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REVIEW 4 major objections 5 minor 72 references

In right-handed neutrino portal freeze-in, ignoring dark-sector internal dynamics underestimates the final dark matter abundance by 30% or more than 95%.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-03 17:01 UTC pith:HIKUQZ3Y

load-bearing objection The 30%/95% freeze-in error claim is plausible but unverifiable as written; the stress-test's number-conservation objection misses the N1N1→χχ channel, so the worry is not fatal, but the paper needs code or explicit matrix elements to be trustworthy. the 4 major comments →

arxiv 2512.10762 v2 pith:HIKUQZ3Y submitted 2025-12-11 hep-ph

Electroweak right-handed neutrino portal dark matter

Wan-Zhe Feng , Ao Li , Zong-Huan Ye , Zi-Hui Zhang This is my paper
classification hep-ph
keywords dark matterright-handed neutrino portalfreeze-infreeze-outType-I seesawrelic abundanceBoltzmann equationsheavy neutral leptons
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that in freeze-in dark matter production through an electroweak right-handed neutrino portal, the internal interactions among dark-sector particles materially change the predicted relic density. Solving the full coupled Boltzmann equations — including a separate dark-sector temperature — gives a final dark matter abundance that is roughly 30% higher than the common shortcut of treating each dark species as an independent freeze-in component and then adding late decays, and more than 95% higher when the three-point decay N↔χφ is forbidden. The authors construct three concrete Type-I seesaw realizations (regular, structure cancellation, and split seesaw) whose parameters are fixed by neutrino oscillation data and experimental constraints, and they show that a fermion–scalar dark sector can reproduce the observed relic density through either secluded freeze-out or freeze-in. The upshot is that relic-density calculations in portal freeze-in models should not decouple the dark species; the coupled system is required.

Core claim

The paper claims that in the right-handed neutrino portal freeze-in framework, internal dark-sector interactions substantially alter the final DM relic abundance. When the dark sector reaches internal thermal equilibrium, one must introduce a hidden-sector temperature T_h distinct from the visible-sector temperature T_v and evolve it together with the comoving number densities. The authors compare their full multi-temperature treatment with the approximate two-step procedure standard in parts of the literature — first compute freeze-in for each species separately, then add late decays — and find that the approximation underestimates the final abundance by about 30% when the three-point proce

What carries the argument

The central machinery is the set of coupled Boltzmann equations for the comoving number densities Y_χ, Y_φ, and Y_Ni, supplemented by the temperature ratio η = T_v/T_h when the dark sector self-thermalizes. The portal is the renormalizable Yukawa y_x χφN, which after seesaw mixing becomes a sum over neutrino mass eigenstates. The three seesaw benchmarks (regular, structure cancellation, and split) provide numerically fixed portal couplings derived from neutrino data, and the full collision terms include all relevant two-to-two, one-to-two, and internal dark-sector processes with real-intermediate-state subtraction.

Load-bearing premise

The dark sector is assumed to couple to the Standard Model only through the right-handed neutrino portal Yukawa y_x χφN; if any additional coupling exists — Higgs portal, kinetic mixing, a light dark photon, or extra Yukawas — the freeze-in/freeze-out yields and the quoted 30%/95% error figures would change.

What would settle it

For Model d (Benchmark-SS1) and Model e (Benchmark-SS2), compute the relic density with an independent Boltzmann solver that includes all internal dark-sector interactions and compare to the paper's full values (Ωχh²=0.12 in both cases). If the independent-species treatment with late decays also gives 0.12 within a few percent for Model d, or if removing the internal interactions changes the abundance by less than a few percent, the paper's 30%/95% discrepancy claim would be falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Freeze-in relic densities in right-handed neutrino portal models must be computed from the coupled system; the independent-species approximation is not reliable when the dark sector self-thermalizes.
  • The paper's benchmarks show that both freeze-out and freeze-in can yield Ωχh²≈0.12 while satisfying current neutrino and collider constraints, giving concrete targets for LHC and future heavy-neutral-lepton searches.
  • In Case-SC, the O(10^-2) portal couplings make the scenario particularly accessible to collider searches and neutrino experiments, so a positive signal would tie dark matter directly to seesaw neutrino mass generation.
  • When the three-point channel N↔χφ is absent, the error from the approximate treatment exceeds 95%, meaning that any study using the shortcut in such models should be revisited.
  • The multi-temperature treatment used here provides a template for other hidden-sector models where the dark sector has its own temperature and internal equilibrium.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same failure of the independent-species approximation likely occurs in other freeze-in portal constructions (Higgs portal, dark photon portal) with self-interacting dark sectors; the magnitude of the bias should scale with the strength of internal number-changing versus number-conserving interactions.
  • A dark sector that reaches internal equilibrium during freeze-in will have a non-thermal or multi-temperature momentum distribution for the dark matter, which could leave imprints in structure formation (e.g., Lyman-α forest) or in the effective number of relativistic species; these are testable with upcoming surveys.
  • The >95% error when N↔χφ is forbidden suggests that the underestimate is driven by the absence of a channel that directly transfers abundance between dark species; mapping this error as a function of the mass splitting m_φ−m_χ and the coupling y_x would give a practical diagnostic for when the shortcut can be used.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies dark matter in a right-handed neutrino portal framework, with a dark fermion χ and complex scalar φ coupled to the SM only through Yukawa interactions with right-handed Majorana neutrinos. Three seesaw realizations (regular, structure-cancellation, split seesaw) are constructed, and Particle Swarm Optimization is used to fit seesaw parameters to neutrino data. The authors solve coupled Boltzmann equations for the dark and neutrino sectors for both freeze-out and freeze-in production, and claim that a common treatment of freeze-in—treating each dark species as an independent freeze-in component and adding late decays afterward—underestimates the final relic abundance by ~30% when N↔χφ is open and by >95% when it is closed. The paper discusses experimental constraints, benchmark models, and the multi-temperature hidden sector formalism.

Significance. If the 30%/95% claim is correct, it provides a useful cautionary result for freeze-in calculations in neutrino portal models, where internal dark sector dynamics can significantly modify the relic density. The systematic construction of viable seesaw benchmarks via PSO is a contribution, and the multi-temperature formalism is published previously. However, the central quantitative claim is not fully reproducible from the manuscript: the approximate procedure is not specified algorithmically, the matrix elements and cross-sections are not given in closed form, and no code is provided. The paper also helpfully identifies that dark-number-changing processes involving the portal neutrino must be included, which resolves a potential paradox in Model d.

major comments (4)
  1. [§4.3.2, Models d and e] The 'common approximate procedure' is described only in prose and never written as an algorithm or equations. The central 30%/95% claim is defined as the difference between this procedure and the full solution, so the procedure must be specified precisely: the list of production processes for each species, how LH→χϕ is apportioned between χ and ϕ, when decays are applied, and which reverse/annihilation processes (e.g., χχ→N1N1) are omitted. Without this, the comparison is not well-defined and cannot be checked.
  2. [§4.2, Eqs. (41)-(52)] The Boltzmann equations are written in terms of thermally averaged cross-sections ⟨σv⟩, decay widths ⟨Γ⟩, and J-functionals whose explicit forms are not given; only references to Feynman diagrams are provided. Since the 30%/95% result is a numerical output of a private solver, the authors should provide the code, the full matrix elements, or at least a validation against a simpler limit. As it stands, the readers cannot independently reproduce the central numerical claim.
  3. [§4.3.2, Model d] In Model d, M1 < mχ + mϕ, so N1→χϕ is closed, but the text states that χχ→N1N1 and ϕϕ→N1N1 occur and N1 subsequently decays to SM particles. These processes do not conserve the total dark number, so the final χ yield is not simply the total dark number produced by LH→χϕ. This should be stated explicitly when comparing with the approximate procedure, and the contribution of N1-mediated channels to the >95% error should be quantified. Otherwise readers may incorrectly infer a number-conservation paradox.
  4. [Table 2 and Fig. 8] The 30%/95% error is demonstrated only for two tuned benchmark points where yx is adjusted so the full calculation gives Ωχh²=0.12. The abstract and conclusion present these numbers as general statements. Please show the error as a function of yx over the viable parameter range, and ideally for more freeze-in benchmarks, to establish that the effect is not an artifact of the selected points. Also clarify that the 'reproduction' of Ω=0.12 is a fit to yx, not a parameter-free prediction.
minor comments (5)
  1. [Throughout] Typos and grammar: 'Majorara' (§2.3.1), 'diaganolization' (App. B), 'šCsterile' (Fig. 3 caption), 'the eventually decrease' (§4.3.2), missing space in 'electroweakright-handedneutrinoportaldarkmatterprovidesarobustframework' (Conclusion).
  2. [Eq. (35)] The definition of yx'_i is ambiguous, as the index contraction with the seesaw matrix U is not written explicitly. Please define yx'_j = Σ_i yx_i U_{i+3,j} (or the equivalent) to clarify which mass eigenstates couple to the dark sector.
  3. [References] Refs. [37] and [63] have placeholder-looking DOIs ('10.1103/vshn-gbdp' and '10.1103/df3g-32t9') and Ref. [38] is 'to appear'. These should be corrected or updated before publication.
  4. [Appendix H, Eq. (128)] The equation contains a garbled expression 'ρ dρh/dTh' that is likely a typo for 'dρh/dTh'. The surrounding derivation should be proofread.
  5. [§4.3.2, Model d paragraph] The sentence 'since the right-handed neutrino N1 cannot decay into dark particles, the freeze-in contribution from N1→χϕ is absent. As a result, the approximate treatment deviates strongly...' is logically unclear. The deviation arises from neglecting internal dark-sector interactions and the treatment of N1 dynamics, not merely from the absence of N1→χϕ. Rephrase for clarity.

Circularity Check

0 steps flagged

No significant circularity: the 30%/95% claim is a controlled Boltzmann-scheme comparison at fixed couplings; relic-density matching is an acknowledged benchmark fit, not a prediction.

full rationale

The central quantitative claim is a comparison between two solution schemes at fixed benchmark parameters: the full coupled multi-temperature Boltzmann system (Eqs. 48-51, Appendix H) and an approximate independent-component plus late-decay procedure. The approximate treatment is an external approximation rather than a rearrangement of the full equations, so the 30%/95% discrepancy is a computed numerical result, not a relation true by construction. The benchmark models do contain a fitted element: yx is chosen so that the full calculation returns Ωχh^2 ≈ 0.12, as acknowledged in the freeze-out table caption ('Parameters are chosen to satisfy ... the observed abundance of the dark matter') and visible in the Ωχh^2 column of Table 2. This makes 'reproducing the observed relic abundance' a benchmark-matching exercise rather than an independent prediction, but it does not infect the 30%/95% comparison, because both numbers are evaluated with the same chosen couplings. The hidden-temperature formalism is drawn substantially from the authors' prior work (Refs [33-37]), but Appendix H re-derives the relevant equations, and the method does not assume or contain the 30%/95% result. No equation in the paper is set equal to its target by construction; no uniqueness theorem is imported; and no fitted parameter is disguised as a prediction. The approximate procedure is described in prose rather than as an explicit algorithm, which is a reproducibility concern but not evidence of circularity. No significant circularity found.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 2 invented entities

The calculation rests on standard seesaw physics plus a self-cited multi-temperature formalism and the pure-portal assumption. The DM abundance is fit by tuning yx and masses; the seesaw parameters are fit to neutrino data. The paper's genuinely new quantitative claim (30%/95% error) is computed but not independently verifiable from the text alone.

free parameters (4)
  • Dark Yukawa coupling yx = e.g., 0.433 (FO Model a), 2.2e-6 (FI Model a), 0.29 (SS1), 1.7e-12 (SS3)
    Adjusted in Tables 1 and 2 so that the calculated Ωχh² equals the observed 0.12 (or the combined 0.12 in Model f). This is a fit to the relic-density target, not a prediction.
  • Dark sector masses mχ, mφ = e.g., (210,350), (240,360), (440,450) GeV
    Benchmark mass points chosen by hand; they set kinematics for decays and annihilations.
  • Seesaw input parameters (yν_ij, Mi) for Benchmarks RS/SC/SS = Appendix I; e.g., Benchmark-RS: yν entries 1e-7–4.6e-7, Mi = 200,220,260 GeV
    Fitted by Particle Swarm Optimization to reproduce |U_PMNS| (Eq. 30), mass-squared differences (Eqs. 32–33), mass-sum bound (Eq. 31), and for Case-SC non-unitarity bounds (Appendix D). Fitted to independent data.
  • Case-SC fine-tuning parameter ε = 6.34546e-10
    Tiny parameter in the structure-cancellation matrix Eq. (19), chosen so that an otherwise rank-1 Yukawa structure generates the observed tiny neutrino masses. Ad hoc small parameter.
axioms (5)
  • standard math Type-I seesaw mixing: light neutrino mass matrix mν ≈ -MD MN^{-1} MD^T via Schur-complement diagonalization (Eq. 24)
    Standard seesaw mechanism, used as background throughout Section 2.
  • domain assumption Multi-temperature hidden-sector Boltzmann formalism (Eqs. 48–52, Appendix H) from Ref [33] is valid
    The coupled evolution with hidden temperature Th is taken from the authors' own previous work (Ref [33]) and applied without re-derivation. Published, peer-reviewed, but not independently reproduced here.
  • ad hoc to paper Dark sector has no additional couplings to the SM besides the right-handed neutrino portal
    Section 3 assumes the dark U(1) gauge boson is heavy/decoupled and the evolutions of χ, φ are purely portal-driven. If other portals exist, the calculated relic densities and the 30%/95% error figures change.
  • domain assumption Classical Maxwell-Boltzmann statistics and standard thermally-averaged cross sections are used in all collision terms
    Standard in DM Boltzmann solvers; quantum statistics and statistical factors are neglected, which can shift freeze-in yields at the O(1–10%) level.
  • domain assumption The 'common approximate procedure' (independent freeze-in components + late decays) is representative of the literature
    The paper critiques this approximation but does not cite a specific reference that uses it, making the exact target of the 30%/95% comparison difficult to verify.
invented entities (2)
  • Dark fermion χ no independent evidence
    purpose: Dark matter candidate; stable under a dark Z2/U(1); couples to right-handed neutrinos via yx χφN.
    Mass and coupling yx are benchmark parameters fitted to the relic density; no independent collider or astrophysical signature is computed for χ itself.
  • Dark complex scalar φ no independent evidence
    purpose: Partner for χ in the portal Yukawa; heavier than χ; decays to χ+ν before BBN or contributes to DM if long-lived.
    Its mass, coupling, and lifetime are chosen to satisfy benchmark conditions; no independent evidence is provided.

pith-pipeline@v1.3.0-alltime-deepseek · 37472 in / 14235 out tokens · 145085 ms · 2026-08-03T17:01:33.361363+00:00 · methodology

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read the original abstract

We study dark matter coupled to the Standard Model via electroweak scale right-handed neutrinos in a Type-I seesaw framework. We consider a minimal dark sector containing a fermion $\chi$ and a complex scalar $\phi$ whose only connection to the Standard Model is through renormalizable Yukawa interactions with right-handed Majorana neutrinos, thus realizing a neutrino portal after seesaw mixing. We discuss three representative realizations of electroweak right-handed neutrinos arising from the Type-I seesaw mechanism, spanning small, tiny, and ultraweak couplings to the Standard Model sector, so that the dark particles can either undergo secluded freeze-out or be produced via freeze-in. Instead of merely estimating the order of magnitude of the seesaw couplings, we use the Particle Swarm Optimization algorithm to obtain viable seesaw parameter sets consistent with neutrino data and other constraints, and then compute the coupled evolution of the dark particles and right-handed neutrinos, reproducing the observed dark matter relic abundance in representative benchmark scenarios. For freeze-out, dark matter depletion is controlled by coupled dark sector dynamics, requiring a full Boltzmann treatment for a reliable relic abundance. For freeze-in, internal dark interactions also alter the relic density: treating hidden particles as independent components with late decays added afterward can misestimate the abundance by $30\%$ or even $95\%$, depending on the interaction structure. Electroweak right-handed neutrino portal dark matter thus provides a robust and predictive framework that tightly connects neutrino physics, heavy neutral lepton phenomenology, and the cosmological dark matter relic density, offering a well-motivated benchmark for complementary collider, neutrino, and cosmological probes at the high energy frontier.

Figures

Figures reproduced from arXiv: 2512.10762 by Ao Li, Wan-Zhe Feng, Zi-Hui Zhang, Zong-Huan Ye.

Figure 1
Figure 1. Figure 1: Exclusion limits at the 95% confidence level on the active-sterile mixing parameter [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Exclusion limits at the 95% confidence level on the active sterile mixing parameter [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Exclusion limits at the 95% confidence level on the activešCsterile mixing parameter [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A schematic illustration of the dark sector evolution via freeze-out. At high [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: An exhibition of the dark sector freeze-out evolution for Models [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: For Case-RS and SC, the right-handed neutrinos reach thermal equilibrium during [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: For Case-SS, the portal right-handed neutrino [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: An exhibition of the freeze-in evolution for Model [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Relevant Feynman diagrams for 2 → 2 scattering processes are shown. The first row depicts dark particle χ, ϕ annihilation into light or heavy neutrinos after seesaw mixing; the inverse processes lead to freeze-in production of the dark sector particles. The second row shows internal dark sector interactions mediated by light or heavy neutrinos. The third row illustrates processes arising from χϕ → LH after… view at source ↗
Figure 10
Figure 10. Figure 10: Feynman diagrams of the heavy neutrino N1,2,3 decaying to dark sector particles χ, ϕ, and to SM particles. 42 [PITH_FULL_IMAGE:figures/full_fig_p042_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Feynman diagram of ϕ decay to χ and light (always present) or heavy (if mϕ > mχ + mNi ) neutrinos. where g v eff (g h eff) is the visible (hidden) effective degrees of freedom. Further one can deduct the derivative of ρv, ρh w.r.t. Th ρ dρh dTh = [PITH_FULL_IMAGE:figures/full_fig_p043_11.png] view at source ↗

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