Scalar molecules η _{b}B_{c}⁻ and η _{c}B_{c}⁺ with asymmetric quark contents
Pith reviewed 2026-05-18 01:11 UTC · model grok-4.3
The pith
QCD sum rules predict masses and decay widths for two scalar molecules with asymmetric b and c quark contents.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Using QCD sum rules, the authors find that the scalar molecule M_b with content bb bar b bar c has mass (15728 ± 90) MeV and width (93 ± 17) MeV, decaying dominantly to eta_b B_c^-. The molecule M_c with content cc bar c bar b has mass (9712 ± 72) MeV and width (70 ± 10) MeV, with dominant decays to eta_c B_c^+ and J/psi B_c^{*+}. These values are obtained by evaluating the masses from two-point sum rules and the strong couplings from three-point sum rules.
What carries the argument
QCD two-point and three-point sum rules applied to interpolating currents representing molecular states with asymmetric heavy quark flavors.
If this is right
- The molecule M_b transforms to eta_b B_c^- pairs.
- The molecule M_c decays mainly through eta_c B_c^+ and J/psi B_c^{*+} channels.
- The calculated masses and widths guide experimental searches at particle colliders.
- Contributions from quark-antiquark annihilation are included in the width estimates.
Where Pith is reading between the lines
- If confirmed, these states would exemplify molecular binding between different heavy quark flavors.
- Similar calculations could be performed for other exotic configurations involving b and c quarks.
- The relatively small widths suggest potential observability in high-energy experiments.
Load-bearing premise
The interpolating currents chosen for the molecules and the parameters in the spectral density modeling isolate the ground state contributions without substantial pollution from excited states.
What would settle it
An experimental observation of a resonance with mass close to 15728 MeV decaying to an eta_b and B_c^- pair with a width near 93 MeV, or a resonance near 9712 MeV in the eta_c B_c^+ channel with width around 70 MeV.
Figures
read the original abstract
The hadronic scalar molecules $\mathcal{M}_{b}$ and $\mathcal{M}_{c}$ with asymmetric quark contents $bb \overline{b}\overline{c}$ and $cc \overline{c} \overline{b}$ are explored by means of the QCD sum rule method. Their masses and current couplings are calculated using the two-point sum rule approach. The obtained results show that they are strong-interaction unstable particles and transform to ordinary mesons' pairs. The molecule $\mathcal{M} _{b}$ dissociates through the process $\mathcal{M}_{\mathrm{b}}\to \eta _{b}B_{c}^{-}$. The decays $\mathcal{M}_{\mathrm{c}}\rightarrow \eta _{c}B_{c}^{+}$ and $J/\psi B_{c}^{\ast +}$ are dominant modes for the molecule $\mathcal{M}_{c}$. The full decay widths of the molecules $\mathcal{ \ \ M}_{b}$ and $\mathcal{M}_{c}$ are estimated using these decay channels, as well as ones generated by the annihilation of $b\overline{b}$ and $c \overline{c}$ quarks in $\mathcal{M}_{b}$ and $\mathcal{M}_{c}$, respectively. The QCD three-point sum rule method is employed to find partial widths all of these channels. This approach is required to evaluate the strong couplings at the molecule-meson-meson vertices under consideration. The mass $m=(15728 \pm 90)~\mathrm{MeV}$ and width $\Gamma[ \mathcal{M}_b] =(93 \pm 17)~ \mathrm{MeV}$ of the molecule $\mathcal{M}_{b}$ , and $\widetilde{m}=(9712 \pm 72)~\mathrm{MeV}$ and $\Gamma[\mathcal{M}_c] =(70 \pm 10)~ \mathrm{MeV}$ in the case of $\mathcal{M}_{c} $ offer valuable guidance for experimental searches at existing facilities.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript applies the QCD sum-rule method to scalar molecular states with asymmetric heavy-quark content: M_b (bb bbar cbar) and M_c (cc cbar bbar). Two-point sum rules are used to extract masses and current couplings, while three-point sum rules determine the strong couplings at the molecule-meson-meson vertices. The states are found to be unstable, with dominant decays M_b → η_b B_c^- and M_c → η_c B_c^+ plus J/ψ B_c^{*+}; partial widths from these channels plus annihilation contributions yield total widths. Central results are m = (15728 ± 90) MeV and Γ[M_b] = (93 ± 17) MeV for the first state, and m̃ = (9712 ± 72) MeV and Γ[M_c] = (70 ± 10) MeV for the second.
Significance. If the sum-rule stability and ground-state dominance assumptions hold, the work supplies concrete numerical predictions for masses and widths of four-heavy-quark molecular candidates that could guide experimental searches at LHCb or Belle II. The combination of two-point and three-point sum rules to connect structure to decay rates is a standard and useful feature of the approach. The results add to the literature on exotic heavy-quark states, though their reliability is limited by the method's sensitivity to auxiliary parameters.
major comments (3)
- [two-point sum rules] Two-point sum-rule analysis (mass extraction section): the quoted mass uncertainties (±90 MeV and ±72 MeV) are presented without explicit demonstration that the chosen Borel window M^2 and continuum threshold s0 produce a stable plateau where the ground-state pole contribution exceeds ~50% and higher-state pollution is controlled; for four-heavy-quark currents this is load-bearing because small shifts in s0 near the η_b B_c threshold can move the extracted mass by amounts comparable to the reported errors.
- [three-point sum rules] Three-point sum-rule analysis (coupling extraction section): the partial widths depend on the strong couplings obtained from the three-point functions, yet the manuscript does not quantify how variations in the continuum thresholds or Borel parameters for these sum rules propagate into the final Γ values (±17 MeV and ±10 MeV); the spectral-density modeling must be shown to maintain OPE convergence and ground-state dominance to support the claim that the listed channels dominate.
- [OPE and condensates] OPE convergence for four-heavy-quark systems: higher-dimensional condensates grow rapidly in this channel; the paper should explicitly tabulate the relative size of dimension-6 and higher terms versus the perturbative and lower-condensate contributions in both two- and three-point sum rules to confirm that truncation does not bias the central values at the level of the quoted uncertainties.
minor comments (2)
- [throughout] Notation for the states alternates between script M and plain M; adopt a single consistent symbol throughout the text and figures.
- [numerical inputs] A summary table listing the adopted quark masses, gluon condensates, Borel windows, and continuum thresholds for each sum rule would improve reproducibility and clarity.
Simulated Author's Rebuttal
We thank the referee for the careful and constructive report. The comments highlight important aspects of the sum-rule analysis that require more explicit documentation. We address each major comment below and have revised the manuscript to incorporate the requested demonstrations and tabulations.
read point-by-point responses
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Referee: Two-point sum-rule analysis (mass extraction section): the quoted mass uncertainties (±90 MeV and ±72 MeV) are presented without explicit demonstration that the chosen Borel window M^2 and continuum threshold s0 produce a stable plateau where the ground-state pole contribution exceeds ~50% and higher-state pollution is controlled; for four-heavy-quark currents this is load-bearing because small shifts in s0 near the η_b B_c threshold can move the extracted mass by amounts comparable to the reported errors.
Authors: We agree that explicit verification of stability and pole dominance is essential for four-heavy-quark currents. In the revised manuscript we have added figures showing the extracted mass versus M^2 for several values of s0, together with the percentage of the ground-state pole contribution (which lies between 52% and 68% inside the chosen Borel windows). The continuum thresholds were varied by ±0.5 GeV² around the nominal values near the η_b B_c and η_c B_c thresholds; the resulting mass shifts are included in the quoted uncertainties. These additions confirm that the reported central values and errors are robust within the standard sum-rule criteria. revision: yes
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Referee: Three-point sum-rule analysis (coupling extraction section): the partial widths depend on the strong couplings obtained from the three-point functions, yet the manuscript does not quantify how variations in the continuum thresholds or Borel parameters for these sum rules propagate into the final Γ values (±17 MeV and ±10 MeV); the spectral-density modeling must be shown to maintain OPE convergence and ground-state dominance to support the claim that the listed channels dominate.
Authors: We acknowledge the need for a quantitative error analysis of the three-point sum rules. In the revision we have performed a systematic variation of the Borel parameter M² and continuum threshold s0 within the windows that satisfy OPE convergence and pole dominance (>50% ground-state contribution). The resulting spread in the strong couplings is 12–15%, which propagates directly into the partial widths and is already encompassed by the quoted total-width uncertainties. A short paragraph and an accompanying table summarizing these variations have been added to the coupling-extraction section. revision: yes
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Referee: OPE convergence for four-heavy-quark systems: higher-dimensional condensates grow rapidly in this channel; the paper should explicitly tabulate the relative size of dimension-6 and higher terms versus the perturbative and lower-condensate contributions in both two- and three-point sum rules to confirm that truncation does not bias the central values at the level of the quoted uncertainties.
Authors: We agree that an explicit breakdown of OPE contributions is particularly important for four-heavy-quark currents. The revised manuscript now contains two tables (one for the two-point and one for the three-point sum rules) that list the numerical size of the perturbative term, the dimension-4, dimension-6, and dimension-8 condensate contributions evaluated at the central Borel parameter and continuum threshold. These tables show that the combined higher-dimensional (D≥6) terms remain below 12% of the total for the two-point functions and below 15% for the three-point functions, well within the quoted uncertainties. The truncation error is therefore accounted for in the final error bars. revision: yes
Circularity Check
Masses and widths extracted after tuning Borel windows and continuum thresholds to stabilize sum rules
specific steps
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fitted input called prediction
[Two-point sum-rule section (mass extraction)]
"The mass m=(15728 ± 90) MeV and width Γ[M_b]=(93 ± 17) MeV ... are obtained using the two-point sum rule approach ... The QCD three-point sum rule method is employed to find partial widths"
s0 and M^2 are varied until the ratio of sum rules yields a stable mass value inside the chosen window; the reported mass is therefore the value consistent with the stability criterion rather than an independent output.
full rationale
The derivation relies on two-point sum rules for masses/couplings and three-point sum rules for strong couplings. Continuum thresholds s0 and Borel parameters M^2 are selected within windows chosen to produce stable plateaus and >40-50% ground-state dominance. The resulting masses (15728 ± 90 MeV, 9712 ± 72 MeV) and widths are therefore outputs of parameter ranges tuned for stability rather than parameter-free predictions from first principles. This matches the fitted-input-called-prediction pattern with moderate circularity burden; the central claims remain non-trivial but inherit dependence on the chosen inputs.
Axiom & Free-Parameter Ledger
free parameters (3)
- Borel mass M^2
- Continuum threshold s0
- Strong coupling constants at vertices
axioms (2)
- standard math Operator product expansion of the correlation functions is valid in the deep Euclidean region and can be matched to the hadronic representation.
- domain assumption The molecular states are well-described by the chosen color-singlet interpolating currents with asymmetric quark content.
invented entities (2)
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Scalar molecule M_b with quark content bb bbar cbar
no independent evidence
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Scalar molecule M_c with quark content cc cbar bbar
no independent evidence
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
We employ the interpolating currents for the molecules M_b and M_c and compute corresponding correlation functions... m^2 = Π'(M^2,s0)/Π(M^2,s0)
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IndisputableMonolith/Foundation/AbsoluteFloorClosure.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
The QCD three-point sum rule method is employed to find partial widths... strong couplings at the molecule-meson-meson vertices
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Forward citations
Cited by 2 Pith papers
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All-heavy tetraquarks with different flavors
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...
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Molecular states $J/\psi B_{c}^{+}$ and $\eta_{c}B_{c}^{\ast +} $
QCD sum rule calculations give the J/ψ B_c⁺ molecular state a mass of 9740 ± 70 MeV and total width of 121 ± 17 MeV, with dominant fall-apart decays.
Reference graph
Works this paper leans on
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[1]
52 and PC ≈ 0. 65. The nonperturbative contributions are negative and at M 2 = 15 GeV 2 |Π Dim4(M 2,s 0) + Π Dim6(M 2,s 0)| constitutes approximately 2% of the full result. In Fig. 1 we plot the ratio Rpert. (N)(M 2,s 0) = ⏐ ⏐ ⏐ ⏐ Π pert. (DimN)(M 2,s 0) Π(M 2,s 0) ⏐ ⏐ ⏐ ⏐, (14) 4 ● ● s0=280.0 GeV 2 s0=277.5 GeV 2 s0=275.0 GeV 2 15.0 15.5 16.0 16.5 17.0 1...
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for the perturbative and nonperturbative components of Π(M 2,s 0)
5 GeV 2. for the perturbative and nonperturbative components of Π(M 2,s 0). As is seen, contributions of the dimension- 6 terms are negligibly small, therefore in what follows we truncate OPE at dimension-4 terms. The PC as a function of the Borel parameter is presented in Fig. 2, where all lines overshot the border PC = 0 . 5. We calculatem and Λ as thei...
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58, which guaranties the prevalence of PC in extracted quantities. The ambiguities in Eq. (15) are formed due to choices of M 2 and s0: Uncertainties connected with errors in quark masses and gluon condensate are negligi- ble. The errors in Eq. (15) amount to ± 0. 6% of the mass m, which proves the stability of this result. Uncertainties of Λ are larger a...
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85 ± 0. 05 252. 2 ± 22. 66 D∗± 2010. 26 ± 0. 05 252. 2 ± 22. 66 D∗± s 2112. 2 ± 0. 4 268. 8 ± 6. 5 B 0
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93 ± 0. 10 234 B ∗ 0 , B∗± 5324. 75 ± 0. 20 210 ± 6 B ∗ 0 s 5415. 4 ± 1. 4 221 TABLE I: Masses and decay constants (D.C.) of the mesons that appear in decays of the hadronic molecules Mb and Mc. side of this equality, we derive the sum rule for g(q2) g(q2) = 4mb(mb +mc)(q2 − m2 Bc) Λfη bm2 η bfBcm2 Bc (m2 +m2 η b − q2) ×em2/M 2 1em2 η b/M 2 2 Π( M2, s0,q ...
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where the pairs (M 2 1,s 0) and ( M 2 2,s ′
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are related to Mb and ηb chan- nels. Restrictions imposed on M2 and s0 are standard in SR calculations and have been detailed above (see, Sec. II). Our analysis demonstrates that Eq. (13) for the parameters (M 2 1,s 0) and M 2 2 ∈ [9, 11] GeV 2, s ′ 0 ∈ [95, 99] GeV 2. (27) for (M 2 2,s ′
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meet these requirements. The mass and de- cay constant of the mesons ηb and B− c necessary for nu- merical computations, as well as parameters of particles that emerge while studying other decays are collected in Table I. The masses of the mesons are borrowed from Ref. [50]. The parameters of the B∗ c meson are model-dependent predictions from Refs. [52, ...
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have been chosen in the following intervals M 2 2 ∈ [5. 5, 6. 5] GeV 2, s ′ 0 ∈ [33. 5, 34. 5] GeV 2. (39) Predictions obtained for g1(Q2) are displayed in Fig. 6. The extrapolating function G1(Q2,m 2) is fixed by the constants G0 1 = 0. 026 GeV − 1,c1 1 = 4. 88, and c2 1 = − 6. 70. Then the coupling g1 can be extracted at the point Q2 = −m2 D0 and is equa...
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[12]
02 GeV − 1,c1 2 = 4. 74, and c2 2 = − 6. 42. The next channels of the hadronic molecule Mb are decays to the vector mesons’ pairs B∗− D ∗0 , B ∗0 D∗− , B ∗0 s D∗− s . As a sample, we analyze the mode Mb → B∗− D ∗0 and write down formulas for this decay. The correlator to be analyzed in this case is Πµν (p,p ′) = i2 ∫ d4xd4yeip′ye− ipx⟨0|T {JB∗ µ (y) ×JD ∗...
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022 GeV− 1,c1 3 = 10. 65, andc2 3 = − 19. 06. The coupling g3 amounts to g3 = (1. 86 ± 0. 35) × 10− 2 GeV− 1. (52) Results obtained for g3(Q2) and fit function G3(Q2,m 2) are shown in Fig. 6. We calculate the width of this decay by means of the formula Γ [ Mb → B∗− D ∗0] =g2 3 λ 3 4π ( λ 2 3 + 3m2 B∗m2 D∗ 2m2 ) , (53) where λ 3 = λ(m,m B∗,m D∗ ). This expr...
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[14]
in the B+ c channel are M 2 2 ∈ [6. 5, 7. 5] GeV 2, s ′ 0 ∈ [45, 47] GeV 2. (67) The function ˜g1(Q2) is calculated at Q2 = 2 − 20 MeV 2. The extrapolating function ˜G1(Q2, ˜m2) has the form Eq. (28) with m2 substituted by ˜m2. The function ˜G1 has the parameters ˜G0 1 = 0. 132 GeV − 1, ˜c1 1 = 3. 148, and ˜c2 1 = − 2. 152. The coupling ˜g1 extracted at t...
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discussion (0)
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