Fully-heavy tetraquarks: $bb\bar{c}\bar{c}$ and $bc\bar{b}\bar{c}$

Xiaoyun Chen

REVIEW 4 major objections 5 minor 40 references

Fully-heavy tetraquarks: $bb\bar{c}\bar{c}$ and $bc\bar{b}\bar{c}$

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Fully-heavy tetraquarks with two bottom and two charm quarks do not bind; the model finds only above-threshold resonances, with specific masses near 13 GeV.

desk verdict Competent no-bound-state calculation, but the resonance masses from the real scaling method are not established and need either much stronger analysis or much softer language. read the letter →

arxiv 1908.08811 v2 pith:QIGAJWS7 submitted 2019-08-22 hep-ph

Xiaoyun Chen This is my paper · ORCID

classification hep-ph

keywords fully-heavytetraquarkschiralquarkmodelGaussianexpansionmethodrealscalingexotichadronsresonancesboundstatesbottom-charm

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Does a four-quark system made only of heavy quarks—two bottom quarks with two anticharm quarks ($bb\bar{c}\bar{c}$), or bottom-charm with their antiparticles ($bc\bar{b}\bar{c}$)—form a bound tetraquark? The paper argues no: in a nonrelativistic chiral quark model, with meson-meson, diquark-antidiquark, and mixed configurations, every low-lying state lands above the corresponding two-meson decay threshold. With no room for a bound state, the model instead predicts a set of resonances, at about 13.1–13.2 GeV for $bb\bar{c}\bar{c}$ and 12.9–13.0 GeV for $bc\bar{b}\bar{c}$. A sympathetic reader would care because it gives concrete mass targets and says where not to look for stable fully-heavy tetraquarks.

What carries the argument

The machinery is the nonrelativistic chiral quark model Hamiltonian $$H=\sum_i m_i+\frac{p_{12}^2}{2\mu_{12}}+\frac{p_{34}^2}{2\mu_{34}}+\frac{p_{1234}^2}{2\mu_{1234}}+\sum_{i<j}\left(V^C_{ij}+V^G_{ij}\right),$$ with a central confinement term $V^C_{ij}=(-a_c r_{ij}^2-\Delta)\lambda^c_i\cdot\lambda^c_j$ and a one-gluon-exchange term containing Coulomb, hyperfine, and smeared delta pieces. Parameters are fixed by meson spectra. The four-body Schrödinger equation is solved by the Gaussian expansion method, expanding each relative coordinate in Gaussians with geometric progression size parameters and including all color, spin, and flavor channels for both meson-meson and diquark-antidiquark structures. Genuine resonances are distinguished from continuum with the real scaling method, which multiplies the Gaussian size parameters of the color singlet-singlet meson-meson channel by a factor $\alpha$; a true resonance keeps its energy stable as $\alpha$ grows, while continuum states fall to thresholds.

What would settle it

A lattice QCD computation of the lowest $bb\bar{c}\bar{c}$ or $bc\bar{b}\bar{c}$ eigenvalue that comes out below the corresponding two-meson threshold—for $bc\bar{b}\bar{c}$ below $\eta_b+\eta_c$ at 12321 MeV in this model—or an experimental discovery of a narrow fully-heavy tetraquark below its two-meson threshold would falsify the paper's no-bound-state claim.

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Extended reading notes

Core claim

The central discovery is a negative result with positive predictions: no bound $bb\bar{c}\bar{c}$ or $bc\bar{b}\bar{c}$ tetraquark exists in this model, because the color matrix elements of the tetraquark and its two-meson decay products are identical and the color-magnetic interaction is never attractive enough to pull the energy below threshold. After solving the four-body Schrödinger equation, all computed eigenvalues sit above the relevant meson-pair thresholds. Using the real scaling method, the paper identifies stable resonance plateaus: for $bb\bar{c}\bar{c}$, 13140, 13180, and 13230 MeV for $0(0^+)$, $0(1^+)$, and $0(2^+)$; for $bc\bar{b}\bar{c}$, 12860, 13020, 13020, and 12910 MeV for $0(0^{++})$, $0(1^{++})$, $0(2^{++})$, and $0(1^{+-})$.

Load-bearing premise

The model's Hamiltonian, with only central confinement and one-gluon-exchange terms and parameters fixed by meson spectra, correctly describes fully-heavy four-quark dynamics, and the real scaling method reliably separates true resonances from continuum in these systems.

Editorial extensions

If this is right

No stable $bb\bar{c}\bar{c}$ or $bc\bar{b}\bar{c}$ tetraquark should exist below the lowest two-meson threshold; experimental searches should target above-threshold resonances instead of stable particles.
The predicted lowest resonances are 13140, 13180, and 13230 MeV for $bb\bar{c}\bar{c}$ $0(0^+)$, $0(1^+)$, $0(2^+)$, and 12860, 13020, 13020, and 12910 MeV for $bc\bar{b}\bar{c}$ $0(0^{++})$, $0(1^{++})$, $0(2^{++})$, $0(1^{+-})$, providing concrete mass targets.
The color interaction contributes no binding energy because the color matrix element of the tetraquark equals that of its two-meson pair; the color-magnetic interaction does not improve binding for these heavy systems.
Mixing between the meson-meson and diquark-antidiquark structures changes the ground-state energies very little, so either structure alone gives nearly the same thresholds and resonance pattern.
For $bc\bar{b}\bar{c}$, charge-parity is a good quantum number, and the model places the $1^{+-}$ resonance below the $1^{++}$ one, giving two C-parity partners with different masses.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the no-bound-state result holds, binding in fully-heavy tetraquarks would require effects beyond central confinement plus one-gluon-exchange, such as tensor forces or explicit coupled-channel dynamics; the predicted resonance masses could shift by tens of MeV under those additions.
The near equality of color matrix elements between tetraquark and two-meson configurations suggests the tetraquark wave function is largely a weakly interacting meson pair; one testable consequence is that decay widths of the predicted resonances should be broad enough to appear as enhancements rather than narrow peaks.
The real scaling method was applied only to meson-meson color singlet-singlet channels; applying it also to diquark-antidiquark or octet-octet channels might reveal additional resonances or shift the lowest ones.
The predicted $bc\bar{b}\bar{c}$ $1^{+-}$ resonance at 12910 MeV lies below its $1^{++}$ partner at 13020 MeV; if produced, C-parity selection rules could distinguish them in final states such as $\eta_b J/\psi$ versus $\Upsilon J/\psi$, offering a test of the model.

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Signed reviews

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear X,

The one-liner: this is a competent variational calculation of the bbcc and bcbc tetraquark spectra; the no-bound-state conclusion is plausible and consistent with several previous potential-model studies, but the resonance masses extracted from the stabilization procedure are not convincing as quantitative predictions.

What is new: the paper extends the author's earlier full-bottom study to the mixed-heavy systems, explicitly including meson-meson and diquark-antidiquark structures and their mixing. The parameters are fixed by the meson spectrum, not fitted to tetraquarks, which is good practice. The numerical ground-state energies sit above the corresponding two-meson thresholds, and the qualitative color-magnetic analysis in Table V supports that. That negative-result part is solid, to the extent a variational upper-bound calculation can be.

The soft spots: first, the Summary's phrase "leaving no space for a bound state" is too strong. A variational energy above threshold does not prove the absence of a bound state; it only shows this basis and Hamiltonian did not find one. The conclusion is probably right, but the wording overstates what a variational calculation demonstrates. Second, the resonance claims rest entirely on the real scaling of only the meson-meson color singlet-singlet configurations. No widths, no pole locations, and no complex-scaling or phase-shift analysis are provided. The claimed states lie 180 to 540 MeV above threshold, where broad resonances are expected, and the paper itself concedes they may be too wide to observe. The plateau in the stabilization plot could be an artifact of the restricted basis. A referee should ask for a justification of the single-channel scaling and for some estimate of the width. Third, all orbital angular momenta are set to zero; D-wave channels are omitted, which can matter at these energies.

Who this is for: hadron spectroscopists interested in LHCb searches for fully-heavy tetraquarks. They get a useful negative result on bound states and a set of mass windows, but the resonance masses should be treated as indicative, not predictive.

Recommendation: this deserves peer review, not a desk reject. The calculation is honest and the negative result is worth having, but the resonance analysis needs either a more rigorous treatment (complex scaling, phase shifts, widths) or a much more cautious statement before publication.

Referee Report

4 major / 5 minor

Summary. This paper studies the fully-heavy tetraquark systems $bb\bar{c}\bar{c}$ and $bc\bar{b}\bar{c}$ within a nonrelativistic chiral quark model, using the Gaussian expansion method. The calculation includes meson-meson and diquark-antidiquark structures, their mixing, all allowed color and spin configurations, and antisymmetrization for $bb\bar{c}\bar{c}$. The variational energies in Tables VI and VII lie above the corresponding two-meson thresholds in every channel, from which the paper concludes that no bound tetraquark exists in these systems. Applying the real scaling (stabilization) method to the meson-meson color singlet-singlet configurations, the paper then identifies resonance candidates: $13140$, $13180$, and $13230$ MeV for $bb\bar{c}\bar{c}$ with $0(0^+)$, $0(1^+)$, and $0(2^+)$, respectively, and $12860$, $13020$, $13020$, and $12910$ MeV for $bc\bar{b}\bar{c}$ with $0(0^{++})$, $0(1^{++})$, $0(2^{++})$, and $0(1^{+-})$, respectively.

Significance. If the results are correct, the paper provides a useful systematic survey of two debated fully-heavy tetraquark systems, supporting the no-bound-state side of the current controversy while making concrete, testable resonance predictions. The variational bound-state part is credible: model parameters are fixed by meson spectra rather than fitted to the tetraquark masses, the color-spin channel space is broad, and the CMI analysis in Table V gives a transparent qualitative consistency check. However, the distinctive new claim of the paper is the resonance spectrum, and that claim rests on a stabilization procedure whose validation is incomplete: no widths or pole positions are computed, only one color configuration is scaled, and all orbital angular momenta are set to zero. The resonance predictions are therefore not yet established at the level required for a definitive statement.

major comments (4)

[III, after Eq. (18), Figs. 3-6] The real scaling method is applied only to the Gaussian size parameters of the meson-meson color singlet-singlet configurations while all other channels are kept fixed. In a truncated coupled-channel basis, a flat energy-versus-α line can also arise from an avoided crossing or a pseudostate, and the standard confirmation of a resonance is the location of a pole in the complex energy plane, for example by complex scaling or phase-shift/Jost-function analysis, together with a width. The paper reports no such pole or width, and in Sec. IV it concedes that the higher states 'may be too wide to be observed.' Given that the claimed resonances lie several hundred MeV above their lowest thresholds, the stabilization plateaus in Figs. 3-6 are insufficient by themselves to establish these states as genuine resonances.
[II, Eq. (14) and following text] All orbital angular momenta are taken to be zero in this calculation. This omits tensor-coupled D-wave meson-meson channels, which can mix with the S-wave basis at excitation energies of several hundred MeV above threshold. The omission affects both the variational energies and the interpretation of the stabilization plateaus, so the resonance positions reported in Sec. IV are not robust against this truncation.
[III and Summary, Tables VI-VII] The conclusion that no bound state exists is based on variational upper bounds $E_{cc}$ that lie above the theoretical thresholds. Because the Gaussian expansion is variational, an energy above threshold in a finite basis does not rigorously exclude a bound state in the full Hilbert space. The summary phrase 'leaving no space for a bound state' is therefore stronger than what Tables VI and VII demonstrate. The CMI argument in Table V makes the absence of a deeply bound state plausible, but it does not close the variational gap.
[II, Eq. (18)] The Gaussian basis parameters $r_1$, $r_{n_{\max}}$, and $n_{\max}$ are never specified, and no convergence study or numerical uncertainty is reported for either the variational energies or the stabilization plateaus. Without this information the reader cannot assess whether the quoted resonance energies, for example 13140 MeV for the $0^+$ $bb\bar{c}\bar{c}$ state, are numerically converged.

minor comments (5)

[Introduction] There is a typographical error: 'lager masses' should be 'larger masses'.
[II, Eq. (4)] The running coupling in Eq. (4) should explicitly state that the argument of the logarithm is dimensionless; as written, $\mu_{ij}$ and $\mu_0$ are in MeV while $\Lambda_0$ is given in fm$^{-1}$ in Table I, so the units should be reconciled.
[III, Figs. 3-6] The captions of the stabilization figures do not identify which horizontal lines correspond to which threshold; adding this information would make the figures substantially easier to interpret.
[Tables VI-VII] The notation for charged mesons is inconsistent, with both $B_c^-$ in the text and $B_c$ in the table column headers; the table should use a single consistent notation.
[References] Some references are given only as arXiv preprints even though published versions exist; updating these would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: parameters are fixed by meson spectra rather than by tetraquark data, and the predicted resonance energies emerge from a stabilization scan rather than from fitted inputs.

full rationale

The paper's derivation chain is self-contained in the relevant sense. The Hamiltonian in Eq. (1) and its parameters in Table I are determined by fitting the meson spectrum, not by any tetraquark mass or resonance input; the tetraquark energies and the resonance candidates are then computed as eigenvalues of the same Hamiltonian in a constructed variational basis. The close agreement between the pure meson-meson color singlet-singlet eigenvalues and the corresponding two-meson thresholds is a consequence of the color algebra (Table IV shows vanishing inter-cluster color matrix elements for the 1 x 1 configuration), so it is a consistency check rather than a circular prediction. The no-bound-state conclusion is a variational result including diquark-antidiquark, meson-meson, and mixed configurations, and it is reinforced by the CMI analysis in Table V. The resonance energies are identified by real scaling of the Gaussian size parameters, and no resonance energy is fitted to any prior tetraquark result. Self-citations to Refs. [24], [33], and [35] supply model details and prior applications, but the central calculation is presented with explicit equations and externally grounded method citations ([29]-[32], [34]); none of these self-citations is load-bearing in the sense of importing an unverified uniqueness theorem or defining the outcome into the input. Concerns about whether the stabilization procedure reliably distinguishes genuine resonances from continuum artifacts are physical and methodological validity questions, not circularity, and therefore do not raise the circularity score.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The calculation uses an established quark model with parameters fixed by ordinary meson spectroscopy, plus a variational basis and a stabilization method. No new particles or forces are introduced. The main extras beyond the model inputs are the numerical choices for the Gaussian basis and the real scaling protocol, which are not fully documented.

free parameters (4)

Constituent quark masses m_c=1728 MeV, m_b=5112 MeV (also m_u=m_d=313, m_s=536) = 1728, 5112, 313, 536 MeV
Fixed by fitting meson spectra; heavy quark masses set the scale of the tetraquark energies.
Confinement parameters a_c=101 MeV fm^-2, Delta=-78.3 MeV = 101 MeV fm^-2, -78.3 MeV
Determined by fitting meson spectrum; controls the long-range potential in the tetraquark Hamiltonian.
OGE running-coupling parameters alpha0=3.67, Lambda0=0.033 fm^-1, mu0=36.98 MeV, s0=28.17 MeV = 3.67, 0.033 fm^-1, 36.98 MeV, 28.17 MeV
Fitted to meson spectrum; these set the strength and scale of the one-gluon-exchange interaction.
Gaussian basis parameters r1, rnmax, nmax (not tabulated)
Chosen by hand for the Gaussian expansion; not specified in the paper, so numerical convergence cannot be independently checked.

assumptions (4)

domain assumption The chiral quark model Hamiltonian of Eq. (1), with linear confinement and one-gluon exchange, is a valid description of fully-heavy tetraquarks.
The entire calculation rests on this model; no QCD first-principles justification is given beyond meson-spectrum fits.
domain assumption Only central parts of the potentials are needed for the low-lying states considered here.
The text states 'only central parts are presented' before Eq. (2) and sets all orbital angular momenta to zero; tensor and spin-orbit effects are not estimated.
standard math The Gaussian expansion method converges to the exact solution of the model Hamiltonian for the chosen basis.
GEM is a standard variational expansion; convergence with nmax is asserted but not demonstrated in this paper.
domain assumption The real scaling method identifies genuine resonances when eigenenergies stay stable as the Gaussian width is scaled.
Stabilization is a standard heuristic; the paper applies it to only the meson-meson color singlet-singlet channel and does not compute widths.

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Cite this review

Pith. "Pith review of Fully-heavy tetraquarks: $bb\bar{c}\bar{c}$ and $bc\bar{b}\bar{c}$." pith.science (2026). https://pith.science/paper/QIGAJWS7

@misc{pith2026190808811,
  author       = {Pith},
  title        = {Pith review of: Fully-heavy tetraquarks: $bb\barc\barc$ and $bc\barb\barc$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QIGAJWS7}},
  note         = {Machine review of arXiv:1908.08811}
}

abstract

In the framework of a nonrelativistic chiral quark model, we continue to study the mass spectra of the fully-heavy $bb\bar{c}\bar{c}$ and $bc\bar{b}\bar{c}$ tetraquarks. In the present calculations, two structures, meson-meson [$\bar{Q}Q$][$\bar{Q}Q$] and diquark-antidiquark [$QQ$][$\bar{Q}\bar{Q}$] ($Q$ = $c$ or $b$), and their mixing, along with all possible color, spin configurations are considered. The calculations suggest that no bound state can be formed for $bb\bar{c}\bar{c}$ and $bc\bar{b}\bar{c}$ systems. However, resonances are possible because of the color structure. Several resonances are predicted and their stabilities are checked using the real scaling method.

Figures

Figures reproduced from arXiv: 1908.08811 by the authors.

**Figure 3.** FIG. 3. The stabilization plots of the energies of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** FIG. 4. The stabilization plots of the energies of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: FIG. 5. The stabilization plots of the energies of [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. The stabilization plots of the energies of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Choi et al

Belle Collaboration (S.-K. Choi et al. ), Phys. Rev. Lett. 91, 262001 (2003)

work page 2003

[2] [2]

BaBar Collaboration (Aubert B et al. ), Phys. Rev. Lett. 95, 142001 (2005)

work page 2005

[3] [3]

C”, for example χ f 3,C m0 , are the charge conjugate of the corresponding wave function withou t superscript “C

are taken into account. For bb¯c¯c state, the wave functions need to be antisymmetrized. All the allowed channels are demonstrated in Table III. Forbc¯b¯c, TABLE III. The allowed channels of bb¯c¯c andbc¯b¯c systems in meson-meson (M-M) and diquark-antidiquark (D-A) struc- tures. χ σ 34f 3c1, 2∓ m = √ 1 2 (χ σ 3 1 χ f 3 m0χ c1, 2 m ± χ σ 4 1 χ f 3,C m0 χ ...

work page

[4] [4]

Ablikim et al

BESIII Collaboration, (M. Ablikim et al. ), Phys. Rev. Lett. 110, 252001 (2013). 8

work page 2013

[5] [5]

Belle Collaboration, (Z. Q. Liu et al. ), Phys. Rev. Lett. 110, 252002 (2013)

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[6] [6]

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coupling, respectively. The detailed coupling process for the color wave functions can refer to our previous work [ 35]. (2) Meson-meson structure. For the spin part, the wave functions are the same as those of the diquark-antidiquark structure, Eq. ( 6). For the ﬂavor part, there are three wave functions, one function for bb¯c¯c system, χ f 1 m0 = (¯cb)(...

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