{"id":"20241b61-85f8-4fd6-bd6d-a0636f436262","arxiv_id":"2509.08964","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Predicted branching fractions for H to fully charmed tetraquark plus photon, with the quark-gluon mechanism dominant at about 2 x 10^-9.","lead":"This paper estimates how often the Higgs boson could decay into a fully charmed tetraquark and a photon, using a relativistic quark model. It finds the largest rate comes from a quark-gluon diagram, around two parts per billion, which future Higgs factories might partially probe.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Headline branching scales as |Ψ(0,0,0)|^2 imported from companion paper; no uncertainty or variational parameters are given, and the same section admits the short-distance wave function is poorly known.","rationale":"The paper is a self-consistent model calculation: the formalism is explicit, the three S-wave mechanisms share the same wave-function normalization, and the quark-gluon dominance is robust because it is independent of the overall |Ψ(0,0,0)|^2 factor. The real soft spot is the absolute normalization of the headline branching. That number is controlled by a single externally imported parameter, Ψ(0,0,0), which is not derived or bounded in this paper and is precisely the short-distance quantity that the paper's own cutoff remark says is poorly constrained. This is not an internal inconsistency or a disagreement with consensus; it is an unvalidated input with no error estimate. The reader's weakest_assumption points to the same issue, and the existing CONDITIONAL verdict captures it. No change to the verdict is therefore needed: the calculation should be viewed as a model-dependent estimate whose numerical normalization is not yet robust, even though the mechanism hierarchy is likely stable.","tokens_in":21486,"tokens_out":16904,"duration_ms":195938,"concrete_test":"Reproduce the companion variational calculation of [36] and evaluate Ψ(0,0,0) via Eq. (45). Then vary (i) the Gaussian basis size K until the variational energy is stable, and (ii) the large-momentum cutoff mentioned after Eq. (48) from m to 2m, refitting α_s within the charmonium spectrum. Additionally, compute Ψ(0,0,0) with an independent potential model that also reproduces the X(6900) mass. If |Ψ(0,0,0)|^2 changes by more than a factor of ~4 across these variations, the Table I branching should be quoted as a range rather than a single number.","verdict_should_be":"UNCHANGED","load_bearing_attack":"All S-wave widths in Table I—Eqs. (29), (54), (64)—are proportional to |Ψ(0,0,0)|^2, with Ψ(0,0,0)=0.10 GeV^{9/2} quoted in Eq. (49) as taken from the companion variational calculation [36]. The present paper does not give the variational parameters A_ij, C_I, the basis size K, or any uncertainty on this number. This matters because Ψ(0,0,0) is a short-distance property: it is the coordinate-space wave function at zero relative separations, i.e., exactly the part of the four-body wave function least constrained by fitting the mass spectrum. The paper itself signals the same short-distance uncertainty when it says that one of the integrals in Eqs. (46)–(48) is cut off at relativistic momentum m because the wave function is poorly known there. If the physical X(6900) is not a compact [cc][cbar cbar] bound state—for example, a hadronic molecule or a state with large six-quark/color-octet components—or even if the compact radius changes by a factor of ~1.5, |Ψ(0,0,0)|^2 and hence the headline branching 0.21×10^-8 and the '~60 events at FCC-hh' estimate change by an order of magnitude or more. The relative ordering of the three S-wave mechanisms is unaffected because all share the same factor, so the 'quark-gluon dominates' statement is robust; the numerical central claim is not. The paper attaches a ≥30% uncertainty to R'(0,0,0) for the 1-- state but gives no analogous estimate for Ψ(0,0,0).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper estimates branching fractions for the exclusive Higgs decay H -> (cc cbar cbar) + gamma in a relativistic quark model. It considers three S-wave mechanisms for the 1+- tetraquark (ZZ-boson, quark-gluon, and quark-gluon loop) and one P-wave mechanism for the 1-- state. The main numerical result is that the quark-gluon mechanism dominates, with Br ~ 0.21 x 10^-8, while the ZZ and loop mechanisms are suppressed by several orders of magnitude. The authors also translate this branching fraction into an estimate of about 60 events at FCC-hh. Relativistic corrections in the amplitudes are retained to second order in internal momenta, with the required momentum moments taken from a companion variational calculation.","tokens_in":22015,"tokens_out":6655,"duration_ms":81102,"significance":"If correct, the paper provides a concrete, testable prediction for a rare exclusive Higgs decay and identifies the dominant production mechanism. Its strengths are the explicit construction of multiple production amplitudes and the clear separation between mechanisms. Importantly, the conclusion that the quark-gluon mechanism dominates among the S-wave channels is robust, because all S-wave widths share the same factor |Psi(0,0,0)|^2. However, the absolute branching fractions depend on a short-distance four-quark wave function whose uncertainty is not quantified, and one of the loop formulas requires a specified analytic continuation. The numerical predictions should therefore be regarded as estimates unless the model dependence is quantified.","major_comments":[{"comment":"All S-wave widths, Eqs. (29), (54), (64), are proportional to |Psi(0,0,0)|^2. The value Psi(0,0,0)=0.10 GeV^{9/2} is taken from companion paper [36], but neither its uncertainty nor the variational parameters are given. The text after Eqs. (46)-(48) itself admits that the wave function is poorly known at relativistic momenta; since Psi(0,0,0) is a short-distance quantity, this uncertainty feeds directly into the headline branching 0.21 x 10^-8 and the ~60-event estimate. Please provide an uncertainty estimate or a sensitivity scan, e.g. +/-50% in Psi(0,0,0). The relative ordering of the S-wave mechanisms is unaffected.","section":"Sec. II, Eq. (49) and Table I"},{"comment":"The loop structure function A_Q(tau) is written in terms of ArcSin(sqrt(tau)), while for the physical b- and t-quark loops tau = M_H^2/(4m_Q^2) > 1. In this regime the expression is complex unless a specific analytic continuation is specified; Eq. (64) then takes an absolute value of the expression. The dispersion representation (60) is the correct route, but the final formula must be given in a manifestly real form, or with an explicit continuation, before the Table I loop entry can be verified.","section":"Sec. II, Eqs. (62) and (64)"},{"comment":"The relativistic parameters omega_p, omega_q, omega_t are obtained with a hard cutoff at |p|=m because, as stated in the text, the wave function is unreliable at relativistic momenta. These parameters are numerically significant: they change the quark-gluon branching from 0.15 x 10^-8 to 0.21 x 10^-8 in Table I. The cutoff prescription and the neglect of mixed moments <pt>, <qt>, <pq> need a sensitivity study (e.g. varying the cutoff between m/2 and 2m) before the error on the central value can be assessed.","section":"Sec. II, Eqs. (46)-(48)"},{"comment":"The calculation assumes a compact diquark-antidiquark [cc][cbar cbar] configuration with each pair in a color antitriplet and spin 1. If the physical X(6900) has a large molecular or six-quark component, Psi(0,0,0) is not the relevant production quantity and the absolute rates can change by orders of magnitude. This is a legitimate model choice, but it should be explicitly stated as a limitation and, if possible, calibrated against alternative wave-function models.","section":"Sec. II, color structure after Eq. (30)"},{"comment":"No uncertainties are propagated into Table I. The only error estimate given in the text is 'not less than 30 percent' for R'(0,0,0) in the 1-- case, but this is not reflected in the table. Since the paper ends with an event-rate estimate for FCC-hh, an error budget for all rows is needed.","section":"Table I"}],"minor_comments":[{"comment":"The entry |2++> is labeled J_T=0; it should be J_T=2.","section":"Eq. (9)"},{"comment":"Notation such as 'sin2 theta MZ' should read 'sin 2 theta_W M_Z', and similar missing/faulty subscripts occur in several amplitudes.","section":"Eqs. (17) and (54)"},{"comment":"There is an apparent typesetting artifact: 'gamma^omega hat r - (3/4 hat T - hat p + ...)' should be cleaned up.","section":"Appendix A, Eq. (A7)"},{"comment":"The phrase 'determined by first half of terms in matrix elements Q' is unclear; please define the tilde-Q elements in this context.","section":"Eq. (45)"},{"comment":"The event-rate estimate 'about 60' at FCC-hh assumes a specific integrated luminosity and should also state assumptions about reconstruction efficiency and acceptance.","section":"Table I and conclusion"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on the companion arXiv paper [36] for the key wave-function input, and that paper is not yet journal-published. For the present work to be self-contained on its central numerical claim, the authors should provide enough variational information (or an appendix) to make Psi(0,0,0) independently checkable. The dominance conclusion is more robust than the absolute branching fractions, which is worth emphasizing in a revision. Overall the paper falls within the scope of the journal if rare Higgs decays are considered in scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid, straightforward extension of the same group's H -> J/psi J/psi calculation [28] to a four-quark final state. The genuinely new result is the branching fraction for H -> T(1+-) + gamma, about 2e-9 for the quark-gluon mechanism, which dominates the ZZ and loop mechanisms by several orders of magnitude. That ordering is the most valuable output, and it should survive scrutiny.\n\nWhat the paper does well: explicit amplitudes for three mechanisms, relativistic corrections via momentum integrals, and a clean factorized form for each width. The authors are transparent about the quark-model input: the tetraquark wave function comes from their variational calculation [36], and they note where the wave function is poorly known. The statement that ~60 events per year at FCC-hh could be expected is useful for planning.\n\nThe soft spots are real but proportionate. All S-wave widths in Table I are proportional to |Psi(0,0,0)|^2, a short-distance quantity. This paper quotes it as 0.10 GeV^{9/2} without error bar, and the companion paper's variational parameters are not reproduced. If the X(6900) is not a compact diquark-antidiquark state, or if its radius differs by a factor of 1.5, the branching changes by an order of magnitude. The paper itself flags the poor knowledge of the wave function in the relativistic region, so this is an acknowledged limitation, but it means the headline number is an estimate, not a prediction. The 1-- P-wave result comes with an admitted 30% uncertainty on R', yet is listed in Table I with no error. Also, Eq. (62) uses ArcSin(sqrt(tau)) for tau > 1 without explicitly showing that the imaginary parts cancel in the modulus squared; this is probably fine but should be stated.\n\nThese are addressable issues, not fatal ones. The relative ordering of the three mechanisms is independent of the common wave-function factor, so the main physics message holds if the model class is accepted.\n\nThis paper is for readers interested in rare Higgs decays and exotic hadron production. It deserves a serious referee. The reviewer should ask for error propagation and more detail on the companion wave-function calculation, but the paper is worth engaging with.\n\nRecommendation: send to peer review with major revision. The calculation is technically involved and the final state is new; it will be a reference point even if future experiments don't see the signal at the quoted rate.","headline":"New estimate for H -> fully charmed tetraquark + gamma: quark-gluon mechanism dominates at ~2e-9 branching, but the absolute rate inherits order-of-magnitude model dependence from the imported tetraquark wave function.","tokens_in":22430,"tokens_out":2881,"would_cite":true,"duration_ms":29974,"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 predicts that the Higgs boson, via a quark–gluon mechanism, decays to a fully charmed tetraquark plus a photon with a branching fraction around 2×10⁻⁹, making the state potentially visible at future Higgs factories.","keywords":["fully charmed tetraquark","Higgs boson decay","relativistic quark model","branching fraction","quark–gluon mechanism","X(6900)","variational wave function","exotic hadrons"],"falsifier":"Search for the exclusive decay H → J/ψ J/ψ γ at a Higgs factory with sensitivity below 10⁻⁹; the model predicts this channel should be fed by the 1⁺⁻ tetraquark at a rate near 2×10⁻⁹, so observing it at a level far below 10⁻¹⁰ would falsify the quark–gluon dominance and the assumed tetraquark wave function.","tokens_in":21423,"feed_emoji":"⚛️","tokens_out":6046,"duration_ms":65143,"temperature":0.7,"pith_summary":"This paper asks whether the Higgs boson, decaying in a rare exclusive channel, can serve as a production site for fully charmed tetraquarks: bound states of two charm quarks and two charm antiquarks, such as the X(6900) candidate. Using a relativistic quark model with a four-particle wave function obtained by variational methods, the authors compute branching fractions for H → T + γ through three mechanisms: quark–gluon, ZZ-boson, and quark–gluon loop. Their central result is that the quark–gluon mechanism dominates, with a branching fraction around 2×10⁻⁹ for the vector 1⁺⁻ tetraquark, several orders of magnitude larger than the other two. If these numbers are right, Higgs factories could observe dozens of such events, making the Higgs a new window on the internal structure of exotic multiquark states.","feed_headline":"Charm tetraquarks should appear in Higgs decays once in 500 million","feed_subtitle":"At 2 parts per billion, the decay is rare but accessible to proposed Higgs factories.","key_machinery":"The central object is the tetraquark quasipotential wave function at zero relative coordinates, Ψ(0,0,0), the probability amplitude for all four constituents to meet at one point; its square sets the overall scale of every S-wave decay width. This wave function is obtained variationally as a Gaussian expansion for a [cc][\\bar c\\bar c] diquark–antidiquark configuration, yielding Ψ(0,0,0) = 0.10 GeV^{9/2} and mean-square relative momenta ω_p = ω_q = 0.18, ω_t = 0.12. The decay amplitudes are built by convoluting the hard H → c\\bar c (+γ) interaction with this wave function, using spin projection operators for spin-1 diquark and antidiquark pairs, Wigner rotations, and relativistic corrections","core_discovery":"The paper claims that among the mechanisms for exclusive production of a fully charmed vector tetraquark T(cc\\bar c\\bar c) in the decay H → T + γ, the quark–gluon mechanism dominates. The authors calculate relativistic amplitudes for three production paths—quark–gluon, ZZ-boson, and quark–gluon loop—by convoluting the hard production amplitude with the tetraquark quasipotential wave function, keeping relativistic corrections up to quadratic order in the internal relative momenta. They obtain a branching fraction of 0.21×10⁻⁸ for H → Q\\bar Q → T(1⁺⁻) + γ, compared with 0.61×10⁻¹⁴ for the ZZ mechanism and 0.50×10⁻¹¹ for the loop mechanism; an orbital-excitation state T(1⁻⁻) is predicted at 0.8","pith_inferences":["If a compact [cc][\\bar c\\bar c] tetraquark exists, the same wave-function machinery could be applied to other Higgs exclusive decays, such as H → T(0⁺⁺) + γ or H → T + Z, testing whether the 1⁺⁻ dominance is a physical pattern or a model artifact.","The extreme sensitivity to |Ψ(0,0,0)|² suggests a clean cross-check: a lattice QCD computation of the four-quark distribution at zero separation for the fully charmed system would either confirm or rule out the diquark–antidiquark assumption underpinning the rate.","Measuring the invariant-mass shape of the cc\\bar c\\bar c system in Higgs decays would help distinguish a compact tetraquark from a J/ψ J/ψ threshold enhancement: the model predicts a narrow resonance, whereas a molecular picture would produce a broader, shifted distribution.","The predicted dominance of the quark–gluon mechanism implies that tetraquark searches at Higgs factories should trigger on charmed final states with an associated photon rather than on Z/Z-associated channels, which are heavily suppressed."],"forward_implications":["The quark–gluon mechanism gives a branching fraction around 2.1×10⁻⁹ for H → T(1⁺⁻) + γ, exceeding the ZZ and loop mechanisms by four to five orders of magnitude.","Because the decay width scales as |Ψ(0,0,0)|², any refinement of the four-quark wave function directly rescales all predicted rates.","The predicted event yield at a 100 TeV Higgs factory is roughly 60 tetraquarks per year at 30 ab⁻¹, making the decay a plausible discovery channel.","The T(1⁻⁻) orbital-excitation state has a branching fraction near 0.88×10⁻¹⁰, smaller than the ground-state vector but still larger than the ZZ-mediated S-wave rate.","The quark–gluon channel is an order of magnitude more probable than the previously studied H → J/ψ J/ψ decay, so tetraquark final states may be a better probe of the Higgs–charm coupling."],"fun_headline_variants":["Tetraquarks from Higgs decays: 2 per billion events","Quark-gluon route dominates Higgs-to-tetraquark decays","Rare Higgs decay yields charmed tetraquarks","Higgs factories could see 2 ppb tetraquark signal"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The calculations assume the fully charmed tetraquark is a compact [cc][\\bar c\\bar c] diquark–antidiquark state whose S-wave wave function at zero separation is Ψ(0,0,0) = 0.10 GeV^{9/2}; if the physical X(6900) is instead a molecule or has large noncompact components, every predicted branching fraction changes by orders of magnitude.","fun_headline_variants_meta":{"raw":{"variants":["Tetraquarks from Higgs decays: 2 per billion events","Quark-gluon route dominates Higgs-to-tetraquark decays","Rare Higgs decay yields charmed tetraquarks","Higgs factories could see 2 ppb tetraquark signal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000461,"raw_usage":{"total_tokens":2092,"prompt_tokens":643,"completion_tokens":1449,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":387,"completion_tokens_details":{"reasoning_tokens":1376}},"tokens_in":387,"tokens_out":1449,"duration_ms":12195,"temperature":1.0,"reasoning_tokens":1376,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T19:53:43.072638+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search for the exclusive decay H → J/ψ J/ψ γ at a Higgs factory with sensitivity below 10⁻⁹; the model predicts this channel should be fed by the 1⁺⁻ tetraquark at a rate near 2×10⁻⁹, so observing it at a level far below 10⁻¹⁰ would falsify the quark–gluon dominance and the assumed tetraquark wave function.","supporting_citations":[],"review_version":1}