{"id":"afacdd27-3c14-4b7f-9c58-d0e7004a671b","arxiv_id":"2603.12735","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"In the Skyrme model, the rare decay Λ_b → p ̅p n has branching fraction (1.10 ± 0.27) × 10⁻⁶, consistent with earlier O(10⁻⁶) estimates.","lead":"This paper computes the rate for a rare heavy-baryon decay, Λ_b → p ̅p n, using the Skyrme model's bound-state picture. It predicts a branching fraction of about one in a million, consistent with earlier estimates and potentially useful for future searches for new sources of CP violation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central branching-fraction prediction rests on an uncontrolled low-order Padé continuation to the time-like region; the paper's own caveat that the approximants are 'for illustration' leaves the numerical result unsupported.","rationale":"The reader's weakest-assumption analysis already identifies the time-like Padé continuation as the fragile load-bearing premise, and I agree. The strongest claim is the numerical branching fraction, and the calculation is a coherent model estimate with independent circumstantial support from earlier O(10⁻⁶) estimates. However, the paper contains no machine-checked derivation or reproducible code, and the numerical input to the Padé continuation is absent. The explicit statement in Sec. IV that the approximants are low-order and illustrative is an in-scope admission that the continuation is not claimed to be converged. The concern is not that the Skyrme model disagrees with consensus; it is that the analytic continuation crosses a regime where the stated H-function representation formally requires Re(α)>0, so the branch/pole structure is uncontrolled. A next-order Padé stability check is the single concrete test that would settle whether the central prediction is robust. Since the reader already returned a conditional verdict and my concern does not require changing that verdict, I recommend leaving it unchanged.","tokens_in":10530,"tokens_out":8824,"duration_ms":82872,"concrete_test":"Require the authors to provide the complete Padé fit data (orders, residues R_aj, and poles z_j for w0 and w1) and recompute Eq. (35) with the next higher Padé order while keeping all other inputs fixed. If B(Λ_b→p ar p n) moves outside the quoted 1σ band by more than a factor of 2, or if the dominant imaginary part of G_A/G_P changes sign or position, the central time-like continuation is not converged and the prediction should be treated as unvalidated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (35) integrates |2M_N G_A(q²)+(q²/2M_N)G_P(q²)|² over q²∈[(2M_N)²,(M_Λb−M_N)²]≈[3.52,21.9] GeV². The form factors G_A and G_P are obtained by replacing Q→iq in the H-function expressions (16)–(19), with the radial w functions fitted by low-order multipoint Padé approximants. The paper does not report the Padé orders, residues z_j, pole positions, or fit quality, and Sec. IV explicitly states that 'only relatively low-order Padé approximants are adopted here for illustration.' Since √q² reaches ~4.7 GeV while the Yukawa tails have scales Mπ and 3Mπ, the continuation violates the condition Re(α)>0 in Eq. (14): for q>Mπ or q>3Mπ, one of the two H-function arguments becomes µ−q with negative real part. The resulting analytic continuation through E1 therefore requires an unspecified branch choice, and the imaginary parts of G_A and G_P depend on the position of Padé poles that are not physical resonances. Consequently the quoted uncertainty, ±0.27×10⁻⁶, does not include the dominant systematic error of the calculation, and B=(1.10±0.27)×10⁻⁶ is not independently verifiable from the manuscript. The dropped nonfactorizable contribution is also an assumption, but the Padé/time-like continuation is the more directly load-bearing gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the purely baryonic weak decay Λ_b → p pbar n within the Skyrme model, treating the heavy baryon as a bound state of a heavy meson and a Skyrmion. The decay amplitude is evaluated via a spectator/factorization approximation, with the vacuum-to-(n,pbar) weak matrix element expressed through time-like axial and pseudoscalar form factors obtained by analytic continuation of numerically computed space-like Skyrme form factors through multipoint Padé approximants. The central result is a branching fraction B(Λ_b → p pbar n) = (1.10 ± 0.27) × 10^-6, compared with previous estimates of about 2 × 10^-6. The paper emphasizes that the calculation is preliminary, that only low-order Padé approximants are used for illustration, and that 1/N_c and 1/M_B uncertainties are not quantified.","tokens_in":10992,"tokens_out":4064,"duration_ms":33377,"significance":"If the result is sound, the paper would be significant in establishing a Standard Model estimate for an unobserved, potentially CP-sensitive class of baryonic decays, and would demonstrate the utility of the Skyrme bound-state method for time-like weak form factors. The explicit prediction of O(10^-6) with a differential distribution is a falsifiable target for LHCb and future colliders. The paper also carefully lists its own caveats. However, because the numerical central value controls the scientific message, the load-bearing steps need to be reproducible and controlled; at present, the manuscript does not provide enough information to verify the continuation or the uncertainty.","major_comments":[{"comment":"The central prediction B=(1.10±0.27)×10^-6 is governed by the integral of |2MN GA + (q^2/2MN) GP|^2 over q^2 up to ~21.9 GeV^2, where GA and GP come from the Padé continuation of the w functions. Yet the manuscript states in Sec. IV that 'only relatively low-order Padé approximants are adopted here for illustration' and does not report the Padé orders, the number and positions of poles zj, the residues Rj, nor the quality of the fit to the space-like w functions. Without this information the continuation cannot be reproduced or checked, so the central numerical result is not independently verifiable. Please provide the Padé poles/residues/orders and a validation plot (e.g. fit vs. exact space-like points; sensitivity to [n/m] order).","section":"IV, Fig. 2 and Eq. (35)"},{"comment":"The analytic continuation step is not fully specified. H(α,β) in Eq. (14) is defined for Re(α)>0, but after the replacement Q→iq the H-functions in Eqs. (16)-(19) are evaluated at α = μ1 − iq or μ2 − iq (and μ1 + iq, μ2 + iq). For the upper part of the q^2 integration range, q exceeds Mπ (and even 3Mπ), so μ − q has negative real part for one of the two terms; the formulas nevertheless employ E1(−α z) outside its principal domain. The manuscript does not state the analytic continuation convention, branch choice, or how the low-order Padé poles cross the integration contour. Since the imaginary parts of GA and GP depend on this, the result depends on an unspecified choice. The authors should spell out the continuation rule or replace this step with a controlled method (e.g. contour deformation, dispersion relation, or multipole over a quoted range of validity).","section":"II.B.4, Eq. (14)"},{"comment":"The quoted uncertainty ±0.27×10^-6 propagates only 'numerical inputs' (Sec. IV). The paper's own caveats identify three larger systematic effects: the unquantified 1/N_c and 1/M_B corrections, the unknown Padé-order dependence, and the neglected nonfactorizable diagram (Fig. 1b). The present error bar therefore strongly understates the theoretical uncertainty of the central claim and cannot be used to validate the O(10^-6) prediction against previous estimates. The paper should provide an estimate of the Padé-order sensitivity (e.g. different [m/n] values) and a statement of the expected size of 1/N_c corrections, or explicitly present the result as an order-of-magnitude estimate with a much larger error.","section":"IV, uncertainty budget"},{"comment":"The factorization/spectator reduction of Eq. (28) to Eq. (31) is asserted rather than derived. In particular, the identification of the initial proton with the final proton removes the initial-state soliton from the matrix element, and the q^2-dependence of the overlap ϕ(k*) is then computed from kinematics alone. The 'proton acts as a pure spectator' assumption is plausible at leading order in 1/N_c but the nonfactorizable topology is dropped. This may be acceptable for a first estimate, but it should be explicitly presented as an assumption (not merely 'assume this contribution to be suppressed'), since direct CP-violating observables would be sensitive to the dropped contributions.","section":"III, Eqs. (28)-(31)"},{"comment":"The relation between the multipoint Padé approximants in Eq. (11) and the Mittag-Leffler partial-fraction form in Eq. (13) is not justified. Eq. (11) contains polynomials P_(n) and Q_(m) of degrees n and m with m = n + p; Eq. (13) is a finite sum of simple poles only, which is not the generic partial fraction of a rational function unless there are no repeated poles and no polynomial part. If the Padé fit generates multiple poles, Eq. (13) is incomplete. This is a technical detail that needs to be fixed or clarified so the derivation is well-defined.","section":"II.B.4, text between Eqs. (10) and (13)"}],"minor_comments":[{"comment":"The notation for the branching fraction is inconsistent: '𝒪(10^{-6})' in the abstract and 'O(10−6)' later. Please unify.","section":"Abstract, Introduction"},{"comment":"The notation 'q^2' is used for both the momentum transfer and the dummy 'Q' in the Fourier-Bessel transforms; define Q^2 = -q^2 clearly to avoid confusion in Eqs. (16)-(19).","section":"II.B.4"},{"comment":"The heavy-meson decay constant is defined via ⟨0|qbar γμ γ5 Q|H(p)⟩, but Eq. (21) does not contain f_H nor the term that would relate it to the current; tie the definition to the actual Lagrangian or state it as external input.","section":"II.C"},{"comment":"The notation 'C_{s',s_B;s}' and the sum over spins are not fully specified; state the CG coefficients for the Λ_b (isospin 0, spin 1/2) to p + heavy meson decomposition.","section":"III, Eq. (26)"},{"comment":"The figure is not visible in the text; please ensure the plot includes axis labels, units, and the region q^2∈[3.52, 21.9] GeV^2 with the kinematic limits marked.","section":"IV, Fig. 2"},{"comment":"The text says 'the uncertainty only reflects the numerical inputs used in the calculation', but the parameter list does not show which input drives the 24% uncertainty; give the breakdown.","section":"IV"},{"comment":"The bound-state wavefunction ϕ(p) contains κ defined via F'''(0), whose numerical value is not given; specify κ or the resulting oscillator parameters.","section":"II.D, Eq. (23)"},{"comment":"Refs. [1,2] are authored by the same group as the present paper; this is not a problem, but flag that 'previous estimates' are not independent checks unless the methods differ materially.","section":"References / IV"},{"comment":"Typos: 'CG coefficiens' (Sec. III), the Källén function definition is missing a closing parenthesis, and 'timelike' vs 'time-like' are used inconsistently.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper has a plausible overall strategy and the result is likely qualitatively in the right ballpark, but the central number is not reproducible as written because the Padé continuation is under-specified and the paper itself disclaims its reliability ('for illustration'). The missing pieces are fixable: provide the Padé pole/residue data, quantify the dependence on Padé order and on the branch choice in Eq. (14), and reframe the error bar as an order-of-magnitude estimate. If those are provided, the paper could be acceptable; with the current level of detail, acceptance would set a bad precedent for an unverifiable numerical claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a clean, honest model estimate, but the headline number is not yet independently checkable because the time-like continuation is deliberately low-order and under-reported. The authors should get referee time, but the referee should send it back for a real treatment of the continuation uncertainty.\n\nWhat is new: they apply the Skyrme bound-state picture to Lambda_b -> p pbar n, a purely baryonic weak decay, and give an explicit factorization formula for the rate, with the vacuum-to-n-pbar matrix element expressed through time-like G_A and G_P. That specific calculation, including the numerical branching fraction (1.10 +- 0.27) x 10^-6, is new. The paper is upfront that this is a first test, that the Padé approximants are adopted \"for illustration,\" and that the nonfactorizable topology is dropped by assumption. Those are the right caveats, and the final number agrees with the O(10^-6) scale in Refs. [1,2], so the conclusion is plausible.\n\nThe soft spot, and it is a serious one, is Eq. (35): the rate is essentially the integral of |2 M_N G_A + (q^2/2M_N) G_P|^2 over q^2 up to about 22 GeV^2. Those form factors come from replacing Q -> i q in the H-function expressions, with radial functions fitted by low-order multipoint Padé approximants. The manuscript reports neither the Padé orders, the pole positions and residues, the fit quality, nor how the branch cut of E_1 is handled. In fact the H-function representation in Eq. (14) is stated for Re(alpha) > 0, and in the time-like region q > M_pi one of the alpha = mu - i q arguments violates that condition. That may be fixable, but it needs to be stated and justified. Because of this, the quoted +-0.27 x 10^-6 covers only the input parameters, not the dominant systematic error. I also think the spectator/factorization treatment is rough, but the continuation gap is the load-bearing one.\n\nThe citation pattern is fine: Refs. [1,2] are overlapping authors, but the quoted earlier rates are used for comparison, not as inputs. The paper does not hide its limitations, and the math is coherent in outline. It is a legitimate proof-of-principle, not a finished prediction.\n\nWho this is for: people working on LHCb or future searches for rare baryonic decays, and practitioners of soliton models who want to see the method pushed toward weak transitions. A serious referee should not desk-reject it, but should request the missing continuation details and an honest systematic uncertainty before the number is quoted. My recommendation: send to peer review, with the expectation of major revision.","headline":"A coherent but preliminary Skyrme-model estimate of Lambda_b -> p pbar n that lands at the known O(10^-6) scale; the central number is plausible but rests on an under-specified time-like continuation, so it needs a serious referee and likely major revision.","tokens_in":11416,"tokens_out":2327,"would_cite":false,"duration_ms":24969,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["12.39.Dc","13.30.-a"],"model":"deepseek-v4-flash","headline":"The Skyrme model, treating Λ_b as a heavy meson bound to a soliton, predicts the purely baryonic decay Λ_b→p̄pn with branching fraction (1.10±0.27)×10⁻⁶, matching earlier order-of-magnitude estimates.","keywords":["Skyrme model","heavy baryons","purely baryonic weak decays","branching fraction","time-like form factors","Padé approximation","bound-state model","CP violation"],"falsifier":"A branching-fraction measurement of Λ_b→p̄pn that falls well outside (1.10±0.27)×10⁻⁶ would contradict the prediction; a lattice-QCD calculation of the time-like axial and pseudoscalar nucleon form factors in that q² window would directly test the continuation step.","tokens_in":10436,"feed_emoji":"⚛️","tokens_out":4048,"duration_ms":36472,"temperature":0.7,"pith_summary":"The paper aims to show that the Skyrme model — in which baryons are topological solitons and heavy baryons are bound states of a heavy meson and a soliton — can compute the rate of an as-yet-unobserved, purely baryonic weak decay, Λ_b → p p̄ n. Working in the large-N_c spectator/factorization approximation, it obtains the Standard Model prediction B(Λ_b → p p̄ n) = (1.10 ± 0.27) × 10⁻⁶, in line with earlier estimates at the 10⁻⁶ level. A sympathetic reader would care because this gives a concrete, testable Standard Model baseline for a channel sensitive to new sources of CP violation in the baryonic sector.","feed_headline":"Skyrme model pegs heavy-baryon decay at one in a million","feed_subtitle":"The pure-baryon channel Λ_b→p̄pn is a standard-model benchmark for future CP-violation searches.","key_machinery":"The heavy baryon as a bound state of a heavy meson and a Skyrmion; the hedgehog soliton solution with collective quantization; time-like form factors built via multipoint Padé approximation of space-like form factors (through exponential-integral kernels H(α, β) from the pion tail); and the spectator/factorization reduction of the four-baryon transition matrix element to a product of a heavy-meson decay constant and a two-baryon current matrix element.","core_discovery":"Within the bound-state Skyrme picture, Λ_b is a proton (Skyrmion) plus a B meson. The weak transition b→u operates on the B component, producing the n p̄ pair through the time-like nucleon weak current, while the proton is a spectator. The decay rate is then driven by the combination |2M_N G_A(q²) + (q²/2M_N) G_P(q²)|² of the time-like axial and pseudoscalar form factors. These form factors are obtained by Padé-continuing space-like Skyrme form factors into the time-like region. Evaluated numerically, the branching fraction comes out at (1.10 ± 0.27) × 10⁻⁶, about half of previous estimates but of the same order of magnitude.","pith_inferences":["The Padé continuation's reliability could be tested directly: compare the continued time-like form factors against independent lattice-QCD results for G_A and G_P in this q² window, since the paper itself flags that only low-order approximants were used.","The near-threshold enhancement of the differential rate (the paper's Fig. 3) is a kinematic plus form-factor effect; if measured, the q² shape would discriminate between the Skyrme form-factor model and other parameterizations.","The smallness of the imaginary parts of G_A and G_P in the time-like region is a distinctive prediction; e⁺e⁻ → n p̄ cross-section data could constrain these imaginary parts and would directly probe the validity of the continuation."],"forward_implications":["If correct, LHCb and other experiments have a sharp Standard Model target: B(Λ_b→p̄pn) ≈ 10⁻⁶, within reach of current hadron-collider data.","The rate pins down a combination of the time-like axial and pseudoscalar nucleon form factors in the q² range from (2M_N)² to (M_Λb−M_N)², a region largely unconstrained by other data.","The factor-of-two discrepancy with earlier estimates indicates either missing contributions (nonfactorizable, relativistic, or higher 1/N_c effects) or overestimated earlier values; both are resolvable by refining the calculation.","The same bound-state machinery extends to other purely baryonic modes such as Λ_b→p̄pΛ, with a predicted suppression from phase space.","Because the final state contains four spin-1/2 baryons, T-odd triple-product correlations can be constructed; this rate provides the Standard Model baseline for those CP-violation searches."],"fun_headline_variants":["Skyrme model predicts Λ_b decay to three baryons at 10⁻⁶","Λ_b→p p̄ n: a one-in-a-million baryonic decay from Skyrme model","Purely baryonic weak decay of Λ_b: rate ~1e-6 in Skyrme model","Skyrme bound-state model yields Λ_b→p p̄ n branching fraction","New baryonic decay channel Λ_b→p p̄ n predicted at 10⁻⁶"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The time-like form factors obtained by low-order Padé continuation of space-like Skyrme results remain accurate over the entire integration range q² ∈ [(2M_N)², (M_Λb−M_N)²] ≈ [3.5, 21.9] GeV², where the decay rate is integrated.","fun_headline_variants_meta":{"raw":{"variants":["Skyrme model predicts Λ_b decay to three baryons at 10⁻⁶","Λ_b→p p̄ n: a one-in-a-million baryonic decay from Skyrme model","Purely baryonic weak decay of Λ_b: rate ~1e-6 in Skyrme model","Skyrme bound-state model yields Λ_b→p p̄ n branching fraction","New baryonic decay channel Λ_b→p p̄ n predicted at 10⁻⁶"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000612,"raw_usage":{"total_tokens":2642,"prompt_tokens":661,"completion_tokens":1981,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":405,"completion_tokens_details":{"reasoning_tokens":1860}},"tokens_in":405,"tokens_out":1981,"duration_ms":16476,"temperature":1.0,"reasoning_tokens":1860,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T18:15:48.039577+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A branching-fraction measurement of Λ_b→p̄pn that falls well outside (1.10±0.27)×10⁻⁶ would contradict the prediction; a lattice-QCD calculation of the time-like axial and pseudoscalar nucleon form factors in that q² window would directly test the continuation step.","supporting_citations":[],"review_version":1}