Forbidden frozen-in dark matter

Andrzej Hryczuk; Dimitrios Karamitros; Leszek Roszkowski; Luc Darm\'e

REVIEW 2 major objections 5 minor 1 cited by

Forbidden frozen-in dark matter

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A mediator that cannot decay into dark matter at zero temperature can do so once its thermal mass grows, and this 'forbidden freeze-in' can dominate dark-matter production.

desk verdict A solid and useful paper that names a real freeze-in regime and derives clean formulas; the main caveat is that part of the model scan leans on an unverified mediator-equilibrium assumption at small portal couplings. read the letter →

arxiv 1908.05685 v2 pith:CYEO3DHX submitted 2019-08-15 hep-ph

Luc Darmé , Andrzej Hryczuk , Dimitrios Karamitros , Leszek Roszkowski This is my paper · ORCID

classification hep-ph

keywords forbiddenfreeze-inthermalmassdarkHiggsportalmatterrelicdensitymechanismlong-livedmediatorearlyuniversecosmology

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 identifies a dark-matter production regime it calls 'forbidden freeze-in': a mediator particle that cannot decay into dark matter at zero temperature can develop a large thermal mass in the hot early Universe, making the decay energetically allowed. The paper argues that this thermally opened decay channel, rather than standard freeze-in processes computed with zero-temperature masses, can dominate the production of dark matter. For decays through dimension-four operators, the resulting relic abundance is nearly independent of the dark-matter mass and requires a larger coupling than ordinary freeze-in; for higher-dimensional operators the production becomes sensitive to the reheating temperature. A scalar 'dark Higgs' portal model provides a concrete realisation, and the paper maps the parameter regions that give the observed relic density and survive current and future experimental constraints.

What carries the argument

The load-bearing object is the thermally corrected mediator mass, $m_{S,T}^2 \simeq m_S^2 + \alpha^2 T^2$, where $\alpha^2 = \lambda_S/24$ comes from the scalar self-interaction. With the vacuum decay width $\Gamma_{S\to\chi\bar\chi} = (y_\chi^2/8\pi)(m_S^2-4m_\chi^2)^{3/2}/m_S^2$, the allowed window is controlled by the ratio $z = 2m_\chi/(\alpha T)$: the decay is open for $z<1$ and shuts off at $z=1$. Substituting the thermally varying mass into the freeze-in Boltzmann equation turns the yield evolution into $dY_{DM}/dz \propto K_1(\alpha)(1-z^2)^{3/2}/\sqrt{g h}$ times the derivative of the entropy degrees of freedom, and integrating from $z=0$ to $z=1$ produces the closed-form abundance. The mechanism is carried by the combined $\alpha^4$ and $K_1(\alpha)$ prefactors: they set the production efficiency and make the final abundance nearly mass-independent for dimension-four operators.

What would settle it

For a scan point with very small mediator–Standard Model couplings (near the lower edge of the scanned ranges, e.g. mixing parameters around $10^{-8}$), solve the coupled Boltzmann equations without assuming $S$ was in equilibrium at early times, tracking $S$'s full phase-space distribution; if the resulting relic abundance departs from the mass-independent $Y_{DM,0}$ formula, the claim that forbidden freeze-in dominates in that region is wrong.

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Extended reading notes

Core claim

The paper's central claim is that 'forbidden frozen-in dark matter' is a generic freeze-in sub-regime. Once a bath particle $S$ in equilibrium with the Standard Model plasma acquires a thermal correction to its mass, $m_{S,T}^2 \simeq m_S^2 + \alpha^2 T^2$ with $\alpha^2 = \lambda_S/24$ for a quartic self-interaction, the decay $S \to \chi\bar\chi$ can proceed at high temperature even when the zero-temperature mass satisfies $m_S < 2m_\chi$. For a dimension-four decay operator the production peaks just before the decay closes, at $z \equiv 2m_\chi/(\alpha T) \approx 1$, and the final yield is approximately $Y_{DM,0} = (\alpha^2 y_\chi / 5\times10^{-9})^2 (1\,\mathrm{GeV}/(2m_\chi)) K_1(\alpha)/\sqrt{g h}$ evaluated at $\langle z\rangle \approx 0.34$. The relic abundance $\Omega h^2$ is then, up to order-one corrections, independent of $m_\chi$, in contrast to the linear mass dependence of standard freeze-in, and the required Yukawa coupling is typically larger. For dimension-five and higher operators the yield is dominated by the highest temperatures, so the abundance depends on the reheating temperature. The paper's dark Higgs portal example produces both regimes in a numerical scan and shows the allowed regions are constrained by and partially testable through long-lived scalar searches.

Load-bearing premise

The argument assumes the mediator $S$ was in equilibrium with the hot plasma in the early Universe, because that is what justifies the thermal-mass formula $m_{S,T}^2 \simeq m_S^2 + \alpha^2 T^2$; if $S$ was never abundant or interacting enough to be in equilibrium, the forbidden decay window would not open.

Editorial extensions

If this is right

Parameter regions with $m_S < 2m_\chi$ that standard freeze-in would rule out can now yield the observed dark-matter abundance, so the viable space of portal models is larger than previously thought.
For dimension-four decay operators, the dark-matter mass and the production coupling are effectively decoupled from the relic abundance, so attempts to match the observed density must use larger couplings than standard freeze-in suggests.
For higher-dimensional operators, the forbidden freeze-in contribution is dominated by the highest temperatures, making the final abundance dependent on the reheating temperature rather than only on particle masses.
Around the transition point $m_S \approx 2m_\chi$, the predictions become sharply mass-sensitive over a width of order $\alpha m_S$, which is a direct consequence of the thermal mass opening the decay.
In the concrete dark Higgs model, the viable forbidden-freeze-in points populate long-lived-mediator regions that upcoming forward and beam-dump experiments can begin to probe.

Reading between the lines

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

Because the mechanism only requires that the mediator be in kinetic equilibrium, it should extend to models where the mediator is chemically decoupled from the Standard Model but self-interacting; the paper notes this possibility but does not include it in the numerical scan.
The near mass-independence for dimension-four decays implies a degeneracy: different dark-matter masses can share the same thermal-mass parameter and coupling while all matching the observed abundance, which would make the mechanism difficult to pin down by relic-density measurements alone.
Vector and fermion mediators with large thermal masses should exhibit the same forbidden window, so the regime is likely not limited to scalar portals; checking this would require only replacing the thermal-mass and decay-rate formulas.

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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

Bottom line: this paper is more than a re-run of the earlier gravitino/axino observation. It isolates a regime that had been noted in passing, gives it a name, derives closed-form yields, and builds a zero-VEV dark-Higgs example that is cleanly different from ref. [14]. The central physics checks out: with mS,T^2 = mS^2 + alpha^2 T^2, decays that are kinematically forbidden in vacuum open at high temperature, and for dimension-four operators the relic abundance is almost mchi-independent. That is a real and useful result, and the paper is honest about the earlier literature.

The strongest part is Sec. 2. The analytic approximation for the yield, eq. (2.23), is simple enough to reproduce and matches the numerical curves. The paper also carefully walks through why d>4 operators behave differently. In the model section, the zero-VEV choice avoids a complication present in earlier constructions, and the experimental constraints are applied sensibly.

The soft spots are real but not fatal. The numerical scan assumes S was in chemical equilibrium for x < 0.1 and solves the coupled Boltzmann equations only for x > 0.1. But the forbidden production peaks at x ~ alpha mS/(2mchi), which is below 0.1 for alpha < 0.3. That means most of the forbidden production is computed under exactly the equilibrium assumption. Footnote 16 explicitly concedes that the thermal mass formula fails once S is chemically frozen out, and it asserts without a scan-level check that this does not affect the studied model. With A and lambdaHS allowed down to 1e-8, that assertion deserves a real check. For the smallest portal couplings, the quoted ychi and relic-density contours are conditional, not wrong in mechanism. There is also a minor tension between that concession and the introduction's claim that the equilibrium assumption can be relaxed. The private code makes verification harder, though the analytic formulas give enough to reconstruct the main claims.

For a benchmark point with adequately large lambdaHS, the forbidden freeze-in mechanism holds up. The paper is aimed at freeze-in model builders and anyone computing relic abundances with thermal-mass corrections; they should read it and revisit earlier models. I would send it to peer review and ask for either a point-by-point verification of S equilibrium in the scanned region or a restriction of the claimed region to where it is verified.

Referee Report

2 major / 5 minor

Summary. This paper introduces and studies a 'forbidden freeze-in' mechanism for dark matter production. The central observation is that a mediator S in thermal equilibrium with the Standard Model bath develops a temperature-dependent mass m_{S,T}^2 ≈ m_S^2 + α^2 T^2; even if the vacuum decay S→χχ is kinematically forbidden (m_S < 2m_χ), it becomes allowed at high temperature. In Sec. 2 the authors derive analytic estimates for the DM yield. For dimension-four operators the yield is nearly independent of m_χ and the required Yukawa coupling y_χ is larger than in standard freeze-in; for higher-dimensional operators production is dominated at high temperatures. In Sec. 3 they implement this in a dark Higgs portal model with Dirac fermion dark matter, solve the coupled S/χ Boltzmann equations, scan the parameter space with MultiNest, and apply BBN, SN1987, CHARM, E949, LHCb, FASER, and SHiP constraints, finding new viable regions in the m_S–m_χ–λ_S–y_χ parameter space.

Significance. The forbidden freeze-in mechanism is a genuinely useful generalization of freeze-in: it is generic whenever a bath particle has a sizable thermal self-mass, and it produces distinctive phenomenology, notably a relic abundance nearly independent of m_χ and a stronger DM–mediator coupling than standard freeze-in. The closed-form limits in Eqs. (2.19) and (2.23) are clearly derived and reproduce the correct limiting behavior in Fig. 3, and the presentation of the experimental constraints is a service to the community. The use of the observed Ωh² as an external constraint is standard and not circular. However, the quantitative results of the scan are conditional on a chemical-equilibrium assumption for S that is not verified for the smallest allowed portal couplings, and the analytic thermal-mass formula omits a contribution that is not negligible in part of the scanned model range; these points must be addressed before the model regions can be considered robust.

major comments (2)

[§3.2, footnote 16; Table 1] The numerical setup assumes S is in chemical equilibrium for x < 0.1 and only solves the coupled S–χ Boltzmann equations for x > 0.1. For the forbidden regime, production peaks at z ≈ ⟨z⟩ = 0.34, i.e. at T_peak ≈ 2m_χ/(α⟨z⟩). With the ranges in Table 1 this can exceed 10^4 GeV for m_χ = 50 GeV and α = 0.01, where number-changing S↔SM rates mediated by A, λHS ~ 10^-8 fall well below the Hubble rate (a rough λHS^2 T/H estimate gives ~10^-3). In such cases S cannot be assumed to track its equilibrium abundance, and footnote 16 itself concedes that the thermal mass formula (2.3) fails once S is chemically frozen out; the statement that this 'has no implications for our results' is not checked point-by-point in the scan. Since the equilibrium initial abundance of S is an input to the yield in Eq. (2.23), the small-portal part of the forbidden region in Fig. 8 and the associated y_χ values are not yet reliable. Please verify the equilibrium condition by computing the relevant rates at each scan point, or extend the numerical integration to x < 0.1 with a well-defined initial condition for S.
[§2.1, Eq. (2.3); §3.1] The analytic treatment uses α^2 = λS/24 in Eq. (2.14), i.e. the thermal mass from the S self-interaction alone. In the model of Sec. 3.1, the quartic portal λHS S^2 H†H also contributes to the thermal mass of S, by an amount of order λHS T^2 (for instance λHS T^2/6 for the four real Higgs degrees of freedom). For λHS = 10^-2 and λS = 10^-4, both within the Table 1 ranges, this contribution is much larger than λS T^2/24, so the 'α' appearing in Eqs. (2.16), (2.22), and (2.23) is not simply the combination of scan parameters used in the model. If the numerical code includes the full thermal mass, the analytic closed forms and the plotted α are not the same quantity; if it uses only Eq. (2.3), a relevant physical effect is omitted. Please specify the full thermal-mass expression used in the numerical computation and either include the λHS contribution in the definition of α or demonstrate that it is negligible throughout the region shown in the plots.

minor comments (5)

[§1] The sentence 'in which the scenario can natually be realised' contains a typo: 'natually' should be 'naturally'.
[§2.3] The phrase 'aboundance' in 'the final dark matter aboundance' and similar places should be 'abundance'.
[Fig. 7 and surrounding text] In the discussion of Fig. 7a, the text says 'In Fig. 6a the final DM abundance is determined by the branching fraction...'; this cross-reference appears to be mislabeled and should refer to Fig. 7a.
[Eq. (2.6)] The relation between the summed |M|² and Γ_χ should clarify whether Γ_χ is the width for S→χχ with χ and χ̄ counted together or separately; the current notation is slightly ambiguous.
[§3.2.2] The 'additional 10% theoretical uncertainty' used with the Planck relic-density constraint is not described; please state how it was implemented in the MultiNest likelihood.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the forbidden-freeze-in yield is derived from the Boltzmann equation with the thermal-mass ansatz, and the relic-density match is an external constraint rather than a fitted prediction.

full rationale

The derivation chain is self-contained. The central analytic results, Eqs. (2.16), (2.22), and (2.23), follow by inserting the thermal-mass ansatz mS,T^2 = mS^2 + alpha^2 T^2 (Eq. 2.14), the Yukawa decay width (Eq. 2.21), and the standard freeze-in Boltzmann equation (Eqs. 2.7-2.11) into a single integral over z. The numerical factors, including 5e-9 and the mean value <z> = 0.34, are computed from the integrals of the phase-space factor (1-z^2)^{3/2} times a Bessel function; they are not fitted to the final relic abundance. The relic density Omega h^2 is then compared with the externally measured Planck value after scanning y_chi, lambda_S, and the other model parameters. Adjusting scanned couplings to satisfy the observed relic abundance is standard model-building, not a fitted input masquerading as a prediction. The assumption that S is in chemical/kinetic equilibrium at early times, including the numerical prescription that S traces its equilibrium value for x < 0.1, is a physical regime assumption; it does not define the DM yield in terms of itself, and the paper explicitly flags the limitation in footnote 16 rather than hiding it. Self-citations such as [1], [7], [37], and [42] appear only as contextual or methodological references; none is invoked as a uniqueness theorem or as the load-bearing justification for the forbidden-freeze-in mechanism. The distinction from prior work [14] is based on the zero-VEV structure and a different mass region, so this is not a renaming of a known result. No equation in the paper reduces to its own input by construction, and no prediction is statistically forced by a prior fit to the same data.

Assumptions & free parameters 6 free parameters · 6 assumptions · 2 invented entities

The central claim depends on the finite-temperature mass formula, the equilibrium history of S, and the subdominance of 2-to-2 channels. The model parameters are scanned rather than derived, and the λHS/A contributions to the thermal mass are not fully accounted for in the analytic treatment.

free parameters (6)

lambda_S (dark Higgs self-coupling) = scanned log-uniform over 1e-4 to 1
Sets the thermal mass α^2 = λS/24 in Eq. (2.14) and controls the efficiency of forbidden freeze-in; the paper selects points via MultiNest to match Ωh^2, so it acts as an adjusted input.
y_chi (DM Yukawa coupling) = scanned log-uniform over 1e-14 to 1e-8; forbidden-region values around 1e-12 to 1e-8
The DM-mediator coupling; it is the main parameter tuned by the relic-density constraint and determines the size of the freeze-in yield in Eq. (2.23).
mu_chi (DM mass parameter) = scanned log-uniform over 0.005 to 50 GeV
The dark matter mass; the paper's mass-independence claim is tested by varying it across this range.
mu_S (dark Higgs mass parameter) = scanned log-uniform over 0.100 to 50 GeV
Sets the zero-temperature mediator mass, which controls whether the vacuum decay is forbidden (mS < 2mχ) or allowed.
A (trilinear Higgs-dark Higgs mixing) = scanned log-uniform over 1e-8 to 1e-2 GeV
Controls S-SM mixing and mediator lifetime; affects experimental constraints and the equilibrium assumption, but not the central yield formula.
lambda_HS (quartic portal coupling) = scanned log-uniform over 1e-8 to 1e-2
Controls S equilibrium with the SM bath and the Higgs-VEV contribution to mS; also enters the consistency condition Eq. (3.6).

assumptions (6)

standard math The thermal self-energy of a scalar with quartic self-interaction gives Π_S^(T) = λS T^2/24 and daisy resummation changes this by at most 20% for λS <= 1 (Eq. 2.2 and Sec. 2.1).
The backbone of the thermal mass used everywhere; the paper explicitly states this is an approximation checked only for λS <= 1.
domain assumption S was in chemical equilibrium at early times and remains in kinetic equilibrium through DM production; for x < 0.1 its abundance tracks equilibrium (Sec. 3.2).
Needed for the initial conditions and for the thermal mass formula; footnote 16 acknowledges the formula fails if S is chemically decoupled.
domain assumption χ is absent after reheating and never reaches chemical equilibrium with the SM plasma (Sec. 2.2 and Sec. 3.2).
Standard freeze-in setup; if χ had an initial abundance or thermalized, the yield calculation would change.
domain assumption 2-to-2 processes (SS ↔ χχ, hh ↔ χχ, Sh ↔ χχ) are subdominant to S -> χχ decays over the scanned parameter space (Sec. 2.3 and Sec. 3.1).
The analytic treatment ignores them and the paper says it checked this numerically for its scan, but not for all possible models.
ad hoc to paper The effective mediator mass is mS,T^2 ≈ mS^2 + α^2 T^2 with α^2 = λS/24, neglecting temperature-dependent Higgs-VEV and λHS/A contributions in the analytic treatment (Eq. 2.14).
Used throughout Sec. 2.3; in the Higgs portal model λHS can be 1e-2 while λS can be 1e-4, so this approximation may not hold over the full scan.
domain assumption The trilinear term induced by the dark Higgs VEV is negligible, satisfying A/µS << 12 mS^2/(λS v^2) (Eq. 3.6).
Imposed for consistency in the scan; it restricts the valid parameter space but is stated in Sec. 3.1.

invented entities (2)

S, dark Higgs scalar mediator independent evidence
purpose: Thermally massive mediator whose decays S -> χχ produce dark matter; mixes with the SM Higgs via A and λHS.
Long-lived scalar searches (CHARM, LHCb, BBN, SN1987A, FASER, SHiP) provide falsifiable handles, as shown in Fig. 9.
χ, Dirac fermion dark matter
purpose: Stable DM candidate produced only through feeble S decays; never in equilibrium.
No direct experimental handle is identified for χ itself; its presence is inferred from the relic abundance and the mediator phenomenology.

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Cite this review

Pith. "Pith review of Forbidden frozen-in dark matter." pith.science (2026). https://pith.science/paper/CYEO3DHX

@misc{pith2026190805685,
  author       = {Pith},
  title        = {Pith review of: Forbidden frozen-in dark matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CYEO3DHX}},
  note         = {Machine review of arXiv:1908.05685}
}

read the original abstract

We examine and point out the importance of a regime of dark matter production through the freeze-in mechanism that results from a large thermal correction to a decaying mediator particle mass from hot plasma in the early Universe. We show that mediator decays to dark matter that are kinematically forbidden at the usually considered ranges of low temperatures can be generically present at higher temperatures and actually dominate the overall dark matter production, thus leading to very distinct solutions from the standard case. We illustrate these features by considering a dark Higgs portal model where dark matter is produced via decays of a scalar field with a large thermal mass. We identify the resulting ranges of parameters that are consistent with the correct dark matter relic abundance and further apply current and expected future collider, cosmological, and astrophysical limits.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Probing Dark Matter freeze-in with long-lived particle signatures: MATHUSLA, HL-LHC and FCC-hh
hep-ph 2019-08 conditional novelty 6.0 of 10

Projected MATHUSLA, HL-LHC, and FCC-hh forward detector sensitivities probe Higgs-mediated freeze-in dark matter across parent masses up to about 10 TeV.

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