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Nearly degenerate dark scalars can produce dark matter through coscattering, avoiding the usual direct-detection and lepton-flavor exclusions.

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2026-08-04 09:46 UTC pith:AO2E25S5

load-bearing objection A credible, carefully done phenomenology paper on scalar coscattering in the inverse scotogenic model; the main caveat—vanishing dark-scalar self-couplings—is real but openly flagged by the authors. the 2 major comments →

arxiv 2510.13231 v2 pith:AO2E25S5 submitted 2025-10-15 hep-ph

Coscattering Dark Matter in the Inverse Scotogenic Models

Ang Liu , Zhi-Long Han , Fei Huang , Feng-Lan Shao , Wei Wang This is my paper
classification hep-ph PACS 95.35.+d12.60.-i
keywords dark mattercoscatteringscotogenic modelscalar dark matterHiggs portalYukawa portaldisplaced vertexnear-degenerate scalars
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 argues that the inverse scotogenic model, a framework that ties dark matter to tiny neutrino masses through loop processes, can produce the observed dark matter abundance by coscattering rather than ordinary thermal freeze-out. The key is to make the two lightest Z2-odd scalars nearly degenerate in mass, mφ1 ≲ mφ2, so the slightly heavier φ2 converts into φ1 through inelastic scattering off standard-model particles (φ2 + SM → φ1 + SM) while φ1 itself barely annihilates. The authors show this works through either the Higgs portal or the Yukawa portal, opening a wide parameter space: dark matter masses up to about 1.1 TeV in the Higgs case and 4 TeV in the Yukawa case, with mass splittings of a few GeV and small couplings that naturally escape existing bounds. If correct, a common-origin explanation of neutrino mass and dark matter need not be tuned to avoid lepton flavor violation and direct detection.

Core claim

In the inverse scotogenic model, when the two lightest dark scalars are nearly degenerate (mφ1 ≲ mφ2), the relic abundance of the lightest scalar φ1 is set not by φ1φ1 annihilation but by the conversion process φ1 + SM → φ2 + SM (coscattering) together with φ1φ2 → SM + SM coannihilation. Through the Higgs portal (λ2 ~ 1, λ12 ~ 10^-4, Δmφ ~ 1-10 GeV) the viable range for φ1 extends to about 1.1 TeV; through the Yukawa portal (y2 ~ 1, y1 ≲ 10^-4, Δmφ up to about 40 GeV) it extends to about 4 TeV. In both portals the required couplings are small enough to satisfy direct detection, Higgs-decay bounds, lepton flavor violation, BBN/CMB limits, and current collider constraints. The heavier partner

What carries the argument

The central object is the inelastic conversion process φ2 + SM → φ1 + SM, enabled by the small mass gap Δmφ ≡ mφ2 − mφ1. For coscattering to set the relic density, three conditions must hold at freeze-out: φ1 remains in kinetic equilibrium with the SM; no chemical potential develops for φ1 (φ2 annihilation is efficient); and the last reactions to decouple are the number-changing exchanges between φ1 and φ2. The coupled Boltzmann equations track both abundances; the near-degeneracy suppresses φ1 pair annihilation while the conversion keeps feeding φ1, so φ1 decouples early and its final abundance is determined by the depletion of the conversion reaction. The same near-degeneracy kinematically

Load-bearing premise

The dark scalar quartic self-couplings λ22, λ13, and λ31 are assumed to be exactly zero; if any of these is of order 0.1, the extra conversion processes they generate would shift or erase the coscattering parameter space claimed in the paper.

What would settle it

A future measurement or upper bound on the dark scalar quartic couplings (e.g., λ22 ≳ 0.1 generating φ2φ2 → φ1φ1) would shrink the coscattering window. A null result from displaced-vertex searches at the LHC/HL-LHC for the predicted Δmφ = 10 GeV parameter space (mφ2 ≲ 200 GeV, cτφ2 ~ 10^-2 to 10^4 m) would exclude the most promising Higgs-portal samples; similarly, DARWIN excluding λ1 ~ 10^-3 at mφ1 around 0.1-1 TeV would rule out the least-coupled regime.

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

If this is right

  • If the coscattering regime is correct, the dark matter mass is tied to the near-degeneracy: φ1 below roughly 1.1 TeV in the Higgs-portal case and 4 TeV in the Yukawa-portal case, with Δmφ of order 1-10 GeV.
  • The large coupling of the heavier partner (λ2 ~ 1 or y2 ~ 1) combined with a tiny φ1 coupling (λ1 or y1) naturally suppresses both direct-detection and lepton-flavor-violating signals.
  • The heavier scalar φ2 is long-lived and decays via three-body φ2 → φ1 f f, so the model predicts displaced-vertex events at the LHC/HL-LHC for Δmφ = 10 GeV, with additional sensitivity in dilepton-plus-missing-energy searches in the Yukawa case.
  • BBN constrains the longest-lived samples (τφ2 ≳ 50 s); for Δmφ = 1 GeV much of the Higgs-portal coscattering region is threatened unless the mass splitting is larger.
  • The two portals are experimentally distinguishable by the φ2 decay products: hadronic final states for the Higgs portal, leptonic final states for the Yukawa portal.

Where Pith is reading between the lines

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

  • If this picture is right, present-day annihilation of φ1 pairs is very slow (⟨σv⟩ ≲ 10^-28 cm^3/s for the Higgs portal and ≲10^-37 for the Yukawa portal), so indirect-detection experiments should see nothing; a null signal is the expectation.
  • A measurement of the φ1-φ2 mass splitting, for instance from displaced-vertex kinematics, would directly confirm the mechanism; finding a splitting above roughly 20-40 GeV would disfavor coscattering in favor of ordinary WIMP or coannihilation regimes.
  • The same near-degenerate two-scalar structure could be imported into other radiative neutrino-mass models; the essential ingredients are a small mass gap and a large coupling of the heavier state to the thermal bath.
  • The assumption of exactly vanishing scalar quartics is pivotal; if λ22 (the φ1^2 φ2^2 coupling) is instead O(0.1), conversion processes like φ2φ2 → φ1φ1 would dilute or destroy the coscattering phase, making an upper bound on these quartics a decisive test.

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

2 major / 6 minor

Summary. The paper studies scalar dark matter in the inverse scotogenic model, a variant of the scotogenic mechanism in which the Z2-odd sector contains two nearly degenerate real singlet scalars φ1, φ2 (plus a heavier φ3), a fermion doublet Ψ and a singlet χ. With φ1 as the lightest Z2-odd particle, the authors consider two limiting interaction structures: the Higgs-portal scenario (λ_i and λ_12 dominate) and the Yukawa-portal scenario (y_i dominate). In each case they write coupled Boltzmann equations for the co-evolution of the φ1 and φ2 number densities, including annihilation, coannihilation, conversion through inelastic scattering off the SM bath (φ2 SM→φ1 SM), and the long-lived decay φ2→φ1 f\bar{f}. They impose the Planck relic-density interval and identify three dynamical phases (coscattering, coannihilation, WIMP) as a function of the mass splitting Δm_φ and the relevant coupling. They then subject the surviving parameter space to constraints from LZ/DARWIN direct detection, invisible and undetected ATLAS Higgs decays, Fermi-LAT/H.E.S.S./CTA indirect detection, BBN and CMB limits on late decays, LFV (μ→eγ), and LHC/HL-LHC displaced-vertex and dilepton searches. The main quantitative claims are a Higgs-portal coscattering window for m_φ1 ≲ 1.1 TeV and a much wider Yukawa-portal window with m_φ1 up to about 4 TeV, with the displaced-vertex signature observable at part of the parameter space.

Significance. The paper is a carefully executed phenomenological study. Its main strength is the systematic treatment of a non-trivial multi-component Boltzmann system and the inclusion of the most relevant experimental and cosmological constraints. The authors are transparent about several approximations: the use of thermally averaged decay widths, the optimistic DV assumptions (zero background, unit efficiency, N_DV=3), and the O(10%) kinetic-equilibrium uncertainty in the Yukawa portal. If the central assumption of vanishing dark-sector scalar self-couplings can be justified or shown to be robust, the paper would establish that scalar coscattering in the inverse scotogenic model opens a broad, partly testable parameter space that is not excluded by current WIMP constraints. The cross-section and constraint evaluation rely on well-established tools (micrOMEGAs, MadGraph) and are reproducible in principle, though no code is provided. The identification of distinct collider/cosmological signatures for the Higgs vs. Yukawa portals is a useful contribution.

major comments (2)
  1. [Section II, Eq. (1) and following text] The assumption λ22=λ13=λ31=0 is load-bearing but is introduced 'for simplicity' without symmetry or naturalness justification. These φ4 and φ3 terms are allowed by the Z2 symmetry and, if O(0.1), would add s- and t-channel contributions to the conversion processes φ2φ2→φ1φ1, φ2φi→φ1φj, which are currently mediated only by the Higgs or the Yukawa exchanges. Since the relic density in the coscattering regime is set by a balance between conversion, annihilation, and coannihilation, turning on these quartics will shift the phase boundaries in Figures 2 and 8 and the relic-density contours in Figures 3 and 9; the paper itself notes that including them 'would contribute to the conversion processes, thus weakening the effect of the Higgs portal interaction.' Please either provide a symmetry or a quantitative estimate of how large these couplings can be before the claimed coscattering windows an
  2. [Section IV.A, after Eq. (20)] The authors acknowledge that in the Yukawa-portal coscattering regime φ1 is not guaranteed to be in kinetic equilibrium and that this gives 'O(10%) difference in the final relic density compared to the complete result [50].' The standard equilibrium Boltzmann equations are nevertheless used to set the relic-density constraint, which determines y1 and hence the phase classification in Figures 8 and 9. A 10% systematic error in the relic density corresponds to an O(10%) shift in the required y1, which can move the coscattering/coannihilation boundaries and change the extracted m_φ1 reach. Please assess the impact of this approximation on the claimed Yukawa-portal reach, e.g., by recomputing at least the benchmark points with the full treatment of Ref. [50] or by showing that the phase boundaries shift by less than the quoted uncertainties.
minor comments (6)
  1. [Abstract] 'alerts the predictions' should be 'affects the predictions'; 'contrastive coannihilation channel' should be 'contrasting coannihilation channel'.
  2. [Section II, Eq. (1)] The potential also contains λ14/12 φ1^4 and λ24/12 φ2^4; the text says 'vanishing self-interactions' but only sets λ22=λ13=λ31=0. Please clarify whether λ14 and λ24 are also set to zero and, if not, whether they affect the relic-density calculation.
  3. [Sections III.C and IV.C] The displaced-vertex sensitivity is estimated under the optimistic assumptions of zero background and unit detection efficiency; the resulting reach is therefore an upper bound. The paper should state this caveat more prominently in the conclusions.
  4. [Section IV.B] The neutrino-mass generation is not checked for the scanned benchmarks. Since the model is scotogenic, it would be helpful to state that the chosen y_i can be embedded in a flavor structure satisfying the observed neutrino masses and mixing (e.g., through the mixing matrix ξ in Eq. (3)).
  5. [Eqs. (23)-(24)] The Fermi constant G_f and the loop function G(a) use the same letter; please use G_F for clarity.
  6. [Eq. (8)] The notation θ′(x) for the Heaviside function is nonstandard; use θ(x) or H(x).

Circularity Check

0 steps flagged

No significant circularity: relic density is imposed as an external constraint, couplings are fitted to it, and the predicted collider/cosmological signatures are derived downstream; self-citations are background only.

full rationale

The paper's central calculation is a standard freeze-out model-building exercise rather than a disguised fit. The relic density is explicitly imposed as an external input: "The DM relic density is required within the 3σ range of the Planck result [3], i.e., Ωφ1h² ∈ [0.117,0.123]". The scan then fixes λ12 or y1 to satisfy this bound, which is not called a prediction. The subsequent constraints (LZ, ATLAS, Fermi-LAT, BBN/CMB, displaced-vertex rates) are evaluated on the surviving parameter points, so they are independent falsifiable checks rather than inputs to the fit. The cross sections and decay widths come from external codes and standard formulas (micrOMEGAs, MadGraph), and the model and coscattering mechanism are attributed to external references ([68], [49,50], [55]). The self-citations ([23]-[27], [35], [36], [62]) appear only in the literature review and are not load-bearing for the derivation. The one notable weakness is the unmotivated assumption λ22=λ13=λ31=0, which is disclosed "for simplicity" with the explicit acknowledgment that including these quartics would weaken the chosen portal; this is an assumption that limits robustness, not an input-output circularity. No equation reduces to its own input, and no fitted parameter is renamed as a prediction.

Axiom & Free-Parameter Ledger

8 free parameters · 8 axioms · 1 invented entities

The paper inherits the inverse scotogenic model and standard thermal cosmology from the literature. Its own choices are the vanishing scalar self-couplings, the large Δm_F benchmark, small λ1/y1 scans, and optimistic displaced-vertex counting. The relic density is imposed as a constraint and fixes λ12/y1 rather than being predicted independently.

free parameters (8)
  • m_phi1 = scanned 10-2000 GeV (Higgs portal), 10-5000 GeV (Yukawa portal)
    Dark matter mass; the central scan variable in all phase diagrams.
  • Delta_m_phi = benchmarks 1 GeV and 10 GeV; phase diagrams up to ~72 GeV
    Mass splitting between φ2 and φ1; determines coscattering vs coannihilation and the φ2 lifetime.
  • lambda_1 = scanned 10^-4 - 1; fixed at 10^-3 for phase diagrams
    Higgs-portal coupling of the dark matter φ1; controls direct-detection scattering.
  • lambda_2 = benchmarks 1 and 0.1; coscattering requires lambda_2 ≳ 0.4
    Higgs-portal coupling of the heavier scalar φ2; controls annihilation and conversion rates.
  • lambda_12 = chosen to satisfy Ω h² ∈ [0.117,0.123]; O(10^-4) in coscattering, up to O(0.1) in coannihilation
    Mixed φ1-φ2 Higgs-portal coupling; effectively fitted to the relic density.
  • y1 = scanned 10^-6 - 1; coscattering typically y1 ≲ 6×10^-4
    Effective Yukawa coupling of φ1; set by relic density and constrained by LFV and φ2 lifetime.
  • y2 = benchmarks 1 and 0.3; coscattering requires y2 ≳ 0.4
    Effective Yukawa coupling of φ2; controls φ2 annihilation and conversion.
  • Delta_m_F = scanned 100-2000 GeV; benchmark 1000 GeV
    Mass splitting between the doublet fermion and φ2; chosen large to suppress Ψ conversion interference.
axioms (8)
  • domain assumption The inverse scotogenic particle content and Z2 symmetry are adopted as the model, including φ1, φ2, φ3, Ψ, χ.
    Section II; the model is taken from Ref. [68] and not re-derived.
  • standard math Standard Friedmann cosmology with g_s, g_*, and Y_SM = 0.238 is used in the Boltzmann equations.
    Section III A; standard thermal freeze-out treatment.
  • ad hoc to paper Dark scalar self-interactions vanish: λ22 = λ13 = λ31 = 0.
    Section II; suppresses conversion channels that would compete with the chosen portals.
  • ad hoc to paper The scalar mass hierarchy m_φ1 ≲ m_φ2 ≪ m_φ3 holds.
    Section II; makes φ3 irrelevant for dark matter phenomenology.
  • ad hoc to paper The doublet-singlet fermion mixing is small, so m_F = m_ψ± ≈ m_ψ0.
    Section II, after Eq. (2); simplifies the fermion sector.
  • domain assumption φ1 is assumed to remain in kinetic equilibrium with the SM; the Yukawa-portal case is acknowledged to violate this and introduce O(10%) relic-density uncertainty.
    Section IV A and Fig. 7 discussion; the paper models relic density without a full solution of kinetic decoupling.
  • domain assumption The nucleon matrix element constant C ≈ 0.3 in the direct-detection cross section.
    Eq. (11); standard nuclear input, not derived in the paper.
  • ad hoc to paper Displaced-vertex sensitivity assumes zero background, unit detection efficiency, and N_DV = 3 for exclusion.
    Eqs. (17) and (28), Section III C and IV C; labeled by the authors as an optimistic estimate.
invented entities (1)
  • Heavier Z2-odd scalar partner φ2 independent evidence
    purpose: Sustains coscattering via φ2 SM → φ1 SM and provides long-lived displaced-vertex signatures through three-body decays.
    φ2 is not invented in this paper — it is part of the inverse scotogenic model of Ref. [68] — but the paper makes falsifiable predictions for its lifetime and production cross section that are not used to fit the couplings.

pith-pipeline@v1.3.0-alltime-deepseek · 32493 in / 15787 out tokens · 139426 ms · 2026-08-04T09:46:53.921240+00:00 · methodology

0 comments
read the original abstract

The Scotogenic mechanism is an appealing pathway to naturally explain the common origin of dark matter and tiny neutrino mass. However, the conventional scotogenic dark matter usually suffers stringent constraints from the non-observation of lepton flavor violation and direct detection. To generate the non-zero neutrino masses, at least two generations of dark particles are required. For example, two real scalar singlets $\phi_1$ and $\phi_2$ are involved in the inverse scotogenic model, which are odd under the $Z_2$ symmetry. In this paper, we consider the masses of dark scalars are nearly degenerate $m_{\phi_1}\lesssim m_{\phi_2}$, which opens new viable pathway for the generation of dark matter $\phi_1$, such as the coscattering process $\phi_1\text{SM}\to \phi_2 \text{SM}$ and coannihilation processes $\phi_1 \phi_2 \to \text{SM~SM}$ via the Higgs portal or Yukawa portal interactions. We explore the parameter space to produce the correct relic density through coscattering, as well as the contrastive coannihilation channel. We then comprehensively study the constraints of dark matter from Higgs decay, direct detection, and indirect detection. For the heavier dark scalar, the three-body decay $\phi_2\to\phi_1 f\bar{f}$ not only alerts the predictions of big bang nucleosynthesis and cosmic microwave background, but also leads to the observable displaced vertex signature at colliders.

Figures

Figures reproduced from arXiv: 2510.13231 by Ang Liu, Fei Huang, Feng-Lan Shao, Wei Wang, Zhi-Long Han.

Figure 1
Figure 1. Figure 1: FIG. 1. The evolutions of various abundances [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Freeze-out phase diagrams in the parameter spaces of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Direct detection and ATLAS constraints on the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Constraints from the indirect detection experiments in the Higgs portal scenario. Panels (a)-(d) represent the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. CMB and BBN constraints on the dark partner [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The sensitive region of the one DV signature at LHC (gray solid lines) and future HL-LHC (gray dashed lines) [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Coscattering (a) and coannihilation (b) benchmarks for the Yukawa portal scenario. The thermal rates in [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Freeze-out phase diagrams in the parameter spaces of [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Sample distribution in the [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Constraints from the indirect detection experiments in the Yukawa portal scenario. The gray solid, black [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The sensitive region of the one DV signature at LHC (gray solid lines) and future HL-LHC (gray dashed [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The sensitivities of dilepton signature [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗

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