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

Hao Sun; XinXin Qi

A two-component dark matter model with Z2×Z4 symmetry produces the observed relic density via freeze-in for both particles, with the scalar coupling λ_ds reaching values as low as 10^-25, roughly fifteen orders below conventional FIMP coupl

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2026-08-03 03:18 UTC pith:6FD6CBW5

load-bearing objection Plausible incremental FIMP scan for the authors' own Z2×Z4 model, but the abstract's headline λds≈10^-25 is not supported by the paper's own figures or scan resolution. the 3 major comments →

arxiv 2602.08359 v3 pith:6FD6CBW5 submitted 2026-02-09 hep-ph

FIMPs in a two-component dark matter model with Z₂ times Z₄ symmetry

XinXin Qi , Hao Sun This is my paper

classification hep-ph PACS 95.35.+d

keywords freeze-inFIMPtwo-component dark matterZ2×Z4 symmetryscalar singletMajorana fermionrelic densitysinglet vacuum expectation value

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.

This paper claims that a two-component dark matter model, containing a singlet scalar S and a Majorana fermion χ stabilized by a Z2×Z4 symmetry, can generate the full observed dark matter abundance through the freeze-in mechanism. The authors show that both dark matter masses remain viable across almost the entire range from 1 GeV to 3 TeV, and that the scalar quartic coupling λ_ds can be as small as 10^-20 to 10^-25. The key reason is that the fermion mass arises from a large singlet vacuum expectation value v0, which strongly amplifies h2-mediated S production, forcing λ_ds to be extremely tiny while still allowing h2 to dominate S production when m_S is below half the h2 mass. If correct, this establishes an unusually large hierarchy in couplings that still yields the right dark matter density.

Core claim

The paper establishes that, in the decoupling limit with sinθ = 10^-4, the freeze-in production of the two dark matter candidates can be treated independently, and the relic density of each is controlled by a small set of parameters: mχ, y_sf, m2 for χ, and mS, λ_ds, λ_dh, plus the ratio mχ/y_sf = v0 for S. Depending on whether mχ and mS lie below or above m2/2, four distinct production regimes arise. By numerically solving the Boltzmann equations, the authors find that the allowed parameter space covers DM masses from 1 GeV to 3 TeV, with y_sf constrained between about 10^-14 and 2.5×10^-6 and λ_ds ranging from about 10^-24 up to 10^-10 depending on the regime. The central new result is tha

What carries the argument

The central object is the singlet vacuum expectation value v0, defined by the ratio mχ/y_sf. It generates the fermion mass after spontaneous symmetry breaking and simultaneously controls the strength of h2-mediated S production: the decay width Γ(h2→SS) scales as λ_ds² v0²/m2 and the annihilation h2h2→SS also grows with v0. Because freeze-in requires y_sf to be tiny, v0 becomes very large, which boosts S production and forces λ_ds to be correspondingly small. This ratio-symmetric mechanism explains both the ultra-feeble coupling and the persistent dominance of h2 channels.

Load-bearing premise

The paper assumes that SM-Higgs-mediated production of χ is negligible by fixing sinθ = 10^-4, and states that it 'fine-tunes' the value to make this contribution negligible, but it never quantitatively demonstrates that the h1→χχ and SM-annihilation channels are subdominant over the entire scanned parameter space.

What would settle it

For a representative point such as mχ = 100 GeV, y_sf = 10^-9, and sinθ = 10^-4, compute the χ yield from h1 decay and SM annihilation using the same Boltzmann equations; if the resulting Ωχ exceeds about 1% of the observed density, the decoupling limit assumption fails, and the claimed viable regions for λ_ds and y_sf would need to be revised.

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

If this is right

The observed relic density can be matched for both DM components across a wide mass range, from 1 GeV to 3 TeV, with no need for a thermal WIMP-like interaction.
For mχ < m2/2, the Yukawa coupling y_sf is bounded to roughly 3×10^-13 to 10^-11 when h2→χχ dominates χ production; for mχ > m2/2, y_sf can be as large as 2.5×10^-6 via h2h2→χχ.
The scalar coupling λ_ds can be as small as 2×10^-24 for light S while remaining the dominant source of S production through h2 decay, contradicting the usual expectation that such tiny couplings would make the channel irrelevant.
The two DM components do not convert into each other appreciably, so their relic densities add independently and each can be tuned separately to achieve the total observed abundance.
The heavy Higgs h2 can play the dominant role in S production even when λ_ds is fifteen orders below conventional FIMP values, which sharpens predictions for hidden-sector searches at colliders.

Where Pith is reading between the lines

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

The same v0-enhancement mechanism should generalize to any freeze-in model where a fermion dark matter mass is generated by a large vacuum expectation value: the effective couplings of scalars to that vev get amplified, driving quartic couplings to ultra-tiny values, and this hierarchy may be a generic feature rather than a special conspiracy.
Because h2→SS with such a tiny λ_ds produces a very narrow decay width, the h2 state could be long-lived on collider timescales; searches for displaced vertices or missing-energy signatures from h2 decay may offer a testable consequence of this parameter region.
A concrete check of the paper's decoupling assumption is to recompute the χ yield keeping h1-mediated processes (proportional to (y_sf sinθ)^2) for representative points; if those become non-negligible at the highest scanned y_sf values, the allowed parameter space may shrink.
The authors fix λ_dh in the range 10^-14 to 10^-11; extending this range to larger values could alter the lower bound on λ_ds, so the reported 10^-24 bound should be understood as conditioned on that choice.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Plausible incremental FIMP scan for the authors' own Z2×Z4 model, but the abstract's headline λds≈10^-25 is not supported by the paper's own figures or scan resolution.

read the letter

The genuinely new thing here is the observation that v0 = mχ/ysf can be enormous for tiny ysf, which boosts h2-mediated S production and forces λds down to ~10^-20 or below while h2 still dominates. That is a real, though incremental, model-building point, and it is worth knowing about.

The paper does several things well: the four mass hierarchies are treated systematically, the analytic freeze-in estimates in Eqs. (20)-(23) are internally consistent, and the paper is honest that the relic-density constraint is being used as a filter rather than as a prediction. The 'large v0 suppression' mechanism is physically sensible and the plots make the case clearly.

The soft spots are real but fixable. The abstract claims λds can reach O(10^-25), while Section V says O(10^-20) and Fig. 6(c) shows a lower bound of 2×10^-24. That is a five-order-of-magnitude spread in the headline number, and the random scan over a 5D log space does not guarantee reaching the extreme corner (ysf ~ 10^-14, mχ ~ 500 GeV) where such a small λds might live. So the true minimum is not demonstrated, and the abstract is simply not supported by the paper's own results. The neglect of SM-mediated χ production through sinθ is asserted rather than quantified; for sinθ = 10^-4 and ysf ≤ 2.5×10^-6 the h1 contribution scales as (ysf sinθ)^2 and is almost certainly negligible, but a one-line estimate would remove any doubt. No Micromegas inputs or code are provided, so the numerical scan is not independently reproducible.

None of this is fatal. The central mechanism — large v0 suppressing λds while h2 stays dominant — holds up. The paper is for people working on two-component FIMP model building, especially scalar + fermion DM with a heavy singlet. It deserves a serious referee, but the referee should insist that the abstract be reconciled with the body and that the scan either be provided or the claimed minimum be softened to what is actually shown.

Referee Report

3 major / 3 minor

Summary. This paper studies freeze-in production of two-component dark matter in a model with Z2×Z4 symmetry, where a singlet scalar S and a Majorana fermion χ are the DM candidates and a singlet S0 with vev v0 generates mχ. The authors fix sinθ=10^-4, identify four mass hierarchies with respect to m2/2, give approximate analytic estimates (Eqs. (20)–(23)), and solve the Boltzmann equations with Micromegas to map parameter space consistent with Ωh2=0.12. They report viable masses in [1 GeV, 3 TeV] and claim λds can be extremely small: the abstract quotes O(10^-25), while Section V quotes O(10^-20) and Fig. 6(c) gives a lower bound of 2×10^-24. The paper concludes that h2-mediated processes still dominate S production even for such tiny λds.

Significance. If the results hold, the model extends the two-component FIMP framework into a regime of ultra-small scalar portal couplings and shows that the new Higgs can still govern DM production. The separation of the parameter space into four mass cases is useful, and the analytic estimates provide a valuable cross-check on the numerics. The main strengths are the systematic treatment of the four hierarchies and the explicit analytic expressions for the yields. However, the abstract's headline value for λds is not supported by the body, and the neglect of SM-mediated χ production is not quantitatively justified. These issues affect the central quantitative claims but appear fixable within a revision.

major comments (3)

[Abstract / Sec. V / Eq. (24) / Fig. 6(c)] The abstract states that λds can reach O(10^-25), 'roughly fifteen orders of magnitude below conventional FIMP values'. The body does not support this value: Sec. V says λds can be as tiny as O(10^-20) level, Eq. (24) scans λds from 10^-24 to 10^-10, and Fig. 6(c) reports an allowed lower bound of 2×10^-24 for case i. Since the abstract's number is the paper's headline novelty and is not reproducible from the plotted data, it must be corrected or demonstrated with a dedicated scan. The random scan cannot establish a true minimum; the lower bound appears set by the scan range, not by physics.
[Sec. IV.A, Eq. (13)] The Boltzmann equation for χ (Eq. 13) includes only h2-mediated terms. The paper states in Sec. II that SM-mediated production is negligible because sinθ is fine-tuned, but no calculation is shown. Production via h1→χχ and XX→χχ scales as (y_sf sinθ)^2; with y_sf up to 2.5×10^-6 (cases iii/iv) and sinθ=10^-4, y_sf sinθ ≈ 2.5×10^-10, which is not obviously negligible compared with the h2 processes considered, particularly where h2→χχ is phase-space suppressed. Please compute the SM-mediated yield (or a rigorous upper bound) over the scanned parameter space and show it is subdominant, or include it in Eq. (13).
[Sec. IV.C, Eq. (24)] The quoted viable parameter spaces are obtained from a random scan over the ranges in Eq. (24). The bounds of these ranges are not derived: in particular, the upper bound y_sf ≈ 2.5×10^-6 is estimated from Fig. 3, and the text says the exact value is not needed. Because the paper's quantitative conclusions (e.g., the allowed λds intervals in Figs. 6–9) depend on the completeness of the scan, the authors should either derive the scan boundaries or state how sampling density affects the reported lower/upper bounds. Without this, the reported intervals are not demonstrated to be complete.

minor comments (3)

[Sec. IV.C] The sentence 'In the case of mχ < m2/2, χ production is obtained via the decay process χ→h2h2' appears to have the initial and final states reversed; it should be h2→χχ.
[Eq. (12)] Eq. (12) uses ar{Y}_X for the SM bath, but ar{Y}_X is not defined in Eq. (14) and X is not a single species. Please define it or specify the sum over SM species.
[General] No Micromegas model files or input parameters are provided, so the numerical results cannot be reproduced. Adding the model implementation or a table of key scan settings would improve reproducibility.

Circularity Check

0 steps flagged

No circular derivation: the relic-density scan is a filter, not a prediction; the only self-reference is to the authors' earlier model papers, which is not load-bearing. The abstract's O(10^-25) lambda_ds claim conflicts with the paper's own scan range and figures, but that is a reproducibility error, not circularity.

full rationale

The paper does not derive a prediction from fitted inputs. It states six free parameters (Eq. (16)), fixes sin(theta)=10^-4, and numerically solves the Boltzmann equations (Eqs. (12)-(15)) with Micromegas, then retains points satisfying Omega h^2 = 0.12 (Figs. 6-9). The allowed lambda_ds values are scan outputs, not inputs renamed as predictions. The model Lagrangian, mass matrix, and decay/annihilation rates are given explicitly (Eqs. (1)-(8), (15), (A1)), so the calculation is self-contained. The citations to the authors' own Refs. [20,22] only identify the Z2 x Z4 framework; the present paper re-derives the scalar potential and Boltzmann equations, so these self-citations are not load-bearing. The weakest quantitative point is not circular: the abstract's headline 'lambda_ds can reach O(10^-25)' is five orders of magnitude below the scan lower bound of Eq. (24) (lambda_ds in [10^-24, 10^-10]) and below the plotted lower limit (2x10^-24) of Fig. 6(c), while Section V states O(10^-20). This means the headline minimum is not reproducible from the paper's own scan, and the 'fifteen orders' number is unsupported; however, this is an internal consistency/support failure, not a reduction of an equation to itself. The assumption that SM-mediated chi production is negligible at sin(theta)=10^-4 is asserted without a quantitative check ('we fine-tune the value so that such a contribution can be negligible'), but again is an unquantified approximation rather than circular reasoning. No step in the derivation chain equates a fitted parameter to its prediction by construction.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 3 invented entities

The calculation rests on standard freeze-in formalism, the assumption that h2 is in thermal equilibrium, and the derived v0 hierarchy. No genuinely new particle or force is introduced by this paper beyond the earlier model; the numerical results are parameter fits to the relic-density constraint.

free parameters (6)

sinθ = 10^-4 (fixed by hand)
Chosen to enforce the decoupling limit and suppress SM-mediated DM production; not scanned.
m2 = 1 TeV (fixed for scans; some plots also show 2 TeV)
Benchmark new-Higgs mass; the four-case classification and all headline bounds assume m2=1 TeV.
λds = Case-dependent allowed ranges, e.g. (2×10^-24, 10^-19] for mχ<m2/2, mS<m2/2
Central tiny-coupling claim; values are selected by requiring Ωh2=0.12 in the scan.
ysf = Case-dependent ranges, e.g. (3×10^-13, 10^-11] for mχ<m2/2; [2×10^-8, 2.5×10^-6] for mχ>m2/2
Determines χ abundance and the derived vev v0=mχ/ysf; fitted to relic density.
λdh = Scanned in [10^-14, 10^-11]
Controls SM-Higgs-mediated S production; lower bound chosen so SM contribution is subdominant in the headline cases.
v0 = mχ/ysf = Derived, up to ~10^14 GeV
Not an input but the load-bearing scale: large v0 enhances h2→SS and h2h2→SS, forcing λds to ultra-small values.

axioms (6)

domain assumption Standard freeze-in Boltzmann equations are valid for S and χ.
Adopted from Refs [3,25]; the paper does not derive the formalism and assumes negligible backreaction.
domain assumption The new Higgs h2 is in thermal equilibrium with the SM bath, so its equilibrium abundance \bar Y_h2 sources DM production.
Needed for Eqs (12)-(14); not quantitatively demonstrated for sinθ=10^-4, though plausible.
domain assumption Initial DM abundances are negligible and S–χ conversion processes are negligible.
Stated in Sec. I and IV A; required for the freeze-in computation to be the whole story.
domain assumption Narrow-width decay approximation for h1,h2 → DM processes.
Implicit in the Γ*Y terms in Eqs (12)-(15); valid when parent particles are in equilibrium.
domain assumption h1 is the SM Higgs with m1=125 GeV and v=246 GeV.
Used throughout for m1 and SM couplings; standard but not stated explicitly in the scan setup.
standard math Vacuum stability and perturbative unitarity constraints in Sec. III are sufficient.
The copositivity conditions and unitarity bounds are standard constraints applied to the scalar potential.

invented entities (3)

Singlet scalar S (DM candidate) no independent evidence
purpose: One of the two dark matter components, stabilized by Z2×Z4.
Borrowed from the authors' earlier model (Refs [20,22]); no new falsifiable handle is provided in this paper.
Majorana fermion χ (DM candidate) no independent evidence
purpose: Second dark matter component; mass generated by S0 vev.
Borrowed from the earlier model; no new observable signal is predicted beyond the relic-density scan.
Singlet scalar S0 with vev v0 no independent evidence
purpose: Spontaneously breaks the dark-sector symmetry and generates mχ=ysf v0; creates the large v0 hierarchy.
From the prior model; the large-v0 regime is central to the tiny-λds effect but has no independent experimental evidence.

pith-pipeline@v1.3.0-alltime-deepseek · 15256 in / 20655 out tokens · 229161 ms · 2026-08-03T03:18:46.052287+00:00 · methodology

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read the original abstract

We study the FIMP-FIMP regime of a two-component dark matter model with $Z_2\times Z_4$ symmetry, where a singlet scalar $S$ and a Majorana fermion $\chi$ are the DM candidates. A singlet $S_0$ with vev $v_0$ generates $\chi$'s mass via $m_\chi=y_{sf}v_0$, linking the two sectors: the same $v_0$ that sets $m_\chi$ also controls $h_2$-mediated $S$ production. The relic density constraint on $\chi$ forces $y_{sf}$ to be tiny, driving $v_0$ to $10^9$--$10^{15}$~GeV and requiring the portal coupling $\lambda_{ds}$ to be correspondingly suppressed. We classify the parameter space into four cases according to whether $m_\chi$ and $m_S$ lie above or below $m_2/2$. Numerically solving the Boltzmann equations with \textsc{micrOMEGAs}, we find viable DM masses in $[1~\mathrm{GeV},\,3~\mathrm{TeV}]$ for $m_2=1$~TeV. The coupling $\lambda_{ds}$ spans $10^{-25}$--$10^{-13}$, up to fifteen orders of magnitude below conventional FIMP values---a consequence of the $v_0\gg v$ hierarchy. We verify the theoretical consistency of these extreme values: the $v_0/v$ hierarchy is technically natural, $\lambda_{ds}$ 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. Together with earlier WIMP-WIMP and mixed WIMP-FIMP studies, this work shows that the same $Z_2\times Z_4$ Lagrangian accommodates DM interaction strengths spanning over twenty-five orders of magnitude.