{"id":"4d9f8900-e517-472d-bd04-fe6dcd64f01c","arxiv_id":"1908.10615","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Di-Higgs production at a future 100 TeV hadron collider can probe Higgs portal couplings of a stable scalar of order one for masses above half the Higgs mass.","lead":"This paper calculates how well future colliders could detect a stable dark scalar particle that interacts with ordinary matter only through the Higgs boson. It shows that a 100 TeV proton collider could probe this interaction, while the LHC cannot, and that the full quantum process changes the predicted reach.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"FCC-hh reach claim depends on an unvalidated recast: the 6% κλ projection is applied to a momentum-dependent portal distortion via an ad hoc band comparison, so the quantitative (mS,λ) exclusion contour may shift substantially.","rationale":"I agree with the reader's weakest assumption: the transfer of the 6% κλ projection to a momentum-dependent portal model is the least secure point in the argument. The one-loop calculation itself is plausible — the renormalized amplitude is UV finite (Eq. 16), the implementation uses established tools (Vbfnlo with FeynArts/FormCalc/LoopTools), and the authors bracket the O(λ^4) term as an uncertainty estimate. But the final reach statement inherits an untested experimental projection, and the binned χ² procedure is described only at the level of a comparison to a band, not a real likelihood. A full recast could easily change the quantitative contour while leaving the qualitative conclusion intact. I also note a minor internal inconsistency: the abstract says the scalar is charged under an unbroken global U(1), whereas Eq. (1) and the body use a real Z2-symmetric scalar; if the U(1) model were intended, the amplitudes would differ (extra scalar degrees of freedom and a |S|^2 portal term), so the abstract should be corrected. This does not alter the main concern. The reader's CONDITIONAL verdict remains appropriate; I recommend no change.","tokens_in":11350,"tokens_out":20010,"duration_ms":215652,"concrete_test":"Perform a binned log-likelihood recast of the FCC-hh di-Higgs search using the portal mHH shapes from the authors' Vbfnlo implementation, the projected HH signal and background rates and systematic errors from Ref. [24], and a fit that includes a free κλ nuisance parameter. Compute the 95% CL exclusion contour in (mS,λ) and compare it with the blue region of Fig. 8. If the contour differs by more than about 30% in λ, or if the blue-region boundary at mS > 200 GeV is not reproduced, the quoted reach is not established by the paper's own method.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim—that a 100 TeV FCC-hh with 30/ab can probe λ ≲ 1 for mS up to a few hundred GeV—is secured by the blue region in Fig. 8, but the statistical procedure behind it is not a derived sensitivity estimate. In Sec. III the authors state that a point (mS,λ) is excluded 'if the binned distribution deviates by more than the band indicated by the self-coupling projection in the sense of a binned χ² test.' That band is the envelope κλ = 1 ± 0.06 from Ref. [24], a projection computed for a momentum-independent rescaling of the trilinear Higgs coupling with specific background, systematic, and correlation assumptions. Transferring this band to a model whose distortion is momentum-dependent and complex (thresholds, absorptive parts, box-like terms) assumes that per-bin uncertainties are identical, that the analysis cannot partially absorb the portal distortion into a fitted κλ, and that the LO, QCD-uncorrected shape used here matches the NLO shape underlying the projection. None of these assumptions is tested in the paper; the caveat that 'the eventual sensitivity yield will obviously depend on the details of the machine' does not quantify the error. Because the central conclusion is a reach contour rather than just the existence of a calculable one-loop effect, this unvalidated transfer is load-bearing. A proper recast could move the contour by a factor of a few in λ, especially at larger mS where the distortion is small.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the sensitivity of di-Higgs boson production to a real singlet scalar S with an unbroken Z2 symmetry, coupled to the Standard Model through the Higgs portal operator λ S² Φ†Φ. The authors compute the one-loop contribution of S to gg → HH, including the full momentum dependence of the off-shell Higgs three-point function, the relevant box counterterms, and a consistent on-shell renormalization scheme. They find that the momentum-dependent amplitude, with its threshold and absorptive parts, differs significantly from the effective-potential estimate of a modified Higgs self-coupling. Using a binned χ² comparison to the projected κλ = 1 ± 0.06 sensitivity from Ref. [24], they derive exclusion contours in the (mS,λ) plane for the LHC and for a 100 TeV FCC-hh with 30/ab. They conclude that FCC-hh can probe the portal coupling down to λ ∼ 1 for mS above mH/2 up to a few hundred GeV, a region complementary to lepton-collider Higgs-strahlung precision and missing-energy searches.","tokens_in":11627,"tokens_out":11744,"duration_ms":110252,"significance":"The one-loop calculation itself is a useful and clearly presented result: the authors provide explicit expressions for Γvirt, the counterterms, and the UV-finite combination in Eq. (16), and they implement the result in a modified Vbfnlo setup. The comparison of the full momentum dependence with the effective-potential approximation is informative, and the authors' decision to bracket the result with and without the |M_virt|² contribution shows appropriate caution about perturbative control. If the sensitivity projection were validated, the FCC-hh reach would be a valuable guide for the unexplored Z2-symmetric Higgs portal. At present, however, the quantitative exclusion contours rest on an unvalidated transfer of the κλ projection to a momentum-dependent new-physics distortion, so the central reach claim is not yet established to the precision that the figures suggest.","major_comments":[{"comment":"The FCC-hh exclusion contours are derived by the procedure stated in Sec. III: a point (mS,λ) is excluded 'if the binned distribution deviates by more than the band indicated by the self-coupling projection in the sense of a binned χ² test', where the band is the κλ = 1 ± 0.06 envelope from Ref. [24]. That projection was obtained for a momentum-independent rescaling of the Higgs trilinear coupling, under specific background, systematic, and correlation assumptions for the inclusive di-Higgs measurement. Transferring it to the portal model assumes, without test, that the per-bin uncertainties of the mHH distribution are the same as for the κλ fit, that the analysis cannot partially reabsorb the momentum-dependent portal distortion into a fitted κλ, and that the leading-order, QCD-uncorrected shape used in this paper is adequate at the FCC-hh sensitivity. The caveat in Sec. III that the sensitivity 'will obviously depend on the details of the machine' does not quantify the resulting uncertainty in the (mS,λ) exclusion. Because the central quantitative claim of the paper is that FCC-hh can probe λ ≲ 1 for mS up to a few hundred GeV, this unvalidated transfer is load-bearing. I ask the authors to either validate the recast (for example by calibrating the binned χ² on the κλ signal itself and checking that the mHH shape uncertainties and correlations are compatible with Ref. [24]) or to present the contours clearly as an illustrative extrapolation with an explicit error band reflecting the transfer assumptions.","section":"Section III, Fig. 8"}],"minor_comments":[{"comment":"The phrase 'massesmS≤mH/2≤ few×100 GeV' in the first paragraph should read 'masses mS ≥ mH/2 and up to a few hundred GeV' to be consistent with the abstract and the rest of the paper.","section":"Introduction"},{"comment":"The statement that the expansion is 'to O(λ²)' is confusing because M_virt in Eq. (7) contains both O(λ) and O(λ²) pieces; please clarify that the expression is linear in M_virt and that |M_virt|² is included as an estimate of factorizable two-loop effects, or adjust the power-counting labels accordingly.","section":"Section II, Eq. (19)"},{"comment":"The legend labels 'O(λ²)' and 'O(λ²) + |M_virt|²' should be defined in the caption (order in the cross section vs. order in the amplitude) to avoid ambiguity.","section":"Fig. 6 legend"},{"comment":"The caption should explicitly state which solid and dashed lines correspond to the calculations without and with the |M_virt|² term; the current text 'solid (dashed line)' is difficult to parse.","section":"Fig. 8 caption"},{"comment":"The phrase 'a coupling extraction of λSM at the ≲ 6% level' uses λSM for the Higgs self-coupling; since λ is used for the portal coupling, please use a different symbol (e.g., κλ or λHHH) to avoid confusion.","section":"Section III"},{"comment":"The expression '√s/2mH/2' appears to contain a typo; please clarify the intended threshold expression.","section":"Fig. 5(b) caption"}],"recommendation":"major_revision","confidential_remarks":"The manuscript contains a solid one-loop calculation, but the headline reach projection rests on an unvalidated transfer of the κλ sensitivity from Ref. [24] to a momentum-dependent new-physics signal. I recommend major revision rather than rejection, because the calculation itself is publishable and the projection issue is fixable by either a genuine recast or a suitably qualified claim. The paper is appropriate for the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nFirst: the calculation is real and the main physics point holds. The paper gives the first full momentum-dependent one-loop S contribution to gg→HH in the Z2-symmetric portal with mS > mH/2, and shows that this matters: threshold effects and absorptive parts change the shape of the di-Higgs invariant mass spectrum relative to a simple rescaling of the trilinear Higgs coupling. The sensitivity to lambda at FCC-hh is genuinely better than the effective-potential estimate in some mass regions, and the comparison with the Coleman-Weinberg approximation is useful. The derivation of Gamma_virt, the counterterm, and the UV-finite combination is laid out carefully; the on-shell renormalization is standard; and checking |M_virt|^2 as a rough estimate of higher-order corrections is a sensible sanity check. The authors also put the di-Higgs result in context by comparing with single-Higgs precision at lepton colliders and missing-energy searches, and their citation pattern is honest.\n\nThe soft spot is the quantitative reach. The blue region in Fig. 8 is built by taking the projected ~6% precision on kappa_lambda from Ref. [24] and applying it as a band in a binned chi^2 test to the portal model, whose distortion is momentum-dependent and complex (thresholds, absorptive parts, box-like terms). That transfer assumes per-bin uncertainties are the same, that the analysis cannot partially absorb the portal distortion into a fitted kappa_lambda, and that a LO, QCD-uncorrected shape is adequate for the comparison. None of those assumptions is tested. The stress-test concern is fair: the contour could shift by a factor of a few in lambda at larger mS. I would add the same caveat to the LHC sensitivity: the red dashed line is also a recast of the CMS projection, though there the conclusion is simply 'not sensitive,' which is robust.\n\nIs this fatal? No. The paper is a projection note, and the authors say explicitly that the eventual yield depends on machine details. The central qualitative claim—that momentum dependence significantly affects the sensitivity compared to effective-potential estimates, and that a future 100 TeV machine could probe lambda of order one—survives moderate shifts in the contour. What should be read with care is the exact boundary of the blue region.\n\nWho this is for: people planning future collider benchmarks and anyone working on the Higgs portal dark scalar. It deserves a serious referee. I would accept it for peer review with the expectation that the statistical transfer be softened or validated, but the calculation is sound and the comparison is a real contribution.\n\nRecommendation: engage, but cite with the caveat that the reach contour is indicative, not a derived sensitivity.","headline":"A clean one-loop calculation for the Z2-symmetric Higgs portal in di-Higgs production, with a genuine momentum-dependent result; the reach projections rest on an unvalidated recast of the kappa_lambda precision band, so treat the contours as indicative, not definitive.","tokens_in":12181,"tokens_out":2576,"would_cite":true,"duration_ms":28213,"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":"At a 100 TeV proton collider, double-Higgs production can test the symmetric Higgs portal coupling down to values around one for dark scalar masses from just above half the Higgs mass to a few hundred GeV.","keywords":["Higgs portal","di-Higgs production","dark scalar","Z2 symmetry","future circular collider","Higgs self-coupling","one-loop amplitudes","gluon fusion"],"falsifier":"A direct check would be a measurement of the di-Higgs invariant-mass spectrum at a 100 TeV hadron collider with 30/ab: if the data agree with the Standard Model within the projected precision in the mass bins where the paper marks $\\lambda\\sim1$ regions as excluded, that part of the claimed reach is wrong. On the calculational side, the chief unknown is the size of higher-order electroweak corrections, so computing the full two-loop $gg\\to HH$ amplitude in this model would show whether the $O(\\lambda^4)$ estimate used here adequately bounds the one-loop result at $\\lambda\\simeq1$.","tokens_in":11135,"feed_emoji":"⚛️","tokens_out":12855,"duration_ms":112347,"temperature":0.7,"pith_summary":"The paper argues that double-Higgs boson production at a future 100 TeV proton collider (FCC-hh) can probe the symmetric Higgs portal — a dark scalar $S$ coupled to the Higgs field through $\\lambda S^2|\\Phi|^2$ — in the region where the dark scalar is heavier than half the Higgs mass, a region with few experimental handles today. The essential step is to compute the full momentum-dependent one-loop contribution of $S$ to $gg\\to HH$, not just the shift of the Higgs self-coupling obtained from the effective potential. With that amplitude, the paper shows that an FCC-hh with $30/\\mathrm{ab}$ could reach portal couplings $\\lambda$ of order one for dark scalar masses between roughly $m_H/2$ and a few hundred GeV, where the LHC cannot. This matters because such a scalar is one of the simplest gateways to a dark sector and can be a dark-matter candidate, and the shape of the di-Higgs mass spectrum would distinguish the portal from a plain change in the Higgs self-coupling.","feed_headline":"Double-Higgs pairs expose hidden scalars at 100 TeV","feed_subtitle":"A full one-loop calculation reaches portal couplings of order one where the LHC cannot.","key_machinery":"The load-bearing object is the renormalised one-loop insertion of the portal scalar $S$ into the $gg\\to HH$ amplitude, specifically the off-shell Higgs three-point vertex function $\\Gamma(s,m_H^2,m_H^2)$ built from the one-loop scalar integrals $A_0$, $B_0$, $C_0$ with on-shell renormalisation of the Higgs mass, wave function, and tadpole counter-term. It enters the $s$-channel amplitude together with a $\\delta Z_H$ counter-term contribution that also modifies the box diagrams. What makes this object carry the argument is its explicit dependence on the invariant di-Higgs mass $m_{HH}=\\sqrt{s}$: the vertex acquires real and imaginary parts with a threshold turn-on at $\\sqrt{s}=2m_S$, so the predicted $m_{HH}$ spectrum has a different shape from a momentum-independent change in the Higgs self-coupling. The paper feeds that spectrum into a binned $\\chi^2$ comparison against the projected self-coupling precision to decide which $(m_S,\\lambda)$ points can be excluded.","core_discovery":"On the paper's own terms, the central discovery is that $gg\\to HH$ is a viable indirect probe of a $\\mathbb{Z}_2$-symmetric Higgs portal scalar in the mass window $m_S \\gtrsim m_H/2$ up to a few hundred GeV — precisely the window where on-shell Higgs decays to two scalars are closed and single-Higgs probes are weak. The full renormalised one-loop amplitude, including threshold and absorptive parts and the tadpole-induced counter-term contribution, makes the predicted $m_{HH}$ spectrum deviate from the Standard Model in a momentum-dependent way. Comparing that spectrum with the projected roughly 6% precision on the Higgs self-coupling at a 100 TeV hadron collider with 30/ab of data, the paper finds sensitivity to $\\lambda \\sim 1$ for $m_S$ up to a few hundred GeV, whereas LHC di-Higgs data can only reach $\\lambda$ noticeably above one. At low $m_S$ the threshold effects reduce sensitivity relative to effective-potential estimates, while at larger masses the full calculation gives a stronger reach, and the shape of the deviation is different from a simple rescaling of the trilinear Higgs coupling.","pith_inferences":["The same one-loop machinery could be applied to other Higgs-production modes with different momentum transfers, such as vector-boson fusion or associated $ZH$ production, where the threshold structure of $\\Gamma(s,m_H^2,m_H^2)$ would appear at different kinematic scales; the paper does not compute these extensions.","Because the amplitude has a threshold at $\\sqrt{s}=2m_S$, a high-statistics measurement of the $m_{HH}$ shape could in principle extract the dark scalar mass, not just delineate an excluded region; the paper notes the fingerprinting idea but does not quantify this extraction.","If the dark scalar is a thermal relic, the $\\lambda\\sim1$ region that FCC-hh could test overlaps the parameter space where relic-density and self-interaction constraints are also relevant, so a positive signal would sharpen dark-matter interpretations; this combination is not made in the paper.","The lepton-collider comparison is made at fixed centre-of-mass energies around 240–380 GeV; a future collider that scans energies near the $ZH$ maximum could map the momentum dependence of the same vertex more directly, an extrapolation the paper leaves implicit."],"forward_implications":["A 100 TeV hadron collider with 30/ab of data can test the symmetric Higgs portal for dark scalar masses just above $m_H/2$ up to a few hundred GeV with portal couplings $\\lambda$ of order one, a region that the LHC cannot reach with di-Higgs production.","The full momentum-dependent amplitude predicts a different $m_{HH}$ spectrum than an effective-potential shift of the Higgs self-coupling, so future shape measurements could distinguish a portal scalar from other modifications of the trilinear Higgs coupling.","Di-Higgs constraints at FCC-hh complement lepton-collider Higgs-strahlung precision and missing-energy searches; together they cover more of the $(m_S,\\lambda)$ plane than any single probe.","If a deviation is seen, the invariant-mass dependence of the di-Higgs spectrum can act as a fingerprint of the portal model rather than just evidence for new physics.","The LHC projection alone is not enough to reach this model: detectable couplings at the LHC are large enough that the one-loop perturbative treatment is no longer fully reliable."],"supporting_citations":[{"why":"supplies the projected roughly 6% precision on the SM Higgs self-coupling at a 100 TeV hadron collider that the paper transfers to the portal model.","marker":"[24]"},{"why":"demonstrates the full momentum-dependent one-loop treatment for a related dark matter model coupled to the Higgs, justifying the calculation approach.","marker":"[25]"},{"why":"provides a related one-loop computation of Higgs-pair production in a non-mixing Higgs portal model, used as a starting point and cross-check for the amplitude.","marker":"[26]"},{"why":"defines the effective potential whose self-coupling shift is the baseline approximation the paper compares against.","marker":"[27]"},{"why":"supplies earlier effective-potential constraints and the vacuum-stability region used as the comparison benchmark.","marker":"[7]"},{"why":"gives the earlier effective-potential estimate of Higgs-self-coupling modifications by the portal scalar, the previous sensitivity estimate the full calculation supersedes.","marker":"[21]"},{"why":"provides the projected LHC di-Higgs sensitivity to the Higgs self-coupling used for the LHC reach estimate.","marker":"[40]"},{"why":"defines the missing-energy search sensitivity used as the hadron-collider comparison channel.","marker":"[8]"},{"why":"calculates the Higgs-strahlung corrections in the portal model that set the lepton-collider precision comparison.","marker":"[64]"}],"fun_headline_variants":["Di-Higgs pairs trace hidden scalar in 100 TeV collisions","Full one-loop di-Higgs calculation extends dark scalar reach","Momentum-dependent di-Higgs probes new scalar beyond Higgs decay","Di-Higgs spectrum distinguishes portal scalar from Standard Model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The FCC-hh reach rests on the assumption that the roughly 6 per cent precision projected for a simple rescaling of the Higgs self-coupling can be transferred, bin by bin, to the differently shaped momentum-dependent distortion a portal scalar produces, with the same backgrounds and systematic errors.","fun_headline_variants_meta":{"raw":{"variants":["Di-Higgs pairs trace hidden scalar in 100 TeV collisions","Full one-loop di-Higgs calculation extends dark scalar reach","Momentum-dependent di-Higgs probes new scalar beyond Higgs decay","Di-Higgs spectrum distinguishes portal scalar from Standard Model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000981,"raw_usage":{"total_tokens":4164,"prompt_tokens":947,"completion_tokens":3217,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":563,"completion_tokens_details":{"reasoning_tokens":3143}},"tokens_in":563,"tokens_out":3217,"duration_ms":22308,"temperature":1.0,"reasoning_tokens":3143,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:38:39.549498+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check would be a measurement of the di-Higgs invariant-mass spectrum at a 100 TeV hadron collider with 30/ab: if the data agree with the Standard Model within the projected precision in the mass bins where the paper marks $\\lambda\\sim1$ regions as excluded, that part of the claimed reach is wrong. On the calculational side, the chief unknown is the size of higher-order electroweak corrections, so computing the full two-loop $gg\\to HH$ amplitude in this model would show whether the $O(\\lambda^4)$ estimate used here adequately bounds the one-loop result at $\\lambda\\simeq1$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the projected LHC di-Higgs sensitivity to the Higgs self-coupling used for the LHC reach estimate."}],"review_version":1}