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arxiv: 2604.03047 · v2 · submitted 2026-04-03 · ✦ hep-ph

All-heavy tetraquarks with different flavors

Pith reviewed 2026-05-13 17:49 UTC · model grok-4.3

classification ✦ hep-ph
keywords all-heavydifferentflavorstetraquarksdecayfall-apartmassstates
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The pith

The 1S states of bb bbar cbar, cc cbar bbar, bb cbar cbar, and bc bbar cbar tetraquarks are predicted to have masses in 16.06-16.14, 9.65-9.74, 12.89-12.94, and 12.75-12.99 GeV with narrow fall-apart decay widths from tenths to several MeV.

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

The authors apply a nonrelativistic potential model to four-quark systems where all quarks are heavy, either bottom or charm. They solve the four-body problem using the explicitly correlated Gaussian method to obtain a complete set of 1S state masses for the four flavor combinations listed. These masses fall into narrow windows around 16 GeV for the bottom-rich system, 9.7 GeV for the charm-rich system, and 12.8-13 GeV for the mixed ones. The wave functions from the mass calculation are then fed into a quark-exchange model to estimate how easily each state can decay by rearranging into two ordinary mesons such as Upsilon plus J/psi or B_c plus J/psi. The resulting widths are small, typically a few MeV or less, which would make the states relatively long-lived and potentially visible as peaks in invariant-mass distributions at hadron colliders. The work stays within the established framework of phenomenological quark models that have been used for decades to describe heavy mesons and baryons, but extends the technique to these particular all-heavy, mixed-flavor tetraquarks.

Core claim

The 1S states of the all-heavy tetraquarks with different flavors may have narrow fall-apart decay widths, ranging from a few tenths to several MeV. Some may have good potentials to be established at LHC in their optimal fall-apart decay channels, such as ΥJ/ψ, ΥB_c^-, and J/ψB_c^+. (abstract)

Load-bearing premise

The nonrelativistic potential quark model with parameters fitted to known mesons remains accurate for these four-body systems, and the quark-exchange model correctly captures the fall-apart decay dynamics without significant relativistic or multi-body effects.

Figures

Figures reproduced from arXiv: 2604.03047 by Jun-Jie Liu, Lin-Qin Xie, Ming-Sheng Liu, Wei-Xiang Wang, Xian-Hui Zhong, Zhi-Biao Liang.

Figure 1
Figure 1. Figure 1: FIG. 1: Mass spectrum of all-heavy tetraquarks with di [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: A comparison of the masses of the lowest 1 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: A comparison of the masses of the lowest 1 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: A comparison of the masses of the lowest 1 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: A comparison of the masses of the lowest 1 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
read the original abstract

In a nonrelativistic potential quark model framework, we carry out a precise calculation of the mass spectrum of the all-heavy tetraquarks with different flavors, $bb\bar{b}\bar{c}$, $cc\bar{c}\bar{b}$, $bb\bar{c}\bar{c}$, and $bc\bar{b}\bar{c}$, by adopting the explicitly correlated Gaussian method. A complete mass spectrum for the $1S$ states is obtained. For the $bb\bar{b}\bar{c}$, $cc\bar{c}\bar{b}$, $bb\bar{c}\bar{c}$, and $bc\bar{b}\bar{c}$ systems, the $1S$ states are predicted to lie in the mass ranges of $ \sim(16.06,16.14)$, $\sim(9.65,9.74)$, $\sim(12.89,12.94)$, and $\sim(12.75,12.99)$~GeV, respectively.Moreover, by using the obtained masses and wave functions, we evaluate the fall-apart decay properties within a quark-exchange model.The results show that the $1S$ states of the all-heavy tetraquarks with different flavors may have narrow fall-apart decay widths,which ranging from a few tenths to several MeV. Some all-heavy tetraquarks with different flavors may have good potentials to be established at LHC in their optimal fall-apart decay channels, such as $\Upsilon J/\psi$, $\Upsilon B_c^-$, and $J/\psi B_c^+$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper calculates the mass spectra of the 1S states for all-heavy tetraquarks with different flavors (bb b-bar c-bar, cc c-bar b-bar, bb c-bar c-bar, bc b-bar c-bar) in a nonrelativistic potential quark model using the explicitly correlated Gaussian method, reporting mass ranges of approximately (16.06-16.14), (9.65-9.74), (12.89-12.94), and (12.75-12.99) GeV respectively. It then evaluates fall-apart decay widths via a quark-exchange model, finding narrow widths from a few tenths to several MeV and suggesting observability at the LHC in channels such as ΥJ/ψ, ΥB_c^-, and J/ψB_c^+.

Significance. If the results hold, the work supplies concrete numerical predictions for masses and decay widths of exotic all-heavy tetraquarks that could directly guide experimental searches at the LHC, helping to test the applicability of quark models to multiquark systems and potentially identifying new states beyond conventional hadrons.

major comments (3)
  1. [§2] §2 (potential model and fitting): The quark potential parameters are described as fitted to known heavy-meson masses, but no explicit parameter values, fitting procedure, or goodness-of-fit metrics are provided. This is load-bearing because the tetraquark masses and wave functions are computed directly from these parameters, preventing independent assessment of the numerical ranges.
  2. [§4] §4 (decay calculation): The quark-exchange model for fall-apart widths is applied to the four-body systems without derivation details, validation on known states, or discussion of relativistic/multi-body corrections. This directly supports the central claim of narrow widths (tenths to several MeV) and LHC accessibility, yet lacks the quantitative checks needed to establish reliability.
  3. [Numerical results] Numerical results section: The reported mass intervals and decay widths are presented without uncertainties, sensitivity analyses to parameter variations, or error propagation from the meson fits. This undermines the precision claims for the 1S states and the assertion that some channels are experimentally promising.
minor comments (2)
  1. [Abstract] Abstract contains a grammatical error: 'widths,which ranging' should be 'widths, ranging'.
  2. [Throughout] Notation for flavor combinations (e.g., bb b-bar c-bar) should be standardized and cross-checked against the tables for consistency.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments highlight important areas for improving transparency and rigor. We address each major comment below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses
  1. Referee: §2 (potential model and fitting): The quark potential parameters are described as fitted to known heavy-meson masses, but no explicit parameter values, fitting procedure, or goodness-of-fit metrics are provided. This is load-bearing because the tetraquark masses and wave functions are computed directly from these parameters, preventing independent assessment of the numerical ranges.

    Authors: We agree that explicit details are necessary for reproducibility. In the revised manuscript, we will add a dedicated subsection in §2 listing all potential parameters (including the values of α, β, and the Coulomb and linear coefficients), describe the χ² minimization procedure used to fit the known heavy-meson spectrum (η_c, J/ψ, B_c, Υ, etc.), and report the resulting goodness-of-fit metric along with the fitted masses for comparison with experiment. revision: yes

  2. Referee: §4 (decay calculation): The quark-exchange model for fall-apart widths is applied to the four-body systems without derivation details, validation on known states, or discussion of relativistic/multi-body corrections. This directly supports the central claim of narrow widths (tenths to several MeV) and LHC accessibility, yet lacks the quantitative checks needed to establish reliability.

    Authors: We acknowledge the need for additional methodological transparency. The revised version will include an expanded §4 with a brief derivation of the quark-exchange transition amplitude for fall-apart decays, validation tests on conventional heavy mesons (e.g., reproducing known widths or selection rules where applicable), and a discussion of the model's limitations, including the neglect of relativistic corrections and multi-body effects, which are consistent with the nonrelativistic framework used for the masses. revision: yes

  3. Referee: Numerical results section: The reported mass intervals and decay widths are presented without uncertainties, sensitivity analyses to parameter variations, or error propagation from the meson fits. This undermines the precision claims for the 1S states and the assertion that some channels are experimentally promising.

    Authors: We agree that quantitative uncertainty estimates would strengthen the presentation. In the revision, we will add a sensitivity analysis by varying the fitted potential parameters within their 1σ ranges from the meson fit and recomputing the tetraquark masses and widths to obtain uncertainty bands. Full analytic error propagation is not feasible within the variational Gaussian method, but the sensitivity study will provide a practical estimate of theoretical uncertainty. revision: partial

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The calculation depends on a nonrelativistic potential whose parameters are fitted to known mesons and on the assumption that the explicitly correlated Gaussian basis accurately solves the four-body Schrödinger equation for these systems.

free parameters (1)
  • Quark potential parameters
    Standard parameters of the nonrelativistic potential (string tension, Coulomb strength, etc.) fitted to known heavy-meson spectra.
axioms (1)
  • domain assumption Non-relativistic treatment is valid for bottom and charm quark systems
    Invoked by adopting the nonrelativistic potential quark model framework.

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Reference graph

Works this paper leans on

108 extracted references · 108 canonical work pages · 4 internal anchors

  1. [1]

    Department of Physics, Hunan Normal University, and Key L aboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Educa tion, Changsha 410081, China

  2. [2]

    Tianjin Key Laboratory of Quantum Optics and Intelligent Photonics, School of Science, Tianjin University of Technology, Tianj in 300384, China and

  3. [3]

    All-heavy tetraquarks with different flavors

    Synergetic Innovation Center for Quantum E ffects and Applications (SICQEA), Hunan Normal University, C hangsha 410081, China In a nonrelativistic potential quark model framework, we ca rry out a precise calculation of the mass spec- trum of the all-heavy tetraquarks with di fferent flavors, bb¯b¯c, cc¯c¯b, bb¯c¯c, and bc¯b¯c, by adopting the explicitly corr...

  4. [4]

    H = 4∑ i=1 (mi + Ti) −TG + ∑ i< j Vi j(ri j), (1) where mi and Ti stand for the mass and kinetic energy of the i- th quark, respectively

    Hamiltonian In this work, to describe the tetraquark system we adopt a nonrelativistic Hamiltonian [9], i.e. H = 4∑ i=1 (mi + Ti) −TG + ∑ i< j Vi j(ri j), (1) where mi and Ti stand for the mass and kinetic energy of the i- th quark, respectively. TG is the center-of-mass kinetic energy. Vi j(ri j) represents the e ffective potentials between the i-th and j...

  5. [5]

    In the flavor space, the available configurations for all all-heavy tetraquark systems with di ffer- ent flavors are bb¯b¯c, cc¯c¯b, bb¯c¯c, and bc¯b¯c

    States classified in the quark model To calculate the spectroscopy of a Q1Q2 ¯Q3 ¯Q4 system, first we construct the configurations in the product space of spa- tial ⊗flavor ⊗color ⊗spin. In the flavor space, the available configurations for all all-heavy tetraquark systems with di ffer- ent flavors are bb¯b¯c, cc¯c¯b, bb¯c¯c, and bc¯b¯c. This implies that the flav...

  6. [6]

    It is a well-established variational method to solve quantum few - body problems

    Numerical method To solve the four-body problem accurately, we adopt the explicitly correlated Gaussian (ECG) method [ 56– 58]. It is a well-established variational method to solve quantum few - body problems. The spatial part of the wave function for the 1S -wave tetraquark system is expanded in terms of ECG basis 3 TABLE II: Configurations of all-heavy t...

  7. [7]

    In contrast, in this work, the main component of the higher-mass state is |bb ¯3 1(¯b¯c)3 1⟩0 0 and that of the lower-mass state is |bb6 0(¯b¯c)¯6 0⟩0

  8. [8]

    The main reason for this di fference is that the trial wave function adopted in this work is more complete than that used previously, as dis- cussed in Sec. II (A3). 6 TABLE V: The numerical results of the mass spectrum (in MeV), the mass contributions of each Hamiltonian part (in MeV), an d the root- mean-square radii (in fm) for the 1 S -wave eigenstates...

  9. [9]

    29 −0. 95 0 . 13 −0. 96 −0. 29 0 . 06                               |{bb}6 0(¯b¯c)¯6 1⟩ 0 1 |{bb}¯3 1(¯b¯c)3 0⟩ 0 1 |{bb}¯3 1(¯b¯c)3 1⟩ 0 1                               16126 16119 16066                800/ 444/ −1159/ 2 0 . 27/ 0. 37/ 0. 30/ 0. 38/ 0. 38/ 0. 30 792/ 427/ −1138/ −...

  10. [10]

    89 −0. 39 0 . 22                               |{cc}6 0(¯c¯b)¯6 1⟩ 0 1 |{cc}¯3 1(¯c¯b)3 0⟩ 0 1 |{cc}¯3 1(¯c¯b)3 1⟩ 0 1                               9723 9722 9659                751/ 578/ −920/ 13 0 . 47/ 0. 39/ 0. 49/ 0. 43/ 0. 43/ 0. 49 750/ 584/ −920/ 7 0 . 48/ 0. 41/ 0. 49/ 0....

  11. [11]

    81 −0. 46 0 . 36 −0. 03 −0. 35 −0. 81 −0. 30 −0. 38                                             |(bc)6 0(¯b¯c) ¯6 0⟩ 0 0 |(bc)6 1(¯b¯c) ¯6 1⟩ 0 0 |(bc)¯3 0(¯b¯c)3 0⟩ 0 0 |(bc)¯3 1(¯b¯c)3 1⟩ 0 0                                             12985 12936 12853 12752    ...

  12. [12]

    39 −0. 91 −0. 06 0 . 15 −0. 67 −0. 27 0 . 59 0 . 35 −0. 62 −0. 30 −0. 61 −0. 40                                             |(bc)6 1(¯b¯c)¯6 1⟩ 0 1 |(bc)¯3 1(¯b¯c)3 1⟩ 0 1 |(bc)6 1(¯b¯c)¯6 0⟩ 0− 1 |(bc)¯3 1(¯b¯c)3 0⟩ 0− 1                                             129...

  13. [13]

    It should be mentioned that the results obtained with complex scaling method [ 25] are systematically ∼450 MeV larger than ours

    Our results are com- patible with the nonrelativistic quark model predictions b ased on dynamic calculations [ 21– 24] and di ffusion Monte Carlo calculations [20], the diquark model predictions [ 27, 30, 31], and the results predicted by the CGAN framework [ 53]. It should be mentioned that the results obtained with complex scaling method [ 25] are system...

  14. [14]

    They are mixed states between two di fferent color configurations 6 ⊗¯6 and ¯3 ⊗3

    0+ states For the two 0 + states T(bb¯b¯c)0+ (16132) and T(bb¯b¯c)0+ (16064), there is a significant mass splitting, ∆M ≃ 70 MeV , which is mainly due to the spin-spin interactions. They are mixed states between two di fferent color configurations 6 ⊗¯6 and ¯3 ⊗3. The high mass state T(bb¯b¯c)0+ (16132) is dominated by the ¯3 ⊗3, while the low mass state T(b...

  15. [15]

    There is a significant mass gap ∆M ≃50 MeV be- tween them and the low-lying state T(bb¯b¯c)1+ (16066)

    1+ states Among the three 1 + states, the two high-lying states T(bb¯b¯c)1+ (16126) and T(bb¯b¯c)1+ (16119) are nearly degenerate to- gether. There is a significant mass gap ∆M ≃50 MeV be- tween them and the low-lying state T(bb¯b¯c)1+ (16066). The configuration mixing in these states is slight. As shown in Table V, the low-lying state T(bb¯b¯c)1+ (16066) i...

  16. [16]

    The ΥB∗ c is the only allowed fall-apart decay channel of T(bb¯b¯c)2+ (16139)

    2+ state For the 2+ state T(bb¯b¯c)2+ (16139), as a pure |bb ¯3 1(¯b¯c)3 1⟩0 2 state, whose mass is very close to that of the high-lying 0 + and 1 + states, T(bb¯b¯c)0+ (16132) and T(bb¯b¯c)1+ (16126). The ΥB∗ c is the only allowed fall-apart decay channel of T(bb¯b¯c)2+ (16139). The partial width is predicted to be Γ[T(bb¯b¯c)2+ (16139) → ΥB∗ c] ≃0. 86 M...

  17. [17]

    0+ states For the two T(cc¯c¯b)0+ (9733) and T(cc¯c¯b)0+ (9650), the mass splitting is predicted to be ∆M ≃ 90 MeV . The mass split- ting between T(bb¯b¯c)0+ (16132) and T(bb¯b¯c)0+ (16064), ∆M ≃ 70 MeV , is lightly smaller than that of the cc¯c¯b system is due to the suppression of the heavy bottom quark. As shown in Table V, the T(cc¯c¯b)0+ (9736) and T...

  18. [18]

    There is a signif- icant mass gap ∆M ≃ 40 MeV between them and the low-lying state T(cc¯c¯b)1+ (9659) originating from the di fference of color structure

    1+ states The two high-lying 1 + states T(cc¯c¯b)1+ (9722) and T(cc¯c¯b)1+ (9723) are highly degenerate. There is a signif- icant mass gap ∆M ≃ 40 MeV between them and the low-lying state T(cc¯c¯b)1+ (9659) originating from the di fference of color structure. Sizeable configuration mixing exists in these 1 + states. As shown in Table V, the low-lying state ...

  19. [19]

    The J/ψB∗ c is the only allowed fall-apart decay channel in all of T(cc¯c¯b)2+ (9738)

    2+ state For the 2 + state T(cc¯c¯b)2+ (9738), as a pure |cc¯3 1(¯c¯b)3 1⟩0 2 state, the mass is very close to that of the high-lying 0+ and 1+ states, T(cc¯c¯b)0+ (9733) and T(cc¯c¯b)1+ (9723). The J/ψB∗ c is the only allowed fall-apart decay channel in all of T(cc¯c¯b)2+ (9738). The partial width is predicted to be Γ[T(cc¯c¯b)2+ (9738) → ΥB∗ c] ≃0. 18 M...

  20. [20]

    Our re- sults are generally compatible with the nonrelativistic qu ark model predictions based on dynamic calculations [ 10, 21– 23] and di ffusion Monte Carlo calculations [ 20], the diquark model predictions [ 27, 30], and the results predicted by the CGAN framework [53]

  21. [21]

    As shown in Table V, they are mixed states between two di ffer- ent color configurations

    0+ states For the two 0 + states, T(bb¯c¯c)0+ (12942) and T(bb¯c¯c)0+ (12888), there is a significant mass splitting of ∆M ∼ 50 MeV . As shown in Table V, they are mixed states between two di ffer- ent color configurations. The high and low mass states are dominated by the |{bb}¯3 1(¯c¯c)3 1⟩0 0 and |{bb}6 0(¯c¯c)¯6 0⟩0 0 components, respectively. The mass o...

  22. [22]

    For the T(bb¯c¯c)1+ (12931) and T(bb¯c¯c)2+ (12944), the allowed fall-apart decay channels are BcB∗ c and B∗ cB∗ c, respectively

    1+ and 2+ states The T(bb¯c¯c)2+ (12944), as the highest mass state in the bb¯c¯c system, only about 13 MeV lies above the 1 + state T(bb¯c¯c)1+ (12931), and is also nearly degenerate with the high- lying 0 + state T(bb¯c¯c)0+ (12942), due to the similar color-spin structures. For the T(bb¯c¯c)1+ (12931) and T(bb¯c¯c)2+ (12944), the allowed fall-apart dec...

  23. [23]

    Our results are generally compatible with the nonrelativistic quark model pre- dictions based on dynamic calculations [ 21– 23], the diquark model predictions [ 27, 28], and the results predicted by the CGAN framework [53]

  24. [24]

    From Ta- ble V, one can find that the dominant color component of the two high-lying states T(bc¯b¯c)0++ (12985) and T(bc¯b¯c)0++(12936) is ¯3 ⊗3

    0++ states For the four JP = 0++ states, there are strong configuration mixings between the 6 ⊗¯6 and ¯3 ⊗3 configurations. From Ta- ble V, one can find that the dominant color component of the two high-lying states T(bc¯b¯c)0++ (12985) and T(bc¯b¯c)0++(12936) is ¯3 ⊗3. While for the two low-lying states T(bc¯b¯c)0++(12853) and T(bc¯b¯c)0++ (12752), the domi...

  25. [25]

    The dominant color component of the two high-lying states T(bc¯b¯c)1+−(12987) and T(bc¯b¯c)1+−(12970) is ¯3 ⊗3

    1+−states For the four JP = 1+−states, there are also strong config- uration mixings between the 6 ⊗¯6 and ¯3 ⊗3 configurations. The dominant color component of the two high-lying states T(bc¯b¯c)1+−(12987) and T(bc¯b¯c)1+−(12970) is ¯3 ⊗3. While for the two low-lying states T(bc¯b¯c)1+−(12826) and T(bc¯b¯c)1+−(12780), the dominant color component is 6 ⊗¯6....

  26. [26]

    It may have sizeable decay rates into the BcB∗ c = B+ c B∗− c + B∗+ c B− c and B∗+ c B∗− c channels with comparable partial widths Γ[T(bc¯b¯c)1+−(12780) → BcB∗ c/ B∗+ c B∗− c ] ≃0

    4 MeV . It may have sizeable decay rates into the BcB∗ c = B+ c B∗− c + B∗+ c B− c and B∗+ c B∗− c channels with comparable partial widths Γ[T(bc¯b¯c)1+−(12780) → BcB∗ c/ B∗+ c B∗− c ] ≃0. 21/ 0. 14 MeV. (41) For the other 1 +− states, the fall-apart decay widths are pre- dicted to be ∼100 keV . These states may be di fficult to ob- serve in their fall-apar...

  27. [27]

    The low-mass state T(bc¯b¯c)1++(12862) and the high-mass state T(bc¯b¯c)1++ (12945) are governed by the 6 ⊗¯6 and ¯3⊗3 com- ponents, respectively

    1++ and 2++ states From Table V, one can find that there is a slight mixing be- tween the 6 ⊗¯6 and ¯3 ⊗3 configurations in the 1 ++ and 2 ++ states. The low-mass state T(bc¯b¯c)1++(12862) and the high-mass state T(bc¯b¯c)1++ (12945) are governed by the 6 ⊗¯6 and ¯3⊗3 com- ponents, respectively. However, for the 2 ++ sector, the case is reversed, the low-ma...

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