Singlet-doublet dark matter beyond freeze-out

Aaron Pierce; Alexander Takla; Prudhvi N. Bhattiprolu

REVIEW 2 major objections 6 minor 96 references

Singlet-doublet dark matter beyond freeze-out

T0 review · 2 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper shows that in the small-Yukawa-coupling regime of the singlet-doublet dark matter model, the relic abundance is set by co-scattering, freeze-in, or SuperWIMP processes, requiring two coupled Boltzmann equations rather than the…

desk verdict Solid, useful map of the small-Yukawa singlet-doublet DM parameter space; the central chemical-equilibrium comparison is right, but a key kinetic-equilibrium estimate is deferred to an unpublished companion. read the letter →

arxiv 2608.12480 v1 pith:AATBIHAE submitted 2026-08-12 hep-ph

Prudhvi N. Bhattiprolu , Aaron Pierce , Alexander Takla This is my paper · ORCID

classification hep-ph

keywords singlet-doubletdarkmatterco-scatteringfreeze-inSuperWIMPmechanismrelicabundanceBoltzmannequationscollidersignatures

topics Dark Matter

open problems Dark Matter

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 argues that in the Singlet-Doublet Majorana dark matter model with tiny Yukawa couplings, the observed dark matter abundance is not generally set by the usual co-annihilation freeze-out. Instead the relic density is controlled by conversion processes between a singlet-like dark matter state and heavier doublet-like states: when the conversion rate falls below the Hubble rate before freeze-out, the dark matter and the doublet sector must be evolved with two coupled Boltzmann equations. Using only the single co-annihilation equation in this regime underestimates the final abundance by roughly a factor of four. The paper maps which production mechanism operates across the mass and coupling plane, and shows that for very small Yukawa couplings freeze-in and the SuperWIMP mechanism can reproduce the observed density. If correct, the result means weak-scale dark matter can hide from direct detection while remaining potentially discoverable through collider signals of the doublet sector.

What carries the argument

The load-bearing object is the pair of integrated Boltzmann equations, Eqs. (9) and (10), for the comoving yields $Y_1$ of the singlet-like dark matter and $Y_2$ of the doublet-like sector ($\chi_2^0$, $\chi_3^0$, $\chi^\pm$). The two equations are connected by the conversion rate $\Gamma_{2\to 1}$ of Eq. (12), which sums doublet decays and co-scattering processes such as $\psi \to \chi\, \mathrm{SM}$ and $\chi\, \mathrm{SM} \to \psi\, \mathrm{SM}$. When $\Gamma_{2\to 1}/H \gg 1$, the sub-sectors stay in chemical equilibrium and the pair collapses to the single co-annihilation equation (13); when $\Gamma_{2\to 1}/H \sim O(1)$ near freeze-out, the full pair must be integrated. The paper's small-coupling expressions show that the dominant co-scattering couplings scale as $v y_+/\Delta m$, so reducing the mass splitting compensates for a smaller Yukawa coupling and keeps the conversion rate able to set the abundance.

What would settle it

Run a full momentum-dependent Boltzmann calculation for a representative co-scattering benchmark, such as $M_S = 300$ GeV, $y_1 = 6\times 10^{-8}$, $y_2/y_1 = 0.5$, with $\Delta m$ chosen to give $\Omega h^2 = 0.12$ in the two-equation treatment. If the resulting relic density differs from the integrated-equation result by more than roughly 10 percent, or if the boundary between the co-annihilation and co-scattering regions in the $(m_{\chi_1}, y_1)$ plane shifts measurably, the paper's regime map and its underestimate claim would need revision.

Watch

Extended reading notes

Core claim

The central claim is that the Singlet-Doublet fermion model can match the observed dark matter relic density through four distinct histories depending on the Yukawa couplings $y_1$ and $y_2$: standard co-annihilation freeze-out when $y_1 \gtrsim 10^{-6}$; co-scattering (conversion-driven freeze-out) when chemical equilibrium between the singlet and doublet sub-sectors breaks before freeze-out; freeze-in for $y_1 \sim 10^{-12}$; and SuperWIMP production when long-lived doublets decay after their own freeze-out. The paper's load-bearing quantitative point is that in the co-scattering regime the two-sector evolution must be obtained from the coupled equations (9) and (10); the single effective equation (13) used for co-annihilation assumes chemical equilibrium that no longer holds. For the benchmark $M_S = 300$ GeV, $y_2/y_1 = 0.5$, $y_1 = 6\times 10^{-8}$, the single-equation treatment underestimates the relic abundance by about a factor of four. With the coupled equations, the paper shows that every point in the shaded region of its Fig. 1 can realize the observed density for some mass splitting $\Delta m = M_D - M_S$, with the required splitting decreasing as the Yukawa coupling shrinks.

Load-bearing premise

The entire analysis rests on the assumption that kinetic equilibrium inside the dark sector is maintained, so that the integrated Boltzmann equations (9) and (10) are valid; the paper cites its own unpublished companion study for the estimate that kinetic decoupling affects the relic density by less than about 10 percent.

Editorial extensions

If this is right

Relic-density calculations for this model in the small-coupling region must solve the coupled equations (9) and (10); results that use only the effective single equation inherit the factor-of-four underestimate in the co-scattering regime.
A single weak-scale model can realize four different production mechanisms with the same observed abundance, and the operative mechanism changes continuously with the Yukawa coupling.
For small Yukawa couplings, direct and indirect detection are strongly suppressed, so collider signatures of the doublet sector—prompt decays, displaced vertices, and soft disappearing tracks—become the primary observational windows.
Freeze-in calculations must include the thermal masses of the electroweak mediators; doing so removes the apparent reheat-temperature dependence and makes production dominated by $T\sim M_D$.
Big Bang nucleosynthesis constraints place significant pressure on three-body-decay freeze-in realizations, especially through the long-lived $\chi_3^0$, while the SuperWIMP contribution grows with $M_D$ and can alone account for the full relic density above a critical singlet mass.

Reading between the lines

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

The same 'break chemical equilibrium between sub-sectors and evolve them separately' logic should apply to other feebly coupled singlet-plus-strongly-coupled-doublet or -triplet models; the exact boundary will depend on the analog of $\Gamma_{2\to 1}$.
If the companion full-Boltzmann solution were to show kinetic-decoupling effects exceeding the quoted roughly 10 percent for parameter points near the co-scattering boundary, the regime maps in Figs. 1, 4, 5, 7, and 8 would need to be recomputed; the authors flag this as the main technical caveat.
The model offers a concrete target for a long-lived-particle trigger at the LHC: the $\chi_2^0/\chi_3^0$ asymmetry, with one state prompt and the other displaced, could be a clean handle, and a future muon collider could cover the full doublet mass range; conversely, colliders alone may never see the singlet, so a positive signal would motivate but not prove a dark matter interpretation.
One can partially test the regime boundary without cosmology: measuring the doublet mass splitting and Yukawa couplings at a collider would predict which production mechanism operates, and future precision electroweak measurements could corroborate the doublet sector up to around 500 GeV.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

What you should know first: this is the most complete map yet of the small-Yukawa regime of the singlet-doublet Majorana dark matter model, and its central comparison—single-ODE vs coupled-ODE relic density—holds up. For y1 = 6e-8 the effective co-annihilation equation underestimates the abundance by roughly a factor of four, because chemical equilibrium between the singlet and doublet sectors is lost before freeze-out. That is clearly demonstrated in Fig. 2, with the rate plots on the right showing why.

What's actually new: the full y2/y1 plane, not just the Z-blind slice of Ref. [31]; the custodial y+ -> 0 behavior; the freeze-in/SuperWIMP regime with BBN constraints; and the classification of chi2/chi3 collider signatures as prompt, displaced, or stable across the parameter space. The analytic lifetime estimates are useful and checkable. The paper is also transparent about its one quantitative disagreement with Ref. [31], over the maximum pure-doublet mass (roughly 850 GeV here vs 1.1 TeV there). That discrepancy is not resolved, and it is worth chasing, but it does not touch the small-coupling results.

Soft spots, in proportion. The weakest load-bearing assumption is kinetic equilibrium within the dark sector. The paper defers the justification to an unpublished companion paper [33], claiming the full Boltzmann solution differs by less than ~10%. No derivation or independent check appears. If that estimate fails at the few-ten-percent level, the co-scattering boundary and the freeze-in yields shift. I do not think this is fatal—it is the standard approximation in most relic calculations, and the claim is consistent with generic estimates in [2] and the singlet-triplet analysis [40]—but it should be pinned down before publication. Second, no code or data artifacts are provided, so the numerics cannot be independently reproduced. For a paper whose output is a set of parameter-space maps, that matters more than usual; I would ask for the micrOMEGAs inputs and scripts. Third, the LHC sensitivity estimates are explicitly rough; the paper does not claim a full recast, so this is a minor caveat, not a flaw.

Bottom line: this is a solid, useful reference for anyone working on singlet-doublet, Higgsino-like, co-scattering, or freeze-in dark matter. I would send it to a serious referee, conditional on the authors either providing support for the kinetic-equilibrium approximation or clearly labeling the maps as approximate to that level, and on an explicit discussion of the [31] discrepancy.

Referee Report

2 major / 6 minor

Summary. This paper studies the singlet-doublet Majorana fermion dark matter model in the regime of small Yukawa couplings, where the relic abundance can be set by co-annihilation, co-scattering, freeze-in, or the SuperWIMP mechanism. The authors derive and solve coupled Boltzmann equations for the singlet-like dark matter and the doublet-like sector, Eqs. (9) and (10), and compare them with the single effective equation (13) that assumes chemical equilibrium between the two sectors. They show that in the co-scattering regime the single-equation treatment underestimates the relic abundance by roughly a factor of four for a benchmark with y1 = 6e-8, map the boundaries between production regimes in the (m_chi1, y1) plane, and compute the mass splittings needed to reproduce Omega h^2 = 0.12. For freeze-in, they decompose the relic density into contributions from decays, co-scattering, and SuperWIMP late decays, and derive BBN constraints on the doublet lifetimes. They also discuss collider signatures, including displaced vertices and disappearing tracks, and estimate current and future search sensitivities.

Significance. If the central results hold, the paper provides a useful unified map of dark matter production mechanisms in an economical weak-scale model, and it makes a specific, falsifiable quantitative claim: the common single-ODE co-annihilation treatment underestimates the relic abundance in the small-Yukawa regime, and the coupled two-ODE system is required. The comparison in Fig. 2 is internally consistent and clearly illustrates the breakdown of chemical equilibrium. The paper is also commendable for using standard public tools (SARAH, SPheno, micrOMEGAs) and for openly stating where its results agree or disagree with previous work. The central physics is not circular: the relic density is computed from Boltzmann evolution, not assumed. The main unresolved issue is that the integrated equations rest on a kinetic-equilibrium approximation whose numerical support is delegated to an unpublished companion paper, which is load-bearing for the quantitative boundaries and contours presented here.

major comments (2)

[Sec. III, around Eq. (8) and footnote 2] The integrated Boltzmann equations (9) and (10), and the freeze-in estimate (16), are derived under the assumption that kinetic equilibrium within the dark sector is maintained. The manuscript states that a full solution of the Boltzmann equation differs from the integrated one by less than about 10%, but this estimate is attributed to the authors' unpublished companion paper [33], with no derivation, no momentum-dependent spectrum, and no independent check. This is load-bearing because the co-annihilation/co-scattering boundary is defined as the point where the 1-ODE and 2-ODE results differ by 10%, and the factor-of-four underproduction claim is computed within the same integrated framework. If kinetic corrections are at the few-ten-percent level, the boundaries in Figs. 1, 4, and 5 and the freeze-in yields could shift. I recommend that the authors either include the estimate in an appendix or otherwise provide an independent quantitative justification, and state the sensitivity of the main maps to this approximation.
[Sec. IV.B and Eq. (16)] The description of the freeze-in calculation is unclear about which equations are actually solved. The text says 'We use Eq. (16) accounting for one-directional processes of decay of doublet states, SM-annihilations, and co-scattering,' but Eq. (16) is derived by fixing Y2 = Y2eq and neglecting Y1 in the first line of Eq. (15). That approximation cannot produce the Y2 freeze-out shown in Fig. 6, the late-time SuperWIMP transfer, or the OmegaCS and OmegaSW decomposition in Fig. 7. Please specify whether the numerical results come from solving the coupled system (15) with momentum-averaged rates, or from Eq. (16) plus Eq. (24), and explain how SM-annihilation contributions are included without double counting. This matters for the reliability of Figs. 6-8 and for the claimed BBN constraints.

minor comments (6)

[Sec. II] The word 'Majorna' in the first paragraph of Sec. II appears to be a typo for 'Majorana'.
[Sec. IV.A] The phrase 'how variation in the ratio of ratio the Yukawa couplings' should be 'how variation in the ratio of the Yukawa couplings'.
[Fig. 5 caption] The caption lists the panel masses as '150, 200, 300, and 250 GeV (clockwise from upper left),' but the panel labels and the text indicate the ordering is 150, 200, 250, 300 GeV clockwise; please correct the caption.
[Fig. 7 caption] The caption says the mass splittings are '70,110,500,200 GeV (clockwise from upper-left),' while the text and panel labels indicate 70, 110, 200, 500 GeV in row-major order; please make the caption and figure consistent.
[Fig. 4 right panel] The legend entry for the lower curve appears to read 'y1 = 10^7' but should be 'y1 = 10^-7'; please check the rendered figure.
[References] Reference [76] appears without a collaboration name; please add the collaboration or authors for completeness.

Circularity Check

1 steps flagged · score 3.0 of 10

Main co-scattering comparison is self-contained; the kinetic-equilibrium premise is anchored to the authors' unpublished companion [33].

self citation load bearing [Sec. III (Dark Matter Production Regimes), paragraph on kinetic equilibrium and footnote 2; Ref. [33]]
"We will present our methods for solving the full Boltzmann equation in the Singlet-Doublet case in a companion paper [33]. In that paper, we will demonstrate that a full solution to the Boltzmann equation differs from the integrated one by less than ≲10% for the relic density, and often much less. Because this effect is small, for the remainder of this paper, we will work in the approximation that kinetic equilibrium within the dark sector is maintained."

The validity of the integrated Boltzmann equations (9)-(10), on which all relic-density maps, the co-annihilation/co-scattering boundary (defined as a 10% difference between the 1-ODE and 2-ODE treatments), and the factor-of-four underestimation at y1=6e-8 are computed, is justified solely by the authors' own unpublished companion paper [33]. The quoted passage explicitly defers the demonstration to that companion; no derivation, momentum-dependent spectra, or independent numerical check appears here. Ref. [33] is not machine-checked, code-reproduced, or externally falsifiable in the present text, so it is a load-bearing self-citation rather than independent evidence.

full rationale

Most of the paper's derivation chain is self-contained. The model Lagrangian (2) and mass matrix (3) are diagonalized to give the couplings (6)-(7). The relic abundance is obtained by solving the coupled integrated Boltzmann equations (9)-(10) with micrOMEGAs, and the claimed co-scattering effect is a direct comparison with the single-ODE approximation (13): for y1=6e-8 the 2-ODE system gives roughly four times the final abundance. That comparison does not rely on any fitted parameter being renamed as a prediction; Δm and y1 are chosen along the Ωh^2=0.12 contour, and the paper presents them as contours, not predictions. Freeze-in and SuperWIMP contributions use standard analytic formulas (16)-(18), (23)-(24). The one load-bearing self-citation is Ref. [33], the authors' unpublished companion, which is invoked to justify the kinetic-equilibrium approximation and the less-than-about-10% error estimate; this is a caveat/assumption rather than an input-output circularity, but under the stated review rules it must be flagged as a self-citation on which the quantitative results rely. The central chemical-equilibrium claim retains independent content; therefore the score is 3, not higher.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The model parameters (MS, MD, y1, y2) are inputs from the model, not derived; the paper chooses Delta m and y1 to match the observed relic density along the Omega h^2 = 0.12 contour. It relies on standard cosmology and on the Standard Model particle content. No new entities are introduced. The most fragile input is the assumption of kinetic equilibrium, which is supported only by the authors' unpublished companion paper [33].

free parameters (3)

Delta m (freeze-out mass splitting) = 2.5 to 17.5 GeV depending on m_chi1, y1, y2/y1 (Figs. 4, 5)
Chosen at each parameter point to reproduce the observed relic density, Omega h^2 = 0.12, along the relic contour; it is a fitted value, not a prediction of the model.
y1 (freeze-in Yukawa coupling) = approximately 1e-12 for the shown freeze-in contours (Fig. 8)
Adjusted per point so that the total freeze-in, co-scattering, and SuperWIMP contributions sum to Omega h^2 = 0.12; the value is fit to the observed dark matter density.
Overall normalization in Eq. (21) = not stated explicitly; 'extracted from the pure doublet (Delta m to 0) limit' (footnote 3)
Used in the semi-analytic fit to the mass splitting as a function of the dark matter mass in the co-annihilation plateau; an empirical normalization rather than a first-principles constant.

assumptions (5)

domain assumption The dark sector is odd under an imposed Z2 symmetry, making the lightest state stable dark matter.
Standard model-building input for dark matter stability; invoked in Sec. II without independent justification.
domain assumption CP is conserved and y1, y2 are real with y2/y1 restricted to [-1,1] after using the interchange symmetry.
Sec. IV states this; it restricts parameter space but is common in this model.
ad hoc to paper Kinetic equilibrium between the dark sector and the SM bath is maintained throughout the cosmological evolution; full Boltzmann corrections are less than or about 10 percent.
Sec. III and footnote 2 rely on the authors' unpublished companion paper [33]; this is load-bearing for the integrated Boltzmann equations (9)-(10).
domain assumption The early universe is described by standard FRW cosmology with equal entropy and energy density degrees of freedom, and radiation domination during the production epochs.
Standard cosmology used in Eqs. (13)-(17); not derived in the paper.
domain assumption The thermal-mass regularization of t-channel singularities with the kappa coefficients of Ref. [45] correctly captures the IR behavior; a p_T cut following [42] is applied.
Sec. III B; the freeze-in calculation depends on this to avoid spurious sensitivity to the reheat temperature.

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Cite this review

Pith. "Pith review of Singlet-doublet dark matter beyond freeze-out." pith.science (2026). https://pith.science/paper/AATBIHAE

@misc{pith2026260812480,
  author       = {Pith},
  title        = {Pith review of: Singlet-doublet dark matter beyond freeze-out},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AATBIHAE}},
  note         = {Machine review of arXiv:2608.12480}
}

read the original abstract

The Singlet-Doublet fermion model of dark matter is an economical weak-scale dark matter model that realizes the dark matter abundance through interactions with the electroweak bosons of the Standard Model. Depending on the size of the Yukawa couplings in the model, the dark matter relic abundance can be produced via freeze-out (including co-annihilation), co-scattering, freeze-in, or the SuperWIMP mechanism. We analyze the ways this model can realize the dark matter density with emphasis on the small Yukawa coupling regime. In this limit, direct and indirect detection are difficult, but a rich collider phenomenology is possible.

Figures

Figures reproduced from arXiv: 2608.12480 by the authors.

**Figure 2.** FIG. 2: Left: Evolution of the comoving abundance, [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** FIG. 3: Dependence of the boundaries between the different dark matter production [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4: Mass splitting ∆ [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]

**Figure 5.** Figure 5: FIG. 5: The color bar indicates the mass splitting ∆ [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]

**Figure 6.** Figure 6: FIG. 6: Evolution of the comoving dark matter and doublet abundances, [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]

**Figure 7.** Figure 7: FIG. 7: Contributions to the dark matter relic density Ω [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]

**Figure 8.** Figure 8: FIG. 8: Top panels: Total Decay widths Γ [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]

**Figure 9.** Figure 9: FIG. 9: Branching ratio of [PITH_FULL_IMAGE:figures/full_fig_p028_9.png]

**Figure 10.** Figure 10: FIG. 10: Left: Contours of [PITH_FULL_IMAGE:figures/full_fig_p031_10.png]

Discussion (0). Continue with ORCID to comment.

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