{"id":"32a030b8-8733-416e-8378-c722f2e5c261","arxiv_id":"2603.02764","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Updated inclusive cross sections for VBF and Vh associated Higgs pair production at N3LO QCD+NLO EW and NNLO QCD respectively are provided for the LHC and HL-LHC.","lead":"This paper lists updated predicted rates for two ways the LHC can make pairs of Higgs bosons, using the latest PDFs and including higher-order corrections. The numbers are meant to be the new reference for ATLAS and CMS searches.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Table 1 states EW uncertainties are 'contained in the PDF uncertainty', but this is not a valid assumption; no EW uncertainty estimate is given for the headline VBF HH cross section.","rationale":"The reader's verdict accepted the paper with moderate confidence, noting only a 'stated but not quantified' EW uncertainty as a secondary weak point. My stress-test focuses on this as the single most load-bearing concern. The VBF factorised approximation is an explicit and standard definition; it is not likely to shift the quoted numbers beyond a small, well-understood extent. In contrast, the EW uncertainty statement in Table 1 is logically flawed and directly affects the headline result. The numbers themselves are presumably correct because they rest on established codes (PROVBFHH, RECOLA), and the paper is transparent about what is included. However, the absence of any EW uncertainty estimate means the quoted uncertainty budget is incomplete. For a community report intended as a reference for LHC experiments, this is a real gap. The concern is testable by varying the EW input scheme; if the scheme dependence is small (<0.2%), the neglect is practically harmless and the verdict could remain unchanged. But until that is demonstrated, the paper should be accepted only conditionally, with a corrected uncertainty statement and an explicitly justified EW uncertainty. The reader's weakest assumption did identify the VBF approximation as primary, but also mentioned the EW uncertainty; my emphasis is on the latter, hence 'partial' agreement.","tokens_in":12586,"tokens_out":12293,"duration_ms":121188,"concrete_test":"Recompute the VBF HH cross section at 13.6 TeV and mh=125.09 GeV using the same code (PROVBFHH v2.1.0 for N3LO QCD, RECOLA+MOCANLO for NLO EW) but with the EW input scheme varied: (a) G_F scheme as in Eq. (2), (b) alpha(MZ) on-shell scheme, (c) alpha(0) scheme. The spread in the final N3LO QCD × NLO EW cross section from this scheme choice provides an estimate of missing higher-order EW uncertainty. If the spread exceeds the quoted scale uncertainty (+0.05%/-0.04%) and is non-negligible relative to the 2.7% PDF uncertainty, then Table 1 should be amended to include an explicit EW uncertainty or a clear justification for its smallness. This test would settle whether the current neglect of EW uncertainty is numerically justified or a genuine omission.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central claim is that Table 1 gives state-of-the-art reference values at N3LO QCD+NLO EW for VBF HH production. The caption of Table 1 states: 'no EW uncertainties are included, but that they are assumed to be fully contained in the PDF uncertainty which dominates.' This statement is not logically sound: PDF uncertainties arise from the parton distribution functions and have no connection to missing higher-order electroweak corrections. The NLO EW correction is included, but the uncertainty due to missing higher-order EW effects (e.g., two-loop EW, scheme dependence) is not quantified. If this EW uncertainty is of order 1%, it is much larger than the scale uncertainty (+0.05%/-0.04%) and could be comparable to the quoted PDF uncertainty (2.7%). For a reference intended for ATLAS and CMS searches, the total uncertainty budget should include an estimate of missing EW higher-order corrections. The cross section numbers themselves may be correct, but the uncertainty statement is misleading: the reader might take the quoted total uncertainty as complete, whereas it omits at least one known source of theoretical error. This weakens the claim that these tables are definitive 'state-of-the-art reference values' with fully quantified uncertainties. The issue is not the VBF approximation (which is clearly defined and standard) but the unquantified EW uncertainty in the main result.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This short community note presents updated inclusive cross sections for electroweak Higgs boson pair production at the LHC and HL-LHC. For vector-boson fusion (VBF) production of hh, the authors quote cross sections at N3LO QCD+NLO EW accuracy, obtained by combining N3LO QCD results with the NLO EW corrections of Ref. [41] in a multiplicative way (Eq. (4)). For associated V hh production (W+hh, W−hh, Zhh), the paper quotes NNLO QCD cross sections. All numbers are given for √s = 13, 13.6, 14 TeV, for mh = 125, 125.09, 125.38 GeV, and for κλ = 1 (SM) together with κλ = 0, 2, 3. The setup is described in Section 2: PDF4LHC21 PDFs, αs(MZ)=0.118, GF scheme, and a scale choice for VBF (Eq. (3)) different from that used for V hh. The tables in Sections 3 and 4 are intended as reference values for experimental searches.","tokens_in":12945,"tokens_out":4067,"duration_ms":41551,"significance":"If the quoted numbers are correct, this paper provides a convenient, up-to-date compilation of the highest-order inclusive cross sections for two important Higgs-pair production modes, and the tables would indeed be valuable reference values for ATLAS and CMS. A strength of the manuscript is that the calculations are based on publicly available codes (PROVBFHH v2.1.0, RECOLA+MOCANLO) with a clearly specified setup, and the κλ dependence is included for anomalous coupling interpretations. The main weakness is that the uncertainty treatment, especially the EW uncertainty in the VBF channel, is not fully satisfactory; this is a local but important caveat for a reference document. The VBF approximation itself is standard and clearly stated, so the numerical central values are credible.","major_comments":[{"comment":"The caption states that 'no EW uncertainties are included, but that they are assumed to be fully contained in the PDF uncertainty which dominates.' This is not a valid assumption: PDF uncertainties reflect the knowledge of parton distribution functions and are unrelated to missing higher-order electroweak corrections. The NLO EW correction is included, but the uncertainty from neglected higher-order EW effects (e.g., two-loop EW and scheme dependence) is not quantified. The scale uncertainty is only about ±0.05%, so an unquantified EW uncertainty of order 1% — which is plausible for an NLO EW correction at the LHC — would be comparable to or larger than the scale uncertainty and could be a sizable fraction of the total quoted uncertainty. For a reference value to be used by experiments, this should be either estimated and included or explicitly flagged as a separate missing uncertainty i","section":"Section 3, Table 1 caption"},{"comment":"The general setup in Eq. (1) sets mb = 0 GeV, but the gg→Zhh contributions (triangle, box, pentagon diagrams) are computed with mb = 4.9 GeV. This is mentioned in the text, but the numerical impact of this choice on the Zhh cross sections is not assessed. Since the gg-initiated contribution is non-negligible for Zhh and the scale uncertainty is larger than in W±hh, the paper should either quantify the effect of the bottom-mass treatment or justify why it is negligible at the reported precision. This is relevant for the consistency of the quoted Zhh numbers.","section":"Section 4.2, Eq. (1) vs. mb=4.9 GeV"}],"minor_comments":[{"comment":"The multiplicative combination of N3LO QCD and NLO EW corrections is clearly stated, but the paper does not comment on the size of the NLO EW correction δVBF_NLO EW/σVBF_LO. A single sentence with the numerical value at, say, √s=13.6 TeV would help the reader judge the importance of the missing EW uncertainty.","section":"Section 3, Eq. (4)"},{"comment":"There are minor formatting issues: 'g g' should be 'gg' in several places, and the sentence 'Those contributions have been computed in [35] on which the numbers presented here are based' is awkward. No scientific content is affected.","section":"Section 4.2, formatting"},{"comment":"The captions state 'including scale uncertainties and PDF uncertainties in percent in this order at NNLO QCD'. The phrase 'in this order' is unclear; consider rephrasing to 'at NNLO QCD in QCD' or similar.","section":"Tables 3–5 captions"},{"comment":"The paper relies on public codes, which is good, but it would be helpful for reproducibility to list the exact version of RECOLA/MOCANLO (MoCaNLO is cited as a 2026 preprint) and, if possible, to provide a short table of input parameters or a link to a repository with the numbers.","section":"General"}],"recommendation":"minor_revision","confidential_remarks":"The reader's report and my own reading agree on the main technical content. The only issue that could escalate to a major revision is the EW-uncertainty statement in Table 1; if the authors cannot provide a quantitative estimate, the paper should be accepted with an explicit caveat that the quoted total uncertainty excludes unknown higher-order EW corrections. I recommend requesting a revised version with this fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: this is a useful update of reference cross sections for EW HH production, with the new content being PDF4LHC21, 13.6 TeV, and κλ=0,2,3. The numbers are credible, but the caption to Table 1 overclaims about EW uncertainties, and that should be fixed before experiments quote these as final.\n\nWhat's new: the underlying calculations are from public codes like PROVBFHH and RECOLA+MOCANLO, and the original papers are cited properly. The setup is clearly specified: masses, widths, scale choice, PDF set. The tables give state-of-the-art QCD+EW values for VBF HH at N3LO QCD+NLO EW, and NNLO QCD for Vhh. The VBF approximation is defined explicitly, and using it for an inclusive cross section is standard practice in this field. The numbers look internally consistent.\n\nThe soft spot: the statement in Table 1's caption that EW uncertainties are 'fully contained in the PDF uncertainty' is not logically sound. PDF uncertainties are determined by the PDF fit; they don't reflect missing higher-order electroweak corrections. The NLO EW correction is included, but the uncertainty due to two-loop EW effects or scheme dependence is not quantified. For a table intended as a reference for ATLAS and CMS, the error budget should either include an estimate of missing EW corrections or explicitly state that they are not included. Right now the caption invites the reader to treat the total uncertainty as complete, which it isn't. This is a local, fixable issue. The cross sections themselves are computed with established codes and are not the concern.\n\nOther notes: the Zhh numbers use mb=4.9 GeV for the gluon-initiated contributions while the rest uses mb=0; the text acknowledges this, so it's not a hidden inconsistency. The κλ tables without EW corrections are clearly labeled.\n\nWho should read this: experimental colleagues who need central values and uncertainties for their HH searches, and anyone who wants an up-to-date reference. It doesn't change physics but it's exactly what a community report should provide.\n\nRecommendation: send to peer review. A referee should ask for a correction to the EW uncertainty statement, but the paper does not require major rework.","headline":"Useful reference numbers for EW HH production, but the EW uncertainty claim in Table 1 is not valid; fix it before quoting.","tokens_in":13386,"tokens_out":3240,"would_cite":true,"duration_ms":30662,"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":"This paper supplies the inclusive cross sections that experiments should use as the reference for electroweak Higgs boson pair production at the LHC and HL-LHC.","keywords":["Higgs boson pair production","vector-boson fusion","associated production","N3LO QCD","NNLO QCD","NLO electroweak corrections","trilinear Higgs self-coupling","LHC cross sections"],"falsifier":"A calculation of inclusive VBF hh at NLO QCD that keeps s-channel diagrams and t/u interference, compared with the factorised N3LO result, would settle the matter: if the difference exceeds the quoted ~0.05% scale uncertainty, the central values are not robust. A second test is to compute photon-induced VBF contributions with a photon PDF; if they exceed 1%, the stated uncertainty budget is incomplete.","tokens_in":12499,"feed_emoji":"⚛️","tokens_out":6627,"duration_ms":61207,"temperature":0.7,"pith_summary":"The paper gives updated inclusive cross sections for the two electroweak Higgs-pair production channels at the LHC and HL-LHC. For vector-boson fusion (VBF) it reports values at third-order QCD combined with next-to-leading-order electroweak corrections, about 1.87 fb at 13.6 TeV; for associated W/Z + hh production it reports second-order QCD values around 0.36-0.40 fb. The numbers are provided at three beam energies, three Higgs masses, and four values of the trilinear Higgs self-coupling. The practical claim is that these tables are the current reference for experimental searches and for translating limits into constraints on the Higgs self-coupling.","feed_headline":"Higgs-pair cross sections: 1.87 fb for VBF at N3LO QCD","feed_subtitle":"Reference values at 13, 13.6 and 14 TeV for LHC and HL-LHC searches, with the trilinear self-coupling dependence.","key_machinery":"For VBF hh, the machinery is the factorised VBF approximation: the two quark lines are treated as independent, s-channel diagrams are excluded, t/u interference is dropped, and the NLO electroweak correction is applied as a multiplicative factor to the N3LO QCD cross section, with the renormalisation/factorisation scale set by the di-Higgs transverse momentum. For Vhh, the machinery is off-shell vector-boson production followed by the splitting V* -> V hh, computed at NNLO QCD; for Zhh, gluon-initiated triangle, box, and pentagon diagrams with a finite bottom mass are included, and the scale is the invariant mass of the V hh system.","core_discovery":"The central claim is that the inclusive cross sections in Tables 1-5 are the most up-to-date theoretical predictions for electroweak Higgs boson pair production. For VBF hh, the paper combines N3LO QCD with NLO electroweak corrections multiplicatively, obtaining sigma = 1.870 fb at a center-of-mass energy of 13.6 TeV for mh = 125.09 GeV, with scale uncertainties below 0.1% and a 2.7% PDF uncertainty. For W+hh, W-hh, and Zhh, the NNLO QCD values at 13.6 TeV are 0.357, 0.187, and 0.396 fb respectively; the Zhh channel carries larger scale uncertainties (about 3%) because of gluon-initiated contributions. The kappa_lambda dependence, spanning kappa_lambda = 0, 2, 3, is provided so that anomalou","pith_inferences":["If full non-factorisable and s-channel contributions were included for inclusive VBF hh, the quoted sub-percent scale uncertainty would be tested; the paper's own numbers give the benchmark for that comparison.","The assertion that electroweak uncertainties are fully contained in the PDF uncertainty is untested; a dedicated estimate would either confirm the 2.7% band or enlarge it.","The tables show the cross section rising steeply for kappa_lambda = 3 and dropping for kappa_lambda = 0 in the associated-production channels, suggesting these modes could act as discriminators for the sign and magnitude of the self-coupling even though their SM rates are small.","The omission of photon-induced contributions is stated to be below 1%, but since the recommended PDF set lacks a photon, an independent check with a photon-containing PDF would settle whether that estimate holds."],"forward_implications":["If these numbers stand, they become the baseline for LHC experiments searching for VBF and Vhh double-Higgs production, replacing older handbook values.","The sub-percent scale uncertainties on VBF hh imply that theory is not the limiting factor for VBF interpretations; PDF uncertainty dominates the error budget.","The separate W+hh and W-hh values allow independent constraints on the couplings of two Higgs bosons to W and Z bosons, something VBF alone cannot provide.","The kappa_lambda rows let any future measurement be mapped directly onto a value or bound for the trilinear self-coupling within the assumed coupling-modifier framework.","The larger scale uncertainty on Zhh means that channel's interpretive power is limited by missing higher-order or top-mass-renormalisation effects, as the paper itself notes."],"fun_headline_variants":["Higgs-pair VBF: 1.87 fb at N3LO QCD+EW for 13.6 TeV","New N3LO QCD+EW Higgs-pair cross sections: VBF 1.87 fb","Updated Higgs-pair: 0.1% scale error, 2.7% PDF at 13.6 TeV","Higgs-pair VBF at N3LO: 1.87 fb, scale uncertainty under 0.1%"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the factorised VBF approximation — no s-channel diagrams, no t/u interference, and QCD and EW corrections multiplying independently — is accurate for the inclusive cross section at the sub-percent level; if it is not, the VBF numbers in Table 1 shift.","fun_headline_variants_meta":{"raw":{"variants":["Higgs-pair VBF: 1.87 fb at N3LO QCD+EW for 13.6 TeV","New N3LO QCD+EW Higgs-pair cross sections: VBF 1.87 fb","Updated Higgs-pair: 0.1% scale error, 2.7% PDF at 13.6 TeV","Higgs-pair VBF at N3LO: 1.87 fb, scale uncertainty under 0.1%"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000252,"raw_usage":{"total_tokens":1366,"prompt_tokens":681,"completion_tokens":685,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":425,"completion_tokens_details":{"reasoning_tokens":563}},"tokens_in":425,"tokens_out":685,"duration_ms":5927,"temperature":1.0,"reasoning_tokens":563,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T19:14:46.268909+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A calculation of inclusive VBF hh at NLO QCD that keeps s-channel diagrams and t/u interference, compared with the factorised N3LO result, would settle the matter: if the difference exceeds the quoted ~0.05% scale uncertainty, the central values are not robust. A second test is to compute photon-induced VBF contributions with a photon PDF; if they exceed 1%, the stated uncertainty budget is incomplete.","supporting_citations":[],"review_version":1}