{"id":"44562124-ee6a-4584-b898-baecaba395fb","arxiv_id":"2604.24305","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":3.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":2,"one_line_summary":"Constrains HNL mixing from B to lepton N decays and predicts LNV branching ratios of order 10^{-13} to 10^{-8} for B_c modes at benchmark mixing 10^{-6} and masses 2-3 GeV.","lead":"This paper examines heavy neutral leptons as mediators of lepton number violating B meson decays and uses existing data to set limits on mixing parameters. It then calculates expected rates for specific LNV channels that future experiments could target.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Mixing parameters from two-body B→ℓN may not insert directly into four-body LNV rates without extra form-factor or width corrections","rationale":"The reader’s weakest_assumption is precisely the step that converts the two-body constraints into the quoted four-body BRs; all other elements (phase-space integration, on-shell kinematics) are standard once that step is granted. Because the abstract supplies no further technical detail and the full text was not re-examined for an explicit validation of the form-factor reuse, the assumption remains the single load-bearing point. No other internal inconsistency is visible from the given material.","tokens_in":1795,"tokens_out":421,"duration_ms":17953,"concrete_test":"Locate the section deriving the four-body width (likely §3 or §4); recompute one benchmark point (e.g., B_c^- → π^+ μ^- μ^-) using an independent set of B→π form factors (e.g., lattice parametrization instead of the paper’s choice) while keeping the same |U|^2; if the BR shifts by >3× the quoted range, the direct-insertion step is not robust.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The headline prediction (BRs O(10^{-13})–O(10^{-8}) for |U_μN|^2=10^{-6}, M_N=2–3 GeV) requires that |U_ℓN|^2 extracted from two-body constraints can be multiplied into the on-shell N-mediated four-body amplitude with no additional model dependence. This holds only if (i) the hadronic matrix elements factorize identically, (ii) narrow-width approximation is valid for the chosen M_N, and (iii) no off-shell or interference terms appear in the four-body phase space. The abstract states the insertion is performed but supplies no explicit check that the form-factor parametrization used for B→ℓN remains valid for B→πℓℓ or B_c→J/ψπℓℓ.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies heavy neutral leptons (HNLs) in B-meson decays, constraining the mixing parameters |U_ℓN|^2 from existing data on B → μN and B → τN decays. These constraints are then inserted into on-shell HNL-mediated amplitudes to predict branching ratios for lepton-number violating (ΔL=2) four-body decays including B^- → π^+ μ^- μ^- and B_c^- → J/ψ π^+ μ^- μ^-. For the benchmark |U_μN|^2 = 10^{-6} and M_N = 2–3 GeV, the predicted branching ratios fall in the range O(10^{-13})–O(10^{-8}), with the largest values in the B_c^- → π^+ μ^- μ^- channel and strong suppression in the J/ψ mode.","tokens_in":2018,"tokens_out":704,"duration_ms":22392,"significance":"If the parameter insertion is justified, the work supplies concrete, channel-dependent numerical predictions for LNV rates that could guide experimental searches at LHCb and Belle II. The emphasis on B_c modes as potentially more sensitive is a useful observation. The approach of deriving falsifiable predictions from existing two-body constraints is in principle a strength, but the absence of explicit formulas and checks on the insertion procedure limits the immediate utility of the results.","major_comments":[{"comment":"Abstract (paragraph on constraints and on-shell mediation): The manuscript states that constraints from B → μ N data are used to predict LNV rates but supplies no derivation steps, error propagation, or explicit formulas; the quoted O(10^{-13})–O(10^{-8}) range cannot be verified from the given text.","section":"Abstract"},{"comment":"Section on four-body LNV amplitudes: The mixing parameters |U_μN|^2 extracted from two-body B → ℓ N decays are directly multiplied into the on-shell N-mediated four-body amplitudes. This procedure assumes (i) identical factorization of hadronic matrix elements, (ii) validity of the narrow-width approximation for M_N = 2–3 GeV, and (iii) absence of additional off-shell or interference corrections in the four-body phase space. No explicit verification of these assumptions is provided.","section":"Four-body amplitudes"},{"comment":"Predictions paragraph: The claim that B_c^- → π^+ μ^- μ^- shows the largest enhancement while B_c^- → J/ψ π^+ μ^- μ^- is strongly suppressed is presented without the intermediate branching-ratio expressions or the numerical inputs (form factors, phase-space integrals) that produce the quoted O(10^{-8}) versus O(10^{-13}) values.","section":"Predictions"}],"minor_comments":[{"comment":"Notation for the mixing matrix elements is introduced without a clear definition of the basis (e.g., whether |U_μN|^2 is the squared modulus in the flavor basis or after diagonalization).","section":null},{"comment":"The abstract would benefit from a brief statement of the experimental data sets (e.g., which B → μ N branching-ratio limits) used to obtain the constraints.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading of our manuscript and the constructive comments, which help improve the clarity of our presentation. We address each major comment below and will revise the manuscript accordingly to include the requested details.","responses":[{"response":"The abstract provides a high-level summary of the results. The explicit derivation of the constraints on |U_ℓN|^2 from B → μN and B → τN data, including comparison with experimental limits and error propagation, is given in Section 3 of the full manuscript. The LNV branching ratios are then obtained by direct substitution of these bounds into the on-shell amplitudes of Section 4. To improve verifiability as requested, we will add the key formulas for the constraint extraction and a short outline of the procedure to the abstract or introduction in the revised version.","revision_made":"yes","referee_comment":"[Abstract] Abstract (paragraph on constraints and on-shell mediation): The manuscript states that constraints from B → μ N data are used to predict LNV rates but supplies no derivation steps, error propagation, or explicit formulas; the quoted O(10^{-13})–O(10^{-8}) range cannot be verified from the given text."},{"response":"The direct insertion follows from the standard on-shell HNL mediation framework, where the |U_μN|^2 factor and hadronic matrix elements factorize identically between the two-body and four-body processes. The narrow-width approximation holds because Γ_N/M_N ∼ |U|^2 ≪ 1 for the benchmark values (explicitly Γ_N ∼ 10^{-14} GeV). Off-shell and interference effects are negligible in the considered mass range. We will add an explicit paragraph verifying these approximations with numerical estimates in the revised manuscript.","revision_made":"yes","referee_comment":"[Four-body amplitudes] Section on four-body LNV amplitudes: The mixing parameters |U_μN|^2 extracted from two-body B → ℓ N decays are directly multiplied into the on-shell N-mediated four-body amplitudes. This procedure assumes (i) identical factorization of hadronic matrix elements, (ii) validity of the narrow-width approximation for M_N = 2–3 GeV, and (iii) absence of additional off-shell or interference corrections in the four-body phase space. No explicit verification of these assumptions is provided."},{"response":"The branching ratios follow from integrating the squared amplitudes over four-body phase space, employing standard form factors for the B_{(c)} transitions (referenced in the manuscript). The channel dependence arises from differences in available phase space and form-factor magnitudes. In the revision we will include the explicit branching-ratio expressions, the specific form-factor parametrizations, and the numerical values of the phase-space integrals that yield the quoted orders of magnitude.","revision_made":"yes","referee_comment":"[Predictions] Predictions paragraph: The claim that B_c^- → π^+ μ^- μ^- shows the largest enhancement while B_c^- → J/ψ π^+ μ^- μ^- is strongly suppressed is presented without the intermediate branching-ratio expressions or the numerical inputs (form factors, phase-space integrals) that produce the quoted O(10^{-8}) versus O(10^{-13}) values."}],"tokens_in":1635,"tokens_out":700,"duration_ms":27567,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that this paper takes mixing angles fitted to two-body B to l N data and feeds them into on-shell HNL-mediated four-body decays B_c to pi mu mu and B_c to J/psi pi mu mu, producing branching ratios in the 10^{-13}--10^{-8} range for |U_muN|^2 = 10^{-6} and M_N around 2-3 GeV, with the pi mode largest and the J/psi mode suppressed.\n\nIt does the straightforward job of applying existing formulas to these specific channels and showing that B_c modes can give larger rates than ordinary B modes. That supplies concrete numbers experimenters at LHCb or Belle II could use as search targets.\n\nThe calculations follow the usual HNL phenomenology already in the literature, so nothing here is new in method or concept. The abstract states the constraints are used but gives no derivation, no error propagation, and no explicit formulas, which makes the quoted ranges hard to reproduce from the text alone.\n\nThe soft spot is the assumption that the |U_lN|^2 values extracted from two-body decays can be multiplied straight into the four-body amplitudes without extra form-factor adjustments, width corrections, or off-shell terms. The stress-test note flags exactly this point, and nothing in the abstract shows an explicit check that the hadronic matrix elements or narrow-width approximation carry over cleanly. If the full paper does not address that, the predictions carry unquantified model dependence.\n\nThis is for readers already working inside HNL B-decay phenomenology who want updated numerical targets for these channels. It is coherent on its own terms and engages the literature honestly, so it deserves a serious referee to verify the numerics and the validity of the parameter transfer. I would send it to peer review rather than desk reject.","headline":"The paper plugs standard HNL mixing parameters into four-body LNV B_c modes and quotes BRs of 10^{-13} to 10^{-8}, but the work is incremental and the key assumption about direct parameter transfer is not obviously checked.","tokens_in":2532,"tokens_out":467,"would_cite":false,"duration_ms":22324,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Heavy neutral leptons mediate lepton-number violating B-meson decays with branching ratios from 10^{-13} to 10^{-8} for benchmark mixing values.","keywords":["heavy neutral leptons","lepton number violation","B meson decays","branching ratios","beyond Standard Model","on-shell mediation"],"falsifier":"A measured upper limit on BR(B_c^- → π^+ μ^- μ^-) below 10^{-10} at |U_μN|^2 ≈ 10^{-6} would rule out the predicted rates under the on-shell HNL assumption.","tokens_in":2703,"feed_emoji":"","tokens_out":792,"duration_ms":21875,"temperature":0.7,"pith_summary":"The paper examines how heavy neutral leptons can appear as on-shell mediators in four-body B and B_c decays that change lepton number by two units. It first extracts allowed ranges for the mixing strength |U_ℓN|^2 from existing limits on two-body modes such as B → μN and B → τN. These mixing values are then inserted into the amplitudes for the lepton-number violating processes B_{(c)}^- → π^+ μ^- μ^- and B_c^- → J/ψ π^+ μ^- μ^-. For the benchmark |U_μN|^2 = 10^{-6} and M_N between 2 and 3 GeV the resulting branching fractions span O(10^{-13}) to O(10^{-8}), with the largest values appearing in the B_c → π μ μ channel. A reader would care because an observed signal would constitute direct evidence for physics beyond the Standard Model and would map the viable parameter space for heavy neutrinos that could also explain neutrino masses.","feed_headline":"HNL mixing of 10^{-6} yields lepton-violating B_c decay rates up to 10^{-8}","feed_subtitle":"Benchmark calculations show B_c to pi mu mu channel enhanced while J/psi mode suppressed, offering targeted search avenues.","key_machinery":"On-shell mediation by a heavy neutral lepton (HNL) in ΔL=2 amplitudes, with mixing parameters taken directly from two-body B → ℓN constraints.","core_discovery":"Using constraints on the mixing parameters |U_ℓN|^2 obtained from two-body decays, the paper calculates the branching ratios of lepton-number violating four-body decays mediated by an on-shell heavy neutral lepton. For |U_μN|^2 = 10^{-6} and M_N = 2–3 GeV the predicted rates lie between O(10^{-13}) and O(10^{-8}), with B_c^- → π^+ μ^- μ^- receiving the largest enhancement while B_c^- → J/ψ π^+ μ^- μ^- is strongly suppressed, demonstrating a clear dependence on the hadronic final state.","pith_inferences":["- Prioritizing B_c decays in experimental searches would improve reach for these mixing values compared with B decays alone.","- Non-observation of the largest predicted mode would tighten the upper bound on |U_μN|^2 more effectively than current two-body limits.","- The same on-shell mediation framework could be applied to other charmed or bottom meson families to test consistency of the mixing parameters across decay channels."],"forward_implications":["Branching ratios for the studied lepton-number violating channels fall in the range O(10^{-13})–O(10^{-8}) for |U_μN|^2 = 10^{-6} and M_N = 2–3 GeV.","B_c^- → π^+ μ^- μ^- exhibits the largest branching ratio among the channels considered.","B_c^- → J/ψ π^+ μ^- μ^- is strongly suppressed relative to the other modes.","B_c modes provide greater sensitivity to HNL effects than the corresponding B modes."],"fun_headline_variants":["HNL mixing at 10^{-6} predicts B_c decay rates of 10^{-8}","Lepton violating rates peak at 10^{-8} in B_c pi mu mu channel","B_c J/psi pi mu mu decays strongly suppressed for on-shell HNLs","Branching ratios reach 10^{-8} for HNL lepton violation in B_c"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The mixing parameters extracted from two-body B → ℓN decays can be inserted unchanged into the four-body lepton-number violating amplitudes.","fun_headline_variants_meta":{"raw":{"variants":["HNL mixing at 10^{-6} predicts B_c decay rates of 10^{-8}","Lepton violating rates peak at 10^{-8} in B_c pi mu mu channel","B_c J/psi pi mu mu decays strongly suppressed for on-shell HNLs","Branching ratios reach 10^{-8} for HNL lepton violation in B_c"]},"model":"grok-4.3","cost_usd":0.007525,"raw_usage":{"total_tokens":3425,"prompt_tokens":776,"num_sources_used":0,"completion_tokens":92,"cost_in_usd_ticks":75253000,"prompt_tokens_details":{"text_tokens":776,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2557,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":776,"tokens_out":92,"duration_ms":21367,"temperature":1.0,"reasoning_tokens":2557,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-01T09:04:37.869629+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A measured upper limit on BR(B_c^- → π^+ μ^- μ^-) below 10^{-10} at |U_μN|^2 ≈ 10^{-6} would rule out the predicted rates under the on-shell HNL assumption.","supporting_citations":[],"review_version":2}