{"id":"98a08eee-7d30-4f80-809d-321e112be143","arxiv_id":"2509.07072","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The paper provides fast, analytic two-loop virtual QCD corrections for gluon-induced ZH and off-shell ZZ production with full top-quark mass dependence, validated against numerical results at the sub-percent level.","lead":"Physicists computed the next-to-leading-order QCD corrections for two gluon-fusion processes that produce a Higgs boson with a Z boson, or a pair of Z bosons, including off-shell states. These corrections matter for LHC precision measurements, especially for indirect constraints on the Higgs boson's width.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Off-shell Z*Z* amplitudes omit the double-triangle contribution, which Eq. (27) admits is unknown off-shell; the abstract's 'complete NLO virtual corrections' claim therefore outruns the computed results.","rationale":"The reader's weakest assumption identifies the same load-bearing gap: the off-shell Z*Z* result lacks the double-triangle contribution, with no independent two-loop off-shell benchmark. My stress-test confirms this is the most serious issue. The on-shell validation is genuinely strong: the t-expansion and high-energy expansions are checked against exact one-loop results and against two-loop numerical results for on-shell ZZ and ZH, with better-than-per-mille agreement in Figs. 8, 9, 16-18. The same machinery is credible for the off-shell box-type contributions. But the explicit admission after Eq. (27) that the off-shell double-triangle term is unknown means the paper's own decomposition is incomplete for off-shell Z*Z*. Since the abstract and conclusions claim complete NLO virtual corrections for these processes, the claim is stronger than the supplied evidence. This does not undermine the computed box, triangle and on-shell results, and the gap may be fillable; hence the reader's CONDITIONAL verdict is appropriate. No verdict change is needed, but the condition should specifically require the off-shell double-triangle contribution and an off-shell benchmark before the abstract's completeness claim is accepted.","tokens_in":27224,"tokens_out":5847,"duration_ms":66133,"concrete_test":"Compute the off-shell double-triangle contribution A^(i),dt at a representative off-shell point, e.g. q3^2=q4^2=(125 GeV)^2, pT=150 GeV, sqrt(s)=400 GeV, from the product of the one-loop gg->H amplitude and the H->Z*Z* decay amplitude (or from a direct two-loop numerical evaluation with pySecDec). If this contribution changes V_ZZ_fin by more than the claimed sub-per-mille accuracy, the completeness claim in the abstract fails. The authors should either provide the off-shell A^(i),dt or explicitly list it as a missing piece and adjust the claim accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the paper provides the NLO virtual corrections to off-shell Z boson pair production in gluon fusion. However, Eq. (27) decomposes the helicity amplitudes into vector, axial-vector, triangle and double-triangle contributions, and the text immediately after Eq. (27) states: 'Analytic results for the double-triangle contribution can be found in Ref. [15] for on-shell Z bosons. To our knowledge, the off-shell result is not yet known.' No off-shell expression for A^(i),dt is provided anywhere in the paper. For on-shell ZZ production the double-triangle diagrams contribute at two loops (their interference is computed in Ref. [15]); by continuity they also contribute when the Z bosons are off-shell, since the s-channel Higgs-mediated vertex H->Z*Z* is non-vanishing for q3^2,q4^2 != 0. Thus the off-shell Z*Z* amplitudes, as presented, are not the complete NLO virtual corrections. The numerical benchmarks in Figs. 8 and 9 are comparisons to existing on-shell results [21,23], so they do not validate the off-shell claim. The convergence tests in Figs. 5-7 demonstrate the consistency of the forward and high-energy expansions for the computed terms, but they cannot detect a systematically omitted contribution. The abstract's statement 'Our results constitute the next-to-leading order virtual corrections to these processes' is therefore not supported for the off-shell ZZ case as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper computes two-loop QCD corrections to gg→ZH and to off-shell Z-boson pair production gg→Z*Z*, mediated by top-quark loops. The method combines a deep high-energy expansion (up to m_t^112) with a forward-limit/t-expansion (up to t^10 for ZZ and t^5 for ZH, including several orders in the final-state virtualities), supplemented by Padé approximants. The resulting helicity amplitudes and form factors are used to construct finite NLO virtual corrections. The authors validate the expansions against exact one-loop results, against agreement of the two expansion regimes, and against pySecDec-based numerical results for on-shell ZZ and for ZH. They also obtain gg→γγ and gg→Zγ amplitudes as by-products. A central caveat is that the double-triangle contribution to off-shell Z*Z* is not computed; the manuscript states explicitly that the off-shell result is not yet known.","tokens_in":27643,"tokens_out":4955,"duration_ms":58001,"significance":"If the computed amplitudes are correct, this is a technically valuable step: it provides an analytic, fast-evaluable representation of the top-quark-mediated two-loop box contributions to gg→ZH and gg→Z*Z*, with full dependence on s, t, q3^2, q4^2 and m_t. The paper benefits from strong cross-checks: the one-loop approximations are compared to exact results, the two independent expansions agree, and the finite remainders agree with independent pySecDec results at the 0.02% level for ZZ and at the per-mille level for ZH. No constants are fitted to the target cross sections. The main limitation is that the off-shell Z*Z* amplitudes omit the double-triangle contribution, so the abstract's claim that the results 'constitute the next-to-leading order virtual corrections' is not supported for off-shell ZZ production.","major_comments":[{"comment":"The decomposition in Eq. (27) includes the double-triangle contribution A^(i),dt, but the text immediately after states: 'To our knowledge, the off-shell result is not yet known.' No off-shell expression is provided anywhere. Since the two-loop double-triangle diagrams contribute to on-shell ZZ via their interference (Ref. [15]) and the H→Z*Z* vertex is non-vanishing for q3^2,q4^2≠0, the off-shell amplitudes are not complete NLO virtual corrections. The abstract's statement 'Our results constitute the next-to-leading order virtual corrections to these processes' therefore overreaches for off-shell ZZ. The authors should either compute and include the off-shell double-triangle term or explicitly rescope the claims to the v_t^2, a_t^2, and triangle contributions and amend the abstract and conclusions accordingly.","section":"Sec. 3.1, Eq. (27) and following text"},{"comment":"The finite remainder V_fin^ZZ is defined from the helicity amplitudes, but since A^(1) lacks the double-triangle part, V_fin^ZZ for off-shell kinematics omits the corresponding two-loop interference with the LO amplitude. The excellent agreement with the on-shell numerical results of Refs. [21,23] does not validate the off-shell region, because those reference points are on-shell. The convergence tests in Figs. 5–7 only assess the terms that were actually computed; they cannot detect a systematically omitted contribution. Please state precisely which contributions enter V_fin^ZZ and provide either an off-shell numerical benchmark or an explicit statement of this limitation.","section":"Sec. 3.4, Eq. (29) and Figs. 8–9"}],"minor_comments":[{"comment":"The statement that the gg→γγ amplitudes can be used off-shell up to q^2 ∼ (125 GeV)^2 is an extrapolation from the behaviour seen for gg→HH and gg→Z*Z*; no direct off-shell check for γγ is provided. Please label this as an expectation or add a numerical test.","section":"Sec. 3.5"},{"comment":"The footnote claims that the ZH form factors can be immediately applied to off-shell Z and Higgs bosons. The double-triangle contribution for ZH is taken from Ref. [13], which is derived for on-shell kinematics; the off-shell extension of that contribution should be justified or qualified.","section":"Sec. 4.1, footnote 6"},{"comment":"The construction of the Padé approximants is described verbally (range of m_t orders, treatment of log(m_t) terms). For reproducibility, please give the precise algorithm or point to a code/ancillary file defining the window and the uncertainty estimate.","section":"Sec. 2.3.3"}],"recommendation":"major_revision","confidential_remarks":"This is a technically strong computation with extensive internal and external checks. The main issue is that the off-shell double-triangle contribution is missing while the abstract claims completeness. I do not see grounds for rejection, because the computed box and triangle contributions are soundly validated on-shell and the completeness claim can be corrected by rescoping. The revision should make the scope of the off-shell Z*Z* result unambiguous and remove the word 'complete' for that process unless the double-triangle term is supplied."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a technically impressive expansion calculation, and the comparisons to pySecDec are convincing. But the abstract overclaims: the off-shell Z*Z* result is not the complete NLO virtual correction, because the paper admits the double-triangle contribution is only known on-shell (Eq. (27) ff.). That is a load-bearing gap for the phenomenology the authors cite, where off-shell ZZ interference is used for Higgs-width constraints.\n\nWhat is genuinely new: first two-loop off-shell Z*Z* box amplitudes (vector and axial-vector parts) with full dependence on s, t, q3^2, q4^2, mt; a deep t-expansion up to t^10 and a high-energy expansion up to m_t^112 whose combination covers phase space. Benchmarks against Agarwal et al. and Chen et al. show agreement below 0.02% for ZZ and below per mille for ZH. The by-products gg->gamma*gamma* and gg->Z*gamma* are useful. Two independent expansion routes for the t-expansion (amplitude-level vs. master-integral-level) agreeing is a strong internal check. The paper is honest about the double-triangle gap in the text, but the abstract's \"complete NLO virtual corrections\" line is not supported by what is computed.\n\nThe central soft spot is exactly that missing off-shell double-triangle contribution. For on-shell ZZ it is known and contributes; for off-shell it is not computed or parametrized anywhere. Anyone wanting to use these amplitudes for off-shell ZZ with the Higgs-mediated interference needs that piece. A second, softer spot is that the off-shell region has no independent two-loop numerical benchmark; validation relies on on-shell numerics plus convergence of the q^2 series. That is reasonable but not airtight. Minor: no public code yet (promised in ggxy), though ancillary files are cited.\n\nFor ZH, the calculation looks complete and the benchmarks are excellent. The problem is specifically the off-shell ZZ claim. This paper deserves a serious referee: it is a substantial step and the expansion technology is well tested. But the authors should either provide the off-shell double-triangle contribution or clearly scope the abstract and conclusions to the box and triangle contributions rather than \"complete NLO virtual corrections.\" I would send it to peer review, not desk-reject.","headline":"Technically impressive expansion calculation with a real completeness gap: the off-shell ZZ claim outruns the computed results because the double-triangle contribution is missing.","tokens_in":28085,"tokens_out":2497,"would_cite":true,"duration_ms":29383,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["12.38.Bx"],"model":"deepseek-v4-flash","headline":"By combining deep high-energy and forward-limit expansions, this paper delivers two-loop NLO virtual corrections for gg->ZH and, for the first time, off-shell gg->Z*Z*, with full dependence on s, t, the two final-state virtualities, and the","keywords":["two-loop QCD corrections","gluon fusion","ZH production","off-shell Z boson pair production","top quark mass dependence","analytic expansion","helicity amplitudes","Higgs boson"],"falsifier":"An independent numerical two-loop evaluation of a single off-shell gg->Z*Z* helicity amplitude away from the on-shell point (for example sqrt(s)=400 GeV, p_T=100 GeV, q3^2=q4^2=(100 GeV)^2), compared directly with the expansion, would settle whether the convergence observed in the q3^2,q4^2 series persists at two loops; disagreement beyond the claimed uncertainty would invalidate the off-shell extrapolation.","tokens_in":27188,"feed_emoji":"⚛️","tokens_out":9197,"duration_ms":89816,"temperature":0.7,"pith_summary":"The paper aims to provide the next-to-leading-order QCD virtual corrections, mediated by a top-quark loop, for gluon fusion producing a Higgs boson with a Z boson (gg->ZH) and producing two off-shell Z bosons (gg->Z*Z*). Its strategy is not to compute the two-loop integrals exactly everywhere, but to compute very deep expansions in two complementary limits—toward the forward-scattering region (small t) and toward high energies (small top-quark mass)—and to show that the two expansions overlap and cover the full phase space when combined. If correct, this yields the first two-loop off-shell Z-pair amplitudes of this kind and a considerably deeper analytic representation of gg->ZH, with all mass and kinematic dependences kept explicit so parameter and scheme changes are immediate. The authors validate against exact one-loop results and against independent numerical two-loop results for on-shell kinematics, reporting agreement below the percent level and usually far better.","feed_headline":"Two expansions cover two-loop ZH and off-shell ZZ corrections","feed_subtitle":"Forward-limit and high-energy series agree below the per-mille level with numerical benchmarks, including off-shell Z bosons.","key_machinery":"The machinery is a pair of complementary analytic expansions of the same two-loop amplitudes expressed through 161 master integrals. (i) A forward-limit expansion: a Taylor expansion in the Mandelstam variable t (equivalently in delta q = q1+q3) that simultaneously expands in q3^2 and q4^2, computed both before and after integration-by-parts reduction as a cross-check. (ii) A high-energy expansion in m_t -> 0, carried to m_t^100-112 and turned into rational-function approximants (Padé approximants) in m_t. The two expansions are stitched in the intermediate p_T region where their agreement is best—the paper quotes p_T = 150 GeV as the switching point for gg->ZH, and uses the forward expansio","core_discovery":"On the paper's own terms, the central claim is that the helicity amplitudes for gg->Z*Z* and the form factors for gg->ZH can be written, over the entire physical phase space, as a combination of a forward-limit Taylor expansion and a high-energy expansion supplemented by Padé approximants, with a switch in the intermediate transverse-momentum region where the two expansions agree to many digits. The forward expansion includes terms up to t^5 (ZH) or t^10 (Z*Z*) in the massless parts, together with simultaneous expansion in the final-state virtualities q3^2 and q4^2; the high-energy expansion reaches up to m_t^112 (ZH) or m_t^100 (Z*Z*). At two loops both expansions are constructed from the s","pith_inferences":["Beyond the paper: the observed convergence as q3^2 and q4^2 increase—roughly one lost digit per 20 GeV of virtuality—suggests a testable map of the approximation's validity ceiling in the off-shell two-loop region.","Beyond the paper: because the off-shell double-triangle contribution is still missing, the current off-shell Z*Z* results are not yet a complete NLO virtual set; a future exact expression for that term could be added linearly, since the present expansions are linear in the amplitude pieces.","Beyond the paper: the same paired-expansion strategy may transfer to other 2->2 gluon-fusion processes with two different final-state virtualities, where exact two-loop evaluation is hardest.","Beyond the paper: the fast evaluation and explicit parameter dependence make these amplitudes well suited for embedding in event generators with dynamic scales and coupling variations, a step the paper flags as planned but does not demonstrate."],"forward_implications":["The analytic expressions evaluate in milliseconds per phase-space point, so scans over masses, kinematics, and renormalization schemes need no precomputed grids.","The gg->Z*Z* amplitudes open the off-shell region up to virtualities near the Higgs mass, where interference between Higgs-mediated and non-resonant amplitudes is used to constrain the Higgs width.","The by-product gg->gamma*gamma* and gg->Z*gamma* amplitudes give heavy-quark-loop contributions to off-shell diphoton and photon-Z production.","For tested phase-space points the ZZ virtual corrections agree with numerical benchmarks below 0.02% (below 0.005% for p_T below 210 GeV), and the ZH corrections are generally far below the percent level.","Combined with existing real-emission contributions, these finite virtual corrections complete NLO predictions for gluon-initiated ZH and ZZ production at the LHC."],"supporting_citations":[{"why":"Supplies the earlier gg->ZH high-energy and large-m_t virtual corrections that this paper extends to 112 m_t terms and merges with a deeper forward expansion.","marker":"[13]"},{"why":"Gives the on-shell double-triangle contribution for gg->ZZ; its off-shell analogue is the piece the paper states is not yet known.","marker":"[15]"},{"why":"Introduces the Padé-improved high-energy expansion for gg->ZZ and the low/high-energy two-loop results that define the starting point for the ZZ analysis.","marker":"[16]"},{"why":"Provides the on-shell p_T-expansion results for gg->ZZ used as a comparison benchmark for the new t-expansion.","marker":"[19]"},{"why":"Provides the numerical two-loop gg->ZH amplitudes used as the reference in the V_fin^ZH comparison plots.","marker":"[20]"},{"why":"Provides numerical two-loop helicity amplitudes for on-shell gg->ZZ used as benchmarks for the ZZ expansion.","marker":"[21]"},{"why":"Provides the complete NLO numerical gg->ZZ result used as the high-precision reference for the V_fin^ZZ ratio.","marker":"[23]"},{"why":"Establishes the combined forward-limit/high-energy expansion method and the 161 two-loop master integrals on which both expansions in this paper are built.","marker":"[24]"}],"fun_headline_variants":["Two expansions cover two-loop ZH and off-shell ZZ","Forward and high-energy series stitch full phase space for ZH and ZZ","Two-loop virtuals for ZH and off-shell ZZ via matched expansions","Pade meets forward limit: two-loop ZH and ZZ across all phase space","High-energy and forward expansions unite for two-loop ZH and off-shell ZZ"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The results stand or fall on the assumption that expansions validated at one loop and for on-shell Z pairs also converge for off-shell Z virtualities—where the paper notes after Eq. (27) in Sec. 3.1 that even the double-triangle contribution is not yet known, and no independent two-loop off-shell benchmark exists.","fun_headline_variants_meta":{"raw":{"variants":["Two expansions cover two-loop ZH and off-shell ZZ","Forward and high-energy series stitch full phase space for ZH and ZZ","Two-loop virtuals for ZH and off-shell ZZ via matched expansions","Pade meets forward limit: two-loop ZH and ZZ across all phase space","High-energy and forward expansions unite for two-loop ZH and off-shell ZZ"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000247,"raw_usage":{"total_tokens":1348,"prompt_tokens":684,"completion_tokens":664,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":428,"completion_tokens_details":{"reasoning_tokens":573}},"tokens_in":428,"tokens_out":664,"duration_ms":6475,"temperature":1.0,"reasoning_tokens":573,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T22:52:02.225273+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An independent numerical two-loop evaluation of a single off-shell gg->Z*Z* helicity amplitude away from the on-shell point (for example sqrt(s)=400 GeV, p_T=100 GeV, q3^2=q4^2=(100 GeV)^2), compared directly with the expansion, would settle whether the convergence observed in the q3^2,q4^2 series persists at two loops; disagreement beyond the claimed uncertainty would invalidate the off-shell extrapolation.","supporting_citations":[{"cited_title":"Analytic approximations of $2\\to 2$ processes with massive internal particles","cited_arxiv_id":"2302.01356","evidence_quote":"Establishes the combined forward-limit/high-energy expansion method and the 161 two-loop master integrals on which both expansions in this paper are built."}],"review_version":1}