{"id":"f953fffb-2fc8-408b-b98f-eb9062970276","arxiv_id":"2608.12480","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"In the small-coupling limit of the singlet-doublet dark matter model, the relic density is set by co-scattering, freeze-in, or SuperWIMP decays rather than ordinary freeze-out, and the observable signals move from direct detection to long-lived particle searches at colliders.","lead":"This paper maps how the singlet-doublet model of dark matter can produce the observed dark matter abundance through co-annihilation, co-scattering, freeze-in, and late decays (SuperWIMP), focusing on tiny Yukawa couplings. A smart generalist should read it because it shows where this economical dark matter candidate leaves collider signatures like displaced vertices and disappearing tracks, which future LHC and muon-collider searches could find.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central quantitative claims rest on an unpublished kinetic-equilibrium approximation; if that approximation breaks down at the few-ten-percent level, the co-scattering maps and freeze-in yields shift.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing concern, and I agree: the integrated equations that generate all the relic-density maps assume kinetic equilibrium, and that assumption is supported only by an unpublished companion paper. There is no internal logical inconsistency in the paper: the 1-ODE versus 2-ODE comparison and the analytic co-annihilation fit are mutually consistent, and the factor-of-four underestimate is a genuine consequence of the equations as written. The unresolved issue is external support for the kinetic-equilibrium premise. This is especially acute in the freeze-in regime, where the dark matter distribution is populated out of equilibrium and an integrated Y_1 implicitly assumes a kinetic-equilibrium shape. Because the paper explicitly flags this as a deferred result, a careful reader cannot verify the central numerical outputs from the manuscript alone. The unexplained disagreement with Ref. [31] and the absence of code and data artifacts add uncertainty but are secondary; the kinetic-equilibrium assumption is the single point on which the quantitative claims most directly depend. The concern is addressable rather than fatal, so the appropriate verdict remains CONDITIONAL, and my read does not change the reader's verdict.","tokens_in":26259,"tokens_out":5177,"duration_ms":51588,"concrete_test":"Run a full momentum-dependent Boltzmann solver (e.g., DRAKE or an independent phase-space code) for the co-scattering benchmark of Fig. 2: M_S = 300 GeV, y_2/y_1 = 0.5, y_1 = 6e-8, with M_D chosen so the integrated 2-ODE system (15) gives Omega h^2 = 0.12. Compare the final relic density and the time-resolved Y_1(x) against the integrated result, and repeat at y_1 = 1e-7 and 1e-6 to probe the CA-CS boundary. If the full solution differs by more than 10% in any benchmark, the kinetic-equilibrium premise is not safe and Figs. 1, 4, and 5 need recomputation; if it differs by less than 10%, the integrated treatment and the companion-paper estimate are confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest load-bearing premise is that kinetic equilibrium inside the dark sector is maintained throughout the evolution, so that the integrated Boltzmann equations (9)-(10) and the freeze-in estimate (16) are valid. The manuscript explicitly conditions on this in Sec. III: 'Because this effect is small, for the remainder of this paper, we will work in the approximation that kinetic equilibrium within the dark sector is maintained,' with footnote 2 attributing the less-than-or-about-10% estimate to the authors' unpublished companion paper [33]. No derivation, no momentum-dependent spectra, and no independent check are provided for that estimate. This matters because the paper's central quantitative claims are defined by comparisons between integrated treatments: the co-annihilation/co-scattering boundary is set where Eqs. (13) and (9)-(10) differ by 10%, and the reported factor-of-four underestimation for y1 = 6e-8 is computed within the same integrated framework. If the true momentum-dependent solution departs from the integrated one at even the few-ten-percent level in the relevant benchmarks, the boundary and the freeze-in yields (where the dark matter distribution is far from equilibrium by construction) could shift. The concern is not that kinetic equilibrium obviously fails; it is that the load-bearing numerical output depends on an unshown, self-cited approximation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the singlet-doublet Majorana fermion dark matter model in the regime of small Yukawa couplings, where the relic abundance can be set by co-annihilation, co-scattering, freeze-in, or the SuperWIMP mechanism. The authors derive and solve coupled Boltzmann equations for the singlet-like dark matter and the doublet-like sector, Eqs. (9) and (10), and compare them with the single effective equation (13) that assumes chemical equilibrium between the two sectors. They show that in the co-scattering regime the single-equation treatment underestimates the relic abundance by roughly a factor of four for a benchmark with y1 = 6e-8, map the boundaries between production regimes in the (m_chi1, y1) plane, and compute the mass splittings needed to reproduce Omega h^2 = 0.12. For freeze-in, they decompose the relic density into contributions from decays, co-scattering, and SuperWIMP late decays, and derive BBN constraints on the doublet lifetimes. They also discuss collider signatures, including displaced vertices and disappearing tracks, and estimate current and future search sensitivities.","tokens_in":26507,"tokens_out":6062,"duration_ms":58150,"significance":"If the central results hold, the paper provides a useful unified map of dark matter production mechanisms in an economical weak-scale model, and it makes a specific, falsifiable quantitative claim: the common single-ODE co-annihilation treatment underestimates the relic abundance in the small-Yukawa regime, and the coupled two-ODE system is required. The comparison in Fig. 2 is internally consistent and clearly illustrates the breakdown of chemical equilibrium. The paper is also commendable for using standard public tools (SARAH, SPheno, micrOMEGAs) and for openly stating where its results agree or disagree with previous work. The central physics is not circular: the relic density is computed from Boltzmann evolution, not assumed. The main unresolved issue is that the integrated equations rest on a kinetic-equilibrium approximation whose numerical support is delegated to an unpublished companion paper, which is load-bearing for the quantitative boundaries and contours presented here.","major_comments":[{"comment":"The integrated Boltzmann equations (9) and (10), and the freeze-in estimate (16), are derived under the assumption that kinetic equilibrium within the dark sector is maintained. The manuscript states that a full solution of the Boltzmann equation differs from the integrated one by less than about 10%, but this estimate is attributed to the authors' unpublished companion paper [33], with no derivation, no momentum-dependent spectrum, and no independent check. This is load-bearing because the co-annihilation/co-scattering boundary is defined as the point where the 1-ODE and 2-ODE results differ by 10%, and the factor-of-four underproduction claim is computed within the same integrated framework. If kinetic corrections are at the few-ten-percent level, the boundaries in Figs. 1, 4, and 5 and the freeze-in yields could shift. I recommend that the authors either include the estimate in an appendix or otherwise provide an independent quantitative justification, and state the sensitivity of the main maps to this approximation.","section":"Sec. III, around Eq. (8) and footnote 2"},{"comment":"The description of the freeze-in calculation is unclear about which equations are actually solved. The text says 'We use Eq. (16) accounting for one-directional processes of decay of doublet states, SM-annihilations, and co-scattering,' but Eq. (16) is derived by fixing Y2 = Y2eq and neglecting Y1 in the first line of Eq. (15). That approximation cannot produce the Y2 freeze-out shown in Fig. 6, the late-time SuperWIMP transfer, or the OmegaCS and OmegaSW decomposition in Fig. 7. Please specify whether the numerical results come from solving the coupled system (15) with momentum-averaged rates, or from Eq. (16) plus Eq. (24), and explain how SM-annihilation contributions are included without double counting. This matters for the reliability of Figs. 6-8 and for the claimed BBN constraints.","section":"Sec. IV.B and Eq. (16)"}],"minor_comments":[{"comment":"The word 'Majorna' in the first paragraph of Sec. II appears to be a typo for 'Majorana'.","section":"Sec. II"},{"comment":"The phrase 'how variation in the ratio of ratio the Yukawa couplings' should be 'how variation in the ratio of the Yukawa couplings'.","section":"Sec. IV.A"},{"comment":"The caption lists the panel masses as '150, 200, 300, and 250 GeV (clockwise from upper left),' but the panel labels and the text indicate the ordering is 150, 200, 250, 300 GeV clockwise; please correct the caption.","section":"Fig. 5 caption"},{"comment":"The caption says the mass splittings are '70,110,500,200 GeV (clockwise from upper-left),' while the text and panel labels indicate 70, 110, 200, 500 GeV in row-major order; please make the caption and figure consistent.","section":"Fig. 7 caption"},{"comment":"The legend entry for the lower curve appears to read 'y1 = 10^7' but should be 'y1 = 10^-7'; please check the rendered figure.","section":"Fig. 4 right panel"},{"comment":"Reference [76] appears without a collaboration name; please add the collaboration or authors for completeness.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central physics claim is credible and the paper is likely publishable after revision. My main concern is the heavy reliance on the unpublished companion paper [33] for the kinetic-equilibrium approximation that underpins the quantitative results. If that paper is not yet available, I would want the authors to provide a self-contained estimate or a sensitivity check in the present manuscript. I do not see a circularity problem, and the disagreement with Ref. [31] about the pure-doublet mass is presented as a quantitative discrepancy rather than as an unsupported dismissal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nWhat you should know first: this is the most complete map yet of the small-Yukawa regime of the singlet-doublet Majorana dark matter model, and its central comparison—single-ODE vs coupled-ODE relic density—holds up. For y1 = 6e-8 the effective co-annihilation equation underestimates the abundance by roughly a factor of four, because chemical equilibrium between the singlet and doublet sectors is lost before freeze-out. That is clearly demonstrated in Fig. 2, with the rate plots on the right showing why.\n\nWhat's actually new: the full y2/y1 plane, not just the Z-blind slice of Ref. [31]; the custodial y+ -> 0 behavior; the freeze-in/SuperWIMP regime with BBN constraints; and the classification of chi2/chi3 collider signatures as prompt, displaced, or stable across the parameter space. The analytic lifetime estimates are useful and checkable. The paper is also transparent about its one quantitative disagreement with Ref. [31], over the maximum pure-doublet mass (roughly 850 GeV here vs 1.1 TeV there). That discrepancy is not resolved, and it is worth chasing, but it does not touch the small-coupling results.\n\nSoft spots, in proportion. The weakest load-bearing assumption is kinetic equilibrium within the dark sector. The paper defers the justification to an unpublished companion paper [33], claiming the full Boltzmann solution differs by less than ~10%. No derivation or independent check appears. If that estimate fails at the few-ten-percent level, the co-scattering boundary and the freeze-in yields shift. I do not think this is fatal—it is the standard approximation in most relic calculations, and the claim is consistent with generic estimates in [2] and the singlet-triplet analysis [40]—but it should be pinned down before publication. Second, no code or data artifacts are provided, so the numerics cannot be independently reproduced. For a paper whose output is a set of parameter-space maps, that matters more than usual; I would ask for the micrOMEGAs inputs and scripts. Third, the LHC sensitivity estimates are explicitly rough; the paper does not claim a full recast, so this is a minor caveat, not a flaw.\n\nBottom line: this is a solid, useful reference for anyone working on singlet-doublet, Higgsino-like, co-scattering, or freeze-in dark matter. I would send it to a serious referee, conditional on the authors either providing support for the kinetic-equilibrium approximation or clearly labeling the maps as approximate to that level, and on an explicit discussion of the [31] discrepancy.","headline":"Solid, useful map of the small-Yukawa singlet-doublet DM parameter space; the central chemical-equilibrium comparison is right, but a key kinetic-equilibrium estimate is deferred to an unpublished companion.","tokens_in":27084,"tokens_out":3475,"would_cite":true,"duration_ms":30619,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that in the small-Yukawa-coupling regime of the singlet-doublet dark matter model, the relic abundance is set by co-scattering, freeze-in, or SuperWIMP processes, requiring two coupled Boltzmann equations rather than the…","keywords":["singlet-doublet dark matter","co-scattering","freeze-in","SuperWIMP mechanism","relic abundance","Boltzmann equations","collider signatures"],"falsifier":"Run a full momentum-dependent Boltzmann calculation for a representative co-scattering benchmark, such as $M_S = 300$ GeV, $y_1 = 6\\times 10^{-8}$, $y_2/y_1 = 0.5$, with $\\Delta m$ chosen to give $\\Omega h^2 = 0.12$ in the two-equation treatment. If the resulting relic density differs from the integrated-equation result by more than roughly 10 percent, or if the boundary between the co-annihilation and co-scattering regions in the $(m_{\\chi_1}, y_1)$ plane shifts measurably, the paper's regime map and its underestimate claim would need revision.","tokens_in":26019,"feed_emoji":"🌌","tokens_out":7154,"duration_ms":60509,"temperature":0.7,"pith_summary":"This paper argues that in the Singlet-Doublet Majorana dark matter model with tiny Yukawa couplings, the observed dark matter abundance is not generally set by the usual co-annihilation freeze-out. Instead the relic density is controlled by conversion processes between a singlet-like dark matter state and heavier doublet-like states: when the conversion rate falls below the Hubble rate before freeze-out, the dark matter and the doublet sector must be evolved with two coupled Boltzmann equations. Using only the single co-annihilation equation in this regime underestimates the final abundance by roughly a factor of four. The paper maps which production mechanism operates across the mass and coupling plane, and shows that for very small Yukawa couplings freeze-in and the SuperWIMP mechanism can reproduce the observed density. If correct, the result means weak-scale dark matter can hide from direct detection while remaining potentially discoverable through collider signals of the doublet sector.","feed_headline":"Co-scattering sets dark matter density where freeze-out fails","feed_subtitle":"The single co-annihilation formula undercounts the density by a factor of four in the new regime.","key_machinery":"The load-bearing object is the pair of integrated Boltzmann equations, Eqs. (9) and (10), for the comoving yields $Y_1$ of the singlet-like dark matter and $Y_2$ of the doublet-like sector ($\\chi_2^0$, $\\chi_3^0$, $\\chi^\\pm$). The two equations are connected by the conversion rate $\\Gamma_{2\\to 1}$ of Eq. (12), which sums doublet decays and co-scattering processes such as $\\psi \\to \\chi\\, \\mathrm{SM}$ and $\\chi\\, \\mathrm{SM} \\to \\psi\\, \\mathrm{SM}$. When $\\Gamma_{2\\to 1}/H \\gg 1$, the sub-sectors stay in chemical equilibrium and the pair collapses to the single co-annihilation equation (13); when $\\Gamma_{2\\to 1}/H \\sim O(1)$ near freeze-out, the full pair must be integrated. The paper's small-coupling expressions show that the dominant co-scattering couplings scale as $v y_+/\\Delta m$, so reducing the mass splitting compensates for a smaller Yukawa coupling and keeps the conversion rate able to set the abundance.","core_discovery":"The central claim is that the Singlet-Doublet fermion model can match the observed dark matter relic density through four distinct histories depending on the Yukawa couplings $y_1$ and $y_2$: standard co-annihilation freeze-out when $y_1 \\gtrsim 10^{-6}$; co-scattering (conversion-driven freeze-out) when chemical equilibrium between the singlet and doublet sub-sectors breaks before freeze-out; freeze-in for $y_1 \\sim 10^{-12}$; and SuperWIMP production when long-lived doublets decay after their own freeze-out. The paper's load-bearing quantitative point is that in the co-scattering regime the two-sector evolution must be obtained from the coupled equations (9) and (10); the single effective equation (13) used for co-annihilation assumes chemical equilibrium that no longer holds. For the benchmark $M_S = 300$ GeV, $y_2/y_1 = 0.5$, $y_1 = 6\\times 10^{-8}$, the single-equation treatment underestimates the relic abundance by about a factor of four. With the coupled equations, the paper shows that every point in the shaded region of its Fig. 1 can realize the observed density for some mass splitting $\\Delta m = M_D - M_S$, with the required splitting decreasing as the Yukawa coupling shrinks.","pith_inferences":["The same 'break chemical equilibrium between sub-sectors and evolve them separately' logic should apply to other feebly coupled singlet-plus-strongly-coupled-doublet or -triplet models; the exact boundary will depend on the analog of $\\Gamma_{2\\to 1}$.","If the companion full-Boltzmann solution were to show kinetic-decoupling effects exceeding the quoted roughly 10 percent for parameter points near the co-scattering boundary, the regime maps in Figs. 1, 4, 5, 7, and 8 would need to be recomputed; the authors flag this as the main technical caveat.","The model offers a concrete target for a long-lived-particle trigger at the LHC: the $\\chi_2^0/\\chi_3^0$ asymmetry, with one state prompt and the other displaced, could be a clean handle, and a future muon collider could cover the full doublet mass range; conversely, colliders alone may never see the singlet, so a positive signal would motivate but not prove a dark matter interpretation.","One can partially test the regime boundary without cosmology: measuring the doublet mass splitting and Yukawa couplings at a collider would predict which production mechanism operates, and future precision electroweak measurements could corroborate the doublet sector up to around 500 GeV."],"forward_implications":["Relic-density calculations for this model in the small-coupling region must solve the coupled equations (9) and (10); results that use only the effective single equation inherit the factor-of-four underestimate in the co-scattering regime.","A single weak-scale model can realize four different production mechanisms with the same observed abundance, and the operative mechanism changes continuously with the Yukawa coupling.","For small Yukawa couplings, direct and indirect detection are strongly suppressed, so collider signatures of the doublet sector—prompt decays, displaced vertices, and soft disappearing tracks—become the primary observational windows.","Freeze-in calculations must include the thermal masses of the electroweak mediators; doing so removes the apparent reheat-temperature dependence and makes production dominated by $T\\sim M_D$.","Big Bang nucleosynthesis constraints place significant pressure on three-body-decay freeze-in realizations, especially through the long-lived $\\chi_3^0$, while the SuperWIMP contribution grows with $M_D$ and can alone account for the full relic density above a critical singlet mass."],"supporting_citations":[{"why":"Defines co-annihilation and the effective single-equation treatment that the paper contrasts with the coupled pair.","marker":"[1]"},{"why":"Introduces conversion-driven freeze-out and co-scattering, the regime the paper shows is missed by co-annihilation.","marker":"[2]"},{"why":"Provides the co-scattering and conversion-driven freeze-out formalism used to identify when chemical equilibrium breaks down.","marker":"[3]"},{"why":"Gives the freeze-in formalism and analytic decay-abundance estimate used in the freeze-in section.","marker":"[4]"},{"why":"Establishes the SuperWIMP mechanism and the abundance formula used to estimate late-decay contributions.","marker":"[5]"},{"why":"Prior study of co-annihilation in this model; supplies the small-Yukawa-limit masses and couplings used throughout.","marker":"[29]"},{"why":"Earlier freeze-in study of the singlet-doublet model for keV-scale dark matter; supplies decay-width and BBN context for the weak-scale case.","marker":"[30]"},{"why":"Companion paper estimating that kinetic decoupling affects the relic density by less than about 10 percent, the basis for the kinetic-equilibrium assumption.","marker":"[33]"},{"why":"Provides the integrated Boltzmann equation formalism and the conversion rate $\\Gamma_{2\\to 1}$ used in Eqs. (9)-(12).","marker":"[40]"},{"why":"Numerical package used for three-body decay widths and relic computations, including the thermal-mass treatment of the $t$-channel singularity.","marker":"[42]"}],"fun_headline_variants":["Co-scattering fixes dark matter density where freeze-out fails","Four paths to dark matter abundance beyond standard freeze-out","Co-scattering rescues dark matter density where freeze-out cannot","Standard co-annihilation formula falls short by factor four in co-scattering"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire analysis rests on the assumption that kinetic equilibrium inside the dark sector is maintained, so that the integrated Boltzmann equations (9) and (10) are valid; the paper cites its own unpublished companion study for the estimate that kinetic decoupling affects the relic density by less than about 10 percent.","fun_headline_variants_meta":{"raw":{"variants":["Co-scattering fixes dark matter density where freeze-out fails","Four paths to dark matter abundance beyond standard freeze-out","Co-scattering rescues dark matter density where freeze-out cannot","Standard co-annihilation formula falls short by factor four in co-scattering"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001004,"raw_usage":{"total_tokens":4234,"prompt_tokens":923,"completion_tokens":3311,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":3240}},"tokens_in":539,"tokens_out":3311,"duration_ms":20485,"temperature":1.0,"reasoning_tokens":3240,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:07:45.514293+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a full momentum-dependent Boltzmann calculation for a representative co-scattering benchmark, such as $M_S = 300$ GeV, $y_1 = 6\\times 10^{-8}$, $y_2/y_1 = 0.5$, with $\\Delta m$ chosen to give $\\Omega h^2 = 0.12$ in the two-equation treatment. If the resulting relic density differs from the integrated-equation result by more than roughly 10 percent, or if the boundary between the co-annihilation and co-scattering regions in the $(m_{\\chi_1}, y_1)$ plane shifts measurably, the paper's regime map and its underestimate claim would need revision.","supporting_citations":[],"review_version":1}