{"id":"e30d4e44-dd4a-4eaf-826e-715124da4ce9","arxiv_id":"2603.03764","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"In a diquark-model calculation, the ρ-mode excitation of Tcc (inside the light diquark) lies below the λ-mode excitation (between diquarks), an 'inverse hierarchy' driven by the light diquark's large size.","lead":"This paper calculates the excitation spectrum of the Tcc tetraquark in a diquark model and finds that the lowest internal excitation of the light quark pair (ρ-mode) sits below the excitation between the two diquarks (λ-mode), opposite to the usual quark-model expectation. The result suggests that identifying the first excited Tcc state may require decay-pattern analysis, not just mass ordering.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Inversion of ρ–λ ordering in Tcc is contingent on the AL1 potential's light-diquark excitation gap; a modest parameter change or the chiral-EFT input reverses it.","rationale":"The reader's weakest-assumption identification is correct and is the most load-bearing issue. The paper's own threshold analysis in §IV shows the ρ-mode lies below the λ-mode only if mud(0−,1−,2−) < 1.225 GeV; AL1 yields 1.121 GeV, while the chiral-EFT model [19] with lattice input gives 1.484 GeV and the normal ordering. Since the AL1 diquark masses are predictions, not calibrated to diquark data, the central claim is not robust. Secondary inconsistencies (text quotes ρ-mode centrifugal energy as 0.368 GeV vs. 0.404 GeV in Table III; the text compares the isolated diquark radius with the inter-cluster radius) weaken the proposed mechanism's quantitative basis, but the dominant issue remains the diquark-mass-gap sensitivity. The conditional verdict is appropriate.","tokens_in":16448,"tokens_out":11707,"duration_ms":101313,"concrete_test":"Compute the light-diquark excitation gap (mud(0−,1−,2−) − mud(0+)) from lattice QCD at physical quark masses. If the gap is ≥0.56 GeV (i.e., mud(0−,1−,2−) ≳ 1.225 GeV for mud(0+)≈0.666 GeV), the AL1 value 0.455 GeV is excluded and the inverted ρ−λ ordering in Tcc disappears, confirming that the central claim is an artifact of the unvalidated AL1 diquark-mass prediction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that the ρ-mode excitation of Tcc lies below the λ-mode is not a robust outcome of the diquark model: it is controlled by the AL1 potential's prediction of the excited light-diquark mass mud(0−,1−,2−)=1.121 GeV. The authors themselves identify the threshold mud(0−,1−,2−)=1.225 GeV at which the ordering reverts to the naive hierarchy (Sec. IV). The chiral-EFT model [19], which uses lattice-QCD input for the ground-state diquark, predicts 1.484 GeV—well above the threshold—and consequently yields the normal ordering. The paper offers no independent validation of the AL1 diquark excitation gap (0.455 GeV), and the AL1 parameters were fit to meson/baryon spectra, not to diquark masses. The proposed centrifugal-energy explanation is also internally inconsistent: the text quotes the ρ-mode centrifugal energy as 0.368 GeV, whereas Table III lists 0.404 GeV, and the argument compares the isolated ud-diquark radius (1.511 fm) with the inter-cluster Tcc λ-mode radius (0.741 fm), rather than the in-medium internal diquark radius. Thus the headline inversion is a model-dependent consequence of an unconstrained input, not a demonstrated physical mechanism.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the doubly-heavy tetraquark Tcc in a diquark model where the system is reduced to a heavy (cc) diquark and a light (ūd) antidiquark interacting via the Silvestre-Brac AL1 potential. The authors solve the two-body Schrödinger equation with the Gaussian Expansion Method, first for the isolated diquarks and then for the tetraquark. Their central numerical result is that the ρ-mode excitation, corresponding to orbital excitation inside the light antidiquark, lies at 0.313 GeV, below the λ-mode interdiquark excitation at 0.395 GeV. This is the opposite of the naive harmonic-oscillator ordering ω_ρ > ω_λ. They attribute the inversion to the smaller centrifugal energy of the ρ-mode compared with the λ-mode, trace it to the larger RMS radius of the light diquark, and extend the calculation to Tbb, Λc, and Λb with qualitatively the same ordering. The paper explicitly notes that adjusting the excited light-diquark mass above 1.225 GeV restores the naive hierarchy and that the chiral-EFT model of Ref. [19] predicts the normal ordering.","tokens_in":16823,"tokens_out":7437,"duration_ms":68440,"significance":"If the result is taken at face value, the paper provides a concrete counterexample to the common harmonic-oscillator intuition for Jacobi-coordinate excitations in hadrons, and it identifies the light-diquark excitation gap as the controlling quantity. The calculation has genuine strengths: GEM convergence is checked over the range-parameter space, the energy decomposition into kinetic, Coulomb, linear, and hyperfine contributions is tabulated, the threshold mass for reversal of the ordering is computed, and the same mechanism is illustrated in several systems. However, the physical significance is substantially limited by the fact that the inversion is not robust across reasonable models: the AL1 potential gives an excited light-diquark mass of 1.121 GeV, whereas the chiral-EFT model cited by the authors gives 1.484 GeV, which lies far above the inversion threshold and yields the normal ordering. The paper is therefore best read as a model study of one potential, not as a robust prediction for Tcc. The claimed 'robustness' applies within the AL1 framework, not across the model uncertainty of the diquark excitation gap.","major_comments":[{"comment":"The central explanation quotes L_ρ = 0.368 GeV for the Tcc ρ-mode centrifugal energy, but Table III lists ⟨l(l+1)/(2μr²)⟩ = 0.404 GeV for the Tcc(1−;...)ρ state; the value 0.368 appears in the isolated mud(0−,1−,2−) row. The text compares L_λ = 0.481 with the isolated-diquark value rather than with the full tetraquark ρ-mode value. This discrepancy is load-bearing because the inversion is attributed to this centrifugal-energy comparison. Please clarify which quantity is meant, correct the inconsistency, and state explicitly that the ρ-mode centrifugal energy is taken from the isolated diquark wavefunction under the frozen-diquark approximation.","section":"§IV and Table III"},{"comment":"The inversion depends on the AL1 prediction m_ud(0−,1−,2−)=1.121 GeV. The authors themselves find that the ordering reverses for m_ud>1.225 GeV, and the chiral-EFT model of Ref. [19] gives 1.484 GeV, which produces the normal hierarchy. The threshold is only 0.104 GeV above the AL1 value, so the result is highly sensitive to the diquark excitation gap. The abstract and conclusion should be tempered: the inversion is a property of the AL1 model, not a robust outcome of the diquark picture. The word 'robustness' should be qualified. No independent constraint on the excited light-diquark mass is provided.","section":"§IV, Table II, and Conclusion"},{"comment":"The statement that the centrifugal energy is 'the main source of the inversion' is supported only by the single comparison L_λ=0.481 > L_ρ=0.368. However, Table III shows that other terms also contribute comparably: for the ρ-mode ΔV_lin=0.267 and ΔV_Coul=0.047, while for the λ-mode ΔV_lin=0.210 and ΔV_Coul=0.137. The interplay of these terms is not discussed. A quantitative decomposition of the excitation energies is needed to justify the claim that the centrifugal term is dominant, rather than merely correlated with the ordering.","section":"§IV, paragraph following Eq. (9)"},{"comment":"The footnote acknowledges that the threshold analysis cannot predict how L and r_rms change when m_ud(excited) is manually varied. This is an important limitation of the proposed mechanism, because it means the 'what-if' calculation does not demonstrate a dynamical mechanism. This statement should be moved into the main text and discussed, as it directly affects the strength of the conclusion.","section":"Footnote 1 in §IV"}],"minor_comments":[{"comment":"The notation for the Tcc ρ-mode state is cumbersome and the parenthetical values are explained only in the caption. It would help to separate the tetraquark excitation energy from the diquark-subsystem excitation energy in a clearer way.","section":"Table III"},{"comment":"The caption states that K is 'arbitrarily chosen'. Since the HO comparison is used as the benchmark for the naive hierarchy, the choice of K should be justified or at least shown not to affect the ordering ω_ρ > ω_λ.","section":"Fig. 5 caption"},{"comment":"The phrase 'the corresponding relative momentump=' has a missing subscript and should read p_ij. Please proofread the introductory Hamiltonian section.","section":"Eq. (1)"},{"comment":"Typo: 'spin-dependent fores' should be 'spin-dependent forces', and 'the true from' should be 'the true form'.","section":"Appendix B"},{"comment":"Reference [12] is a CERN news release; the primary LHCb publication is Ref. [13]. Consider citing the LHCb paper directly in the introduction, as the current citation style mixes news items with peer-reviewed works.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is transparent about its model dependence, but the abstract and conclusion overstate the robustness of the central claim. The threshold at 1.225 GeV is close to the AL1 value of 1.121 GeV, and the chiral-EFT value of 1.484 GeV gives the normal ordering, so the headline 'inversion' is not a robust prediction. The centrifugal-energy explanation also contains an internal inconsistency between the text and Table III. These issues are fixable by reframing the paper as a study of the AL1 diquark model and by correcting the numerics, so major revision rather than rejection seems appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know up front: the advertised inversion of the ρ–λ hierarchy in Tcc is a genuine result of this diquark-model calculation, but it is not a robust one. The authors show themselves that if the mass of the excited light diquark rises from the AL1 value of 1.121 GeV to 1.225 GeV, the ordering flips back to the naive one. The chiral-EFT diquark mass of 1.484 GeV sits well above that threshold, so the competing model gives the normal hierarchy. That's the load-bearing soft spot.\n\nWhat the paper does well: it's a self-contained, reproducible two-body calculation. The GEM setup is described in enough detail to rerun; the parameter scan and convergence check (nmax=20, r1=0.1 fm, rmax=6 fm) are explicit. The extension to Tbb, Λc, Λb is a nice consistency check, and the discussion of η vs ππ decay as a way to distinguish ρ- and λ-modes is practically useful and appropriate for LHCb. The authors also deserve credit for flagging the threshold and admitting the mechanism can't be predicted once you tune the excited-diquark mass inside the Tcc.\n\nThe soft spots, in proportion: the central claim depends entirely on the AL1 potential's prediction of the light-diquark excitation gap (0.455 GeV). AL1 parameters were fit to meson/baryon spectra, not to diquark masses, so that gap is not independently constrained. The centrifugal explanation is plausible but the text and Table III disagree — 0.368 GeV vs 0.404 GeV for the ρ-mode centrifugal energy — and the argument compares the isolated ud-diquark radius (1.511 fm) with the inter-cluster λ-mode radius (0.741 fm), not the in-medium internal radius. That weakens the mechanistic story, though it doesn't invalidate the numerical result within the model.\n\nWho this is for: hadron spectroscopists and anyone tracking Tcc exotics. It will not reshape the field, but it is a concrete, falsifiable prediction with a suggested experimental discriminant.\n\nRecommendation: send it to peer review. It deserves a serious referee. The referee should insist on fixing the text/table mismatch and softening the 'robustness' language, but the calculation is transparent and the question is well-posed.","headline":"Inverted ρ–λ ordering in Tcc is a genuine but fragile model prediction—the authors are honest about the threshold, yet the text/table mismatch and a shaky radius argument need fixing before publication.","tokens_in":17284,"tokens_out":3657,"would_cite":true,"duration_ms":36217,"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":"The paper argues that in the doubly charmed tetraquark Tcc, excitation inside the light antidiquark (the ρ-mode) costs less energy than excitation between the two diquarks (the λ-mode), reversing the harmonic-oscillator ordering, and that t","keywords":["Tcc tetraquark","diquark model","excitation hierarchy","centrifugal energy","Gaussian expansion method","heavy quark symmetry","chiral partner","hadron spectroscopy"],"falsifier":"Measure or compute the mass of the excited light (ūd) diquark: the paper's own threshold is 1.225 GeV, below which the inversion survives and above which it disappears. Alternatively, observe the decay pattern of the excited Tcc(1−): a dominant η signal supports the ρ-mode assignment, a dominant ππ signal supports the λ-mode; if the decay pattern contradicts the energy ordering, the mechanism is wrong.","tokens_in":16363,"feed_emoji":"⚛️","tokens_out":5492,"duration_ms":50343,"temperature":0.7,"pith_summary":"This paper studies the double-charm tetraquark Tcc as a bound state of a heavy charm-charm diquark and a light anti-up–anti-down diquark. It claims that the first excitation inside the light diquark (the ρ-mode) lies 0.313 GeV above the ground state, below the 0.395 GeV excitation between the two diquarks (the λ-mode), opposite to the ordering expected from a harmonic-oscillator picture. The inversion is traced to a centrifugal-energy comparison: even though the light diquark has the smaller reduced mass, its much larger root-mean-square radius (about 1.5 fm versus 0.74 fm) suppresses its centrifugal term. The same pattern appears in Tbb, Λc, and Λb, and the paper argues that identifying excited Tcc states by mass alone is unreliable; decay patterns (η versus ππ emission) should distinguish the modes.","feed_headline":"Light diquark's size inverts Tcc excitation order","feed_subtitle":"The ρ-mode lands below the λ-mode, so decays — not masses — will tell the two excited states apart.","key_machinery":"The load-bearing object is the light antidiquark (ūd) treated as a color-triplet substructure inside Tcc. Its excitation (the ρ-mode) is governed by the centrifugal energy L = ⟨l(l+1)/(2μ_ud r²)⟩ evaluated at l=1. The inversion mechanism is the competition between the small reduced mass μ_ud, which would raise L, and the mean inverse-square radius ⟨1/r²⟩ set by the nearly twofold larger RMS radius of the light diquark compared with the λ-mode configuration; the quadratic radius dependence wins. The Gaussian Expansion Method supplies the wave functions and radii, and the AL1 potential supplies the diquark masses (m_ud(0+)=0.666 GeV, m_ud(excited)=1.121 GeV) that enter the comparison.","core_discovery":"The central discovery is that the ρ-mode excitation energy of Tcc is not the highest of the three low-lying modes, as a harmonic-oscillator model with reduced masses would predict, but lies between the ρcc- and λ-modes. The authors compute diquark masses from the AL1 potential and then solve the two-body tetraquark Schrödinger equation with the Gaussian Expansion Method. They find excitation energies of 0.313 GeV for the ρ-mode, 0.395 GeV for the λ-mode, and 0.199 GeV for the ρcc-mode. They attribute the inversion to the centrifugal term ⟨l(l+1)/(2μr²)⟩: for the λ-mode this is 0.481 GeV, while for the ρ-mode it is only 0.368 GeV, because the light antidiquark's RMS radius is roughly twice as","pith_inferences":["[Editorial inference] If the inversion survives a more realistic three-body treatment, it would imply that chiral-partner identification from tetraquark spectra is generally ambiguous, not just for Tcc, and that other exotic states with loosely bound light diquarks may show the same reversed ordering.","[Editorial inference] The radius-versus-reduced-mass competition is a general mechanism: any two-body Coulomb-plus-linear system in which the lighter composite has a larger spatial extent could invert excitation ordering. It would be worth testing the same ratio L_ρ / L_λ in other diquark models or on lattice data for diquark correlation lengths.","[Editorial inference] A direct lattice-QCD calculation of the excited light-diquark mass would settle the model dependence; the paper's threshold gives a concrete target (below 1.225 GeV for the inversion, above for the normal order)."],"forward_implications":["The ρ-mode of Tcc is the chiral partner of the ground state but cannot be identified from its energy alone, because it sits below the λ-mode; mass ordering is not a reliable quantum-number tag.","Decay selection rules give an experimental handle: dominant S-wave η emission indicates the ρ-mode, while dominant P-wave ππ emission indicates the λ-mode.","The same inverted hierarchy appears in the bottom counterparts Tbb and Λb, so the mechanism is not specific to the charm sector and should persist in heavy-quark systems.","The ordering is sensitive to the mass of the excited light diquark: above 1.225 GeV the naive hierarchy is restored, so the prediction is falsifiable by an independent determination of that mass."],"fun_headline_variants":["Centrifugal force upends tetraquark excitation order","Light antidiquark size flips Tcc mode hierarchy","Lambda-mode outranks rho-mode in Tcc: centrifugal cause","How light diquark size inverts Tcc excitation energies","Tcc's excitation tiers: centrifugal force reorders levels"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The inversion depends on the AL1 potential's prediction that the excited light antidiquark weighs about 1.12 GeV; if its true mass exceeds 1.225 GeV, the ρ-mode climbs above the λ-mode and the naive ordering returns.","fun_headline_variants_meta":{"raw":{"variants":["Centrifugal force upends tetraquark excitation order","Light antidiquark size flips Tcc mode hierarchy","Lambda-mode outranks rho-mode in Tcc: centrifugal cause","How light diquark size inverts Tcc excitation energies","Tcc's excitation tiers: centrifugal force reorders levels"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001063,"raw_usage":{"total_tokens":4282,"prompt_tokens":717,"completion_tokens":3565,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":461,"completion_tokens_details":{"reasoning_tokens":3480}},"tokens_in":461,"tokens_out":3565,"duration_ms":25784,"temperature":1.0,"reasoning_tokens":3480,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T19:01:08.516611+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure or compute the mass of the excited light (ūd) diquark: the paper's own threshold is 1.225 GeV, below which the inversion survives and above which it disappears. Alternatively, observe the decay pattern of the excited Tcc(1−): a dominant η signal supports the ρ-mode assignment, a dominant ππ signal supports the λ-mode; if the decay pattern contradicts the energy ordering, the mechanism is wrong.","supporting_citations":[],"review_version":1}