{"id":"5db6cf79-c5fa-419a-ad33-f68593ce4104","arxiv_id":"2412.19220","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A two-step diagonalization plus a dummy-particle trick lets standard event generators simulate GeV-scale sterile neutrino oscillations with crossing-width terms, and a QFT derivation gives displaced-vertex distances.","lead":"The paper presents a way to simulate collider events of oscillating sterile neutrinos using standard event generator software even when the neutrinos' decay widths mix. It also provides a quantum-field-theory method to compute how far the neutrinos fly before decaying.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (68) equates the summed two-resonance cross section with Γ22 + Γ33 + 2Γ23, but ordinary Breit-Wigner interference is not the same as a crossing width; the extraction is at best an unstated first-order approximation.","rationale":"The reader's weakest assumption flagged Eq. (68); I agree it is the load-bearing point and sharpen it: the relation is not merely underived, it is the wrong analytic structure unless a first-order approximation is made. However, the paper might still work perturbatively, and the model's physics is presented with enough detail that a targeted check can decide. Because the requested validation and reproducibility artifacts are already part of the reader's conditional verdict, my stress-test does not move the verdict.","tokens_in":26505,"tokens_out":7963,"duration_ms":88320,"concrete_test":"Set up a toy two-resonance model with a known inverse propagator D(s) = diag(s − m_i² + i m_i Γ_ii) + i m Γ_23 off-diagonal. Compute the exact integrated cross section for a common production/decay source, ∫ds |(1,1) D(s)^{-1} (1,1)^T|², and compare it with the right-hand side of Eq. (68) using the same Γ22, Γ33. Vary Γ23/ΔM from 0.1 to 5 and check whether σ_sum − σ_2 − σ_3 equals 2Γ23 to the claimed accuracy; a nonzero deviation at any point settles that Eq. (68) is not generally exact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section VI.A, the dummy-particle trick rests on Eqs. (67)-(69): the coherent two-resonance cross section is asserted to be Γ22 + Γ33 + 2Γ23, so Γ23 can be read off from the difference of generator-computed cross sections. This is the only input that carries the crossing-width effect into the second-step diagonalization (28), so if it is not exact the whole simulation chain is not validated.\n\nThe problem is that, in the resummed propagator (24), Γ23 is an off-diagonal element of the inverse of (p̸ − M + iΓ/2), before inversion. The physical amplitude is a sum over the two complex poles of that matrix, with residues set by the diagonalization (28); it is not the sum of two independent Breit-Wigner propagators with widths Γ22 and Γ33. An unmodified event generator computing 'N2+N3' uses diagonal Breit-Wigner denominators, and its interference term, shown in Eq. (71), depends only on Δm and the diagonal widths; it contains no Γ23. The equality (68) can hold at best as a first-order statement in Γ23 (with equal, real dummy couplings), but the paper applies it over the whole parameter plane, including R̂DM/SM = 5 where Γ23 can exceed ΔM. No derivation or numerical validation of (68) is given, so the extracted Γ23, and therefore the masses and widths fed into the simulation, are not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a simulation method for collider events with nearly degenerate GeV-scale sterile neutrinos when crossing (off-diagonal) widths are non-negligible. The proposed pipeline is: diagonalize the neutrino mass matrix; compute the diagonal and off-diagonal widths; build the non-Hermitian matrix M - iΓ/2 and rotate it with a complex orthogonal matrix V; generate events with standard tools treating the rotated fields N'_2, N'_3 as Majorana fermions; extract Γ23 using a dummy-particle s-channel cross-section trick; use cross-section rescaling for narrow resonances; and reweight event probabilities by oscillation/displaced-distance functions derived in Section V. Numerical examples are given for pp → W±* → μ±N → μ±μ±jj, with R_LNV/R_LNC contours and one displacement probability plot.","tokens_in":26868,"tokens_out":10308,"duration_ms":104629,"significance":"If valid, the method is practically valuable: it would allow unmodified MadGraph-based pipelines to include crossing widths and displaced-vertex distributions for pseudo-Dirac sterile neutrinos, with testable predictions for Rll and oscillation probabilities. The paper is unusually explicit about its approximations (neglect of γ5, B², and δMΓ5 terms in Section IV), and the proposed event-reconstruction criterion using the .lhe mother-particle information is concrete. The main limitation is that the crucial Γ23 extraction and the overall simulation chain are not validated independently; the asserted relation in Eq. (68) is not derived from the resummed propagator, and no comparison with a direct two-pole calculation is provided. The central claim is therefore plausible but not yet established.","major_comments":[{"comment":"The relation σ_{N′φ′→(N2+N3)→everything} ∝ Γ22 + Γ33 + 2Γ23 is the only step that feeds Γ23 into the second-step diagonalization, but it is asserted without proof. In the resummed propagator (24), Γ23 is an off-diagonal element of M - iΓ/2 before inversion; the physical amplitude is a sum over the complex poles of (p̸ - M + iΓ/2)^{-1}, with residues fixed by the complex orthogonal matrix V of Eq. (28). It is not the sum of two independent Breit-Wigner amplitudes with widths Γ22 and Γ33. An unmodified generator uses diagonal Breit-Wigner denominators, and its interference term has the form shown in Eq. (71), which depends on Δm and the diagonal widths but contains no Γ23. Equation (68) can therefore hold at best in a special first-order or fine-tuned kinematic limit, while the paper applies it over the whole parameter plane, including R^{DM}_{SM}=5 in Section VII, where Γ23 can exceed ΔM. A derivation of Eq. (68) from the resummed propagator, or an independent numerical validation of the extracted Γ23, is required.","section":"Section VI.A, Eqs. (66)-(69)"},{"comment":"The numerical results do not validate the algorithm. The Rll contours in Figs. 2 and 3 and the oscillation probabilities in Fig. 4 are produced with the proposed pipeline, but no comparison is made with a direct calculation using the full 2×2 resummed propagator of Eq. (24). In particular, the Γ23=0 limit is a natural cross-check: Eq. (68) then predicts σ_{N′φ′→(N2+N3)} ∝ Γ22+Γ33, whereas the two independent Breit-Wigner amplitudes produced by an unpatched generator have a nonzero interference term for Δm≠0. A benchmark with Γ23=0, a small-Γ23 benchmark, and a direct pole calculation would show whether the dummy-particle extraction and the rescaled event generation reproduce the correct cross sections. Without such a test, the \"proof of validity\" announced in the abstract is not substantiated by the numerical section.","section":"Section VII"},{"comment":"The rotated fields N′_i are not self-conjugate, as Eq. (30) shows, yet the manuscript instructs the generator to treat them as Majorana 4-spinors. This identification is a substantive assumption: standard generators use the Majorana Feynman rules and the diagonal propagator (25), while the actual object is a complex combination defined through the non-Hermitian rotation V. The statement that this is necessary \"for a self-consistent simulation\" is not a proof. Since the central claim is that unmodified tools can be used, this point needs either a derivation showing that all S-matrix elements of the rotated Lagrangian coincide with those of Eq. (24), or a numerical cross-check.","section":"Section IV, Eqs. (29)-(30)"},{"comment":"The step from the S-matrix expression in Eq. (41) to the probability distribution P_{|Δx|} in Eq. (42) is not derived. Equation (41) yields an amplitude; taking the modulus squared of only the oscillating exponential part ignores the integration over external phase space and the spinor structure of the \"...\" factors, and the paper gives no wave-packet justification for factorizing a probability in |Δx|. The final formulas (59)-(64) may be a reasonable practical reweighting, but they are presented as a QFT derivation. I ask for either a clear derivation, including the treatment of the unobserved final state and the normalization, or an explicit demonstration that the reweighting reproduces the known exponential decay law in the single-resonance limit.","section":"Section V, Eqs. (33)-(42)"}],"minor_comments":[{"comment":"The text defines v1=(1,0,0)^T, v2=(0,1,0)^T, and then repeats v3=(1,0,0)^T; the third vector should be (0,0,1)^T.","section":"Section III, text near Eq. (8)"},{"comment":"There are typos: \"Feynmann\" in the abstract and \"THe\" in several section headings; please proofread.","section":"Abstract and headings"},{"comment":"The completeness-like relation should use \\bar u and \\bar v, not u(p,λ)u(p,λ) and v(p,λ)v(p,λ), unless an unusual spinor normalization is intended; as written the expression is not Lorentz invariant and is inconsistent with the subsequent use of uu = 2mδ in Eq. (58).","section":"Section V, Eq. (54)"},{"comment":"All four panels in each figure have the same caption text and no (a)-(d) labels, so the reader cannot map the panels to R^{DM}_{SM}=0.1, 0.5, 1, 5; please label the panels.","section":"Figures 2 and 3"},{"comment":"The dummy couplings y′_i and the precise meaning of \"everything\" are not specified; the proportionality constants in Eq. (68) depend on both, and the conditions under which the narrow-width approximation is valid should be stated.","section":"Section VI.A, Eq. (68)"},{"comment":"The rescaling argument should specify the common rescaling of m_i, Γ_i and Δm that leaves the two-resonance cross section invariant; in particular, the transformation of the interference term is not written out.","section":"Section VI.B, Eq. (71)"},{"comment":"Several parameter planes are called \"plain\" instead of \"plane\"; also, no MadGraph/FeynRules model files or event-generation scripts are provided, which limits reproducibility.","section":"Section VII"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the journal's scope as a phenomenological methods paper, but the lack of a companion repository and the missing validation benchmarks make it difficult to judge reproducibility. The manuscript would be strengthened by an appendix with the explicit Feynman rules for the dummy-particle process and a Γ23=0 sanity check."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: the dummy-particle trick and the two-step diagonalization are genuinely useful, but the paper's central extraction of Γ23 is asserted, not derived, and the numerical validation is too thin to certify the method. The paper deserves a serious referee, not because it is error-free, but because the idea is new and the fix may be a manageable revision.\n\nWhat is new: a way to get the crossing width Γ23 out of a standard event generator by adding two dummy fields (N′ and φ′), then feeding the resulting non-Hermitian mass matrix through a second diagonalization so MadGraph can handle the nearly-degenerate sterile neutrinos without patches. The QFT treatment of the displaced vertex distance as a reweighting of the generated .lhe events is also a step beyond the usual on-shell/oscillation hybrid. These are concrete, practical contributions, and the paper walks through the steps clearly.\n\nThe soft spots are real. Equation (68) is the linchpin: σ(N2+N3) ∝ Γ22 + Γ33 + 2Γ23. The paper just asserts it. As your stress-test note says, an unmodified generator sees two diagonal Breit-Wigner resonances; its interference term depends on Δm and the diagonal widths, not on Γ23. The equality can only hold under some first-order or equal-coupling approximation, and the paper never states that condition. It then applies the trick over the whole parameter plane, including R̂DM/SM = 5, where Γ23 can be larger than ΔM. That is an unquantified leap. I also could not find a cross-check against an explicit two-pole calculation, and no code or data are provided, so the numerical results are hard to assess. The paper says it 'proves the validity of our algorithm' in the abstract, but no proof of (68) appears.\n\nThe other approximations (neglect of γ5 terms and off-diagonal mass corrections) are stated in the text, so I am not counting those as hidden flaws, but their numerical impact is not examined.\n\nWho it is for: HNL phenomenologists who want to include crossing-width effects in collider simulations with existing tools. If (68) holds in the needed regime, this is a useful tool. As it stands, I would not build on it yet. I would send it to peer review with a request for a derivation or explicit approximation status of (68), an independent numerical validation, and a reproducibility package. Then it could become a solid methods paper.","headline":"A genuinely useful simulation trick with a load-bearing extraction that is asserted, not proven, and thin numerical validation.","tokens_in":27347,"tokens_out":2922,"would_cite":false,"duration_ms":29441,"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 claims that crossing-width effects between nearly degenerate sterile neutrinos can be included in standard collider event generators through a two-step diagonalization and a dummy-particle trick, with displaced-vertex distances…","keywords":["sterile neutrino oscillations","pseudo-Dirac neutrino","crossing-width","lepton number violation","displaced vertex","event generation","quantum field theory"],"falsifier":"Compute the same benchmark process, for example pp→W±*→μ±N→μμjj with a chosen Γ23/ΔM, by an independent diagrammatic calculation that resums the full 2×2 self-energy matrix, and compare the resulting LNV/LNC ratio and displaced-vertex distribution with the dummy-particle-generated events. A mismatch between Γ23 obtained from the subtraction formula and the value of Im[Σ23] computed directly from the Lagrangian would falsify the method.","tokens_in":26304,"feed_emoji":"⚛️","tokens_out":7068,"duration_ms":64294,"temperature":0.7,"pith_summary":"This paper aims to show that collider events for GeV-scale oscillating sterile neutrinos, including the crossing-width term that couples two nearly mass-degenerate states, can be generated with standard, unmodified event-generation tools. The authors prove the validity of a two-step diagonalization procedure: first diagonalize the real part of the mass matrix, then diagonalize the non-Hermitian matrix built from the masses and the full width matrix including off-diagonal entries. They supply a dummy-particle trick to extract the crossing width, and a quantum-field-theory rewriting that turns the propagator denominator into an exponential phase, so the event files carry enough information to simulate the displaced-vertex distance. If correct, this gives a practical way to predict lepton-number-violating versus lepton-number-conserving rates and long-lived signatures for pseudo-Dirac sterile neutrinos without hand-patching simulation code.","feed_headline":"Dummy-particle trick folds sterile-neutrino crossing widths into stock tools","feed_subtitle":"Two-step diagonalization plus QFT phase reweighting yields LNV/LNC ratios and displaced-vertex distances from ready-made generators.","key_machinery":"The load-bearing object is the non-Hermitian matrix M′ = M − iΓ/2 whose off-diagonal entries are the crossing widths between the nearly degenerate fermionic states; diagonalizing it with a complex orthogonal matrix V defines the mass-and-width eigenstates that are put into the simulator. Two auxiliary manipulations carry the practical argument: the dummy-particle process φ′N′→Ni→everything, whose s-channel cross sections yield Γ23 by a linear subtraction, and the replacement of each propagator pole factor by an exponential phase to get the displaced-vertex probability.","core_discovery":"The central claim is that a nearly degenerate pseudo-Dirac sterile-neutrino pair with off-diagonal width Γ23 can be simulated by treating the combination M − iΓ/2 as the effective mass matrix, diagonalizing it with a complex orthogonal matrix, and feeding the resulting eigenstates into an event generator as ordinary Majorana propagators. The crossing width, which the generator cannot compute directly, is obtained by introducing two dummy fields with a Yukawa coupling φ′N′Ni: the cross sections for s-channel N2, N3 and their coherent sum are proportional to Γ22, Γ33, and Γ22+Γ33+2Γ23 respectively, so Γ23 follows by subtraction. The paper further claims that a pure QFT treatment of the intermediate internal lines allows the squared S-matrix to be reorganized into a probability distribution in the spatial separation |Δx|, replacing each Breit-Wigner denominator by exp(i√((p0)2−m_i2+i m_i Γ_i)|Δx|). This yields explicit oscillation probabilities between ND and its antiparticle, and the authors verify that the event files preserve enough information to assign each event to one of the four patterns and to sample its flying distance.","pith_inferences":["The key test of the method is the linearity assumption behind the Γ23 extraction; a direct derivation of that proportionality from the resummed propagator would strengthen the algorithm, and a dedicated Monte-Carlo cross-check against the unpatched full calculation would settle it.","The same dummy-particle subtraction could be repurposed to extract off-diagonal widths in other contexts, such as vector or scalar mixing, wherever the generator cannot input a non-diagonal width matrix.","Because the crossing width couples production and decay stages, phenomenological scans that ignore Γ23, for example in inverse or linear seesaw models with a dark sector, may misestimate both LNV rates and displaced-vertex lengths.","The QFT phase-replacement method sidesteps wave-packet arguments and might be adapted to other long-lived nearly degenerate resonances whose flying distance is currently added by hand after event generation."],"forward_implications":["The LNV/LNC ratio Rll can be evaluated over the mD–μ2 plane with crossing-width effects included, rather than relying on patched or hybrid oscillation codes.","For small SM widths, the event files contain enough mother-particle information to generate displaced-vertex distances through the QFT probability distributions (59), (62)–(64).","The crossing width Γ23 can be large enough to 'knead' the two resonances back together, suppressing Rll relative to the well-separated two-Majorana limit.","The same two-step diagonalization and dummy-particle tricks apply to any nearly degenerate s-channel mediators, not only sterile neutrinos."],"supporting_citations":[{"why":"Establishes the bosonic analogue of the two-step diagonalization with crossing widths that the paper extends to the fermionic sterile-neutrino case.","marker":"[55]"},{"why":"Supplies the single-fermion Breit-Wigner propagator that is generalized here to multiple nearly degenerate fields.","marker":"[65]"},{"why":"Derives the resummed propagator for nearly degenerate heavy Majorana neutrinos, the basis of the second-step diagonalization.","marker":"[56]"},{"why":"Applies the non-Hermitian mass-matrix resummation to resonant leptogenesis, supporting the treatment of M−iΓ/2 as effective masses and widths.","marker":"[57]"},{"why":"Includes the γ5-dependent absorptive parts in the multi-fermion propagator, from which the paper drops subdominant terms to reach its simplified diagonal form.","marker":"[58]"},{"why":"The existing patched simulation of heavy neutrino-antineutrino oscillations that the present method replaces with unmodified generator output.","marker":"[12]"},{"why":"Provides the quantum-field-theory treatment of heavy neutrino-antineutrino oscillations whose propagator-to-exponential replacement the paper adapts for displaced vertices.","marker":"[42]"},{"why":"Introduces the dark-sector sterile-neutrino-portal model that produces a non-vanishing crossing width Γ23 in the reference calculation.","marker":"[63]"},{"why":"The automated event generator whose native width computation and output files the numerical examples rely on.","marker":"[66]"},{"why":"The model-file preparation tool used to input the four-spinor definitions of the diagonalized fields.","marker":"[64]"}],"fun_headline_variants":["Dummy fields unlock crossing-width simulation for sterile neutrinos","Simulate sterile-neutrino oscillations with a dummy-field subtraction trick","QFT phase reweighting gives flying distances for oscillating neutrinos","Sterile-neutrino crossing widths from dummy-field event generation","Pseudo-Dirac trick: simulate sterile neutrinos with stock tools"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The extraction of the crossing width rests on the asserted proportionality σ_{N′φ′→(N2+N3)→everything} ∝ Γ22+Γ33+2Γ23; if the interference between the two dummy resonances is not exactly linear in Γ23, the whole simulation loses its quantitative meaning.","fun_headline_variants_meta":{"raw":{"variants":["Dummy fields unlock crossing-width simulation for sterile neutrinos","Simulate sterile-neutrino oscillations with a dummy-field subtraction trick","QFT phase reweighting gives flying distances for oscillating neutrinos","Sterile-neutrino crossing widths from dummy-field event generation","Pseudo-Dirac trick: simulate sterile neutrinos with stock tools"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000203,"raw_usage":{"total_tokens":1355,"prompt_tokens":885,"completion_tokens":470,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":380}},"tokens_in":501,"tokens_out":470,"duration_ms":5033,"temperature":1.0,"reasoning_tokens":380,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:48:58.033455+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the same benchmark process, for example pp→W±*→μ±N→μμjj with a chosen Γ23/ΔM, by an independent diagrammatic calculation that resums the full 2×2 self-energy matrix, and compare the resulting LNV/LNC ratio and displaced-vertex distribution with the dummy-particle-generated events. A mismatch between Γ23 obtained from the subtraction formula and the value of Im[Σ23] computed directly from the Lagrangian would falsify the method.","supporting_citations":[{"cited_title":"Electroweak Precision Measurements of a Nearly-Degenerate $Z^\\prime$-$Z$ System","cited_arxiv_id":"2309.16794","evidence_quote":"Establishes the bosonic analogue of the two-step diagonalization with crossing widths that the paper extends to the fermionic sterile-neutrino case."},{"cited_title":"Fermion resonance in quantum field theory","cited_arxiv_id":"hep-ph/0611314","evidence_quote":"Supplies the single-fermion Breit-Wigner propagator that is generalized here to multiple nearly degenerate fields."},{"cited_title":"Heavy Neutrino-Antineutrino Oscillations in Quantum Field Theory","cited_arxiv_id":"2012.05763","evidence_quote":"Provides the quantum-field-theory treatment of heavy neutrino-antineutrino oscillations whose propagator-to-exponential replacement the paper adapts for displaced vertices."}],"review_version":1}