FIMPs in a two-component dark matter model with $Z_2 \times Z_4$ symmetry

Hao Sun; XinXin Qi

FIMPs in a two-component dark matter model with $Z_2 \times Z_4$ symmetry

T0 review · 3 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper argues that two dark matter species can both freeze in with portal couplings as small as 10^-25, because the large vacuum expectation value that gives the fermion mass suppresses the couplings needed to match the observed relic a

desk verdict A standard freeze-in parameter scan for a two-component Z2×Z4 model; the useful map of ultra-small λ_ds is undermined by an abstract that promises consistency checks the body never performs. read the letter →

arxiv 2602.08359 v4 pith:6FD6CBW5 submitted 2026-02-09 hep-ph

XinXin Qi , Hao Sun This is my paper

classification hep-ph

keywords two-componentdarkmatterfreeze-inFIMPZ2×Z4symmetryMajoranafermionsingletscalarrelicdensityultra-smallcouplings

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 tries to show that two different dark matter particles—a singlet scalar and a Majorana fermion—can both be produced by the freeze-in (FIMP) mechanism in a single model with Z2×Z4 symmetry. The key move is that the same vacuum expectation value that gives the fermion its mass also controls the new-Higgs interactions that produce the scalar; because the fermion's Yukawa coupling must be extremely small to match the observed dark matter abundance, this vev is enormous, which pushes the required portal coupling λ_ds down to values as small as 10^-25—fifteen or more orders below ordinary FIMP couplings. The paper reports that solving the Boltzmann equations leaves viable dark matter masses from 1 GeV to 3 TeV for a 1 TeV new Higgs, and that even at such tiny couplings the new Higgs can still dominate production of the scalar. It further argues that these extreme values are theoretically consistent: the hierarchy is technically natural, the couplings stay above the gravitational freeze-in floor, and the small couplings could arise from higher-dimensional operators, radiative generation, or a flavor-symmetry mechanism. If right, the model shows one Lagrangian can host dark matter interactions spanning more than twenty-five orders of magnitude.

What carries the argument

The load-bearing object is the identity v0 = mχ/y_sf, which links the fermion mass to the vacuum expectation value of the singlet S0. After spontaneous symmetry breaking, this same v0 appears in the h2-S-S and h2-h2-S-S couplings, so a tiny y_sf simultaneously makes χ light and makes v0 enormous. The freeze-in Boltzmann equations for the abundances Y_S and Y_χ, including decays and 2→2 annihilations of h2 (with SM-Higgs processes suppressed by the decoupling limit sinθ→0), then convert the large v0 into a strong constraint on λ_ds. The Z2×Z4 symmetry is what stabilizes both dark matter candidates and lets their production be treated independently.

What would settle it

Compute the one-loop correction to λ_ds in this model (with loops of S0, h2, χ, and SM states) at a renormalization scale of order m2; if δλ_ds/λ_ds is of order unity or larger, the technical-naturalness premise collapses.

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Extended reading notes

Core claim

The paper's central claim is that in the FIMP-FIMP regime of the Z2×Z4 two-component dark matter model, the observed relic density Ωh² ≈ 0.12 can be reproduced for fermion and scalar masses in [1 GeV, 3 TeV] with m2 = 1 TeV, with portal couplings λ_ds spanning roughly 10^-25–10^-13. The mechanism is the v0 = mχ/y_sf relation: relic-density constraints force y_sf to be tiny (down to ~10^-13 or smaller when mχ < m2/2), making v0 enormous (10^9–10^15 GeV); since the new Higgs h2 couples to the scalar S through λ_ds v0² vertices, this large vev boosts h2-mediated S production, so λ_ds must be correspondingly suppressed to avoid overproduction. The paper verifies the consistency of this extreme r

Load-bearing premise

The load-bearing premise is that the ultra-small Yukawa and portal couplings are technically natural, i.e., quantum corrections do not lift them by order unity, and that the decoupling limit sinθ→0 is chosen so that SM-mediated production of the fermion is negligible.

Editorial extensions

If this is right

Both dark matter candidates can be FIMPs in one model, with the observed relic density reproduced for DM masses from 1 GeV to 3 TeV when the new Higgs mass is 1 TeV.
The portal coupling λ_ds can be as small as 10^-25, far below conventional FIMP values, because the large vev v0 = mχ/y_sf enhances h2-mediated production.
The new Higgs h2 remains the dominant source of scalar dark matter for most of the viable parameter region, even at these ultra-small couplings.
Depending on whether each DM mass is above or below m2/2, production is governed by different channels (h2 decays vs h2h2 annihilations), giving four distinct viable regions.
The same Z2×Z4 Lagrangian can accommodate dark matter interaction strengths spanning over 25 orders of magnitude when combined with the previously studied WIMP and mixed regimes.

Reading between the lines

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

The v0-driven suppression is generic: any model in which a FIMP mass comes from m = y v0 with tiny y inherits a large production cross section, so similarly small portal couplings should appear in other multi-component freeze-in constructions, not just this one.
If h2 is light enough to be produced at colliders, the ultra-small λ_ds makes its decays to dark matter effectively invisible and may give displaced or long-lived signatures; searches for such signatures could probe the parameter region directly.
The technical-naturalness claim is asserted rather than demonstrated; a one-loop computation of the radiative corrections to y_sf and λ_ds, or an explicit UV completion, would be the natural next check of whether the 10^-25 couplings are stable.
The paper's scan assumes a specific decoupling limit; relaxing sinθ to small but nonzero values would open a SM-mediated production channel for χ and raise the allowed y_sf range, which is a testable extension of the parameter scan.

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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,

Quick take: the paper is a straightforward micrOMEGAs scan of the FIMP-FIMP regime of a two-component DM model the same authors introduced earlier. The genuinely new bit is the consequence of v0 = mχ/y_sf. Because relic density forces y_sf down to 10^-11 or below, v0 is pushed to 10^9–10^15 GeV, and since v0 also controls h2-mediated S production, the allowed λ_ds drops to 10^-20 or lower. That mapping, including the four mass-hierarchy cases, is the real content. The analytic estimates in Sec. IV track the numerical curves, and the authors honestly frame the allowed ranges as constraints from Ωh²=0.12, not predictions.

The soft spots are real. The abstract makes three consistency claims: the v0/v hierarchy is technically natural, λ_ds sits 15–20 orders above the gravitational freeze-in floor, and the small couplings can originate from higher-dimensional operators, radiative generation, or Froggatt–Nielsen. None of that appears in the body. Section III only lists perturbativity, unitarity, and vacuum stability. There is no RGE analysis, no naturalness argument, no gravitational freeze-in calculation, no FN construction. This is not a minor omission. The whole point of pushing λ_ds to 10^-24 is that the extreme region is theoretically consistent, and the paper just asserts it. To make it worse, the Introduction says the decoupling limit sinθ→0 is fine-tuned by hand to suppress SM-mediated χ production, which sits awkwardly next to the 'technically natural' claim. The Appendix is also in bad shape: the cross-section formula for h2h2→SS is garbled to the point of being unusable, and no code or parameter files are released, so a reader cannot reproduce the scans independently.

My overall read agrees with the conditional verdict: strip away the overclaiming abstract, and the parameter-space map is probably correct and useful for model-builders. But the abstract's verification statement is load-bearing and unsupported. A referee should either demand the consistency analysis or insist the claims be removed, and should require a cleaned-up appendix or released scan files. In its current form, I would not cite it.

Recommendation: send to peer review, with a clear ask to fix the abstract/body gap and the appendix. The numerical relic-density work itself is worth a serious look.

Referee Report

3 major / 3 minor

Summary. The paper studies the FIMP-FIMP regime of a two-component dark matter model with a Z2×Z4 symmetry, containing a singlet scalar S and a Majorana fermion χ as DM candidates, plus a singlet S0 whose vev v0 generates mχ = y_sf v0. Working in the decoupling limit sinθ→0, the model has six free parameters. The authors use micrOMEGAs to solve the Boltzmann equations and identify viable parameter regions for four mass hierarchies relative to m2/2, reporting DM masses in [1 GeV, 3 TeV] for m2 = 1 TeV and portal couplings λ_ds as small as ~10^-24 because v0 = mχ/y_sf can be 10^9–10^15 GeV. The abstract further claims that these extreme couplings are theoretically consistent through technical naturalness, the gravitational freeze-in floor, and UV mechanisms such as Froggatt–Nielsen, but the body does not carry out these verifications.

Significance. If the advertised consistency claims hold, this would demonstrate a very broad FIMP-FIMP parameter region in a two-component DM model, with h2-mediated production dominating even for extraordinarily small couplings. The numerical analysis has real strengths: the analytic estimates in Eqs. (20)–(23) match the numerical curves, the channel decomposition is physically transparent, and the scan explicitly separates the four mass-hierarchy cases. However, the paper's headline novelty depends on ultra-small couplings being radiatively stable and on h2 being in thermal equilibrium in the decoupling limit; neither is established. The manuscript is therefore not yet ready in its present form.

major comments (3)

[Abstract ¶3; Sec. III] The abstract asserts that 'we verify the theoretical consistency of these extreme values: the v0/v hierarchy is technically natural, λ_ds remains 15–20 orders of magnitude above the gravitational freeze-in floor, and the suppression can originate from higher-dimensional operators, radiative generation, or the Froggatt–Nielsen mechanism.' Sections II–IV contain none of this. Section III (Eqs. 9–11) imposes only perturbativity, unitarity, and vacuum stability. There is no RGE analysis, no symmetry-restoration argument for the limits y_sf→0 or λ_ds→0, no estimate of the gravitational freeze-in contribution, and no explicit UV operator or Froggatt–Nielsen charge assignment. Since the paper's central quantitative result is precisely the extreme smallness of these couplings, this missing verification is load-bearing. I request either adding the missing derivations or explicitly softening the a
[Sec. II after Eq. (7); Eqs. (12)–(14)] In the decoupling limit sinθ→0, Eq. (8) gives λ_sh→0, so the h2 state has no coupling to the SM. Yet Eqs. (12)–(13) use the equilibrium abundance Ȳ_h2 (Eq. 14) as the source for S and χ production. If h2 is not in thermal contact with the SM bath, the freeze-in calculation is not self-consistent. The text says 'we fine-tune the value so that such a contribution can be negligible' (SM contribution to χ), but the scan has no sinθ parameter and the decoupling limit is treated as exact. Please specify the sinθ (or, equivalently, the λ_sh) value used in the micrOMEGAs runs, show that h2 remains in equilibrium down to T∼m2, and verify that the SM-mediated χ production is simultaneously negligible. Otherwise the central numerical results are not supported.
[Abstract; Sec. IV C, Eq. (24), Figs. 6–9] The abstract's headline range 'λ_ds spans 10^-25 to 10^-13' does not match the scan. Equation (24) scans λ_ds ∈ [10^-24, 10^-10], and the smallest viable value shown is λ_ds ≈ 2×10^-24 (Fig. 6(c)); Sec. V summarizes the result as 'O(10^-20) level.' Either extend the scan to actually reach 10^-25 or correct the abstract and summary to state the range that is obtained. This is a central quantitative claim, not merely a typographical issue.

minor comments (3)

[Sec. IV C, paragraph before Fig. 6] The sentence 'χ production is obtained via the decay process χ→h2h2' should read 'h2→χχ' in the case mχ < m2/2.
[Appendix A, Eq. (A1)] The cross-section formula has unmatched parentheses and the logarithmic argument is difficult to parse. Please re-typeset and check the expression.
[Sec. V vs Abstract] The abstract quotes a lower bound 10^-25 for λ_ds, while Sec. V states 'λ_ds can be as tiny as O(10^-20) level.' These should be made consistent.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the FIMP parameter regions are relic-density constraints, not predictions; the abstract's naturalness verification is asserted but missing.

full rationale

The paper's central derivation is self-contained. The Lagrangian (Eq. 1), mass relations (Eq. 3), and Boltzmann equations (Eqs. 12–13) are stated explicitly, and the viable regions in Figs. 6–9 are obtained by imposing the external Planck relic-density datum Ω_DM h² = 0.12, with micrOMEGAs as the numerical solver. The coupling ranges for y_sf and λ_ds are presented as constraints that reproduce the observed abundance, not as independent predictions, so this is standard constraint-satisfaction rather than fitted-input-called-prediction. The relation v0 = mχ/y_sf is used openly to interpret the smallness of λ_ds, and the analytical estimates (Eqs. 20–23) are explicitly described as checks, with micrOMEGAs providing the actual results. Self-citations to Refs. [9,20,22] define the parent model, but the model is restated in full in Section II, so no load-bearing result is imported from those papers; no uniqueness theorem or ansatz is smuggled in via citation. I therefore find no circular step. Two manuscript-internal weaknesses should be flagged, but they are not circularity: (i) Abstract ¶3 claims 'We verify the theoretical consistency of these extreme values: the v0/v hierarchy is technically natural, λds remains 15–20 orders of magnitude above the gravitational freeze-in floor, and the suppression can originate from higher-dimensional operators, radiative generation, or the Froggatt–Nielsen mechanism,' but Sections II–V contain no RGE, symmetry-protection, or UV-completion computation; Section III only imposes perturbativity, unitarity, and vacuum-stability inequalities. This is an omitted proof, not a circular derivation. (ii) Section II admits 'we fine-tune the value' for the decoupling limit sinθ→0, which sits uneasily with the naturalness claim; again this is a consistency gap, not a circular one. The score of 1 reflects only the minor, non-load-bearing self-citation of the parent model and the otherwise self-contained nature of the freeze-in calculation.

Assumptions & free parameters 7 free parameters · 5 assumptions · 4 invented entities

The model is a hand-built Lagrangian with six free parameters; the headline coupling ranges are consequences of imposing Planck Ωh². The main unproven input is the technical naturalness of the resulting extreme hierarchy, asserted in the abstract but not demonstrated in the body.

free parameters (7)

y_sf = Allowed ranges: ~3e-13–1e-11 for mχ < m2/2; ~2e-8–2.5e-6 for mχ > m2/2 (m2 = 1 TeV)
Yukawa coupling fitted to Ωχ h² ≈ 0.12; through mχ = y_sf v0 it sets the large v0 that drives λ_ds suppression.
λ_ds = Allowed ranges roughly 2e-24–5e-11 depending on mass case
Portal coupling fitted by requiring ΩS h² ≈ 0.12; its tiny allowed values are the headline result.
λ_dh = 1e-14–1e-11 (chosen scan range; allowed values)
SM-Higgs portal coupling; random-scanned within a range chosen by hand 'for simplicity'.
mχ = 1 GeV–3 TeV (chosen scan grid)
DM mass scanned; allowed region constrained by relic density.
mS = 1 GeV–3 TeV (chosen scan grid)
DM mass scanned; allowed region constrained by relic density.
m2 = 1 TeV (and 2 TeV for comparison) fixed
New Higgs mass chosen by hand; all main scans use m2 = 1 TeV as benchmark.
sinθ = 0 (decoupling limit)
Mixing angle fine-tuned to zero so SM-mediated χ production is negligible; the text admits 'we fine-tune the value'.

assumptions (5)

domain assumption Initial DM abundance is negligible and χ/S never reach thermal equilibrium (freeze-in initial condition).
Used throughout Sec. IV to justify the Boltzmann equations with no back-reaction from DM.
domain assumption The observed DM relic density is entirely composed of χ and S.
Equations (12)-(13) sum to Ωχ + ΩS = 0.12; no other DM component is assumed.
standard math Standard FRW cosmology with standard entropy and Hubble rates.
Boltzmann equations (12)-(13) and Eq. (22) assume radiation-dominated expansion with g*S.
ad hoc to paper Decoupling limit sinθ→0 suppresses SM-Higgs production of χ.
Sec. II, 'we consider the decoupling limit with sinθ→0'; the introduction says the value is fine-tuned for simplicity. If not taken, χ receives additional SM-mediated production and the parameter maps change.
ad hoc to paper The extremely small y_sf and λ_ds are radiatively stable / technically natural.
Abstract asserts technical naturalness and possible UV origins, but no naturalness calculation or UV completion appears in the body; the claim is load-bearing for the consistency conclusion.

invented entities (4)

S (singlet scalar dark matter)
purpose: One of the two DM components, stabilized by Z2.
Inherited from refs. [20,22]; no direct detection signal because it is a FIMP; no external falsifiable handle used in this paper.
χ (Majorana fermion dark matter)
purpose: Second DM component, stabilized by Z4; mass from y_sf v0.
Its Yukawa is tuned to 1e-11 or below, making it invisible to current experiments; production is via h2 decays/annihilations internal to the model.
S0 (scalar with large vev v0 ~ 1e9–1e15 GeV)
purpose: Generates mχ and controls h2-mediated S production through v0.
No experimental signature in the decoupling limit; its vev is not anchored to any external observable.
h2 (second physical Higgs, mass m2 = 1–2 TeV)
purpose: Dominant mediator for χ and S freeze-in production.
With sinθ→0 it decouples from the SM, so no search channel is predicted; it is an internal mediator.

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Cite this review

Pith. "Pith review of FIMPs in a two-component dark matter model with $Z_2 \times Z_4$ symmetry." pith.science (2026). https://pith.science/paper/6FD6CBW5

@misc{pith2026260208359,
  author       = {Pith},
  title        = {Pith review of: FIMPs in a two-component dark matter model with $Z_2 \times Z_4$ symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6FD6CBW5}},
  note         = {Machine review of arXiv:2602.08359}
}

abstract

We investigate the FIMP-FIMP regime in a two-component dark matter model with a $Z_2\times Z_4$ symmetry, where a singlet scalar $S$ and a Majorana fermion $\chi$ serve as the dark matter candidates. A singlet scalar $S_0$ with vacuum expectation value $v_0$ generates the fermion mass through the relation $m_\chi=y_{sf}v_0$. We show that the tiny Yukawa coupling $y_{sf}$ needed to reproduce the observed relic abundance naturally leads to a large symmetry-breaking scale $v_0$, which induces an ultra-feeble portal coupling $\lambda_{ds}$ responsible for the production of $S$. We find that $\lambda_{ds}$ can reach values of $10^{-25}\lesssim\lambda_{ds}\lesssim10^{-13}$, while gravitational freeze-in provides an irreducible contribution at extremely small couplings. Our results demonstrate that the relic abundance constraint, combined with symmetry breaking and freeze-in dynamics, naturally drives the portal interaction responsible for scalar dark matter production into the ultra-feeble regime.