{"id":"e4a66214-f46c-444f-b7c3-39a013df1848","arxiv_id":"2602.10075","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"QCD sum rules predict tensor molecules ΥB_c*− and J/ψB_c*+ with masses (15864±85) and (9870±82) MeV and full widths 120 and 71 MeV, decaying to constituent heavy mesons and B/D pairs.","lead":"This paper predicts the masses and decay widths of two hypothetical 'tensor molecules' built from heavy quark–antiquark pairs, ΥB_c*− and J/ψB_c*+. The predictions give experimental collaborations concrete new states to search for in the fully-heavy tetraquark spectrum.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Broad-width two-point sum rule is internally inconsistent: Γ≈120 MeV is ~1.8× the 65 MeV gap to the nearest threshold, so the single-pole ansatz of Eq. (3) is not self-consistent.","rationale":"The reader’s weakest_assumption correctly identifies the narrow-pole/continuum separation as the fragile point. I focused on the narrow-pole ansatz rather than the form-factor extrapolation because it is more fundamental: if the state is as broad as the computed width, the two-point sum-rule extraction of m and Λ is not self-consistent, and the three-point width calculation inherits that error. The paper’s own lower-limit result—where the leading channel closes and the total width drops from 120 to 65 MeV—shows the central prediction is not robust under the quoted mass uncertainty. This does not automatically make the paper worthless; QCD sum rules are approximate and broad states are sometimes treated with a single-pole assumption. But the central claim that these are tensor molecules with well-defined masses and widths should be accepted only conditionally, pending a demonstration that the pole-continuum separation remains valid when the width is comparable to the threshold gap. The reader already reached CONDITIONAL, so my read does not change the verdict; it sharpens the specific test needed.","tokens_in":17415,"tokens_out":7901,"duration_ms":95663,"concrete_test":"Re-evaluate the two-point sum rule with a finite-width ground state: replace the zero-width pole in Eq. (3) by a Breit-Wigner spectral density with Γ fixed to 120 MeV, and re-derive Eqs. (7)–(8). Check whether any Borel window satisfying PC ≥ 0.5 and |Π^{Dim4}| ≤ 5%|Π| still exists within M² ∈ [17,20] GeV², s0 ∈ [280,285] GeV², and whether the extracted m shifts by more than the quoted ±85 MeV. If no such window exists, the narrow-pole extraction is invalid and all subsequent widths inherit uncontrolled systematics.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central mass and width predictions are entangled by an unaddressed self-consistency defect. The two-point sum rule for M_T^b extracts m and Λ by assuming an isolated narrow ground-state pole: Eq. (3) writes Π_Phys ~ Λ²/(m²−p²), Eq. (7) then defines m² = Π′/Π, and the PC ≥ 0.5 condition (Eq. (10)) is used to select the Borel window. But the paper’s own width calculation gives Γ[M_T^b] = 120^{+17}_{−12} MeV (Eq. (70)), while the mass is only 65 MeV above the leading threshold m_Υ + m_{B_c^*} = 15799 MeV. Thus Γ exceeds the available phase-space gap by roughly a factor of two. A state this broad cannot be represented as a delta-function pole at s = m² with a well-separated continuum; its spectral strength is smeared over an interval ~mΓ ≈ 1.9 GeV², comparable to the 2.1 GeV² distance to threshold. The extracted m is therefore an effective centroid, not the pole mass that enters the Breit-Wigner used implicitly when translating couplings into widths. The paper itself demonstrates the instability: at the lower-limit mass 15779 MeV the leading channel closes and the width drops to (65±7) MeV, a nearly factor-of-two change from the central value. The quoted uncertainties do not include this model-dependence. The three-point form-factor extrapolation (Eq. (26)) adds further, unquantified systematic error, but the more load-bearing issue is that the two-point analysis and the width calculation rest on mutually incompatible descriptions of the same state. This is not an external-model disagreement; it is an internal consistency problem in the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17973,"tokens_out":7203,"duration_ms":75499,"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":[{"comment":"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","section":"§II, Eqs. (3)–(10) and Eq. (70)"},{"comment":"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.","section":"§III, Eq. (26) and subsequent g_i determinations"},{"comment":"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.","section":"Abstract and §§III–IV"}],"minor_comments":[{"comment":"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.","section":"Eq. (21)"},{"comment":"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.","section":"Eq. (2) and Eq. (13)"},{"comment":"The abbreviation 'l.l.' is used without a definition; it should be defined as 'lower limit of the mass' on first use.","section":"Abstract"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a close methodological continuation of the authors' extensive program on heavy molecular states; the new results are incremental but still of interest if the technical issues are resolved. My main concern is the self-consistency of the narrow-pole treatment for a state whose predicted width exceeds its mass gap to threshold. This is a technical issue that can be addressed with a more careful treatment of the spectral function or an honest systematic-error estimate, so I do not recommend rejection at this stage. The form-factor extrapolation issue is also fixable by comparing alternative fit functions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nThe 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.\n\nBottom 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.","headline":"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.","tokens_in":18444,"tokens_out":2142,"would_cite":true,"duration_ms":25394,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["tensor hadronic molecules","fully heavy tetraquarks","QCD sum rules","masses and decay widths","bottom-charm exotics","spin-2 states","ΥB_c*−","J/ψB_c*+"],"falsifier":"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.","tokens_in":17303,"feed_emoji":"⚛️","tokens_out":5106,"duration_ms":53880,"temperature":0.7,"texified_at":"2026-08-05T20:54:38.673814+00:00","pith_summary":"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.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":8711,"prompt_tokens":867,"completion_tokens":7844,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":867,"completion_tokens_details":{"reasoning_tokens":6972}},"feed_headline":"Two heavy tensor molecules predicted at 15.86 and 9.87 GeV","feed_subtitle":"QCD sum rules give broad resonances that should show up in future heavy-meson spectra.","key_machinery":"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","core_discovery":"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 $M_T^b \\to \\Upsilon B_c^{*-}$","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Heavy tensor molecules at 15.86 and 9.87 GeV are unstable","QCD sum rules predict two unstable tensor molecules","Tensor states predicted with widths ~120 and ~71 MeV","New tensor hadrons decay into meson pairs, QCD predicts","Unbound tensor molecules: masses and decay widths computed"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Heavy tensor molecules at 15.86 and 9.87 GeV are unstable","QCD sum rules predict two unstable tensor molecules","Tensor states predicted with widths ~120 and ~71 MeV","New tensor hadrons decay into meson pairs, QCD predicts","Unbound tensor molecules: masses and decay widths computed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000325,"raw_usage":{"total_tokens":1839,"prompt_tokens":1104,"completion_tokens":735,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":848,"completion_tokens_details":{"reasoning_tokens":649}},"tokens_in":848,"tokens_out":735,"duration_ms":7345,"temperature":1.0,"reasoning_tokens":649,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T01:17:13.269938+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}