{"id":"8045c2c6-45b2-4a1f-8a4e-485b07fd0195","arxiv_id":"2607.14210","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":9,"one_line_summary":"In the SU(6)/Sp(6) composite Higgs model with a scotogenic Z2, three of four neutral pseudo-Goldstone dark matter candidates can reproduce the observed relic density, while the SU(2)L triplet candidate fails; spin-1 resonances matter in part of the parameter space.","lead":"This paper maps where dark matter could hide in a class of composite-Higgs models in which the dark matter particle is a pseudo-Goldstone boson stabilized by a leftover Z2 symmetry. It finds that three of the four candidate particles can explain the observed dark matter abundance, and that heavy spin-1 resonances from the new strong force affect the relic-density calculation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'only three' claim overreaches: Δ0 is excluded only within a simplified scan with Mχ≤1.5 TeV and no mixing; with Δ0–η3 mixing or higher masses it could plausibly enter the relic-density band.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing issue: the simplifying assumptions (free masses, decoupling of most pNGBs, no mixing, truncated potential) are not quantitatively justified, and the Δ0 exclusion is the part of the central claim most sensitive to those assumptions. My analysis agrees, and I further note the Mχ range cutoff as an independent amplifier of the concern. The paper itself flags the simplifying nature of the potential (§2.1) and asserts robustness without proof (§5), which is precisely the kind of limitation the reviewing rule asks to flag. No additional independent concern outweighs this one; the spin-1 resonance treatment is detailed and the three viable candidates are supported within the scan. Since the reader already recommended a conditional verdict with the same rationale, my stress-test does not change the verdict: the paper should be accepted only if the Δ0-exclusion claim is softened to 'no viable region in the simplified scan' or made robust against mixing and higher Mχ. Releasing the numerical code would also help, but the core correctness issue is the overstated uniqueness claim.","tokens_in":24321,"tokens_out":6102,"duration_ms":64673,"concrete_test":"Run a dedicated scan for the Δ0 candidate with Mχ extended to 5000 GeV and a Δ0–η3 mixing angle δ added as a free parameter (e.g., δ∈[0,π/4]), using the same micrOMEGAs pipeline as in §3.2. If any sampled point yields Ωh²=0.120±0.012 and passes LZ2025, then the abstract's 'only three' claim is falsified; if the posterior remains empty, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's headline claim that only three of the four Z2-odd pNGBs can explain the relic density is not established by the presented evidence. The Δ0 exclusion rests on the MCMC scan in §3.2, which restricts Mχ to [250,1500] GeV (Table 5) and assumes no pNGB mixing (assumption (iii), §2.1) together with a truncated seven-operator potential (eq. (2.4)). A pure SU(2)L triplet annihilates efficiently via electroweak gauge bosons, so its relic density is below the observed value at Mχ=1 TeV; however, the one-dimensional scan (fig. 2b) shows Ωh² increasing with Mχ, so the observed band could plausibly be reached above the scanned upper edge. More importantly, Δ0–η3 mixing is allowed by the Z2 and would dilute the triplet fraction of the DM state, weakening the gauge-annihilation channels that make the pure triplet under-abundant and thereby raising Ωh². The paper's assertion in §5 that such omissions 'would change details but not overall features' is unsupported; footnote 1 also concedes that vacuum stability is not fully checked. Thus 'only three' is better read as 'three within the scanned simplified slice', and the Δ0 exclusion needs a robustness argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies dark matter in a class of composite Higgs models with a residual Z2 symmetry, focusing on the SU(6)/Sp(6) coset. Four neutral Z2-odd pNGB states (η3, Δ0, η4, φ0) are identified as potential DM candidates. Using FeynRules/CalcHEP/micrOMEGAs and MCMC scans, the authors conclude that only three of these (η3, η4, φ0) have parameter regions reproducing the observed relic density, while Δ0 does not. They further find that spin-1 resonances can significantly affect the relic abundance, apply LZ 2025 direct-detection limits, and sketch LHC signatures.","tokens_in":24801,"tokens_out":8242,"duration_ms":85527,"significance":"If the exclusions are robust, the paper provides a valuable systematic survey of DM candidates in a concrete composite Higgs setup and highlights an often-neglected role of spin-1 resonances. The numerical pipeline is described in detail and is reproducible in principle; the use of current LZ 2025 limits and the explicit MCMC methodology are strengths. However, the headline claim 'only three of them can produce a relic density' is stronger than the presented scan evidence: it depends on the finite scan range and on the assumption of no pNGB mixing. The Δ0 exclusion in particular needs a robustness argument before the statement can be taken as a model-level result.","major_comments":[{"comment":"The abstract and §5 state that 'only three' of the four Z2-odd pNGBs can explain the relic density. This is not established by the evidence in §3.2. The Δ0 exclusion comes from an MCMC scan with Mχ ∈ [250,1500] GeV (Table 5) under assumption (iii), which excludes all pNGB mixing. Fig. 2b shows Ωh² for Δ0 increasing with Mχ, so the observed band may simply lie above the scanned upper edge. More importantly, Δ0–η3 mixing is not forbidden by the Z2 and could arise from terms omitted from eq. (2.4); such mixing would reduce the triplet fraction of the DM state, weaken the electroweak gauge-annihilation channels, and raise Ωh². The assertion in §5 that omitted mixing 'would change details but not overall features' is unsupported by any quantitative estimate. The claim should be qualified as 'within the simplified slice considered', or the Δ0 exclusion should be backed by an extended scan and","section":"§3.2, Table 5; §2.1 assumption (iii); Abstract"},{"comment":"Unlike the other three candidates, no corner plot, chain length, acceptance rate, or posterior information is shown for Δ0. The paper reports 'we did not find any valid parameter combination' but gives no details on how the Δ0 chain was initialized ('Each chain was initialized from a phenomenologically viable point'), how many points were generated specifically for Δ0, or why the chain did not explore higher Mχ. Since the claim is an exclusion, the absence of this documentation makes the result difficult to verify. A dedicated Δ0 scan over an extended mass range, or at least explicit statistics for the Δ0 chain, is needed.","section":"§3.2, 'We did not find any valid parameter combination for the Δ0 DM candidate'"}],"minor_comments":[{"comment":"ξ = M_A/M_V is listed as an independent parameter in the spin-1 sector, but it does not appear in the MCMC scan vector Θ of eq. (3.2) nor in Table 5. The value used in the scans should be stated explicitly.","section":"§2.3 and Table 5"},{"comment":"The caption's sentence 'The gray band is considered as consistent with the observed DM relic density within the adopted' appears incomplete. Also, the curves for Δ0 in panels (a)-(d) should be identified more clearly, since the text discussion refers to all four candidates.","section":"Fig. 2 caption"},{"comment":"The footnote admits that vacuum stability, including charge-breaking minima, is not fully checked. This caveat should be carried into the conclusions so that the quoted viable regions are not over-interpreted.","section":"Footnote 1, §5"},{"comment":"The implementation is said to be 'available from the authors upon request'. For reproducibility, a public repository with the FeynRules model files and the MCMC scan inputs would be preferable.","section":"§3.2, code availability"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope and the numerical study is useful, but the central 'only three' claim is currently a scan-range/assumption-dependent statement rather than a robust model prediction. The revision should either extend the Δ0 scan and include a mixing robustness test, or carefully qualify the abstract and conclusions. With that change, the paper would be suitable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a look if you work on composite Higgs dark matter. The new content is a first dedicated MCMC scan of the SU(6)/Sp(6) scotogenic model of ref. [42], carried out with a sensible pipeline (FeynRules, CalcHEP, micrOMEGAs) and with the 2025 LZ limit imposed explicitly. The spin-1 resonance contribution to the relic density is demonstrated concretely, not just mentioned. The paper is markedly honest about its simplifying assumptions — free pNGB masses, decoupled extra states, no mixing, a seven-operator truncation of the potential — and it even concedes, in footnote 1, that vacuum stability is not fully checked. That transparency is a real plus.\n\nWhere I part company is the abstract's strongest claim: 'only three of them can produce a relic density consistent with observations.' The evidence is that no valid MCMC point was found for the Δ0 candidate in a scan with Mχ between 250 GeV and 1.5 TeV and with no mixing. That is a statement about the scanned slice, not about the model. Fig. 2b shows Ωh² for Δ0 rising with Mχ, so the observed band could plausibly be reached above the scan edge. More importantly, Δ0–η3 mixing is allowed by the Z2 and would dilute the triplet's gauge-annihilation efficiency, pushing Ωh² upward. The paper's hand-wave that such effects 'would change details but not overall features' is unsupported. That is the one place a referee should push.\n\nThe rest of the scan is plausible within the stated assumptions. The η3, η4, and φ0 viable regions hang together, and the direct-detection projections are useful. Two smaller gripes: the numerical implementation is only 'available from the authors upon request,' which is weak for a scan paper, and the relic density is used as a likelihood target, so the viable regions are fits rather than predictions. Neither is fatal, but the first is easy to fix.\n\nBottom line: this is a competent and useful paper for model-builders interested in composite Higgs DM, but it should not be the last word on Δ0. I would send it to peer review. The right referee will ask for a softer wording or a robustness check (extend the mass range, add Δ0–η3 mixing), plus a code release. With those changes it would be a solid reference.","headline":"New and useful scan of the SU(6)/Sp(6) scotogenic DM sector, but the 'only three' claim overreaches the Δ0 exclusion.","tokens_in":25254,"tokens_out":4502,"would_cite":true,"duration_ms":47792,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A residual Z2 symmetry in a composite Higgs model leaves exactly three viable dark matter candidates.","keywords":["composite Higgs","scotogenic model","dark matter relic density","pseudo-Nambu-Goldstone boson","SU(6)/Sp(6)","Z2 symmetry","spin-1 resonances","direct detection"],"falsifier":"Repeat the relic-density calculation with the full 14-pNGB effective potential, including mixing among Δ0, η3, η4, φ0 and non-decoupled heavier pNGBs, and look for a point where Δ0 is the lightest Z2-odd state, Ωch² = 0.120 ± 0.012, and the direct-detection cross section lies below the current limit. If such a point exists, the paper's conclusion that only three candidates work is false.","tokens_in":24186,"feed_emoji":"⚛️","tokens_out":6097,"duration_ms":53928,"temperature":0.7,"pith_summary":"The paper argues that the SU(6)/Sp(6) composite Higgs model, with a fermionic ultraviolet completion, naturally contains a stable Z2-odd sector whose lightest pseudo-Nambu-Goldstone boson can be dark matter. Among the four neutral candidates—η3, the electroweak-triplet component Δ0, η4, and the doublet component φ0—Markov-Chain-Monte-Carlo scans find that only η3, η4, and φ0 can reproduce the observed relic density while surviving current direct-detection limits; Δ0 annihilates too efficiently through electroweak gauge bosons and always undershoots the relic density. The scans also show that heavy spin-1 composite resonances, not just the Higgs and electroweak bosons, contribute substantially to (co-)annihilation in a large fraction of the viable parameter space. The paper's point is that this model class offers concrete, testable dark matter candidates with a shared origin in the same strong dynamics that produces the Higgs, and that the candidates can be distinguished by future direct-detection and collider measurements.","feed_headline":"Only three of four dark matter candidates pass the relic-density test","feed_subtitle":"In the SU(6)/Sp(6) composite Higgs model, spin-1 resonances drive freeze-out and the triplet candidate fails.","key_machinery":"The argument rests on the SU(6)/Sp(6) coset, whose spontaneous breaking yields fourteen pNGBs; among them the Z2-odd sector contains a triplet Δ, a bi-doublet Φ, and singlets η3 and η4, giving four neutral candidates (Δ0, φ0, η3, η4). The effective potential is truncated to seven quartic operators, with pNGB masses treated as free parameters. Co-annihilation is included by pairing η3 with Δ and η4 with φ, and spin-1 resonances are incorporated through the hidden-symmetry approach as vector and axial-vector states that mix with the electroweak gauge bosons. The MCMC scan explores an eight-dimensional parameter space (Mχ, ΔM, cχχhh, cππhh, θ, gVππ, g̃, MV) and imposes the relic density within","core_discovery":"In the concrete SU(6)/Sp(6) model studied, the unbroken Z2 symmetry embedded in the custodial Sp(6) subgroup stabilizes the lightest of four neutral pNGB states. Treating pNGB masses, a set of quartic couplings, and the overall mass scale of spin-1 vector and axial-vector resonances as free parameters, the authors compute relic density and direct-detection observables with an eight-dimensional MCMC scan. The result: η3, η4, and φ0 each have parameter regions with Ωch² = 0.120 ± 0.012 after imposing current direct-detection bounds, while Δ0 does not, because its SU(2)L triplet nature makes annihilation into W and Z pairs too strong. Spin-1 resonances are not a minor correction: resonant s-cha","pith_inferences":["The Δ0 exclusion looks like a structural feature rather than a scan artifact: because the triplet couples directly to electroweak bosons, its annihilation is too efficient for a thermal relic; the same reasoning would apply to any SU(2)L-triplet pNGB in this class.","The decisive unexamined hinge is the no-mixing assumption. Extending the scalar potential to include the full NLO operator set and mixing among the four Z2-odd states could shift the surviving regions or even revive Δ0; this is the most direct test of the paper's central claim.","Top-partner resonances, which the authors leave to future work, should be expected to participate in t-channel annihilation for the heavier candidates and could alter the preferred mass ranges or open new resonant features.","A testable extension would be to scan with more than two co-annihilating multiplets: the sharp ΔM bounds found here may reflect the truncated spectrum rather than a robust prediction."],"forward_implications":["The electroweak-triplet candidate Δ0 cannot alone match the observed relic density; any model that wants a pNGB triplet as dark matter needs additional annihilation-suppressing structure.","Relic-density calculations for this model class that omit spin-1 resonances miss a substantial portion of the allowed parameter space; resonant and threshold effects near the resonance mass can change the density by a large factor.","For η4 and φ0 dark matter, viable freeze-out requires the co-annihilating partners to be nearly degenerate (ΔM ≲ 10 GeV) and masses above roughly 500 GeV; for η3, the preferred splitting is larger, around 25 GeV.","Direct-detection experiments at the level of about 10^-48 to 10^-49 cm² would probe Higgs-portal couplings as small as about 10^-3, mapping out much of the surviving parameter space.","The residual Z2 symmetry forces every sector—pNGBs, spin-1 resonances, and top-partners—to contain Z2-odd states, which gives the model class a generic collider signature of large missing transverse momentum."],"fun_headline_variants":["Three of four dark matter candidates survive in composite Higgs","Spin-1 resonances enable freeze-out in scotogenic composite Higgs","Triplet pNGB fails relic density, leaving three viable in SU(6)/Sp(6)","Composite Higgs dark matter: spin-1 resonances lead to three survivors"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire relic-density map, including the Δ0 exclusion, rests on the assumption that the four Z2-odd pNGBs do not mix with each other and that all other pNGBs are too heavy to participate; if those states mix or are not decoupled, the viable regions—and the list of viable candidates—could change.","fun_headline_variants_meta":{"raw":{"variants":["Three of four dark matter candidates survive in composite Higgs","Spin-1 resonances enable freeze-out in scotogenic composite Higgs","Triplet pNGB fails relic density, leaving three viable in SU(6)/Sp(6)","Composite Higgs dark matter: spin-1 resonances lead to three survivors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00107,"raw_usage":{"total_tokens":4306,"prompt_tokens":721,"completion_tokens":3585,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":465,"completion_tokens_details":{"reasoning_tokens":3504}},"tokens_in":465,"tokens_out":3585,"duration_ms":23047,"temperature":1.0,"reasoning_tokens":3504,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T02:45:48.801702+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the relic-density calculation with the full 14-pNGB effective potential, including mixing among Δ0, η3, η4, φ0 and non-decoupled heavier pNGBs, and look for a point where Δ0 is the lightest Z2-odd state, Ωch² = 0.120 ± 0.012, and the direct-detection cross section lies below the current limit. If such a point exists, the paper's conclusion that only three candidates work is false.","supporting_citations":[],"review_version":1}