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REVIEW 4 major objections 7 minor 97 references

Simulations of the Sterile Neutrino Oscillations with a Crossing-Width Term

T0 review · 4 major / 7 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict A genuinely useful simulation trick with a load-bearing extraction that is asserted, not proven, and thin numerical validation. read the letter →

arxiv 2412.19220 v2 pith:P4PFJPAD submitted 2024-12-26 hep-ph hep-th

Qiankang Wang , DaYun Qiu , Yi-Lei Tang This is my paper · ORCID
classification hep-phhep-th
keywords sterileneutrinooscillationspseudo-Diraccrossing-widthleptonnumberviolationdisplacedvertexeventgenerationquantumfieldtheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 7 minor

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.

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 (4)
  1. [Section VI.A, Eqs. (66)-(69)] 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.
  2. [Section VII] 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.
  3. [Section IV, Eqs. (29)-(30)] 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.
  4. [Section V, Eqs. (33)-(42)] 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.
minor comments (7)
  1. [Section III, text near Eq. (8)] 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.
  2. [Abstract and headings] There are typos: "Feynmann" in the abstract and "THe" in several section headings; please proofread.
  3. [Section V, Eq. (54)] 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).
  4. [Figures 2 and 3] 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.
  5. [Section VI.A, Eq. (68)] 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.
  6. [Section VI.B, Eq. (71)] 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.
  7. [Section VII] Several parameter planes are called "plain" instead of "plane"; also, no MadGraph/FeynRules model files or event-generation scripts are provided, which limits reproducibility.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the simulation chain is self-contained, with the main caveat being an unproven linear relation in the dummy-particle extraction of Γ23, which is a correctness risk rather than a circular reduction.

full rationale

The paper's derivation chain is not circular. The model Lagrangian fixes masses and couplings; the first diagonalization gives N2,N3; the widths Γ22, Γ33, Γ23 are defined from self-energies in Eq. (18), i.e., Γij = 2ImΣij(m); the second-step diagonalization of M − iΓ/2 in Eqs. (27)-(28) is a standard non-Hermitian mass-matrix rotation; and the oscillation probabilities (59)-(64) and the displaced-vertex recipe follow from the resummed propagator by an explicit QFT calculation. The final predicted observables (RLNV/LNC and the |Δx| distributions) are different quantities from the intermediate dummy-particle cross sections in Section VI.A, so the dummy trick is a parameter-extraction shortcut, not a fit of the target result to itself. The load-bearing weak point is Eq. (68), which asserts σ_{N'φ'→(N2+N3)→everything} ∝ Γ22 + Γ33 + 2Γ23 without derivation; Eq. (71) shows ordinary Breit-Wigner interference depends on Δm and the diagonal widths, so Eq. (68) is at best an unproven first-order identification. This is a missing proof or a possible inconsistency, not a circular step. Self-citations (Refs. [55] and [63]) provide a bosonic analogue and a model Lagrangian but are not load-bearing: the fermionic propagator derivation, the two-step diagonalization, and the simulation recipes are new and independently checkable. Overall, no step reduces by construction to its own inputs; the score of 2 reflects the minor non-load-bearing self-citation and the flagged unproven assumption.

Assumptions & free parameters 4 free parameters · 4 assumptions · 1 invented entities

The central claim rests on several modeling choices: a simplified seesaw benchmark, approximations in the resummed propagator, the macroscopic-distance limit in the oscillation formalism, and the treatment of non-self-conjugate fields as Majorana in event generators. The dummy particles are pure computational artifacts. The free parameters are either controllable input ratios (R_DM_SM, temporary widths) or ad hoc auxiliary couplings. None of these are fitted to experimental data, but several are chosen by hand to make the numerical examples work.

free parameters (4)
  • |yχD| = Adjusted per panel to realize R_DM_SM = 0.1, 0.5, 1, 5
    The dark-sector Yukawa coupling is tuned so that the ratio Γ23_DM / ΔM takes the prescribed values in the numerical examples of Section VII.
  • R_DM_SM = 0.1, 0.5, 1, 5
    Controlled input ratio of the dark-sector crossing width to the mass splitting, chosen by hand to illustrate the algorithm; it is not independently measured.
  • Temporary width Γ' = 0.005 mA to 0.05 mA
    Artificial width used to avoid numerical instability around narrow poles in the cross-section rescaling trick of Section VI.B.
  • Dummy couplings y'_i = Unspecified
    Ad hoc Yukawa couplings for the dummy particles N' and φ' in Eq. (66); they cancel in the cross-section ratio used to extract Γ23.
assumptions (4)
  • domain assumption Approximate U(1)L symmetry with m'_D = 0 and mixing only with νμ
    Simplifies the mass matrix to Eq. (6); this is a benchmark choice, not a general property of the model.
  • ad hoc to paper Neglect of Σ5, B^2, and δM Γ5 terms in the resummed propagator
    Used in Section IV to reduce the propagator to i(/p - (M - iΓ/2))^{-1}; validity requires small off-diagonal γ5 terms and near-degeneracy.
  • domain assumption Macroscopic Δx and nearly on-shell approximations in the oscillation derivation
    In Section V, the derivation requires |Δx| much larger than the inverse momentum spread and p^2 approximately equal to m_i^2, so the propagator becomes an oscillating exponential.
  • ad hoc to paper N'_i treated as Majorana spinors in event generators despite non-self-conjugacy
    After the second diagonalization, N'_i are not self-conjugate according to Eq. (30), but the authors instruct that they be input as Majorana fields for consistency with the diagonalized propagator.
invented entities (1)
  • Dummy fermion N' and dummy scalar φ'
    purpose: To compute the crossing width Γ23 using standard event generators through three cross-section measurements
    Introduced in Section VI.A with artificial Yukawa couplings in Eq. (66); they are not physical particles and have no external experimental evidence.

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Pith. "Pith review of Simulations of the Sterile Neutrino Oscillations with a Crossing-Width Term." pith.science (2026). https://pith.science/paper/P4PFJPAD

@misc{pith2026241219220,
  author       = {Pith},
  title        = {Pith review of: Simulations of the Sterile Neutrino Oscillations with a Crossing-Width Term},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P4PFJPAD}},
  note         = {Machine review of arXiv:2412.19220}
}
read the original abstract

In this paper, we present an algorithm to generate the collider events of the GeV-scale oscillating sterile neutrinos with the ready-made event generation tools in the case that the crossing-widths among the nearly-degenerate fermionic fields arise. We prove the validity of our algorithm, and adopt some tricks for practical calculations. The formulations of the particle oscillation processes are also improved in the framework of the quantum field theory, offering us the ability to simulate the flying distances of the oscillating intermediate sterile neutrinos while regarding them as the internal lines in the Feynmann diagrams.

Figures

Figures reproduced from arXiv: 2412.19220 by the authors.

Figure 1
Figure 1. FIG. 1. (Crossed) self-energy diagram connecting the [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Probabilities of the oscillations between the [PITH_FULL_IMAGE:figures/full_fig_p027_4.png]

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