Tensor states $\Upsilon B_{c}^{\ast -}$ and $J/\psi B_{c}^{\ast +}$

H. Sundu; K. Azizi; S. S. Agaev

REVIEW 3 major objections 4 minor 55 references

Tensor states $\Upsilon B_{c}^{\ast -}$ and $J/\psi B_{c}^{\ast +}$

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper uses QCD sum rules to predict, for the first time, the masses and full widths of the tensor hadronic molecules ΥB_c*− and J/ψB_c*+, finding that both are broad, unbound states that decay promptly to their constituent heavy mesons

desk verdict First QCD sum rule predictions for these two tensor molecules; useful and mostly sound, but the large computed width makes the single-pole mass extraction less clean than the paper admits. read the letter →

arxiv 2602.10075 v2 pith:FD3XXM6Y submitted 2026-02-10 hep-ph hep-exhep-lat

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

classification hep-phhep-exhep-lat

keywords tensorhadronicmoleculesfullyheavytetraquarksQCDsumrulesmassesanddecaywidthsbottom-charmexoticsspin-2statesΥB_c*−J/ψB_c*+

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

The paper predicts, for the first time, the mass and decay width of two fully heavy tensor molecules: $M_T^b = \Upsilon B_c^{*-}$ and $M_T^c = J/\psi B_c^{*+}$. Using two- and three-point QCD sum rules, it obtains masses $m=(15864\pm 85)$ MeV and $\tilde{m}=(9870\pm 82)$ MeV, both above the respective two-meson thresholds. The paper therefore concludes that these structures are unstable against dissociation to their constituent mesons, with dominant decay modes to $\Upsilon B_c^{*-}$, $\eta_b B_c^{-}$ and $J/\psi B_c^{*+}$, $\eta_c B_c^{+}$, plus subleading channels in which b bbar or c cbar quarks annihilate and recombine into $B_{(s)}D_{(s)}$ pairs. The resulting total widths, $\Gamma[M_T^b]=120^{+17}_{-12}$ MeV and $\Gamma[M_T^c]=(71\pm 9)$ MeV, characterize them as relatively broad structures and give future experimental searches a concrete target.

What carries the argument

The central object is the interpolating tensor current $I_{\mu\nu}(x)=[\bar{b}_a \gamma_\mu b_a][\bar{c}_b \gamma_\nu b_b]$ (and its c-flipped counterpart), which creates a $J^P=2^+$ molecule out of two heavy vector mesons. The argument runs through two correlation functions: a two-point correlator whose Borel-transformed spectral density fixes the mass and current coupling, and three-point correlators with meson currents inserted, whose Borel-transformed amplitudes give strong form factors $g_i(q^2)$. Because those form factors are only computable at spacelike $q^2$, they are extrapolated to the physical decay point using a three-parameter exponential fit; the on-shell couplings then feed standard partial-width formulas

What would settle it

Search for a spin-2 enhancement in the $\Upsilon B_c^{*-}$ and $\eta_b B_c^{-}$ invariant-mass spectra near 15.86 GeV with a width of about 120 MeV, and in $J/\psi B_c^{*+}$ and $\eta_c B_c^{+}$ near 9.87 GeV with a width of about 71 MeV; the absence of such broad peaks in existing or future hadron-collider data would falsify the predictions.

Watch

Extended reading notes

Core claim

The central claim is that the tensor states $\Upsilon B_c^{*-}$ and $J/\psi B_c^{*+}$ form hadronic molecules with spin-parity $2^+$ and masses $m=(15864\pm 85)$ MeV and $\tilde{m}=(9870\pm 82)$ MeV. Those masses lie above the thresholds for their constituent mesons, which the paper takes as proof that the molecules are not bound; they decay through fall-apart modes to $\Upsilon B_c^{*-}$, $\eta_b B_c^{-}$ and $J/\psi B_c^{*+}$, $\eta_c B_c^{+}$, while subleading modes arise from b bbar and c cbar annihilation into light quark-antiquark pairs followed by recombination into $B_{(s)}D_{(s)}$ pairs. Working from these decay channels, the paper obtains total widths of $\Gamma[M_T^b]=120^{+17}_{-12}$ MeV and $\Gamma[M_T^c]=(71\pm 9)$ MeV. In the lower mass limit $m=15779$ MeV, the decay $M_T^b \to \Upsilon B_c^{*-}$

Load-bearing premise

The sum-rule extraction treats each molecule as a single narrow ground-state pole; with a width of 120 MeV for $M_T^b$, nearly twice its ~65 MeV gap above threshold, the pole-plus-continuum ansatz is not self-consistent.

Editorial extensions

If this is right

If the masses are right, both molecules sit above their two-meson thresholds, so they will not be bound and must show up as broad resonances in the corresponding four-quark channels.
The predicted mass 15864 MeV and width ≈120 MeV give a concrete search window for a spin-2 state decaying to ΥB_c*− and η_bB_c− in existing hadron-collider data.
The predicted mass 9870 MeV and width ≈71 MeV define an analogous window for J/ψB_c*+ and η_cB_c+.
At the lower end of the mass uncertainty (15779 MeV), the leading decay to ΥB_c*− shuts off, so the state would be visible mainly through η_bB_c− and annihilation-induced B_(s)D_(s) modes—a distinctive branching-fraction signature.
The molecular masses are several hundred MeV removed from diquark-antidiquark model estimates, so future mass measurements can discriminate between internal structures.

Reading between the lines

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

A natural next step would be a coupled-channel or unitary treatment that lets the broad resonance decay self-consistently, rather than treating it as a single pole; this could shift masses and widths by an amount on the order of the total width.
The exponential form-factor extrapolation is an unproven functional ansatz; a lattice calculation of the same three-point functions at spacelike momenta could test whether the on-shell couplings are reliable.
If these states are as broad as predicted, they may be hard to observe as sharp peaks amid continuum backgrounds; angular distributions of the tensor decay could help separate spin-2 signal from S-wave background.
For the lower mass limit of M_T^b, the closing of the dominant fall-apart channel makes the annihilation modes B_(s)D_(s) the only decay path—a sharp prediction that could be tested by searching for an absence of ΥB_c*− production near threshold.

Share X Bluesky LinkedIn Reddit HN

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

Worth a look if you work on fully-heavy exotics. The genuinely new thing is that nobody had calculated masses and full widths for these asymmetric tensor molecules in a hadronic-molecule picture; the paper does it with the standard two- and three-point QCD sum rule machinery, including both fall-apart and annihilation channels. As QCD sum rule papers go, this one is competently executed: OPE convergence is checked, pole dominance is enforced (PC ≥ 0.5), and the Borel windows look sensible. The concrete predictions — m = 15864 ± 85 MeV, Γ = 120^{+17}_{-12} MeV for Mb_T, and 9870 ± 82 MeV, 71 ± 9 MeV for Mc_T — are the kind of numbers experimental colleagues can put on a search plot.

The main soft spot is the one the stress-test flags, and I think it lands. The two-point sum rule treats Mb_T as a single narrow pole in Eq. (3), but the computed width is about 120 MeV while the state sits only 65 MeV above its leading threshold. That is internally inconsistent: a state that broad cannot be represented as a delta-function pole with a clean continuum separation, and the extracted m is probably a centroid rather than the pole mass that goes into the width formula. The paper actually shows the problem itself: at the lower mass limit 15779 MeV the leading channel closes and the width drops to ~65 MeV. The quoted uncertainties don't include that sensitivity. This is a real methodological caveat, not a nitpick.

The second soft spot is the extrapolation of the form factors via the three-parameter exponential in Eq. (26). It fits the spacelike data well, but the couplings at q² = m_final² depend on the assumed functional form, and no systematic error is attached. That's a known limitation of the three-point sum rule approach, but it should be stated more honestly.

The citation pattern is heavy on the authors' own prior papers, but that's mostly because this is a direct extension of their previous scalar and axial-vector molecule work; I don't see an attempt to bury competing results. The diquark-model masses from Refs [14,15] are cited and differ, which is fine.

Bottom line: the paper deserves a serious referee. The central claim is plausible and the method is standard; the broad-width issue should be addressed — either by justifying the pole-continuum separation, or by presenting a more careful spectral treatment — before the numbers are taken at face value. I'd send it to peer review with that request.

Referee Report

3 major / 4 minor

Summary. The paper computes, for the first time, the masses and full widths of two fully-heavy tensor hadronic molecules, M_T^b = Υ B_c*^- (quark content bb̄b̄c̄) and M_T^c = J/ψ B_c*^+ (content cc̄c̄b̄), using QCD sum rules. The two-point sum rules give m=(15864±85) MeV and m̃=(9870±82) MeV. The widths are obtained from three-point sum rules for the dominant fall-apart channels and subleading annihilation channels: Γ[M_T^b]=120^{+17}_{-12} MeV and Γ[M_T^c]=(71±9) MeV. The authors conclude that the states are relatively broad and unstable against strong decays, while noting that at the lower mass limit 15779 MeV the leading channel ΥB_c* becomes kinematically forbidden and the width decreases to (65±7) MeV.

Significance. If correct, these are the first quantitative predictions for asymmetric fully-heavy tensor molecular states, and they provide concrete targets for future experimental searches. The paper follows the standard QCD sum-rule protocol: it enforces pole dominance (PC≥0.5), checks OPE convergence, and gives Borel-window stability plots. The new quantitative results are the masses, couplings, and partial widths for the six main decay channels of each molecule, which could be useful for interpreting future LHCb/CMS data. The main weakness is that the central object M_T^b is predicted to have a width larger than its mass gap above the nearest threshold, which calls into question the narrow-pole approximation used in the two-point sum rule; additionally, the on-shell couplings depend on an ad hoc extrapolation of spacelike form factors.

major comments (3)

[§II, Eqs. (3)–(10) and Eq. (70)] For M_T^b the extracted mass m=15864 MeV is only Q≈65 MeV above the leading threshold m_Υ+m_Bc*=15799 MeV, while the computed total width is Γ≈120 MeV. The two-point sum rule uses a single narrow ground-state pole plus continuum, with PC≥0.5 selecting the Borel window. The product mΓ≈1.9 GeV² is comparable to the squared-mass gap to threshold, Δs≈2.1 GeV², and to the separation from the continuum threshold (s0≈280 GeV², far above). This means the spectral strength of the ground state is smeared over an interval comparable to the distance to the continuum, so the pole-plus-continuum ansatz is not self-consistent. The extracted mass is likely an effective centroid rather than a pole mass. The authors should quantify the effect by repeating the two-point analysis with a finite-width (e.g., Breit-Wigner) ground-state spectral function, or at least provide an estimate of the systematic error
[§III, Eq. (26) and subsequent g_i determinations] The strong couplings g_i are obtained by fitting a three-parameter exponential ansatz Z_i(Q²)=Z₀ exp[z₁ Q²/m² + z₂ (Q²/m²)²] to spacelike Q²>0 sum-rule data and evaluating it at the timelike point Q²=−m_final². This functional form is introduced ad hoc; no theoretical argument or alternative fit is given. The on-shell couplings and hence the partial widths can be sensitive to the choice of parameterization, especially for extrapolations far outside the fitted Q² interval (e.g., for g1, the fitted window is Q²=2–50 GeV², while the on-shell point is Q²=−40.2 GeV²). The quoted uncertainties from the fit do not include this model dependence. The paper should compare at least two independent extrapolating functions (e.g., polynomial or Padé) and fold the spread into the final width errors.
[Abstract and §§III–IV] The mass uncertainty for M_T^b, m=(15864±85) MeV, straddles the leading threshold m_Υ+m_Bc*=15799 MeV. The paper itself states that at the lower mass limit 15779 MeV the ΥB_c* decay is forbidden and the total width drops to (65±7) MeV — almost a factor of two smaller than the quoted central value. The central width Γ=120^{+17}_{-12} MeV is computed only for the central mass and does not incorporate the threshold-sensitivity of the leading decay. As a result, the claim that these states are 'unstable against dissociation to constituent mesons' is not robust within 1σ. The authors should either present the lower-limit scenario as a distinct prediction or add a systematic uncertainty to the width that reflects this threshold sensitivity.

minor comments (4)

[Eq. (21)] The physical side of the ΥB_c* vertex is written with f_{J/ψ} m_{J/ψ} in the numerator, but the final-state meson is B_c*, not J/ψ. Eq. (23) correctly uses f_{B_c*} m_{B_c*}, suggesting a typographical error in Eq. (21) that should be corrected.
[Eq. (2) and Eq. (13)] The interpolating currents are written in a form that appears to be missing Dirac adjoint symbols (e.g., 'ba(x)γµba(x)' instead of '\bar b_a(x)γµ b_a(x)'). This makes the quark content of the currents unclear. Please rewrite the currents using explicit fermion fields and conjugation.
[Abstract] The abbreviation 'l.l.' is used without a definition; it should be defined as 'lower limit of the mass' on first use.
[General] The exponential fitting procedure and the form of the tensor-vector-vector vertex are taken from the authors' previous papers without an explanatory derivation. Since this is a standalone submission, a brief justification of Eq. (26) and Eq. (20), or a reference to the derivation, would improve clarity.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: masses and widths are independent QCDSR outputs; no step reduces a prediction to a fitted input.

full rationale

The paper's central results are not circular by construction. The masses m and \tilde m are extracted from two-point sum rules, Eq. (7) m^2 = \Pi'(M^2,s0)/\Pi(M^2,s0) and Eq. (8), using OPE spectral densities and continuum subtraction; they are outputs, not inputs. The widths are obtained from three-point sum rules Eq. (23) etc., with the same masses and couplings as inputs, and the resulting partial widths are summed to give Eq. (70) and Eq. (78). The only fitted functions are the exponential Z_i(Q^2) in Eq. (26), whose parameters are adjusted to the spacelike SR data (e.g., Eqs. (27), (63)); evaluation at Q^2=-m_final^2 is an extrapolation, and the final width is not used in the fit. Thus no output reduces by definition to a fitted value of the same quantity. The broad-width/narrow-pole tension (\Gamma[\mathcal{M}_T^b]=120 MeV vs. the 65 MeV gap to \Upsilon B_c^{*-}) is a genuine self-consistency and systematic-error concern for the two-point pole ansatz, but it is not circularity: the mass and width are computed independently and then compared. Self-citations (e.g., [47] for the tensor-vector-vector vertex, [42] for propagators) supply standard or previously developed technology, but the load-bearing numerical content does not reduce to a self-citation chain. Therefore the appropriate finding is no significant circularity.

Assumptions & free parameters 5 free parameters · 5 assumptions · 2 invented entities

The analysis rests on standard QCD sum rule technology: quark-hadron duality, a single-pole ansatz, heavy-quark condensate relations, and external meson parameters. The only genuinely ad hoc elements are the Borel/s0 windows and the exponential extrapolation functions, which are chosen to satisfy internal criteria or to fit spacelike form factors. No new fundamental constants are introduced; the two predicted molecules are the only new entities.

free parameters (5)

Borel parameter windows M^2 = [17,20] GeV^2 (M_T^b); [9,11] GeV^2 (M_T^c); meson-channel windows vary
Chosen by hand inside SR criteria; masses and couplings are averages over these windows, so the central values depend on the chosen ranges.
Continuum thresholds s0 = [280,285] GeV^2 (M_T^b); [112,115] GeV^2 (M_T^c); meson windows vary
Chosen by hand to satisfy pole dominance; variation of s0 and M^2 feeds directly into the quoted mass/width errors.
Extrapolation fit Z1 (g1) = Z0=0.379 GeV^-1, z1=3.072, z2=1.850
Fitted to SR data for g1(Q^2); evaluated at Q^2=-m_Bc*^2. The dominant width 24.2 MeV is set by this fit.
Extrapolation fit Z2 (g2) = Z0=31.73 GeV^-1, z1=2.36, z2=3.88
Fitted to SR data for g2(Q^2); evaluated at Q^2=-m_Bc^2; controls Γ(→ηbBc)=30.7 MeV.
Extrapolation fits Z3-Z6 and M_T^c analogues = z1/z2 values in Secs. IV-V and Table I
All subleading couplings and the M_T^c couplings are obtained from 3-parameter exponential fits to spacelike SR points, then continued to the timelike mass shell; all partial widths inherit the fit-form dependence.

assumptions (5)

domain assumption Quark-hadron duality: the hadronic dispersion relation is saturated by a single ground-state pole plus a continuum that is dual to the OPE above s0
Invoked in Eqs. (3), (9), and in every two- and three-point sum rule; this is the standard SR approximation and is not proved within the paper.
domain assumption The interpolating currents I_μν and \tilde I_μν (Eqs. (2), (13)) couple dominantly to the tensor JP=2+ molecular states M_T^b and M_T^c
The sum rule results depend on this coupling being non-zero and the ground state being the molecule; no rigorous current-hadron duality proof is given.
standard math Heavy-quark condensate relation ⟨b̄b⟩ = -(1/(12m_b))⟨α_sG^2/π⟩ (Eq. (50))
Standard heavy-quark expansion from Ref [37]; used to compute all subleading B(s)D(s) width OPEs.
ad hoc to paper Exponential ansatz Z_i(Q^2)=Z0 exp[z1 Q^2/m^2 + z2 (Q^2/m^2)^2] (Eq. (26)) gives the correct analytic continuation of the form factors to the timelike region
This functional form is not derived from QCD; it is chosen to fit the spacelike SR points and then evaluated at q^2=m_final^2. The extracted couplings g_i depend on this choice.
domain assumption External meson masses and decay constants (m_b, m_c, f_Bc, f_B, f_D*, etc.) from PDG and prior SR analyses
Standard external inputs; the quoted uncertainties propagate to the results and are not independently derived in this paper.

invented entities (2)

M_T^b = ΥB_c*− tensor hadronic molecule (JP=2+, quark content bb b̄ c̄) independent evidence
purpose: Predicted state; paper computes mass m=15864±85 MeV and width Γ=120+17/−12 MeV.
The predicted mass/width and decay channels are falsifiable in collider searches for fully-heavy tetraquarks (e.g., LHCb/CMS/ATLAS); no direct experimental evidence yet.
M_T^c = J/ψB_c*+ tensor hadronic molecule (JP=2+, quark content cc c̄ b̄) independent evidence
purpose: Predicted state; paper computes mass 9870±82 MeV and width 71±9 MeV.
Predicted mass/width could be searched in collider data; falsifiable but not yet observed.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Tensor states $\Upsilon B_{c}^{\ast -}$ and $J/\psi B_{c}^{\ast +}$." pith.science (2026). https://pith.science/paper/FD3XXM6Y

@misc{pith2026260210075,
  author       = {Pith},
  title        = {Pith review of: Tensor states $\Upsilon B_c^\ast -$ and $J/\psi B_c^\ast +$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FD3XXM6Y}},
  note         = {Machine review of arXiv:2602.10075}
}

abstract

Tensor states $\mathcal{M}_{\mathrm{T}}^{\mathrm{b}}=\Upsilon B_{c}^{\ast -}$ and $\mathcal{M}_{\mathrm{T}}^{\mathrm{c}}=J/\psi B_{c}^{\ast +}$ are explored using techniques of QCD sum rule method. These hadronic molecules, composed of only heavy quarks, have asymmetric quark contents $bb\overline{b} \overline{c}$ and $cc\overline{c}\overline{b}$, respectively. The masses $ m=(15864 \pm 85)~\mathrm{MeV}$ and $\widetilde{m}=(9870 \pm 82)~\mathrm{MeV} $ prove that these structures are unstable against dissociations to constituent mesons. Full widths of molecules $\mathcal{M}_{\mathrm{T}}^{ \mathrm{b}}$ and $\mathcal{M}_{\mathrm{T}}^{\mathrm{c}}$ are calculated by considering their dominant and subleading decay channels. The subleading channels are processes generated by annihilations of $\overline{b}b$ and $ \overline{c}c$ quarks. For the molecule $\mathcal{M}_{\mathrm{T}}^{\mathrm{b} }$ dominant decays are $\mathcal{M}_{\mathrm{T}}^{\mathrm{b}} \to \Upsilon B_{c}^{\ast -}$ and $\mathcal{M}_{\mathrm{T}}^{\mathrm{b}} \to \eta_b B_{c}^{-}$, whereas subleading channels are transformations to $\mathcal{M} _{ \mathrm{T}}^{\mathrm{b}}\rightarrow B^{(\ast )-}\overline{D}^{(\ast )0}$ and $\overline{B}_{(s)}^{(\ast )0}D_{(s)}^{(\ast )-}$ mesons. In the lower limit ($\mathrm{l.l.}$) of the mass $m=15779~\mathrm{MeV}$ for $\mathcal{M}_{ \mathrm{T}}^{\mathrm{b}}$ decay to $\Upsilon B_{c}^{\ast -}$ mesons is forbidden. In the case of $\mathcal{M}_{\mathrm{T}}^{\mathrm{c}}$ we explore decays to $J/\psi B_{c}^{\ast +}$, $\eta_{c}B_{c}^{+}$, $B^{(\ast)+}D^{(\ast )0}$ and $B_{(s)}^{(\ast )0}D_{(s)}^{(\ast )+}$ mesons. Predictions $\Gamma[ \mathcal{M} _{\mathrm{T}}^{\mathrm{b}}]=120^{+17}_{-12}~ \mathrm{MeV}$, $ \Gamma[\mathcal{M}_{\mathrm{T}}^{\mathrm{b}}]_{\mathrm{l.l.}}=(65 \pm 7)~ \mathrm{MeV}$ and ...

Figures

Figures reproduced from arXiv: 2602.10075 by the authors.

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

**Figure 3.** FIG. 3: Dependence of the mass [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4: QCD data and fit function for [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5: SR data and fit functions for the form factors [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 5.** Figure 5: The partial width of the decay Mb T → B−D 0 is equal to Γ h Mb T → B −D 0 i = (13.5 ± 3.4) MeV. (64) The width of the process Mb T → B 0 D−, as it has been explained above, is approximately equal to Γ h Mb T → B−D 0 i . C. Channels Mb T → B ∗0 s D ∗− s and B 0 sD − s E…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

55 extracted references · 3 linked inside Pith

[1]

where the pairs (M 2 1,s 0) and ( M 2 2,s ′
[2]

It is determined in the following form Π1(M2, s0,q 2) = ∫ s0 (3mb+mc)2 ds ∫ s′ 0 4m2 b ds′ρ1(s,s ′,q 2) ×e−s/M2 1 −s′/M2

correspond to the channels of the molecule Mb T and meson Υ , respectively. It is determined in the following form Π1(M2, s0,q 2) = ∫ s0 (3mb+mc)2 ds ∫ s′ 0 4m2 b ds′ρ1(s,s ′,q 2) ×e−s/M2 1 −s′/M2
[3]

(24) Restrictions applied to ﬁnd parameters M2 and s0 are usual for all sum rule studies and have been explained in Sec. II. Our computations demonstrate that windows for the parameters (M 2 1,s 0) Eq. (11) and M 2 2 ∈ [9, 11] GeV 2, s′ 0 ∈ [98, 100] GeV 2. (25) for ( M 2 2,s ′
[4]

Note that s′ 0 is bounded by the mass mΥ(2S) = (10023.4 ± 0.5) MeV of the meson Υ(2S), i.e., s′ 0<m 2 Υ(2S)

obey these restrictions. Note that s′ 0 is bounded by the mass mΥ(2S) = (10023.4 ± 0.5) MeV of the meson Υ(2S), i.e., s′ 0<m 2 Υ(2S). The sum rule for the form factor g1(q2) is applicable in the region q2 < 0. But g1(q2) determines the cou- pling g1 at the mass shell q2 = m2 B∗c . For that reason, we introduce the function g1(Q2) where Q2 = −q2 and utiliz...

2006
[5]

The mass and decay constant of B∗+ c have been presented in Sec

(75) In numerical computations we employ the masses and decay constants of the mesons B∗+ c and J/ψ. The mass and decay constant of B∗+ c have been presented in Sec. III. As the spectroscopic parameters of the vector char- moniumJ/ψ we employmJ/ψ = (3096.900±0.006) MeV, and fJ/ψ = (411 ± 7) MeV [43, 45]. Computations of the form factor ˜g1(q2) are perform...
[6]

from Eq. (52). The coupling ˜g1 is obtained at Q2 = −m2 BJ/ψ by means of the extrapolating function ˜Z1(Q2) which is given by Eq. (26) after replacement m → ˜m. This function has the parameters ˜Z 0 1 = 0.0676 GeV −1, ˜z1 1 = 1.403, and ˜z2 1 = 0.301 and leads to the prediction ˜g1 ≡ ˜Z1(−m2 BJ/ψ ) = (1.91 ± 0.38) × 10−2 GeV−1. (76) Then the width of the ...
[7]

Heller and J

L. Heller and J. A. Tjon, Phys. Rev. D 32, 755 (1985)

1985
[8]

R. J. Lloyd and J. P. Vary, Phys. Rev. D 70, 014009 (2004)

2004

Show all 55 references

[9]

Karliner, S

M. Karliner, S. Nussinov, and J. L. Rosner, Phys. Rev. D 95, 034011 (2017)

2017
[10]

M. N. Anwar, J. Ferretti, F. K. Guo, E. Santopinto, and B. S. Zou, Eur. Phys. J. C 78, 647 (2018)

2018
[11]

J. Wu, Y. R. Liu, K. Chen, X. Liu, and S. L. Zhu, Phys. Rev. D 97, 094015 (2018)

2018
[12]

M. S. Liu, Q. F. Lu, X. H. Zhong, and Q. Zhao, Phys. Rev. D 100, 016006 (2019)

2019
[13]

Chen, Phys

X. Chen, Phys. Rev. D 100, 094009 (2019)

2019
[14]

M. A. Bedolla, J. Ferretti, C. D. Roberts, and E. San- topinto, Eur. Phys. J. C 80, 1004 (2020)

2020
[15]

M. C. Gordillo, F. De Soto, and J. Segovia, Phys. Rev. D 102, 114007 (2020)

2020
[16]

X. Z. Weng, X. L. Chen, W. Z. Deng, and S. L. Zhu, Phys. Rev. D 103, 034001 (2021)

2021
[17]

G. J. Wang, L. Meng, M. Oka, and S. L. Zhu, Phys. Rev. D 104, 036016 (2021)

2021
[18]

C. Deng, H. Chen, and J. Ping, Phys. Rev. D 103, 014001 (2021)

2021
[19]

Z. H. Yang, Q. N. Wang, W. Chen, and H. X. Chen, Phys. Rev. D 104, 014003 (2021)

2021
[20]

V. O. Galkin, and E. M. Savchenko, Eur. Phys. J. A 60, 96 (2024)

2024
[21]

H. T. An, S. Q. Luo, Z. W. Liu, and X. Liu, Eur. Phys. J. C 83, 740 (2023)

2023
[22]

Hoﬀer, G

J. Hoﬀer, G. Eichmann, and C. S. Fischer, Phys. Rev. D 109, 074025 (2024). 11

2024
[23]

S. S. Agaev, K. Azizi, and H. Sundu, Phys. Lett. B 858, 139042 (2024)

2024
[24]

S. S. Agaev, K. Azizi, and H. Sundu, Phys. Lett. B 864, 139404 (2025)

2025
[25]

S. S. Agaev, K. Azizi, and H. Sundu, Phys. Rev. D 111, 074025 (2025)

2025
[26]

S. S. Agaev, K. Azizi, and H. Sundu, Eur. Phys. J. A 61, 118 (2025)

2025
[27]

A. Ali, A. Y. Parkhomenko, Q. Qin, and W. Wang, Phys. Lett. B 782, 412 (2018)

2018
[28]

A. Ali, Q. Qin, and W. Wang, Phys. Lett. B 785, 605 (2018)

2018
[29]

Carvalho, E

F. Carvalho, E. R. Cazaroto, V. P. Gonsalves, and F. S. Navarra, Phys. Rev. D 93, 034004 (2016)

2016
[30]

L. M. Abreu, F. Carvalho, J. V. C. Cerquera, and V. P. Goncalves, Eur. Phys. J. C 84, 470 (2024)

2024
[31]

Aaij et al

R. Aaij et al. (LHCb Collaboration), Sci. Bull. 65, 1983 (2020)

1983
[32]

Aad et al

G. Aad et al. (ATLAS Collaboration), Phys. Rev. Lett. 131, 151902 (2023)

2023
[33]

Hayrapetyan et al

A. Hayrapetyan et al. (CMS Collaboration), Phys. Rev. Lett. 132, 111901 (2024)

2024
[34]

Hayrapetyan et al

A. Hayrapetyan et al. (CMS Collaboration), arXiv:2602.02252 [hep-ex]
[35]

W. Y. Liu and H. X. Chen, Eur. Phys. J. C 85, 636 (2025)

2025
[36]

S. S. Agaev, K. Azizi, and H. Sundu, Phys. Rev. D 112, 054001 (2025)

2025
[37]

S. S. Agaev, K. Azizi, and H. Sundu, Phys. Lett. B 870, 139885 (2025)

2025
[38]

S. S. Agaev, K. Azizi, and H. Sundu, Phys. Lett. B 871, 140014 (2025)

2025
[39]

W. Y. Liu and H. X. Chen, Universe 11, 36 (2025)

2025
[40]

S. S. Agaev, K. Azizi, and H. Sundu, arXiv:2511.03541 [hep-ph]

arXiv
[41]

S. S. Agaev, K. Azizi, and H. Sundu, arXiv:2512.23030 [hep-ph]

arXiv
[42]

S. S. Agaev, K. Azizi, and H. Sundu, arXiv:2512.06513 [hep-ph]
[43]

M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147, 385 (1979)

1979
[44]

M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147, 448 (1979)

1979
[45]

Becchi, A

C. Becchi, A. Giachino, L. Maiani, and E. Santopinto, Phys. Lett. B 806, 135495 (2020)

2020
[46]

Becchi, A

C. Becchi, A. Giachino, L. Maiani, and E. Santopinto, Phys. Lett. B 811, 135952 (2020)

2020
[47]

S. S. Agaev, K. Azizi, B. Barsbay, and H. Sundu, Phys. Rev. D 109, 014006 (2024)

2024
[48]

S. S. Agaev, K. Azizi, and H. Sundu, Turk. J. Phys. 44, 95 (2020)

2020
[49]

Navas et al

S. Navas et al. [Particle Data Group], Phys. Rev. D 110, 030001 (2024)

2024
[50]

Godfrey, Phys

S. Godfrey, Phys. Rev. D 70, 054017 (2004)

2004
[51]

Lakhina, and E

O. Lakhina, and E. S. Swanson, Phys. Rev. D 74, 014012 (2006)

2006
[52]

E. J. Eichten, and C. Quigg, Phys. Rev. D 99, 054025 (2019)

2019
[53]

S. S. Agaev, K. Azizi, and H. Sundu, Phys. Lett. B 856, 138886 (2024)

2024
[54]

Z. G. Wang, Chin. Phys. C 48, 103104 (2024)

2024
[55]

J. L. Rosner, S. Stone, and R. S. Van de Water, arXiv:1509.02220

arXiv

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

where the pairs (M 2 1,s 0) and ( M 2 2,s ′

[2] [2]

It is determined in the following form Π1(M2, s0,q 2) = ∫ s0 (3mb+mc)2 ds ∫ s′ 0 4m2 b ds′ρ1(s,s ′,q 2) ×e−s/M2 1 −s′/M2

correspond to the channels of the molecule Mb T and meson Υ , respectively. It is determined in the following form Π1(M2, s0,q 2) = ∫ s0 (3mb+mc)2 ds ∫ s′ 0 4m2 b ds′ρ1(s,s ′,q 2) ×e−s/M2 1 −s′/M2

[3] [3]

(24) Restrictions applied to ﬁnd parameters M2 and s0 are usual for all sum rule studies and have been explained in Sec. II. Our computations demonstrate that windows for the parameters (M 2 1,s 0) Eq. (11) and M 2 2 ∈ [9, 11] GeV 2, s′ 0 ∈ [98, 100] GeV 2. (25) for ( M 2 2,s ′

[4] [4]

Note that s′ 0 is bounded by the mass mΥ(2S) = (10023.4 ± 0.5) MeV of the meson Υ(2S), i.e., s′ 0<m 2 Υ(2S)

obey these restrictions. Note that s′ 0 is bounded by the mass mΥ(2S) = (10023.4 ± 0.5) MeV of the meson Υ(2S), i.e., s′ 0<m 2 Υ(2S). The sum rule for the form factor g1(q2) is applicable in the region q2 < 0. But g1(q2) determines the cou- pling g1 at the mass shell q2 = m2 B∗c . For that reason, we introduce the function g1(Q2) where Q2 = −q2 and utiliz...

2006

[5] [5]

The mass and decay constant of B∗+ c have been presented in Sec

(75) In numerical computations we employ the masses and decay constants of the mesons B∗+ c and J/ψ. The mass and decay constant of B∗+ c have been presented in Sec. III. As the spectroscopic parameters of the vector char- moniumJ/ψ we employmJ/ψ = (3096.900±0.006) MeV, and fJ/ψ = (411 ± 7) MeV [43, 45]. Computations of the form factor ˜g1(q2) are perform...

[6] [6]

from Eq. (52). The coupling ˜g1 is obtained at Q2 = −m2 BJ/ψ by means of the extrapolating function ˜Z1(Q2) which is given by Eq. (26) after replacement m → ˜m. This function has the parameters ˜Z 0 1 = 0.0676 GeV −1, ˜z1 1 = 1.403, and ˜z2 1 = 0.301 and leads to the prediction ˜g1 ≡ ˜Z1(−m2 BJ/ψ ) = (1.91 ± 0.38) × 10−2 GeV−1. (76) Then the width of the ...

[7] [7]

Heller and J

L. Heller and J. A. Tjon, Phys. Rev. D 32, 755 (1985)

1985

[8] [8]

R. J. Lloyd and J. P. Vary, Phys. Rev. D 70, 014009 (2004)

2004