arxiv: 2512.23030 · v2 · submitted 2025-12-28 · ✦ hep-ph · hep-ex· hep-lat

Molecular states J/psi B_{c}⁺ and η_{c}B_{c}^{ast +}

S. S. Agaev , K. Azizi , H. Sundu This is my paper

Pith reviewed 2026-05-16 19:12 UTC · model grok-4.3

classification ✦ hep-ph hep-exhep-lat

keywords hadronic moleculesQCD sum rulesJ/ψ B_cexotic mesonsdecay widthsheavy quarkoniaB_c mesonmolecular states

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The pith

QCD sum rules predict a J/ψ B_c⁺ molecular state at 9740 MeV that decays strongly to ordinary meson pairs.

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

The paper uses QCD sum rules to examine possible molecular states with quark content cc c-bar b-bar and spin-parity 1+. It computes masses for J/ψ B_c⁺ and η_c B_c*⁺ and finds them numerically close, so focuses on the first. The calculated mass lies above thresholds for strong decays, and the method yields a total width of 121 MeV from dominant fall-apart channels plus subdominant annihilation processes. A reader would care because the result supplies concrete mass and width values that experiments can test to see whether such four-quark molecular configurations exist among heavy mesons.

Core claim

The authors claim that the axial-vector hadronic molecules M = J/ψ B_c⁺ and ~M = η_c B_c*⁺ both have masses near 9740 MeV, that M decays readily through strong interactions, and that its total width is 121 MeV. The fall-apart mechanism produces the leading channels M → J/ψ B_c⁺ and M → η_c B_c*⁺ while c c-bar annihilation opens six subdominant channels involving B and D mesons; all partial widths are obtained from three-point sum rules.

What carries the argument

QCD sum rule method applied to interpolating currents constructed for the molecular states, used to extract masses, couplings, and decay amplitudes.

If this is right

The mass prediction places the state above the J/ψ B_c⁺ and η_c B_c*⁺ thresholds, enabling strong decays.
Dominant partial widths arise from the fall-apart mechanism into J/ψ B_c⁺ and η_c B_c*⁺.
Six subdominant decay channels are generated by c c-bar annihilation into final states containing B and D mesons.
The predicted total width of 121 MeV supplies a definite target for experimental searches.
The two molecular states are predicted to be nearly degenerate within the accuracy of the method.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Confirmation of the state would strengthen the case that molecular binding occurs in other heavy-quark four-quark systems with similar flavor content.
The specific decay channels identified can be used to design targeted searches in current and future collider data sets.
The numerical closeness of the two states suggests they could appear as a single broadened structure or require angular analysis to separate.

Load-bearing premise

The chosen interpolating current for the molecular state together with the stability window of Borel parameter and continuum threshold accurately capture the bound-state dynamics without large higher-order corrections.

What would settle it

Observation of a resonance in the J/ψ B_c⁺ or η_c B_c*⁺ mass spectrum whose mass lies outside 9670-9810 MeV or whose width lies outside 100-140 MeV would falsify the prediction.

Figures

Figures reproduced from arXiv: 2512.23030 by H. Sundu, K. Azizi, S. S. Agaev.

**Figure 2.** Figure 2: FIG. 2: Dependence of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: SR data and fit functions [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 3.** Figure 3: For the strong coupling g2, we find g2 ≡ Z2(−m2 B∗ c ) = (2.5 ± 0.4) × 10−1 GeV−1 . (40) The width of the channel M →ηcB∗+ c is Γ [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: SR data for [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

read the original abstract

Hadronic molecules $\mathfrak{M}=J/\psi B_{c}^{+}$ and $\widetilde{\mathfrak{ \ M}}=\eta _{c}B_{c}^{\ast +}$ are investigated in the framework of QCD sum rule method. These particles with spin-parities $J^{\mathrm{P}}=1^+$ have the quark contents $cc \overline{c}\overline{b}$. We compute their masses and current couplings and find that they are numerically very close to each other coinciding within accuracy of the sum rule method. Therefore, we concentrate on the molecule $J/\psi B_{c}^{+}$ and explore features of this state in a detailed form. Our prediction $m=(9740 \pm 70)~\mathrm{MeV}$ for its mass means that $\mathfrak{M}$ easily decays to pairs of ordinary mesons through strong interactions. There are two mechanisms responsible for transformations of $\mathfrak{M}$ to conventional mesons. The fall-apart mechanism generates the dominant decay channels $\mathfrak{M} \to J/\psi B_{c}^{+}$ and $\mathfrak{M} \to \eta _{c}B_{c}^{\ast +}$. Annihilation of $ \overline{c}c$ quarks triggers subdominant processes with various final-state $B$ and $D$ mesons: Six of such channels are considered in this work. The partial widths all of decays are computed using the three-point sum rule approach. The width $\Gamma[ \mathfrak{M}]=(121 \pm 17)~ \mathrm{MeV }$ of the hadronic axial-vector molecule $\mathfrak{M}$, as well as its mass provide valuable information for running and future experiments.

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.

Desk Editor's Note private letter to a colleague

Standard QCD sum-rule calculation for two new molecular states gives mass 9740 MeV and width 121 MeV, with the usual dependence on Borel and threshold choices.

read the letter

The paper applies two-point and three-point QCD sum rules to the molecular states J/ψ B_c+ and η_c B_c*+ with cc c-bar b-bar content. It reports masses near 9740 MeV for both, a total width of 121 MeV for the first, and partial widths for fall-apart decays plus six annihilation channels to B and D mesons. The two states come out nearly degenerate, which is consistent with the method. The work is new in targeting these exact combinations and in computing the decay widths from three-point correlators rather than stopping at masses and couplings. That adds concrete numbers experiments could test around 9.7 GeV. The calculations follow the standard framework, with quoted errors from varying the Borel window and continuum threshold, and the authors focus on one state for the detailed decay analysis after noting the similarity. The soft spot is the parameter dependence the stress-test flags. For a four-quark current the OPE includes sizable continuum contributions, and stability of the chosen M^2 and s0 windows against α_s corrections or dimension-8 terms is not guaranteed to hold inside the quoted 70 MeV and 17 MeV uncertainties. If those windows shift under fuller OPE, both the bound-state interpretation and the extracted widths move. The paper checks stability plots, but the circularity in tuning against the same correlator remains a moderate limitation of the approach. This is useful for readers working on heavy-quark exotics or QCD sum-rule applications to molecules. It supplies specific predictions that can be compared with data or other models, even if the method carries its known systematics. I would send it to peer review. The technical steps are laid out clearly enough for referees to assess the numerics and the handling of the windows.

Referee Report

3 major / 1 minor

Summary. The paper uses QCD sum rules to study two hadronic molecules with J^P=1^+ and quark content cc c-bar b-bar: M = J/ψ B_c^+ and its close partner η_c B_c^{*+}. It extracts masses via two-point correlators, finds them numerically similar within method accuracy, and focuses on M with predicted mass (9740 ± 70) MeV. Three-point sum rules are then applied to compute partial widths for fall-apart decays (dominant) and annihilation channels to B and D mesons (subdominant), yielding total width Γ[M] = (121 ± 17) MeV.

Significance. If the sum-rule results are robust, the work supplies concrete, falsifiable predictions for the mass and decay width of a potential exotic axial-vector state in the cc c-bar b-bar sector. The detailed branching ratios for both fall-apart and annihilation modes provide experimental signatures that could be tested at LHCb or future facilities, helping distinguish molecular interpretations from conventional charmonium-bottomonium hybrids.

major comments (3)

[two-point sum-rule mass extraction] In the two-point QCD sum-rule analysis for the mass of M (the section presenting the numerical results for the Borel window and continuum threshold), the quoted mass (9740 ± 70) MeV and residue rely on post-hoc choices of M² and s0 that are tuned for stability of the same correlation function used to extract the final value. For the four-quark molecular current, the OPE receives sizable two-meson continuum contributions; without explicit variation of dimension-8 condensates or inclusion of α_s corrections, the extracted pole mass can shift outside the stated uncertainty, undermining the bound-state claim.
[three-point sum-rule decay widths] The three-point sum-rule calculations of the partial widths (the sections on fall-apart and annihilation decay channels) reuse the identical Borel window and s0 values chosen for the mass. Because the residue and coupling constants are sensitive to these parameters, a modest shift in the stability window propagates directly into the dominant widths (e.g., M → J/ψ B_c^+ and M → η_c B_c^{*+}), making the total Γ = (121 ± 17) MeV dependent on the same fitting procedure that defines the mass.
[interpolating current and OPE] The central assumption that the chosen interpolating current J_μ ~ (c-bar γ_μ c)(b-bar γ^μ γ5 c) accurately represents a loosely bound molecular state is not tested against possible large higher-order OPE terms or mixing with other four-quark structures. If the ground-state dominance is not preserved once those terms are restored, both the mass prediction and the subsequent decay phenomenology rest on an unverified approximation.

minor comments (1)

[numerical analysis] The manuscript would benefit from an explicit table listing the chosen Borel windows, continuum thresholds, and the resulting pole contributions for each correlator, allowing readers to reproduce the stability criteria.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive report. We address each major comment below with point-by-point responses, providing additional checks and clarifications where appropriate. Revisions have been made to strengthen the presentation of the Borel window selection, uncertainty propagation, and OPE convergence.

read point-by-point responses

Referee: In the two-point QCD sum-rule analysis for the mass of M (the section presenting the numerical results for the Borel window and continuum threshold), the quoted mass (9740 ± 70) MeV and residue rely on post-hoc choices of M² and s0 that are tuned for stability of the same correlation function used to extract the final value. For the four-quark molecular current, the OPE receives sizable two-meson continuum contributions; without explicit variation of dimension-8 condensates or inclusion of α_s corrections, the extracted pole mass can shift outside the stated uncertainty, undermining the bound-state claim.

Authors: The Borel window (M² = 4.5–6.5 GeV²) and s0 = 110 GeV² were fixed by the standard requirements of OPE convergence (higher-dimensional terms < 10% of the total) and pole dominance (> 50%). The ±70 MeV uncertainty already incorporates variations of M² and s0 inside the stability plateau. To address the referee’s concern, we have now performed an explicit variation of the dimension-8 condensate by ±30% around its central value; the resulting mass shift remains inside the quoted error band. We have added a new paragraph and an auxiliary table documenting this test. Regarding α_s corrections, the present calculation is performed at leading order; NLO effects in comparable four-quark sum rules typically shift the mass by 5–10 MeV, which is covered by our uncertainty estimate. We have inserted a short discussion of this point in the revised text. revision: partial
Referee: The three-point sum-rule calculations of the partial widths (the sections on fall-apart and annihilation decay channels) reuse the identical Borel window and s0 values chosen for the mass. Because the residue and coupling constants are sensitive to these parameters, a modest shift in the stability window propagates directly into the dominant widths (e.g., M → J/ψ B_c^+ and M → η_c B_c^{*+}), making the total Γ = (121 ± 17) MeV dependent on the same fitting procedure that defines the mass.

Authors: This is the conventional procedure: the residue λ extracted from the two-point function is inserted into the three-point sum rules. The quoted ±17 MeV uncertainty on the total width is obtained by propagating the allowed variations of M² and s0 over the entire stability window. We have added a dedicated sensitivity subsection that tabulates the partial widths for three representative points inside the window; the total width changes by at most 14%, remaining consistent with the reported error. The revised manuscript now explicitly states that the width uncertainty already accounts for the shared parameter choice. revision: yes
Referee: The central assumption that the chosen interpolating current J_μ ~ (c-bar γ_μ c)(b-bar γ^μ γ5 c) accurately represents a loosely bound molecular state is not tested against possible large higher-order OPE terms or mixing with other four-quark structures. If the ground-state dominance is not preserved once those terms are restored, both the mass prediction and the subsequent decay phenomenology rest on an unverified approximation.

Authors: The current is constructed to carry the exact J^P = 1^+ quantum numbers of the J/ψ B_c^+ molecule. Within the chosen Borel window the OPE series converges rapidly, with dimension-6 and higher condensates contributing less than 8% to the correlator. We have strengthened the manuscript by adding an explicit plot of the relative size of each OPE term versus M², confirming ground-state dominance. While a multi-current mixing analysis would be valuable, it lies outside the scope of the present work; we have added a brief remark noting this limitation and estimating that mixing effects are suppressed by the observed pole dominance. revision: partial

Circularity Check

0 steps flagged

No significant circularity; standard QCD sum-rule extraction is self-contained

full rationale

The paper derives masses and widths from two- and three-point correlation functions via OPE, Borel transformation, and dispersion relations. Borel window M² and continuum threshold s₀ are selected for stability of the ground-state contribution, which is a methodological validity check rather than a fit that forces the numerical output by construction. No quoted step equates the final mass or width to the chosen parameters; the sum-rule equations remain independent. Self-citations (if present) support the general technique but are not load-bearing for the specific predictions. The chain is therefore non-circular.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central predictions rest on the operator product expansion, choice of molecular interpolating currents, and phenomenological parameters tuned for sum-rule stability; no independent evidence for the molecular interpretation is provided beyond the calculation itself.

free parameters (2)

Borel parameter M^2
Chosen within a stability window to suppress higher states and continuum contributions
continuum threshold s0
Determined by requiring flatness of the mass sum rule in the Borel window

axioms (2)

standard math Validity of the operator product expansion for the two-point and three-point correlation functions at the chosen scale
Invoked to equate the phenomenological and QCD sides of the sum rules
domain assumption The chosen interpolating current projects onto the molecular state with sufficient overlap
Standard assumption in molecular-state sum-rule studies

invented entities (1)

hadronic molecule M = J/ψ B_c⁺ no independent evidence
purpose: To represent the bound state with J^P=1^+ and quark content cc c-bar b-bar
Postulated as the object whose mass and decays are computed; no independent experimental evidence cited

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discussion (0)

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