{"id":"820ce125-998a-4c51-9181-e2c72a8cc91c","arxiv_id":"1908.08811","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"For fully-heavy bbcc and bcbc tetraquarks, the chiral quark model finds no bound states but predicts several resonances above the two-meson thresholds.","lead":"Using a nonrelativistic quark model, the author computed the masses of tetraquarks made of two b quarks and two c antiquarks, and of the b-c-b-c combination. No stable bound states are found, but several above-threshold resonances are predicted, giving experimental search targets.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Resonance predictions rest on an unvalidated stabilization procedure: only one color component is scaled, no widths or pole positions are computed, and the plateau energies lie hundreds of MeV above threshold; the no-bound-state part is less vulnerable.","rationale":"The claim that carries the paper is the resonance spectrum. The abstract and Summary present seven specific masses, and these are the falsifiable predictions. The no-bound-state part is plausible and agrees with several independent studies, whereas the resonance part depends on a nontrivial identification procedure. My objection is not that the model disagrees with other calculations; it is that the stabilization method as implemented (scaling only one color component, with no pole extraction and no width) has a validation gap. The proposed complex-scaling/all-channel scaling test directly targets that gap. I therefore support the Reader's CONDITIONAL verdict and would not change it. Agreement with the Reader's weakest assumption is full: the real scaling method applied to only singlet-singlet Gaussians was already flagged as the risky link.","tokens_in":13488,"tokens_out":8136,"duration_ms":90328,"concrete_test":"Recompute the bbcc 0(0+) channel with the scaling factor alpha applied to every Gaussian coordinate in every color configuration (meson-meson 1x1 and 8x8, and diquark-antidiquark), and then apply complex scaling to the same Hamiltonian to search for an S-matrix pole. If a pole survives with an energy close to 13140 MeV and a width below roughly 100 MeV, the resonance claim is supported; if the plateau shifts by more than 100 MeV or no pole appears, the summary numbers are artifacts of the partial stabilization. The analogous check should be repeated for at least the bcbc 0(0++) 12860 MeV line.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's distinctive output is the resonance masses announced in Sec. IV, not the no-bound-state statement. Those resonance claims rest on the stabilization step in Sec. III: after Eq. (18), the Gaussian size parameters r_n are multiplied by alpha only for the meson-meson color singlet-singlet configurations, and a line that stays flat in the energy-versus-alpha plots is read as a genuine resonance. Stabilization in a truncated coupled-channel basis can produce flat avoided crossings even when no physical pole exists, and the standard way to confirm a resonance is to locate the pole, e.g. by complex scaling or a phase-shift/Jost-function analysis, and to give its width. None of that is done here. The claimed energies lie 180-540 MeV above the nearest two-meson threshold; at such separations broad states are expected, and the paper itself concedes the states may be too wide to be observed. The single-channel plateau is therefore insufficient to establish the predicted resonances. In addition, all orbital angular momenta in Eq. (14) are taken to be zero, so tensor-coupled D-wave continuum channels that can matter hundreds of MeV above threshold are absent. The no-bound conclusion is less vulnerable because it is a variational study in a large GEM basis whose sign is reinforced by the CMI analysis in Table V, although the upper-bound character of the calculation means the summary phrase leaving no space is stronger than what a variational energy above threshold proves.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the fully-heavy tetraquark systems $bb\\bar{c}\\bar{c}$ and $bc\\bar{b}\\bar{c}$ within a nonrelativistic chiral quark model, using the Gaussian expansion method. The calculation includes meson-meson and diquark-antidiquark structures, their mixing, all allowed color and spin configurations, and antisymmetrization for $bb\\bar{c}\\bar{c}$. The variational energies in Tables VI and VII lie above the corresponding two-meson thresholds in every channel, from which the paper concludes that no bound tetraquark exists in these systems. Applying the real scaling (stabilization) method to the meson-meson color singlet-singlet configurations, the paper then identifies resonance candidates: $13140$, $13180$, and $13230$ MeV for $bb\\bar{c}\\bar{c}$ with $0(0^+)$, $0(1^+)$, and $0(2^+)$, respectively, and $12860$, $13020$, $13020$, and $12910$ MeV for $bc\\bar{b}\\bar{c}$ with $0(0^{++})$, $0(1^{++})$, $0(2^{++})$, and $0(1^{+-})$, respectively.","tokens_in":13870,"tokens_out":4068,"duration_ms":46046,"significance":"If the results are correct, the paper provides a useful systematic survey of two debated fully-heavy tetraquark systems, supporting the no-bound-state side of the current controversy while making concrete, testable resonance predictions. The variational bound-state part is credible: model parameters are fixed by meson spectra rather than fitted to the tetraquark masses, the color-spin channel space is broad, and the CMI analysis in Table V gives a transparent qualitative consistency check. However, the distinctive new claim of the paper is the resonance spectrum, and that claim rests on a stabilization procedure whose validation is incomplete: no widths or pole positions are computed, only one color configuration is scaled, and all orbital angular momenta are set to zero. The resonance predictions are therefore not yet established at the level required for a definitive statement.","major_comments":[{"comment":"The real scaling method is applied only to the Gaussian size parameters of the meson-meson color singlet-singlet configurations while all other channels are kept fixed. In a truncated coupled-channel basis, a flat energy-versus-α line can also arise from an avoided crossing or a pseudostate, and the standard confirmation of a resonance is the location of a pole in the complex energy plane, for example by complex scaling or phase-shift/Jost-function analysis, together with a width. The paper reports no such pole or width, and in Sec. IV it concedes that the higher states 'may be too wide to be observed.' Given that the claimed resonances lie several hundred MeV above their lowest thresholds, the stabilization plateaus in Figs. 3-6 are insufficient by themselves to establish these states as genuine resonances.","section":"III, after Eq. (18), Figs. 3-6"},{"comment":"All orbital angular momenta are taken to be zero in this calculation. This omits tensor-coupled D-wave meson-meson channels, which can mix with the S-wave basis at excitation energies of several hundred MeV above threshold. The omission affects both the variational energies and the interpretation of the stabilization plateaus, so the resonance positions reported in Sec. IV are not robust against this truncation.","section":"II, Eq. (14) and following text"},{"comment":"The conclusion that no bound state exists is based on variational upper bounds $E_{cc}$ that lie above the theoretical thresholds. Because the Gaussian expansion is variational, an energy above threshold in a finite basis does not rigorously exclude a bound state in the full Hilbert space. The summary phrase 'leaving no space for a bound state' is therefore stronger than what Tables VI and VII demonstrate. The CMI argument in Table V makes the absence of a deeply bound state plausible, but it does not close the variational gap.","section":"III and Summary, Tables VI-VII"},{"comment":"The Gaussian basis parameters $r_1$, $r_{n_{\\max}}$, and $n_{\\max}$ are never specified, and no convergence study or numerical uncertainty is reported for either the variational energies or the stabilization plateaus. Without this information the reader cannot assess whether the quoted resonance energies, for example 13140 MeV for the $0^+$ $bb\\bar{c}\\bar{c}$ state, are numerically converged.","section":"II, Eq. (18)"}],"minor_comments":[{"comment":"There is a typographical error: 'lager masses' should be 'larger masses'.","section":"Introduction"},{"comment":"The running coupling in Eq. (4) should explicitly state that the argument of the logarithm is dimensionless; as written, $\\mu_{ij}$ and $\\mu_0$ are in MeV while $\\Lambda_0$ is given in fm$^{-1}$ in Table I, so the units should be reconciled.","section":"II, Eq. (4)"},{"comment":"The captions of the stabilization figures do not identify which horizontal lines correspond to which threshold; adding this information would make the figures substantially easier to interpret.","section":"III, Figs. 3-6"},{"comment":"The notation for charged mesons is inconsistent, with both $B_c^-$ in the text and $B_c$ in the table column headers; the table should use a single consistent notation.","section":"Tables VI-VII"},{"comment":"Some references are given only as arXiv preprints even though published versions exist; updating these would help the reader.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper would be strengthened considerably by a pole-based resonance analysis or at least a multi-channel stabilization study that includes widths, by reporting the Gaussian basis parameters and a convergence test, and by softening the no-bound-state wording in the summary. As it stands, the central new claim (the resonance masses) is not yet supported with the required rigor, although the variational bound-state part is reasonable and worth publishing after revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear X,\n\nThe one-liner: this is a competent variational calculation of the bbcc and bcbc tetraquark spectra; the no-bound-state conclusion is plausible and consistent with several previous potential-model studies, but the resonance masses extracted from the stabilization procedure are not convincing as quantitative predictions.\n\nWhat is new: the paper extends the author's earlier full-bottom study to the mixed-heavy systems, explicitly including meson-meson and diquark-antidiquark structures and their mixing. The parameters are fixed by the meson spectrum, not fitted to tetraquarks, which is good practice. The numerical ground-state energies sit above the corresponding two-meson thresholds, and the qualitative color-magnetic analysis in Table V supports that. That negative-result part is solid, to the extent a variational upper-bound calculation can be.\n\nThe soft spots: first, the Summary's phrase \"leaving no space for a bound state\" is too strong. A variational energy above threshold does not prove the absence of a bound state; it only shows this basis and Hamiltonian did not find one. The conclusion is probably right, but the wording overstates what a variational calculation demonstrates. Second, the resonance claims rest entirely on the real scaling of only the meson-meson color singlet-singlet configurations. No widths, no pole locations, and no complex-scaling or phase-shift analysis are provided. The claimed states lie 180 to 540 MeV above threshold, where broad resonances are expected, and the paper itself concedes they may be too wide to observe. The plateau in the stabilization plot could be an artifact of the restricted basis. A referee should ask for a justification of the single-channel scaling and for some estimate of the width. Third, all orbital angular momenta are set to zero; D-wave channels are omitted, which can matter at these energies.\n\nWho this is for: hadron spectroscopists interested in LHCb searches for fully-heavy tetraquarks. They get a useful negative result on bound states and a set of mass windows, but the resonance masses should be treated as indicative, not predictive.\n\nRecommendation: this deserves peer review, not a desk reject. The calculation is honest and the negative result is worth having, but the resonance analysis needs either a more rigorous treatment (complex scaling, phase shifts, widths) or a much more cautious statement before publication.","headline":"Competent no-bound-state calculation, but the resonance masses from the real scaling method are not established and need either much stronger analysis or much softer language.","tokens_in":14353,"tokens_out":1909,"would_cite":false,"duration_ms":19750,"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":"Fully-heavy tetraquarks with two bottom and two charm quarks do not bind; the model finds only above-threshold resonances, with specific masses near 13 GeV.","keywords":["fully-heavy tetraquarks","chiral quark model","Gaussian expansion method","real scaling method","exotic hadrons","resonances","bound states","bottom-charm tetraquarks"],"falsifier":"A lattice QCD computation of the lowest $bb\\bar{c}\\bar{c}$ or $bc\\bar{b}\\bar{c}$ eigenvalue that comes out below the corresponding two-meson threshold—for $bc\\bar{b}\\bar{c}$ below $\\eta_b+\\eta_c$ at 12321 MeV in this model—or an experimental discovery of a narrow fully-heavy tetraquark below its two-meson threshold would falsify the paper's no-bound-state claim.","tokens_in":13295,"feed_emoji":"⚛️","tokens_out":9355,"duration_ms":84843,"temperature":0.7,"pith_summary":"Does a four-quark system made only of heavy quarks—two bottom quarks with two anticharm quarks ($bb\\bar{c}\\bar{c}$), or bottom-charm with their antiparticles ($bc\\bar{b}\\bar{c}$)—form a bound tetraquark? The paper argues no: in a nonrelativistic chiral quark model, with meson-meson, diquark-antidiquark, and mixed configurations, every low-lying state lands above the corresponding two-meson decay threshold. With no room for a bound state, the model instead predicts a set of resonances, at about 13.1–13.2 GeV for $bb\\bar{c}\\bar{c}$ and 12.9–13.0 GeV for $bc\\bar{b}\\bar{c}$. A sympathetic reader would care because it gives concrete mass targets and says where not to look for stable fully-heavy tetraquarks.","feed_headline":"No bound state for four-quark bottom-charm systems","feed_subtitle":"Model predicts only above-threshold resonances at 12.9 to 13.2 GeV, giving concrete search targets.","key_machinery":"The machinery is the nonrelativistic chiral quark model Hamiltonian $$H=\\sum_i m_i+\\frac{p_{12}^2}{2\\mu_{12}}+\\frac{p_{34}^2}{2\\mu_{34}}+\\frac{p_{1234}^2}{2\\mu_{1234}}+\\sum_{i<j}\\left(V^C_{ij}+V^G_{ij}\\right),$$ with a central confinement term $V^C_{ij}=(-a_c r_{ij}^2-\\Delta)\\lambda^c_i\\cdot\\lambda^c_j$ and a one-gluon-exchange term containing Coulomb, hyperfine, and smeared delta pieces. Parameters are fixed by meson spectra. The four-body Schrödinger equation is solved by the Gaussian expansion method, expanding each relative coordinate in Gaussians with geometric progression size parameters and including all color, spin, and flavor channels for both meson-meson and diquark-antidiquark structures. Genuine resonances are distinguished from continuum with the real scaling method, which multiplies the Gaussian size parameters of the color singlet-singlet meson-meson channel by a factor $\\alpha$; a true resonance keeps its energy stable as $\\alpha$ grows, while continuum states fall to thresholds.","core_discovery":"The central discovery is a negative result with positive predictions: no bound $bb\\bar{c}\\bar{c}$ or $bc\\bar{b}\\bar{c}$ tetraquark exists in this model, because the color matrix elements of the tetraquark and its two-meson decay products are identical and the color-magnetic interaction is never attractive enough to pull the energy below threshold. After solving the four-body Schrödinger equation, all computed eigenvalues sit above the relevant meson-pair thresholds. Using the real scaling method, the paper identifies stable resonance plateaus: for $bb\\bar{c}\\bar{c}$, 13140, 13180, and 13230 MeV for $0(0^+)$, $0(1^+)$, and $0(2^+)$; for $bc\\bar{b}\\bar{c}$, 12860, 13020, 13020, and 12910 MeV for $0(0^{++})$, $0(1^{++})$, $0(2^{++})$, and $0(1^{+-})$.","pith_inferences":["If the no-bound-state result holds, binding in fully-heavy tetraquarks would require effects beyond central confinement plus one-gluon-exchange, such as tensor forces or explicit coupled-channel dynamics; the predicted resonance masses could shift by tens of MeV under those additions.","The near equality of color matrix elements between tetraquark and two-meson configurations suggests the tetraquark wave function is largely a weakly interacting meson pair; one testable consequence is that decay widths of the predicted resonances should be broad enough to appear as enhancements rather than narrow peaks.","The real scaling method was applied only to meson-meson color singlet-singlet channels; applying it also to diquark-antidiquark or octet-octet channels might reveal additional resonances or shift the lowest ones.","The predicted $bc\\bar{b}\\bar{c}$ $1^{+-}$ resonance at 12910 MeV lies below its $1^{++}$ partner at 13020 MeV; if produced, C-parity selection rules could distinguish them in final states such as $\\eta_b J/\\psi$ versus $\\Upsilon J/\\psi$, offering a test of the model."],"forward_implications":["No stable $bb\\bar{c}\\bar{c}$ or $bc\\bar{b}\\bar{c}$ tetraquark should exist below the lowest two-meson threshold; experimental searches should target above-threshold resonances instead of stable particles.","The predicted lowest resonances are 13140, 13180, and 13230 MeV for $bb\\bar{c}\\bar{c}$ $0(0^+)$, $0(1^+)$, $0(2^+)$, and 12860, 13020, 13020, and 12910 MeV for $bc\\bar{b}\\bar{c}$ $0(0^{++})$, $0(1^{++})$, $0(2^{++})$, $0(1^{+-})$, providing concrete mass targets.","The color interaction contributes no binding energy because the color matrix element of the tetraquark equals that of its two-meson pair; the color-magnetic interaction does not improve binding for these heavy systems.","Mixing between the meson-meson and diquark-antidiquark structures changes the ground-state energies very little, so either structure alone gives nearly the same thresholds and resonance pattern.","For $bc\\bar{b}\\bar{c}$, charge-parity is a good quantum number, and the model places the $1^{+-}$ resonance below the $1^{++}$ one, giving two C-parity partners with different masses."],"supporting_citations":[{"why":"The author's earlier full-bottom tetraquark calculation that supplies the model setup and real scaling approach extended here.","marker":"[24]"},{"why":"A potential-model study of fully-heavy tetraquarks that also found no bound states; the present result agrees with it.","marker":"[27]"},{"why":"A study claiming a possible $bc\\bar{b}\\bar{c}$ bound state; the present no-bound conclusion directly contradicts it.","marker":"[22]"},{"why":"A study suggesting bound $bc\\bar{b}\\bar{c}$ may be favorable; provides a contrasting result the paper's calculation speaks against.","marker":"[28]"},{"why":"The Gaussian expansion method used to solve the four-body Schrödinger equation.","marker":"[29]"},{"why":"The real scaling (stabilization) method applied in quark models; used here to identify resonances.","marker":"[30, 31]"},{"why":"The chiral quark model formalism and potential forms adopted in Eqs. (1) and (2).","marker":"[33]"},{"why":"The source of model parameters and the scale-dependent running coupling in Eq. (4).","marker":"[34]"}],"fun_headline_variants":["All bottom-charm tetraquarks lie above threshold","No bound bbcc or bccb tetraquarks in chiral model","Resonances only: four-quark bottom-charm states predicted","Above 12.8 GeV: no bound tetraquarks, only resonances","Tetraquark search targets: resonances near 13 GeV"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The model's Hamiltonian, with only central confinement and one-gluon-exchange terms and parameters fixed by meson spectra, correctly describes fully-heavy four-quark dynamics, and the real scaling method reliably separates true resonances from continuum in these systems.","fun_headline_variants_meta":{"raw":{"variants":["All bottom-charm tetraquarks lie above threshold","No bound bbcc or bccb tetraquarks in chiral model","Resonances only: four-quark bottom-charm states predicted","Above 12.8 GeV: no bound tetraquarks, only resonances","Tetraquark search targets: resonances near 13 GeV"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00059,"raw_usage":{"total_tokens":2753,"prompt_tokens":917,"completion_tokens":1836,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":1744}},"tokens_in":533,"tokens_out":1836,"duration_ms":11832,"temperature":1.0,"reasoning_tokens":1744,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:42:21.668207+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A lattice QCD computation of the lowest $bb\\bar{c}\\bar{c}$ or $bc\\bar{b}\\bar{c}$ eigenvalue that comes out below the corresponding two-meson threshold—for $bc\\bar{b}\\bar{c}$ below $\\eta_b+\\eta_c$ at 12321 MeV in this model—or an experimental discovery of a narrow fully-heavy tetraquark below its two-meson threshold would falsify the paper's no-bound-state claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The author's earlier full-bottom tetraquark calculation that supplies the model setup and real scaling approach extended here."},{"cited_title":"Hughes, E","cited_arxiv_id":null,"evidence_quote":"A potential-model study of fully-heavy tetraquarks that also found no bound states; the present result agrees with it."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"A study suggesting bound $bc\\bar{b}\\bar{c}$ may be favorable; provides a contrasting result the paper's calculation speaks against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Gaussian expansion method used to solve the four-body Schrödinger equation."},{"cited_title":"Hiyama, Y","cited_arxiv_id":null,"evidence_quote":"The chiral quark model formalism and potential forms adopted in Eqs. (1) and (2)."},{"cited_title":"Hiyama, M","cited_arxiv_id":null,"evidence_quote":"The source of model parameters and the scale-dependent running coupling in Eq. (4)."}],"review_version":1}