Phases of Dark Matter from Inverse Decays

Eric Kuflik; Hitoshi Murayama; Ronny Frumkin; Yonit Hochberg

REVIEW 2 major objections 5 minor 79 references

Phases of Dark Matter from Inverse Decays

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Dark matter can be produced by inverse decays of a heavier partner at couplings far below WIMP values, with the production phase surviving kinetic decoupling.

desk verdict Solid, careful extension of INDY dark matter; the NKE analysis is the real technical contribution, but the in-CE scaling's α_ann→∞ derivation needs a finite-coupling sanity check before you trust Eq. (19) in the unitarity-allowed window. read the letter →

arxiv 2504.16981 v1 pith:U7FWUCYW submitted 2025-04-23 hep-ph

Ronny Frumkin , Yonit Hochberg , Eric Kuflik , Hitoshi Murayama This is my paper · ORCID

classification hep-ph PACS 95.35.+d98.80.Cq

keywords inversedecaydarkmatterINDYBoltzmannequationschemicalequilibriumkineticphotonrelicabundancefreeze-out

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

The paper shows that inverse decays, in which a heavier dark particle turns into the dark-matter candidate plus a bath particle, can by themselves set the dark matter relic abundance. It maps the parameter space into distinct phases and identifies the INDY branch where the relic is set by the freeze-out of inverse decays, requiring decay couplings as small as $\alpha_{\rm decay}\sim m_\chi/m_{\rm pl}$ out of chemical equilibrium and $\alpha_{\rm decay}\sim m_\chi^{1+\Delta}/(T_{\rm eq}^{\Delta}m_{\rm pl})$ in chemical equilibrium. It then shows that relaxing kinetic equilibrium changes these couplings by at most about 20 percent for small mass splittings, so the phase structure survives. A renormalizable kinetically mixed dark photon and dark Higgs model realizes these phases and predicts visible dark photon signals in the mass range upcoming accelerator searches will probe.

What carries the argument

The central object is the linearized Boltzmann equation for $Y_\chi$ in the regime where $\psi$ is pinned to chemical equilibrium: $dY_\chi/dx + a(x)Y_\chi = b(x)$, with $a(x)$ the inverse decay rate and $b(x)$ the decay production term. Its closed solution, $Y_{\chi,\infty} = e^{-A_\infty}Y_\chi(1) + \int_1^\infty e^{A(\eta)-A_\infty} b(\eta)\,d\eta$, splits the INDY phase into two branches according to the ratio $w$ between the late-decay and initial-abundance contributions, and saddle-point evaluation of the two terms yields the two scaling laws. For the kinetic-equilibrium question, the non-integrated Boltzmann equation for the distribution function $\bar f_{\chi,q}$ plays the same role, with the same integrable structure in comoving momentum $q$.

What would settle it

Take the full coupled system (1) with $\alpha_{\rm ann}$ capped at the unitarity bound of order 100, use equilibrium initial conditions at $x=1$, and compute the $\alpha_{\rm decay}$ that reproduces the observed abundance for $m_\chi=1$ TeV and $\Delta=0.5$; if it deviates from Eq. (19) by more than the stated kinetic-equilibrium error budget of about 20 percent, the in-chemical-equilibrium INDY branch fails exactly where the paper relies on it. Observationally, a null result from a visibly decaying dark photon search covering the kinetic mixing values of Fig. 8 would exclude the benchmark model.

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Extended reading notes

Core claim

For a two-species dark sector governed by $\psi \leftrightarrow \chi+\phi$ and $\psi\psi\to\tilde\phi\tilde\phi$, the relic abundance can be fixed by the freeze-out of inverse decays rather than by annihilations. In this INDY regime, the heavier state $\psi$ stays in chemical equilibrium with the Standard Model bath while $\chi$ freezes out, and the relic abundance is set either by the $\chi$ present at early times (out-of-chemical-equilibrium branch, $w<1$) or by later decays of $\psi$ (in-chemical-equilibrium branch, $w>1$). These branches give the scaling laws $\alpha_{\rm decay}\sim m_\chi/m_{\rm pl}$ and $\alpha_{\rm decay}\sim m_\chi^{1+\Delta}/(T_{\rm eq}^{\Delta}m_{\rm pl})$, which are many orders of magnitude below standard WIMP couplings. The authors verify that departure from kinetic equilibrium shifts the required coupling by at most roughly 20 percent for $\Delta\le 0.2$, and they show that the same phases appear in a renormalizable $U(1)_d$ dark photon model with a small Yukawa coupling, where visible dark photon decays provide an experimental target.

Load-bearing premise

The central scaling laws assume the heavier particle $\psi$ stays in chemical equilibrium with the ordinary-matter bath for the whole epoch that sets the dark matter abundance, which requires its annihilation coupling to be large enough (effectively infinite) that the simple linearized equations apply.

Editorial extensions

If this is right

If inverse decays control the relic abundance, the required dark matter couplings can be orders of magnitude below WIMP values, making the dark matter naturally weak at direct and indirect detection.
The abundance can depend on initial conditions, but the required coupling changes only logarithmically when the initial yield at $x=1$ is varied over many orders of magnitude.
Departure from kinetic equilibrium changes the required $\alpha_{\rm decay}$ by no more than about 20 percent for $\Delta \le 0.2$, so the integrated-Boltzmann predictions remain valid in the relevant parameter space.
With zero initial abundance, the same inverse-decay system produces freeze-in and freeze-in-freeze-out phases, with FIFO couplings close to the INDY couplings.
The dark photon and dark Higgs realization predicts visibly decaying dark photons in the $\mathcal{O}(10\text{--}1000)$ MeV range, a window that upcoming accelerator searches can cover.

Reading between the lines

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

If the $\alpha_{\rm decay}\sim m_\chi/m_{\rm pl}$ scaling is generic, then any model with a bath-coupled heavier partner and a lighter dark matter state with small mass splitting is pushed to the same coupling line, giving a model-independent target for accelerator searches regardless of the annihilation details.
The logarithmic sensitivity to initial conditions implies that early-universe histories that either produce or deplete $\chi$ before $T=m_\chi$ will barely move the coupling prediction, so the experimental target is stable across many cosmological scenarios.
Falsifying the specific dark photon model would not kill the inverse-decay mechanism itself; a different way of diluting $\psi$, for example a chain of inverse decays, could keep the vertical branch alive at heavier masses.
A direct test of the in-chemical-equilibrium branch would be to solve the full coupled system with $\alpha_{\rm ann}$ capped at the unitarity bound for large $\Delta$ and masses above $10^4$ MeV, where the paper's approximation is most strained.

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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

Quick take: this is a careful, mostly convincing follow-up to the authors' INDY PRL. The genuinely new content is the w-criterion (Eq. 13) separating in- and out-of-chemical-equilibrium INDY, the closed-form scaling laws (16) and (19), and the momentum-resolved kinetic-equilibrium analysis of Section IV. The derivations are explicit and internally cross-checked; I checked the main algebraic steps and they hold up. The NKE section is a real technical contribution: the collision term (25) is derived rather than guessed, and the argument for why α_NKE ≈ α_KE, culminating in Eq. (41), is plausible and backed by Fig. 7. The model section maps the generic couplings onto a renormalizable dark photon/dark Higgs model with tree-level formulas, and the co-scattering check (52) is a nice touch. The citation base is thorough, and the heavy reliance on the authors' own Refs. [31] and [33] is honest self-extension, not circularity.

Where are the soft spots? The stress-test flags a genuine gap: Eq. (8), and therefore the in-CE scaling (19), is derived in the limit Y_ψ = Y_ψ^eq, i.e., α_ann → ∞. The paper does not quantify finite-α_ann corrections, and Fig. 4 indicates the in-CE region (w > 1) overlaps substantially with the dashed α_ann > 100 region. So there is a real chance Eq. (19) describes mostly unitarity-excluded parameter space. That said, it is not fatal: the out-of-CE branch (Eq. 16) carries the model phenomenology (Fig. 8 uses small Δ with w < 1), and the phase map is numerically supported. But a referee should ask for a finite-α_ann check in the solid region.

Other issues are minor. No public solver is released, so Figs. 2–7 can't be independently reproduced; a description of the numerical method would help. Section III.B.1 claims the log of Y_χ^eq(1)/Y_obs is of order unity; the log is actually large (order 10–20 or more), though the scaling (16) is logarithmic so the conclusion survives. The Fig. 1 caption has an inequality typo (α_decay > 10^{-24} should be <).

Bottom line: this deserves a serious referee. It's a solid, honest extension of a known mechanism, with one genuinely useful new tool (the NKE analysis) and one analytic branch whose unitarity-allowed validity is not fully pinned down. I'd send it to review, with a request for the numerical details and a finite-α_ann sanity check on the in-CE scaling.

Referee Report

2 major / 5 minor

Summary. The manuscript studies the two-channel Boltzmann system (Eq. (1)) describing the decay/inverse-decay process ψ↔χ+φ together with the ψψ→φ̃φ̃ annihilation, and maps the regions of (α_decay, α_ann, m_χ, Δ) that reproduce the observed DM abundance. It identifies and analyzes five phases: coannihilation via decays, INDY DM (both out-of-chemical-equilibrium and in-chemical-equilibrium), freeze-in, freeze-in/freeze-out, and freeze-out-and-decay. The central analytical results are the INDY coupling scalings α_decay∼m_χ/m_pl (Eq. (16)) and α_decay∼m_χ^{1+Δ}/(T_eq^Δ m_pl) (Eq. (19)), obtained from the linearized equation (8) and saddle-point integrals. The paper also studies the dependence on initial conditions, relaxes the kinetic-equilibrium assumption in Section IV, and constructs a renormalizable dark-photon/dark-Higgs model with phenomenological prospects, summarized in Fig. 8.

Significance. If the results hold, inverse-decay freeze-out is a self-consistent thermal origin of DM with couplings far below canonical WIMP values and with a falsifiable visible-dark-photon signature. The paper's strengths are the explicit analytic approximations, the numerical cross-checks in Figs. 3–7, the initial-condition analysis showing only logarithmic sensitivity of the coupling, and the non-kinetic-equilibrium treatment. The main caveat is that the in-chemical-equilibrium scaling and the kinetic-decoupling corrections are derived in the α_ann→∞ limit, and their validity in the unitarity-allowed finite-α_ann region is not demonstrated.

major comments (2)

[Section III.B, Eqs. (8) and (19), Fig. 4] The linearized equation (8) is obtained by setting Y_ψ=Y_ψ^eq, i.e., taking the α_ann→∞ limit of the second Boltzmann equation in (1). This approximation underlies the in-CE INDY scaling (19). In Fig. 4 the in-CE branch (w>1 in the right panel) is located in the large-m_χ, large-Δ portion of the left panel, exactly where the curves are dashed because the required α_ann exceeds 100, the unitarity-motivated cutoff used in the paper (Section III.A). For the unitarity-allowed solid portion of the in-CE branch, the paper does not quantify how finite α_ann≤100 modifies Y_ψ and hence the extracted α_decay, so the validity of Eq. (19) in the physical region is not established. Please add a numerical comparison of the full coupled system (1) with the Y_ψ=Y_ψ^eq reduction in the solid region, or explicitly state that the in-CE branch is a limiting-case result and clarify what remains of Eq. (19) in the allowed parameter space.
[Section IV, Eqs. (24)–(25)] The non-kinetic-equilibrium calculation is performed in the limit Y_ψ=Y_ψ^eq (α_ann→∞), as stated at the beginning of Section IV. The conclusion that NKE corrections to α_decay are small (up to 20% for Δ≤0.2) is therefore demonstrated only in the same limit in which the chemical-equilibrium reduction (8) is used, and not for the finite-α_ann points that populate the model parameter space, such as m_χ=1 GeV, Δ=0.15 in Fig. 8. Without a finite-α_ann NKE check, the robustness claim of Section IV is not fully supported for the parameter region used in Section V. Please extend the NKE comparison to finite α_ann≤100 or justify why the α_ann→∞ ratio is representative in that region.

minor comments (5)

[Eq. (17) and Appendix A] Eq. (17) and the definitions of a0 and b0 in Appendix A appear with missing superscripts in the rendered text (e.g., '3353/2' and '22π9/2'); please ensure the published version reads 3^3 5^{3/2}/(2^2 π^{9/2}).
[Eq. (6)] Eq. (6) appears to have the coannihilation factor inverted: for n=n_ψ+n_χ with n_χ/n_ψ=n_χ^eq/n_ψ^eq, the annihilation term should be ⟨σv⟩(n^2-n_eq^2)/(1+n_χ^eq/n_ψ^eq)^2, not the square of (1+n_χ^eq/n_ψ^eq). Please verify and correct.
[Section III.B and IV headings] There are several typos: 'equilirubium' after Eq. (10), 'equilbrium' in the Section III.B.2 heading, 'F reezeout' in the Section III.D heading, 'DEPAR TURE' in the Section IV heading, and 'direct direction' in Section V.C.
[Fig. 4 caption] The caption states that the dashed region indicates where α_ann exceeds 100 'along the horizontal phase,' but Fig. 4 displays only the vertical INDY branch; please clarify which curves are dashed and what 'horizontal phase' refers to.
[Section IV.B] The mass bound in the text is written as m_χ∼< 10^4 MeV; please write 10^4 MeV (or 10 TeV) explicitly for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the INDY scalings are derived from the stated Boltzmann equations and matched to the observed abundance, with the alpha_ann -> infinity assumption explicit and checked numerically.

full rationale

The derivation chain is self-contained. The central scaling laws, Eqs. (16) and (19), are obtained by (i) writing the two-channel Boltzmann equations (1) with stated definitions of alpha_decay and alpha_ann; (ii) in the large-alpha_ann limit reducing to the linear equation (8) with a(x) and b(x) explicitly defined in (9); (iii) integrating to (12); (iv) approximating a and b in the nonrelativistic limit, Eqs. (14) and (17); and (v) matching Y_chi,infinity to the observed Y_obs = c T_eq/m_chi. This is a normal inversion of a solved abundance formula, not a fit of a parameter to a closely related predicted quantity. The numerical full-BE solutions in Figs. 1 and 4 do not presuppose the analytic scalings; the scalings are compared with, not extracted from, those solutions. The NKE analysis (Eqs. 23-41) relaxes kinetic equilibrium and derives the same power-law structure from the saddle-point approximation, so it is also not a renaming. Citations to the authors' prior work [31] supply the original INDY proposal and the form of the BEs, but the equations and computations are stated in full here, and the new phase structure, initial-condition dependence, and model mapping are derived independently. The assumption that psi remains in chemical equilibrium (alpha_ann -> infinity) underlying Eq. (8), and the unitarity concern for the dashed part of Fig. 4, is an approximation-validity issue rather than a circular reduction: the paper states the assumption explicitly and checks the analytic results against numerical solutions of the full BEs. No load-bearing step reduces by construction to its own inputs.

Assumptions & free parameters 7 free parameters · 6 assumptions · 3 invented entities

The mechanism is defined by the two-channel Boltzmann system (1); the paper contributes the phase map, scaling laws, the kinetic-equilibrium robustness check, and the model realization, but everything downstream inherits the Maxwell-Boltzmann and detailed-balance statistics, the chemical equilibrium of psi with the bath, the equilibrium yield at x = 1, and the unitarity bound, none of which are derived here. The free parameters are the couplings and masses of the toy system, scanned to reproduce Y_obs; they are the content of the mechanism rather than hidden fits. The model introduces no new entities beyond the standard dark photon, dark Higgs, and two Dirac fermions, all with accelerator handles.

free parameters (7)

alpha_decay = scanned over roughly 10^-27 to 10^-7 depending on phase (Fig. 1)
Dimensionless decay coupling in Eq. (2). The central results are the values of alpha_decay that reproduce Y_obs; it is the scanned parameter of the mechanism, not a fit hidden inside a derivation.
alpha_ann = scanned over roughly 10^-6 to above 10^2; dashed regions of Fig. 4 exceed 100
Dimensionless annihilation coupling in Eq. (3). Sets the coannihilation versus INDY phase boundary; unitarity limits the viable range.
mass splitting Delta = (m_psi - m_chi)/m_chi = 0.05 to 0.50 in the figures
Controls the exponent in Eq. (19) and the rate maximum x* = 1/Delta. Scanned, not derived.
r = m_phi/m_chi = 0 for the KE results; up to 0.75 r_max in Fig. 7
The phi mass enters the decay rate; the KE phase map sets r = 0 and rescales alpha_decay by the phase-space ratio.
dark degrees of freedom g_chi = g_psi = 4 = 4
Fixed by hand in Section II; enters the rate normalizations and equilibrium yields at the O(1) level.
observed abundance parameterization = Y_obs = 0.54 T_eq/m_chi with T_eq = 0.8 eV
External input from Ref. [1]; every coupling requirement is defined relative to this value.
model parameters (y, e_d, lambda, mu, epsilon) = benchmarks such as lambda = 1, mu = 2 (m_psi - m_chi) or 1.1 (m_psi - m_chi), e_d = 0.32 or 0.58 in Fig. 8
Chosen to map onto alpha_decay and alpha_ann via Eqs. (50) and (51). These are construction choices for the benchmark, not fits to data.

assumptions (6)

domain assumption Maxwell-Boltzmann statistics and detailed balance for the rate equations (1)-(4)
The entire phase map uses MB Boltzmann equations; quantum-statistical corrections are not quantified (Section II).
domain assumption psi and the SM bath are in chemical equilibrium at all relevant times in the INDY phase
Basis for the linear equation (8); introduced in Section III.B with alpha_ann effectively infinite.
domain assumption chi is in kinetic equilibrium with the SM for Sections II and III
Stated in Section IV as an assumption, then relaxed; the relaxation is treated only for constant g* and alpha_ann to infinity.
domain assumption Equilibrium initial yields at x = 1 (Y_chi(1) = Y_chi^eq(1), Y_psi(1) = Y_psi^eq(1))
The out-of-CE INDY relic is dominated by the Y_chi(1) term of Eq. (12), so the pre-x = 1 history is an input; Fig. 5 shows the coupling is logarithmically insensitive to it.
standard math Unitarity restricts alpha_ann to order 100 or less
Used in Section III to bound the valid INDY mass range (Fig. 4 dashed regions); from Refs. [4] and [33].
standard math Saddle point validity with eta_s > 1 for the in-CE scalings (18) and (38)
Appendix A extends the lower integration limit to minus infinity, which requires the saddle point to lie inside the integration domain.

invented entities (3)

U(1)_d dark photon A_d independent evidence
purpose: Mediates the psi -> chi A_d decay and the 2 psi -> 2 A_d annihilation; kinetic mixing epsilon connects it to the SM for detection.
The (m_A_d, epsilon) plane is directly probed by visible dark photon searches (LHCb, NA62, Belle II, FASER, SHiP, DarkQuest), as mapped in Fig. 8.
Dark Higgs h_d independent evidence
purpose: Breaks U(1)_d, sets m_A_d via Eq. (46), and is kept massive enough that psi -> chi h_d is forbidden (Eq. 49).
Indirectly probed through the same dark photon searches; its decay rates set a lower bound on epsilon (Section V.A). No direct search exists, hence the weaker independent handle.
Dirac fermions chi (dark matter) and psi (unstable parent) independent evidence
purpose: chi is the dark matter candidate; inverse decays of psi control its abundance.
The authors note direct production of psi at accelerators as a future signal (Section V.C); chi's own elastic scattering (Eq. 53) is far below current direct-detection reach.

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Cite this review

Pith. "Pith review of Phases of Dark Matter from Inverse Decays." pith.science (2026). https://pith.science/paper/U7FWUCYW

@misc{pith2026250416981,
  author       = {Pith},
  title        = {Pith review of: Phases of Dark Matter from Inverse Decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U7FWUCYW}},
  note         = {Machine review of arXiv:2504.16981}
}

read the original abstract

Inverse decays are an interesting avenue for producing dark matter in the early universe. We study in detail various phases of dark matter parameter space where inverse decays control its abundance, expanding on our work of INDY dark matter and going beyond. The role of initial conditions and the impact of departure from kinetic equilibrium are investigated as well. We show how these inverse decay phases can arise in theories of a kinetically mixed dark photon and dark Higgs, with promising prospects for detection at upcoming experiments.

Figures

Figures reproduced from arXiv: 2504.16981 by the authors.

**Figure 1.** A. Coannihilating via decays First, we consider the phase obtained for large αdecay, which is demonstrated by the horizontal curves of constant αann towards the right side of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: intersect. Numerically we find that the condition for either FI or FIFO is not fulfilled when ∆ < 0.1 and mχ ∼ keV. A comparison between the decay couplings yielding the FIFO case αFIFO and those yielding INDY DM αINDY is shown in the right panel of [PITH_FULL_IMAGE:f…

**Figure 7.** Figure 7: decreases with the increase of mϕ. We learn that the NKE corrections to the coupling values are rather small. The following paragraphs are dedicated to explaining the proximity of the obtained couplings along with other features in [PITH_FULL_IMAGE:figures/full_fig_p0…

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

Discussion (0). Continue with ORCID to comment.

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INDYs out of chemical equilibrium ( w< 1) Here ψ decays can be neglected, and the relic abun- dance of χ is determined by the Yχ (1) term in Eq. (12). Yχ departs from chemical equilibrium early on, and the reaction rate is maximal at x∗ = ∆−1. This can be seen by approximating a(x) in the non-relativistic (NR) limit as a(x)≈ 3 π r10 g∗ gψ gχ ¯β (1 + ∆)5/2...

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Light-Dark

MeV currently constrain ϵ ≲ 10−3− 10−4 [38–47], and most of the relevant parameter space is expected to be probed by future experiments, as demonstrated in the plot [39, 47–57] (see Ref. [66] for a recent compilation of existing searches and future projections for visibly deca...

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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INDYs out of chemical equilibrium ( w< 1) Here ψ decays can be neglected, and the relic abun- dance of χ is determined by the Yχ (1) term in Eq. (12). Yχ departs from chemical equilibrium early on, and the reaction rate is maximal at x∗ = ∆−1. This can be seen by approximating a(x) in the non-relativistic (NR) limit as a(x)≈ 3 π r10 g∗ gψ gχ ¯β (1 + ∆)5/2...

work page

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This can be seen for example in the right panel of Fig

INDYs in chemical equilbrium ( w> 1) For large values of mχ and ∆, the coupling αdecay is large enough such that the inverse decay rate a(x) keeps the abundanceYχ close to its equilibrium value atx∼ 1. This can be seen for example in the right panel of Fig. 2. 6 10-19 10-18 10-17 10-16 10-15 αdecay 10-10 10-9 10-8 Y∞ mχ = 1 GeV ∆ = 0.05 Yχ(1) = 10−2Y eq χ...

work page

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(12) and (27) dominate the relic abundance

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work page

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In this case, ψ’s decays dominate the relic abundance

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work page

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