{"id":"88b7061d-7123-44b3-a843-78d948a9a15b","arxiv_id":"2504.16981","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"Dark matter produced by the freeze-out of inverse decays has a full phase structure, with couplings as small as m_chi over the Planck mass, that is robust to kinetic decoupling and testable through dark photon searches.","lead":"This paper maps the regions of dark matter parameter space where inverse decays, rather than annihilations, set the relic abundance, and checks that the map survives when dark matter is out of kinetic equilibrium with the early-universe plasma. It then builds a dark photon model realizing these phases and shows where current and future accelerator searches could see them.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"INDY scalings rely on ψ staying in chemical equilibrium (α_ann → ∞), but the in-CE branch (large m_χ, Δ) falls in the α_ann > 100 region of Fig. 4; Eq. (19) may only describe unitarity-excluded parameter space unless finite-α_ann corrections are small.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing condition: ψ must remain in chemical equilibrium with the SM bath, which is implemented as α_ann → ∞ in Eq. (8) and again in Section IV's NKE check. My closer reading of Fig. 4 confirms that the in-CE branch (w > 1) is populated in the dashed part of the curves, where the effective α_ann exceeds 100, i.e., outside the unitarity bound quoted by the paper itself. This makes the finite-α_ann robustness a genuinely load-bearing concern rather than a technical footnote: if the full two-channel system at α_ann ≤ 100 gives a substantially different α_decay, the central scaling (19) and the model's predicted parameter space (Fig. 8) would shift. The paper's internal cross-checks, the saddle-point derivations, and the momentum-resolved NKE treatment are strong independent support, but they all inherit the same α_ann → ∞ limit, so they do not resolve this concern. The proposed numerical test directly probes the gap. Because the reader already conditioned acceptance on this assumption, I do not change the verdict; I would make the finite-α_ann check an explicit condition of acceptance.","tokens_in":22223,"tokens_out":5079,"duration_ms":47284,"concrete_test":"Take representative (m_χ, Δ) points on the solid (α_ann ≤ 100) parts of the Fig. 4 INDY curves, especially where w > 1, and solve the full two-channel BEs (1) with α_ann = 10 and α_ann = 100. Compare the α_decay that reproduces Y_obs with the α_decay from the linearized Eq. (8)/Eq. (12) (Y_ψ = Y_ψ^eq). If α_decay(full)/α_decay(lin) exceeds 1.2 anywhere in the allowed region, the in-CE scaling (19) and the derived vertical branch are not robust to finite annihilation couplings.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (8) is derived in Section III.B by setting Y_ψ = Y_ψ^eq, i.e., α_ann → ∞ in the second Boltzmann equation. This is the backbone of both INDY scaling relations: Eq. (16) (out-of-CE) and Eq. (19) (in-CE) both follow from the linearized solution (12). The in-CE branch (w > 1) is the one where the b(η) decay term dominates, and it is precisely the large-m_χ, large-Δ portion of Fig. 4 that is drawn dashed, i.e., where the α_ann needed to keep ψ in equilibrium along the phase boundary exceeds 100 and would violate the unitarity bound. For the solid (unitarily allowed) part of the same curves, the paper does not demonstrate that finite α_ann leaves Eq. (19) numerically accurate; Section IV's kinetic-equilibrium relaxation also explicitly works in the α_ann → ∞ limit. If the deviation is significant at α_ann ≤ 100, the 'vertical branch' is not described by Eqs. (16)/(19) in the allowed window, and the central mechanism's phenomenology (Fig. 8) would need re-mapping.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22406,"tokens_out":17262,"duration_ms":150752,"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":[{"comment":"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":"Section III.B, Eqs. (8) and (19), Fig. 4"},{"comment":"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.","section":"Section IV, Eqs. (24)–(25)"}],"minor_comments":[{"comment":"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}).","section":"Eq. (17) and Appendix A"},{"comment":"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":"Eq. (6)"},{"comment":"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.","section":"Section III.B and IV headings"},{"comment":"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":"Fig. 4 caption"},{"comment":"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.","section":"Section IV.B"}],"recommendation":"major_revision","confidential_remarks":"The finite-α_ann issue in major comment 1 is the main obstacle to acceptance. If the authors can demonstrate numerically that Eq. (19) and the NKE corrections remain accurate for α_ann≤100, the paper would be suitable for publication after revision. If not, the physical claims should be restricted to the out-of-CE branch, which still includes the model benchmarks shown in Fig. 8. The paper is a genuine extension of the authors' earlier PRL [31], with the phase classification, initial-condition analysis, and NKE treatment adding nontrivial value."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhere 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.\n\nOther 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 <).\n\nBottom 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.","headline":"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.","tokens_in":23164,"tokens_out":10162,"would_cite":true,"duration_ms":84256,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["95.35.+d","98.80.Cq"],"model":"deepseek-v4-flash","headline":"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.","keywords":["inverse decay dark matter","INDY","Boltzmann equations","chemical equilibrium","kinetic equilibrium","dark photon","relic abundance","freeze-out"],"falsifier":"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.","tokens_in":21851,"feed_emoji":"🌌","tokens_out":8430,"duration_ms":78746,"temperature":0.7,"pith_summary":"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.","feed_headline":"Inverse decays can set dark matter's abundance with tiny couplings","feed_subtitle":"Two freeze-out branches set the relic density, with dark-photon signals visible to upcoming experiments.","key_machinery":"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 $q$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Proposes the INDY mechanism and defines the two-channel Boltzmann system with $\\alpha_{\\rm decay}$ and $\\alpha_{\\rm ann}$ that this paper extends.","marker":"[31]"},{"why":"Supplies the coannihilation formalism and Griest-Seckel freeze-out approximation used for the horizontal phase and Eq. (7).","marker":"[4]"},{"why":"Establishes the freeze-in framework used for the FI and FIFO phases with $Y_\\chi(0)=0$.","marker":"[26]"},{"why":"Provides the non-integrated Boltzmann treatment for departure from kinetic equilibrium that underlies Section IV.","marker":"[6]"},{"why":"Gives the kinetic-equilibrium-relaxing Boltzmann equation and phase-space treatment for decay processes used in the NKE analysis.","marker":"[28]"},{"why":"Shows how a chain of inverse decays can raise the mass bound, clarifying where the INDY branch applies.","marker":"[33]"},{"why":"Gives the instantaneous freeze-out approximation used to derive Eq. (7) and compare with numerical results.","marker":"[34]"},{"why":"Provides the temperature-dependent $g_*$ and $g_{*s}$ values used in the numerical solutions.","marker":"[32]"}],"fun_headline_variants":["Inverse decays alone set dark matter's abundance","Two freeze-out branches let inverse decays fix dark matter","Dark matter from inverse decays: no annihilation required","Tiny couplings suffice: inverse decays set dark matter","Inverse decay freeze-out sets dark matter with tiny couplings"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Inverse decays alone set dark matter's abundance","Two freeze-out branches let inverse decays fix dark matter","Dark matter from inverse decays: no annihilation required","Tiny couplings suffice: inverse decays set dark matter","Inverse decay freeze-out sets dark matter with tiny couplings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00093,"raw_usage":{"total_tokens":3939,"prompt_tokens":861,"completion_tokens":3078,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":477,"completion_tokens_details":{"reasoning_tokens":3003}},"tokens_in":477,"tokens_out":3078,"duration_ms":22686,"temperature":1.0,"reasoning_tokens":3003,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:54:52.625151+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"In this case, ψ’s decays dominate the relic abundance","cited_arxiv_id":null,"evidence_quote":"Supplies the coannihilation formalism and Griest-Seckel freeze-out approximation used for the horizontal phase and Eq. (7)."}],"review_version":1}