{"id":"c9ffc879-32ba-4be0-bcaa-ec36358eec35","arxiv_id":"2512.02999","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"All nine planar three-loop four-point integral families for two massive external legs are cast into canonical differential equations and evaluated numerically, completing the set needed for leading-colour N3LO diboson production.","lead":"This paper computes all planar three-loop Feynman integrals needed for the most precise QCD predictions of vector-boson pair production at the LHC. It provides the missing mathematical building blocks in both equal- and different-mass cases, opening the way to next-to-next-to-next-to-leading order (N3LO) calculations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The nine-family classification is asserted without proof; if a tenth planar topology exists, the headline 'all planar three-loop integrals' fails. Independent verification is needed.","rationale":"The reader's weakest assumption correctly identifies the classification in Sec. 2.2. I agree this is the most load-bearing gap because the abstract's primary claim is completeness ('all planar three-loop master integrals'). The internal computations for each of the nine families are well supported: canonical DEs, pure bases, 16-digit AMFlow cross-checks, and the analytic equal-mass comparison for PL1/PL2. However, these validations only establish correctness for the families considered; they do not establish that no other planar family exists. The authors state the nine-family taxonomy as a fact without a proof or reference. While it is likely true from known classifications of planar three-loop topologies, the absence of a rigorous enumeration means the central claim is not yet fully established. A conditional accept is appropriate: the nine-family results can be accepted as correct, but the 'all' claim should be contingent on an independent topology check. I also note the different-mass numerical coverage issue (the boundary point (4.3) is not in the physical region for m4/m3 > ~0.80, which includes realistic WZ mass ratios), but that is secondary to the classification question and could be addressed by the authors in a revision; I focus on the completeness claim as the single load-bearing concern.","tokens_in":30366,"tokens_out":9874,"duration_ms":90156,"concrete_test":"Run an independent topology enumeration (e.g., with QGRAF or a graph-theory script) of all planar three-loop four-point graphs with two massive external legs and massless internal propagators, and check that every top-level sector reduces via IBPs to one of the nine families (or equivalently, can be embedded in F123/F132). If a tenth top-level topology is found, the completeness claim fails; if none is found, the classification is confirmed.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central claim is that the nine families in Sec. 2.2 cover *all* planar three-loop four-point integrals with two massive external legs and massless internal propagators. This is stated in the Introduction ('Based on their propagator structure, all planar three-loop integrals with two external massive particles can be organized into nine integral families') and in Sec. 2.2 ('all planar three-loop Feynman integrals ... belong to the nine integral families'), but no derivation, citation, or algorithmic enumeration is provided. The families are defined through two 15-propagator superfamilies F123 and F132, with top sectors shown in Figs. 1–3. If any planar graph cannot be mapped onto one of these superfamilies (or a sub-sector thereof), the computed set is incomplete and the abstract's 'all planar' claim is false. The equal-mass case is obtained by a limit, so a missing different-mass topology would also affect that. This is not a criticism of the nine individual computations, which are cross-checked, but of the exhaustive coverage required by the headline.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper presents the computation of all planar three-loop four-point Feynman integrals with two massive external legs, in both equal- and different-mass configurations, intended for the leading-colour N3LO QCD corrections to vector-boson pair production. The integrals are organized into nine families (two reducible, three ladder-box, four tennis-court), grouped into two 15-propagator superfamilies. For each family, the authors construct pure master integrals, canonical ε-factorised differential equations, and the associated dlog alphabets, using finite-field IBP reduction and a bottom-up basis construction. Numerical values are obtained by generalised power series expansions with boundary conditions from AMFlow. Validation is performed against independent AMFlow evaluations at three phase-space points per mass configuration, against the analytic results of [23] for the equal-mass PL1 and PL2 families, and by a 16-digit precision self-consistency check. The paper provides ancillary files with the master-integral definitions, the connection matrices, boundary values, and evaluation scripts.","tokens_in":30667,"tokens_out":8629,"duration_ms":80741,"significance":"The combination of a canonical-DE database for nine two-mass planar three-loop four-point families with a working numerical evaluation pipeline is a substantial technical contribution. If the completeness of the nine-family set and the coverage of the physical region are established, this work supplies the last missing ingredient for the purely virtual part of N3LO diboson production in leading colour. The paper ships machine-readable ancillary files and a reproducible evaluation workflow; the extensive numerical cross-checks (16-digit agreement) are a notable strength. The observed enlargement of the alphabet relative to two loops is also of independent interest for the analytic structure of Feynman integrals.","major_comments":[{"comment":"The classification of all planar three-loop four-point integrals with two massive external legs into the nine families is stated without proof or citation: 'all planar three-loop Feynman integrals relevant to the production of two vector bosons belong to the nine integral families' (Sec. 2.2). This assertion is load-bearing for the abstract's 'all planar' claim. A missing topology would invalidate the headline even though the nine individual computations are correct. Please add a rigorous justification: either an explicit enumeration of planar three-loop graphs showing that every relevant graph is a subgraph of one of the 15-propagator superfamilies F123/F132, or a citation to a published classification. If such a proof is not available, the claims should be softened to 'the nine families considered here'.","section":"§2.2, Figs. 1–3"},{"comment":"The claim that the different-mass physical region is covered by the stated evaluation protocol rests on two unproven statements: (i) every point on the m4-axis segment from (4.1) to (4.3) is physical, and (ii) for fixed masses the physical region is star-shaped with respect to (4.3). The text says these were verified with Reduce, but no script or output is included. Because this underpins the advertised 'any physical point' evaluation, please make the verification reproducible (e.g., include the Reduce command and result as an ancillary file) or give an analytic argument. Also state explicitly the caveat about the letter m3^2-m4^2 and the recommended mass ordering.","section":"§4, eqs. (4.1)–(4.3)"}],"minor_comments":[{"comment":"The section appears twice in the manuscript text, and the second occurrence contains an embedded editorial note '[DC: Notation a bit unclear...]'. The reference for the equal-mass sector is inconsistent: the text mentions [23] in one place and [24] in the duplicate, and the footnote listing integrals T59, T61, T62, T63, T66, T67 also switches between [23] and [24]. Please remove the duplicate and editorial note and reconcile the citations.","section":"§3.1.1 (and text overall)"},{"comment":"The completeness of the 37-letter alphabet is justified by assumptions (q_j=1, c=±4, restriction (3.18)) that are described as observations from [101]. Since the 'new letters' and 'new square roots' conclusions in Section 5 rest on this alphabet, please state explicitly that the alphabet is determined by this heuristic (with numerical validation) rather than by a proven classification.","section":"§3.2"},{"comment":"Typo 'prefect agreement' should be 'perfect agreement'. Also, the claim about the m4-axis connectivity should include the explicit Reduce query or a statement that the verification script is provided in the ancillary files.","section":"§4"}],"recommendation":"major_revision","confidential_remarks":"The main concern is whether the 'all planar' claim is adequately supported. In my view this is fixable with an added enumeration/citation, so I recommend major revision rather than rejection. The numerical work appears solid; the uncertainties are in the completeness and coverage arguments, not in the per-family computations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the planar two-mass three-loop four-point integral program is now essentially complete in canonical form. The paper constructs pure bases and canonical DEs for all nine families in both equal- and different-mass configurations, provides the full alphabet, and delivers a working numerical evaluation pipeline. The cross-checks are real: 16-digit agreement with independent AMFlow runs at three points per mass configuration, and agreement with Long's analytic equal-mass results for PL1 and PL2. The Zenodo deposit has the bases, d matrix, boundary values, and both DiffExp and AMFlow drivers, so the results are usable as supplied. That is reproducible evidence, not just a writeup.\n\nWhat is genuinely new beyond the authors' earlier first-computation paper and Long's ladder-box paper: the four tennis-court families, the missing ladder-box families, the four new square roots, and the enlarged alphabet (37 letters in the general case, 22 in the equal-mass case). The observation that the planar alphabet grows from two to three loops in the two-mass case is an interesting structural data point.\n\nSoft spots, in proportion. The one that matters is the classification: the statement that all planar three-loop two-mass four-point integrals belong to these nine families is asserted without proof or citation. That assertion is load-bearing for the word 'all' in the title. I don't think it is false—the authors clearly know the topology space and the two-loop case—but a referee should ask for an explicit enumeration or a citation to a classification. The second issue is minor: coverage of the different-mass physical region from the boundary point relies on a Reduce-based check rather than a proof, and the paper says so. The proposed workaround (first evolve m4, then move in s12, s23) looks sound, though it deserves a one-line verification in a revised version.\n\nThe DE construction is standard, with no fitted parameters and no sign of circularity. The numerical evaluation is validated, not just self-consistent.\n\nWho should read it: amplitude practitioners working toward N3LO diboson production, and anyone interested in the analytic structure of two-mass integrals. It deserves a serious referee. I would want the referee to do a spot-check of the ancillary files and press for a proof or reference for the nine-family claim. Accept after revision.","headline":"Completes the planar two-mass three-loop four-point integral set with validated numerics; the 'all' claim rests on an unproved nine-family classification that should be tightened.","tokens_in":31063,"tokens_out":3558,"would_cite":true,"duration_ms":34098,"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 computes all planar three-loop Feynman integrals needed for the leading-colour N3LO QCD corrections to the production of two massive vector bosons at hadron colliders.","keywords":["Feynman integrals","master integrals","three-loop","vector-boson pair production","canonical differential equations","planar integrals","N3LO QCD","two massive external legs"],"falsifier":"Finding a planar three-loop four-point integral with two massive external legs and massless internal propagators whose top sector is not contained in any of the nine families would disprove the completeness claim; alternatively, exhibiting a physical phase-space point that cannot be connected to the boundary point by a straight line inside the physical region would break the evaluation pipeline.","tokens_in":30250,"feed_emoji":"⚛️","tokens_out":8395,"duration_ms":79008,"temperature":0.7,"pith_summary":"The paper sets out to compute every planar three-loop Feynman integral that enters the leading-colour N3LO QCD corrections to the production of two massive or off-shell vector bosons at hadron colliders. It organises these integrals into nine four-point families with massless internal propagators and two external massive legs, covering both equal and different masses. For each family it constructs a pure basis of master integrals whose differential equations take canonical logarithmic form, and it evaluates the integrals numerically by generalised power series expansions. If the classification is complete, this supplies the last missing planar virtual ingredient for three-loop diboson amplitude computations.","feed_headline":"All planar three-loop integrals for vector-boson pairs computed","feed_subtitle":"This completes the planar virtual integrals needed for next-to-next-to-next-to-leading-order diboson predictions.","key_machinery":"The load-bearing mechanism is the canonical differential equation for a pure basis of master integrals: a choice of integrals that makes the dimensional-regulator dependence factor out and leaves only logarithmic one-forms, or letters. This reduces evaluation of the integrals to iterated integrals over a finite alphabet. The paper builds these pure bases sector by sector, using finite-field reconstruction to manage the algebraic complexity, and determines the alphabet from the singularity structure of the differential equations.","core_discovery":"The central discovery is a complete set of canonical differential equations for all planar three-loop four-point integrals with two massive external legs. The nine families—two reducible, three ladder-box, four tennis-court topologies—are each equipped with a pure basis of master integrals and a connection matrix of logarithmic one-forms; the alphabet contains 37 letters in the general mass case and 22 in the equal-mass case, including four square roots not present at two loops. Boundary values are fixed at a physical phase-space point, and the integrals are evaluated by solving the differential equations as generalised power series expansions, with independent numerical checks at several ph","pith_inferences":["The nine-family classification is asserted without derivation or citation; a tenth planar topology with two massive legs would invalidate the word 'all'. This is an editorial caution, not a failure of the construction.","The paper explicitly stops at planar virtual integrals: non-planar contributions, top-quark loops, and real-emission pieces are outside its scope, so a full N3LO cross-section still requires those ingredients.","A self-inserted note in the manuscript flags unclear notation in the figure for the 15-integral sector; the ancillary files supply the definitions, so this is a presentation gap rather than a mathematical one.","Since the five square roots are not simultaneously rationalisable, a fully analytic multiple-polylogarithm solution may require splitting the integration domain and rationalising subsets at each stage—a testable direction the paper itself suggests."],"forward_implications":["The nine families, in both equal- and different-mass configurations, are now available in canonical form with a numerical evaluation pipeline for physical kinematics.","The planar virtual master integrals required for leading-colour N3LO diboson amplitudes are, on this paper's claim, complete; three-loop amplitude computations can proceed on this basis.","Because the solutions are iterated integrals of logarithmic one-forms, the results enable further formal study of the integrals' algebraic structure.","The alphabet contains more letters and square roots than the two-loop case, showing that analytic complexity grows with loop order when two external masses are present.","Independent numerical checks at multiple phase-space points agree to 16 digits, and the equal-mass ladder-box families match an existing analytic solution."],"fun_headline_variants":["All planar three-loop integrals for dibosons solved","Nine planar three-loop families for dibosons fully done","Complete set of planar three-loop diboson integrals","Diboson N3LO: planar three-loop integrals complete","Planar three-loop integrals for vector-boson pairs obtained"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The completeness claim rests on the unproven assertion that every planar three-loop integral with two massive external legs and massless propagators belongs to one of the nine listed families, and the numerical pipeline assumes the chosen boundary point reaches the whole physical region by straight lines, a property verified numerically rather than proven.","fun_headline_variants_meta":{"raw":{"variants":["All planar three-loop integrals for dibosons solved","Nine planar three-loop families for dibosons fully done","Complete set of planar three-loop diboson integrals","Diboson N3LO: planar three-loop integrals complete","Planar three-loop integrals for vector-boson pairs obtained"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000417,"raw_usage":{"total_tokens":1912,"prompt_tokens":597,"completion_tokens":1315,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":341,"completion_tokens_details":{"reasoning_tokens":1233}},"tokens_in":341,"tokens_out":1315,"duration_ms":12983,"temperature":1.0,"reasoning_tokens":1233,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T18:51:40.667377+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Finding a planar three-loop four-point integral with two massive external legs and massless internal propagators whose top sector is not contained in any of the nine families would disprove the completeness claim; alternatively, exhibiting a physical phase-space point that cannot be connected to the boundary point by a straight line inside the physical region would break the evaluation pipeline.","supporting_citations":[],"review_version":1}