopt nonrelativistic or relativized quark models, QCD sum rules, or relativistic quasipotential approaches (Wang et al., 2019, Lü et al., 2020, Faustov et al., 2022, Wu et al., 26 Jan 2024, Wu et al., 15 Aug 2025). The Hamiltonian typically includes the kinetic energies of the heavy quarks, a color-dependent confinement potential, and spin–spin (chromomagnetic) interactions. For four-body systems, the spatial wavefunction is expanded variationally (often with Gaussian bases over Jacobi coordinates), and complex scaling methods are employed to distinguish resonant from bound states.

Three main structural assignments coexist in the literature:

• Diquark–antidiquark compact tetraquarks: Treated as $[QQ][\bar{Q}'\bar{Q}']$, with color antitriplet ($\bar{3}_c$) or sextet ($6_c$) diquark components (Wang et al., 2019, Xia et al., 27 Aug 2025). Pauli symmetry restricts allowable spin–color combinations.
• Molecular (dimeson) states: Constructed in the color-singlet–singlet ($1_c\otimes1_c$) basis, analogous to two weakly bound quarkonia ($Q\bar Q'$), typically disfavored for S-wave fully heavy systems due to large kinetic energy and short range (Wu et al., 26 Jan 2024).
• Hidden color octet–octet: Explicitly considered within QCD sum rules (Yang et al., 2020), featuring color-octet clusters coupled to a singlet via gluons.

The Gaussian Expansion Method (GEM) and Complex Scaling Method (CSM) combined with various potential models (e.g., AL1, AP1, BGS) are established tools for finely resolving both compact and spatially extended (molecular) configurations (Wu et al., 26 Jan 2024, Wu et al., 15 Aug 2025, Meng et al., 1 Apr 2024).

2. Mass Spectra, Quantum Numbers, and Internal Color Structure

Fully heavy tetraquark mass calculations are highly model-dependent but exhibit qualitative consistency across approaches:

• S-wave ($J^{PC}=0^{++},1^{+-},2^{++}$) $cc\bar c\bar c$ and $bb\bar b\bar b$: Ground-state masses robustly lie above the lowest ($1S$–$1S$) meson–meson thresholds (e.g., $2M_{J/\psi}$ or $2M_\Upsilon$) by 200–500 MeV (Wang et al., 2019, An et al., 2022, Faustov et al., 2022, Meng et al., 1 Apr 2024). For $cc\bar c\bar c$, predicted masses for resonant states cluster around 6.9–7.3 GeV with widths 30–80 MeV; for $bb\bar b\bar b$ they lie near 19.7–20.0 GeV.
• Excited states (radial or orbital): For $cc\bar c\bar c$, candidates for $X(6900)$ and $X(7200)$ manifest as 1$^{++}$, 2$^{++}$ (resonant) excitations at expected positions (Faustov et al., 2022, Wu et al., 26 Jan 2024, Wu et al., 15 Aug 2025). The ground state often aligns more closely with observed $X(6600)$, with the $X(6900)$ interpreted as a radially or orbitally excited tetraquark (Zhang et al., 2022).
• Color structure: Unlike mesons and baryons, tetraquarks exhibit a rich color configuration space. Both $\bar3_c\otimes3_c$ and $6_c\otimes\bar6_c$ diquark–antidiquark bases contribute, with the sextet usually lower in energy for ground states due to overall color–colorelectric (Coulomb-like) attraction, while chromomagnetic interactions induce mixing and further fine structure (Wang et al., 2019, Weng et al., 2020). Octet–octet (hidden color) configurations are also relevant in QCD sum rule analyses (Yang et al., 2020).
• Stability and decay thresholds: Theoretical consensus is that fully heavy S-wave tetraquarks are unbound with respect to strong decays, typically lying above open heavy-quarkonium ($J/\psi J/\psi$, $\Upsilon\Upsilon$) or mixed-flavor ($B_cB_c$) thresholds, except possibly in some mixed $bc\bar b\bar c$ channels for specific quantum numbers (Yang et al., 2021, Chen et al., 2022, Wu et al., 25 Jun 2024).

A summary of key mass predictions can be organized as:

System	Ground $J^{PC}$ (MeV)	First Excited (MeV)	Threshold (MeV)	Status
$cc\bar c\bar c$	6400–6500 (Zhang et al., 2022)	6900–7200 (Faustov et al., 2022, Wu et al., 26 Jan 2024)	6194 ($J/\psi J/\psi$)	Above threshold
$bb\bar b\bar b$	19200 (Zhang et al., 2022)	19770–20000	18800 ($\Upsilon\Upsilon$)	Above threshold
$bc\bar b\bar c$	12200–12400 (Yang et al., 2021, Wu et al., 25 Jun 2024)	13200–13500	12500–13300 ($B_c$)	Marginal/stable in some $J^{PC}$

3. Dynamical Mechanisms: Color, Spin, and Symmetry

The color–electric interaction (i.e., the static color Coulomb potential) is the leading term controlling the fully heavy tetraquark energy scale, favoring color–sextet diquark–antidiquark configurations as the tightly bound core. Chromomagnetic (spin–spin) interactions, which are suppressed by the large heavy-quark mass, lift degeneracies among $J^{PC}$ multiplets and drive mixing between color configurations (Weng et al., 2020). The role of color–spin mixing is particularly prominent in states with identical flavors, where Pauli symmetry restricts available spin–color combinations.

Heavy-quark symmetry underpins the systematics of the tetraquark spectrum, especially in relating doubly heavy and fully heavy configurations (Quigg, 2018). For instance, the axial-vector $bb\bar u\bar d$ state is predicted to be stable against strong decays by virtue of heavy-quark symmetry relations relating its mass to the sum of heavy–light meson masses, with the kinetic suppression making strong strong decays kinematically forbidden for sufficiently large heavy-quark mass.

4. Resonance Structure, Decay Patterns, and Spatial Configuration

Fully heavy tetraquark states above decay thresholds are predicted to be resonances with widths typically $30$–$80$ MeV (Wu et al., 26 Jan 2024, Wu et al., 15 Aug 2025). S-wave decays dominate, for instance:

• $cc\bar c\bar c$ resonances decay to $J/\psi J/\psi$ or $\eta_c \eta_c$;
• $bc\bar b\bar c$ can decay to $J/\psi\,\Upsilon$ or $B_cB_c$ depending on quantum numbers.

Exotic $C$-parity channels such as $0^{+-}$, $2^{+-}$ are characterized by the absence of open S-wave quarkonium thresholds, resulting in "zero-width" states that can only decay into P-wave quarkonia (e.g., $h_c h_c$), thus theoretically much narrower (Wu et al., 26 Jan 2024).

Calculations of root mean square (rms) interquark distances robustly indicate these states are predominantly “compact,” with all pairwise separations of order $0.2$–$0.7$ fm, distinguishing them from loosely bound hadronic molecules, which would have spatial extents $\gtrsim 1.0$ fm (Wu et al., 26 Jan 2024, Wu et al., 15 Aug 2025, An et al., 2022). Exceptions can arise in higher excitations or in cases dominated by dimeson spatial correlations, as found for certain $X(7200)$ candidates (Wu et al., 26 Jan 2024).

5. Extensions: Mixed-Flavor States and Experimental Implications

Mixed-flavor fully heavy tetraquarks such as $bc\bar b\bar c$, $bb\bar c\bar c$, $cc\bar c\bar b$, and $bb\bar b\bar c$ have been systematically analyzed using the same four-body frameworks (Wu et al., 25 Jun 2024). The lowest-lying $bc\bar b\bar c$ S-wave resonant states (in particular the $1^{++}$ and $2^{++}$ near $13.3$ GeV) are highlighted as especially promising for discovery in $J/\psi\Upsilon$ channels due to favorable kinematics and production mechanisms (Wu et al., 25 Jun 2024).

Production cross sections at high-energy colliders are evaluated via double parton scattering (DPS) models, where the cross section scales as $g(x,\mu^2)^4$ and can reach the nanobarn level for $T_{4c}$ at LHC energies; mixed-flavor tetraquarks are down by approximately two orders of magnitude (Abreu et al., 2023). Exclusive production in $e^+e^-$ ($\gamma^*$-initiated, NRQCD factorization) calculations indicate that the compact $2^{++}$ $cc\bar c\bar c$ state could be observed at Belle 2 with $50\,ab^{-1}$, with other configurations suppressed in rate (Liang et al., 24 Feb 2025). For $Z$-factory energies, the cross sections for any fully heavy tetraquark are predicted to be extremely small, rendering detection unlikely (Liang et al., 24 Feb 2025).

6. QCD Sum Rules and Lattice Results

QCD sum rule methodologies (moment and Laplace sum rules) yield mass predictions for both S-wave and exotic $J^{PC}$ manifestly consistent with quark model calculations, reinforcing the assignments of the observed $X(6900)$ as a $0^{-+}$ or $1^{-+}$ compact $cc\bar c\bar c$ tetraquark (Yang et al., 2020, Chen et al., 2022, Albuquerque et al., 2023). Sum rule analyses distinguish two mass “bands”: a broad structure at 6.2–6.7 GeV attributed to $0^{++}$ ($\overline{\eta_c}\eta_c$-like), and a narrower one at 6.8–6.9 GeV corresponding to $0^{-+}$ or $1^{-+}$ states ($\overline{\chi_{c0}}\chi_{c0}$ or $P_cP_c$). In the bottom sector, fully bottomed tetraquarks are predicted to lie below some open-bottom decay thresholds, suggesting possible strong-interaction stability in selected quantum numbers (Chen et al., 2022).

Laplace sum rule studies, incorporating Factorized Next-to-Leading Order (FNLO) QCD corrections and stability criteria, provide precise results for both fully heavy and doubly heavy tetraquarks, confirming the strong flavor and quantum number dependence of binding and resonance formation (Albuquerque et al., 2023).

7. Synthesis and Implications for QCD Spectroscopy

The landscape established by constituent quark models, QCD sum rules, and potential models indicates that:

• Fully heavy tetraquarks favor compact spatial configurations in the color–sextet channel, dominated by color–electric attraction, with secondary chromomagnetic mixing (Weng et al., 2020, Wang et al., 2019).
• All S-wave ground states for both the $cc\bar c\bar c$ and $bb\bar b\bar b$ systems lie above open-flavor or quarkonium decay thresholds, predicting resonance rather than bound state behavior (An et al., 2022, Faustov et al., 2022, Meng et al., 1 Apr 2024).
• Exceptions exist for certain color–spin configurations in mixed-flavor ($bc\bar b\bar c$) systems, where kinematic suppression of decays could yield relatively narrow, possibly observable states (Yang et al., 2021, Wu et al., 25 Jun 2024).
• The observed $X(6900)$ and $X(7200)$ are best interpreted as higher-lying $cc\bar c\bar c$ resonances with quantum numbers $0^{++}$ or $2^{++}$ (Wu et al., 26 Jan 2024, Wu et al., 15 Aug 2025); lower structures such as the $X(6600)$ may correspond to S-wave ground states in some models (e.g., (Zhang et al., 2022)) or to P-wave excitations in others (Xia et al., 27 Aug 2025).
• Experimental verification will allow direct probes of the color structure and binding mechanisms of heavy multiquark systems, providing critical tests of both color-confining dynamics and the role of diquarks as QCD constituents (Quigg, 2018).

A plausible implication is that with increasing heavy-quark mass, fully heavy tetraquark stability improves due to enhanced color–electric attraction, but only for configurations and quantum numbers where the decay phase space vanishes or is kinematically suppressed. The confirmation of compact fully heavy tetraquarks and mapping of their spectra will serve as benchmarks for disentangling confining forces and color correlations in exotic QCD matter.