{"id":"99571cd4-b74e-47e8-8b9a-3f46325d3d60","arxiv_id":"2603.10922","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Triply heavy tetraquarks cc¯c¯q and bb¯b¯q are compact, unstable states with ground masses 5.2–5.5 GeV and 15.0–15.3 GeV; narrow resonances arise from amplitude cancellation and should appear in J/ψDs*/ηcDs and ΥB* channels.","lead":"The paper predicts masses, sizes and decay widths for four families of unobserved triply-heavy tetraquarks using a non-relativistic quark model. The results supply concrete mass windows and preferred decay channels that experiments can target.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the reader's already-flagged model-extrapolation risk.","rationale":"The manuscript is a self-contained, carefully tabulated application of a standard non-relativistic quark model to an experimentally empty sector. All numerical ingredients (masses, eigenvectors, RMS radii, partial widths) are given, the Pauli-allowed state counting is transparent, and the experimental search windows are concrete. The only load-bearing assumption is precisely the one the reader already identified: that a Hamiltonian tuned exclusively to two-body mesons remains adequate for four-body systems above threshold. No further technical flaw—incorrect matrix elements, inconsistent color-spin algebra, or numerical instability—appears. Consequently the CONDITIONAL verdict with high confidence is appropriate and needs no adjustment.","tokens_in":28614,"tokens_out":486,"duration_ms":5004,"concrete_test":"Recompute the two narrow 0+ masses and their partial widths (Tc2¯c¯s(5360) and Tb2¯b¯n(15052)) after (i) increasing the Gaussian basis size nmaxi from 5 to 8 and (ii) varying the hyperfine range parameters r0ij, eta by ±10 % around the Table-I values; if either mass shifts by more than ~30 MeV or the cancellation that keeps Γ < 1 MeV is lifted, the narrow-resonance claim weakens.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption correctly isolates the central soft spot: the eight-parameter non-relativistic Hamiltonian of Eq. (1), fitted only to ordinary mesons (Table I), is assumed to remain quantitatively reliable for four-body systems that sit above open-flavor thresholds. Within that framework the calculation is internally consistent—GEM solution of the four-body Schrödinger equation, color-spin mixing (Table III), RMS radii (Table IV) all supporting compact configurations, and the quark-interchange widths that produce the narrow states via diagram cancellation. No additional internal inconsistency, hidden circularity, or unstated assumption that would independently undermine the strongest claim (state counting, mass windows, compactness, and the existence of narrow resonances) was found. The continuum-coupling and relativistic-correction issues are real but already acknowledged by the reader as standard model dependence for the field.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper applies a nonrelativistic quark model (confinement plus color-spin hyperfine Hamiltonian with eight parameters fixed exclusively to ordinary meson masses) to the four triply heavy systems cc¯c¯n, cc¯c¯s, bb¯b¯n and bb¯b¯s. The four-body Schrödinger equation is solved with the Gaussian expansion method in Jacobi coordinates, color-spin configuration mixing is diagonalized, and the resulting wave functions are used to extract masses, RMS radii, radiative widths (tree-level quark-photon coupling) and OZI-allowed rearrangement widths (quark-interchange model). Both flavor sectors are predicted to contain exactly two 0+, three 1+ and one 2+ S-wave states, with ground-state masses in the windows 5.2–5.5 GeV and 15.0–15.3 GeV; all states lie above open-flavor thresholds, are compact according to the RMS analysis, and decay dominantly by rearrangement, a subset remaining narrow because of destructive interference among the four quark-interchange diagrams. Concrete experimental search channels (J/ψD*_s / η_c D_s near 5.3–5.4 GeV and ΥB* near 15.0–15.1 GeV) are proposed.","tokens_in":28844,"tokens_out":1372,"duration_ms":25053,"significance":"If the predictions hold, the work supplies the first fully consistent spectroscopic-plus-decay survey of the experimentally unexplored triply heavy sector within a single Hamiltonian and wave-function framework. The explicit color-spin mixing matrices (Table III), the RMS radii that quantitatively support compact rather than molecular configurations (Table IV), and the falsifiable partial-width ratios that arise from diagram cancellation constitute concrete, testable guidance for LHCb, Belle II and BESIII. The calculation is parameter-free once the meson fit is fixed, and the same wave functions feed both spectra and decays—an internal consistency that is a genuine strength relative to many earlier estimates.","major_comments":[{"comment":"Abstract versus Sec. IV B / Table IV: the abstract lists T_{b^{2}¯b¯n}(15148,2+) as a narrow-resonance example arising from Feynman-amplitude cancellation, yet Table IV and the explicit discussion in Sec. IV B assign that state a total width of 13–14 MeV while the genuinely narrow state (Γ < 1 MeV) is T_{b^{2}¯b¯n}(15052,0+). The same mismatch appears in the proposed search window (15.1–15.2 GeV versus 15.0–15.1 GeV). This is a load-bearing misidentification of the paper’s own central claim about narrow states and must be corrected throughout.","section":"Abstract, Sec. IV B, Table IV"},{"comment":"Sec. II A and Table I: the eight Hamiltonian parameters are determined solely from two-body meson spectra and then applied without adjustment to four-body systems that sit 100–300 MeV above open-flavor thresholds. Continuum coupling, possible molecular admixtures and relativistic corrections are therefore uncontrolled. While this is a standard model assumption, the paper’s strongest claim (existence and narrowness of specific resonances) rests on it; a quantitative estimate of the associated uncertainty, or a systematic comparison with the lattice and QCD-sum-rule results already cited in the Introduction, is required for the claim to be robust.","section":"Sec. II A, Table I"},{"comment":"Sec. III C, Eqs. (23)–(27) and the four diagrams of Fig. 2: the narrow widths are attributed to destructive interference among the C1, C2, T1 and T2 amplitudes. Because the relative weights of those diagrams are fixed by the color-spin mixing angles of Table III and by the GEM wave functions, the cancellation is model-dependent. A short sensitivity check (varying the mixing angles within the range allowed by the off-diagonal ⟨H⟩ entries, or rescaling the relative diagram strengths) would demonstrate that the narrowness is not an accidental numerical cancellation.","section":"Sec. III C, Fig. 2, Table III"}],"minor_comments":[{"comment":"Table III (bb¯b¯s 1+ block): the off-diagonal entry is printed as “15503”; comparison with Table II shows it must be 15203. Correct the typographical error.","section":"Table III"},{"comment":"Notation for the states oscillates between T_{c2¯c¯s}, T_{c^{2}¯c¯s} and T_{c^{2}¯c¯s}; adopt one consistent superscript convention throughout the text, tables and figures.","section":"Throughout"},{"comment":"Figs. 3 and 4 are information-dense; the partial-width ratios printed beneath the mass lines are hard to parse at journal scale. Consider moving the numerical ratios into a supplementary table or enlarging the font.","section":"Figs. 3–4"},{"comment":"The RMS-radius discussion (Sec. III A) correctly notes the equality ⟨r_{13}⟩ = ⟨r_{23}⟩ etc. forced by identical heavy quarks, but never states the numerical precision with which the GEM basis respects that equality; a one-sentence check would be reassuring.","section":"Sec. III A, Table IV"},{"comment":"Several references to earlier quark-model work on the same systems (e.g., the extended chromomagnetic and relativized-quark-model papers already cited) give only mass ranges; a short side-by-side mass table would help the reader gauge the model dependence without hunting through the literature.","section":"Sec. I, Sec. IV"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a competent, self-contained application of methods the authors have already validated on related systems. The factual error in the abstract is easily fixed and does not indicate deeper problems. The work is appropriate in scope and technical level for a solid hep-ph journal; I see no novelty or citation-pattern concerns that would require editorial intervention beyond ordinary review."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that this is a clean, complete application of the non-relativistic quark model to the four triply-heavy systems that still have no experimental candidates. They solve the four-body Schrödinger equation with GEM, diagonalize color-spin mixing (Table III), extract RMS radii that support compact configurations, and compute both rearrangement and radiative widths from the same wave functions. The headline numbers are concrete: two 0+, three 1+, one 2+ states per system, masses 5.2–5.5 GeV (cc¯c¯q) and 15.0–15.3 GeV (bb¯b¯q), all above threshold, with a few narrow states (Tc2¯c¯s(5360,0+), Tb2¯b¯n(15052,0+)) produced by cancellation among the four quark-interchange diagrams. Search windows in J/ψDs*/ηcDs and ΥB* are given with partial-width ratios that experimental groups can actually use.\n\nWhat is new is the simultaneous treatment of spectroscopy + decays for all four flavor combinations with the same Hamiltonian and the same basis. Prior work either stopped at masses or treated different systems. The calculation is internally consistent: parameters are fixed once on ordinary mesons (Table I) and never re-tuned; the free-parameter list is transparent; RMS radii and width tables are complete. Self-citations are to earlier applications of the same machinery, not circular re-fits.\n\nThe soft spot is exactly the one the reader flagged and the stress-test confirmed: the eight-parameter Hamiltonian fitted only to two-body mesons is assumed to remain quantitative for four-body states sitting above open-flavor thresholds. Continuum coupling, relativistic corrections and possible molecular components are neglected. That is standard model dependence for this literature, not a hidden flaw, and the paper does not over-claim. No internal inconsistency or invented entities appear.\n\nThis is for people who need numbers and channels for LHCb/Belle-II/BESIII searches, or who want a benchmark for other models. It deserves a serious referee. I would cite the mass windows and the narrow-state mechanism if I were writing on heavy tetraquarks this year, and I would bring the tables to reading group.","headline":"Solid, usable systematics for the empty triply-heavy sector: concrete masses, widths and search channels from a standard GEM + quark-interchange calculation, with the usual model-extrapolation caveat.","tokens_in":29439,"tokens_out":564,"would_cite":true,"duration_ms":5554,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Triply heavy tetraquarks are compact, all unstable against rearrangement, and a few stay narrow from amplitude cancellation, with masses fixed in two clear windows.","keywords":["triply heavy tetraquarks","nonrelativistic quark model","Gaussian expansion method","color-spin mixing","rearrangement decays","compact tetraquarks","quark-interchange model"],"falsifier":"A dedicated scan of the J/ψ D*s and ηc Ds invariant-mass spectra in the 5.3–5.4 GeV window that either finds a narrow peak near 5360 MeV with roughly equal branching fractions to those two channels, or rules it out at the few-MeV level.","tokens_in":29514,"feed_emoji":"⚛️","tokens_out":1020,"duration_ms":15317,"temperature":0.7,"pith_summary":"The paper sets out to map the low-lying S-wave spectrum and decay pattern of the four triply heavy systems cc¯c¯n, cc¯c¯s, bb¯b¯n and bb¯b¯s, which have never been seen experimentally. Using a non-relativistic quark model whose parameters were fixed on ordinary mesons, it solves the four-body Schrödinger equation with Gaussian bases and includes color-spin mixing. Both the charm and bottom families produce exactly two 0+, three 1+ and one 2+ states, sitting in the 5.2–5.5 GeV and 15.0–15.3 GeV windows respectively. Root-mean-square radii show the states are compact, not loosely bound molecules. All of them lie above open-flavor thresholds, so rearrangement strong decays dominate while radiative widths are tiny; a handful of states remain narrow because the four quark-interchange diagrams cancel. The concrete mass windows and branching ratios give experiment a clear target list in J/ψ D*s/ηc Ds and Υ B* channels.","feed_headline":"Triply heavy tetraquarks: compact, unstable, some narrow","feed_subtitle":"Masses sit in two windows; amplitude cancellation keeps a few resonances searchable","key_machinery":"An effective non-relativistic Hamiltonian (confinement plus color-spin hyperfine terms) is diagonalized for the four-body system by the Gaussian expansion method in Jacobi coordinates, with full color-spin configuration mixing; the same wave functions then feed a quark-interchange model that evaluates OZI-allowed two-body widths.","core_discovery":"Both the cc¯c¯q and bb¯b¯q systems each contain precisely two J^P = 0+, three J^P = 1+ and one J^P = 2+ ground states whose masses fall in 5.2–5.5 GeV and 15.0–15.3 GeV; every state is a compact tetraquark that decays mainly by quark rearrangement, and selected resonances (for example Tc^{2}¯c¯s(5360,0+) and Tb^{2}¯b¯n(15052,0+)) stay narrow because the four Feynman amplitudes interfere destructively.","pith_inferences":["The same Hamiltonian and Gaussian machinery can be applied immediately to the mixed-flavor triply heavy systems (e.g., cc¯b¯q) that are still missing from the present survey.","A null result in the proposed 5.3–5.4 GeV window would force a re-examination of how much continuum coupling must be restored once the states sit above threshold.","The pattern of amplitude cancellation suggests that other four-body systems with identical heavy-quark pairs may also hide unexpectedly narrow resonances."],"forward_implications":["Experiments can target two compact mass windows—5.3–5.4 GeV in charm and 15.0–15.1 GeV in bottom—with specific final states that already have good reconstruction efficiency.","Partner states that are nearly mass-degenerate can still be told apart by their total widths and dominant branching ratios.","The same cancellation mechanism that produces the narrow 0+ states will appear in any other rearrangement calculation that keeps the four quark-interchange diagrams.","If the predicted compact radii are confirmed, molecular interpretations of these particular quantum numbers become disfavored."],"fun_headline_variants":["Triply heavy tetraquarks: compact unstable states, some narrow","cc¯c¯q and bb¯b¯q: six states each in 5.2-5.5 and 15.0-15.3 GeV","All triply heavy tetraquarks decay mainly by quark rearrangement","Narrow triply heavy resonances from Feynman amplitude cancellation","Search triply heavy tetraquarks in J/ψDs and ΥB channels"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The same simple potential fitted only to ordinary meson masses remains accurate enough for four-body systems that contain three heavy quarks and sit above open-flavor thresholds, with continuum and molecular effects neglected.","fun_headline_variants_meta":{"raw":{"variants":["Triply heavy tetraquarks: compact unstable states, some narrow","cc¯c¯q and bb¯b¯q: six states each in 5.2-5.5 and 15.0-15.3 GeV","All triply heavy tetraquarks decay mainly by quark rearrangement","Narrow triply heavy resonances from Feynman amplitude cancellation","Search triply heavy tetraquarks in J/ψDs and ΥB channels"]},"model":"grok-4.5","effort":"low","cost_usd":0.008904,"raw_usage":{"total_tokens":2150,"prompt_tokens":951,"num_sources_used":0,"completion_tokens":96,"cost_in_usd_ticks":89040000,"prompt_tokens_details":{"text_tokens":951,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1103,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":951,"tokens_out":96,"duration_ms":12250,"temperature":1.0,"reasoning_tokens":1103,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T23:15:08.558971+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"A dedicated scan of the J/ψ D*s and ηc Ds invariant-mass spectra in the 5.3–5.4 GeV window that either finds a narrow peak near 5360 MeV with roughly equal branching fractions to those two channels, or rules it out at the few-MeV level.","supporting_citations":[],"review_version":1}