{"id":"6163c9d5-415b-482c-9e4a-0847280d31c4","arxiv_id":"2502.09132","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Full NLO QCD predictions for pp -> hhjj production in VBF with HEFT couplings cλ, cV, c2V, including non-factorising and Higgs-strahlung diagrams, show that some allowed coupling combinations reshape key observables.","lead":"This paper computes the next-to-leading-order QCD corrections to double-Higgs production through vector-boson fusion at the LHC, allowing anomalous Higgs couplings within the Higgs Effective Field Theory framework. It is a read for Higgs physicists because it removes an approximation used by earlier public tools and shows that certain experimentally allowed coupling combinations visibly distort key distributions.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sec. 2.2 asserts without enumeration that no dχ=4 HEFT operator contributes to pp→hhjj at O(g_ew^4 g_s^2); an overlooked operator would undermine the 'leading HEFT' claim and the shape results.","rationale":"I agree with the reader that the weakest link is the completeness of the operator set (2.9). The rest of the calculation is well supported: the amplitude-level validation against OpenLoops over 10^6 phase-space points with O(10^-8) agreement in the virtual finite part is strong, the polynomial (4.6) is an exact monomial decomposition for the three coupling structures in the Born subdiagrams, and the scale variation is handled conventionally. The L4 issue is distinct and load-bearing because the abstract claims 'leading operators in HEFT'; if a missing dχ=4 operator contributes at the same order, the cross-section parametrisation and the shape observables in Sec. 4.3 could be numerically affected. The paper's exclusion argument in Sec. 2.2 is asserted rather than demonstrated, and it explicitly covers only part of the L4 basis. This does not invalidate the NLO QCD machinery for the given Lagrangian, but it means the 'leading' qualifier is not yet fully justified. The reader's CONDITIONAL verdict already reflects this, so no verdict change is needed; the condition is to make the L4 power-counting check explicit or to soften the claim to the operators of eq. (2.9).","tokens_in":19448,"tokens_out":33703,"duration_ms":367153,"concrete_test":"Generate the complete L4 operator list for the HEFT basis of Refs. [38, 57, 58] (e.g., via the Hilbert-series database or a FeynRules/UFO implementation of the NLO HEFT Lagrangian), and for each operator test whether it can be attached to four light-quark legs and two Higgs legs in a tree diagram with coupling order O(g_ew^4 g_s^2) under the assignment used in Sec. 2.2. Concretely, write a short script or hand table that enumerates all L4 operators and computes the formal power of g_s and g_ew for their lowest-order contribution to q q → q q h h; if any operator yields exactly O(g_s^2 g_ew^4), the exclusion fails. If none does, the 'leading' claim is supported and the conditional can be relaxed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that the calculation captures the leading HEFT effects in pp→hhjj at NLO QCD rests on the assertion in Sec. 2.2 that no L4 (dχ=4) operator can be inserted at tree level with the required external states and coupling order O(g_ew^4 g_s^2). The argument is not shown: 'One can check' is the full derivation. The discussion explicitly covers only the X^2 U h and X U h D^2 bosonic classes; it does not address the fermionic operators in the complete L4 basis of Refs. [38, 57, 58], e.g., four-fermion contact terms with two Higgs legs or ψ^2 X h operators. The coupling-power assignment used to exclude gluon-Higgs operators (assigning g_s^2 to loop-generated ggh/gghh vertices) is an assumption about the UV origin of the coefficients, not a consequence of HEFT chiral counting itself. If any L4 operator contributes at the same order as the NLO QCD corrections, the restricted Lagrangian (2.9) is not the full leading set, and the polynomial (4.6) and the shape statements in Sec. 4.3 would be incomplete. The NLO QCD computation for the given Lagrangian is not put in question, but the interpretation as 'leading HEFT' is.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents the next-to-leading order QCD corrections to Higgs pair production in vector boson fusion, pp -> hhjj, within the Higgs Effective Field Theory framework. The calculation uses an automated Whizard-GoSam interface with a UFO model file containing the three leading anomalous couplings c_lambda, c_V and c_2V. It includes non-factorisable one-loop diagrams and s/u-channel Higgs-strahlung type topologies, going beyond the usual structure-function approximation. The authors validate their setup at LO against MadGraph5 and Whizard+OpenLoops, and at the amplitude level against OpenLoops over a large phase-space sample. They present total cross sections and K-factors for eleven benchmark points, a polynomial parametrisation of the total cross section in the anomalous couplings, and differential distributions for p_T(h), m_hh, Delta_eta(h,h) and Delta_R(h,h). The central phenomenological claim is that combinations of anomalous couplings that are still allowed by current constraints can produce characteristic shape distortions in these observables.","tokens_in":19717,"tokens_out":15251,"duration_ms":169235,"significance":"If the operator-completeness claim is secured, this is a valuable calculation: it provides the first full NLO QCD prediction for HEFT VBF double-Higgs production without the structure-function approximation, and it demonstrates that combined anomalous couplings can generate shape features that are absent when each coupling is varied individually. The technical validation is strong: amplitude-level checks over 10^6 phase-space points show relative differences of 10^-15 for Born and real amplitudes and 10^-8 for the virtual finite part, and the authors make their UFO model and example run files publicly available. The shape observables, especially the dip-to-peak change in Delta_eta(h,h), are plausible and experimentally relevant for the HL-LHC. The main reservation concerns the derivation of the 'leading HEFT operator' set, which is the basis for interpreting the results as the leading EFT effects.","major_comments":[{"comment":"The claim that the Lagrangian (2.9) contains the leading HEFT operators for pp -> hhjj at NLO QCD rests on the statement in Sec. 2.2 that 'one can check' that no L4 operator can be inserted in a tree topology at O(g_ew^4 g_s^2). This is not demonstrated. The subsequent discussion covers only the bosonic classes X^2 U h and X U h D^2; it does not enumerate the fermionic operators in the complete L4 bases of Refs. [38, 57, 58] (for example four-fermion contact terms with two Higgs fields, or psi^2 X h operators), nor does it show that their coefficients cannot be of comparable size under a weakly or strongly coupled UV. The coupling-power assignment that excludes gluon-Higgs operators is an assumption about UV origins rather than a consequence of HEFT chiral counting. Because the abstract and Sec. 2.2 use 'leading' to characterise the restricted operator set, this is load-bearing for the shape conclusions in Sec. 4.3 and for the polynomial (4.6). Please either provide a complete enumeration of the L4 operators and a stated power-counting assumption, or explicitly restrict the claims to the L2 Lagrangian and adjust the abstract and conclusions accordingly.","section":"Sec. 2.2, Eq. (2.9)"}],"minor_comments":[{"comment":"There is a typo in the Introduction: 'in the in the so-calledstructure function approximation' should read 'in the so-called structure function approximation'.","section":"Sec. 1"},{"comment":"The fitted polynomial (4.6) should give sigma/sigma_SM = 1 at c_lambda = c_V = c_2V = 1, but the central coefficients in Table 2 sum to about 1.04 at that point. Please clarify whether the SM point was included in the fit and consider constraining the parametrisation to pass through (1,1,1).","section":"Sec. 4.2, Table 2"},{"comment":"The conclusions state that the NLO QCD corrections reduce the SM cross section by about 7%, but the SM row of Table 1 gives sigma_NLO/sigma_LO = 0.904(6), which is a reduction of about 9.6%. Please reconcile this statement with the tabulated result.","section":"Sec. 5"},{"comment":"The GoSam version used is described as 'to appear' [42], and the Whizard-GoSam interface details are referenced to a Master thesis [88]. Please provide version identifiers or a more detailed public documentation of the interface so that the calculation is reproducible beyond the provided example files.","section":"Sec. 3.1"},{"comment":"In Figures 3 and 4, the uncertainty bands are attributed to the uncertainties of the fit coefficients. Please state explicitly whether the full covariance matrix of the fit was propagated, since the coefficients in Table 2 are strongly correlated.","section":"Sec. 4.2"},{"comment":"Several entries in Table 1 have a lower scale-variation error of exactly 0.000 while the upper error is non-zero. A brief comment on why the scale dependence is one-sided for those benchmark points would be helpful.","section":"Sec. 4.1, Table 1"}],"recommendation":"major_revision","confidential_remarks":"The technical NLO QCD calculation appears solid and well validated. The main issue is the HEFT operator-completeness claim in Sec. 2.2: the text asserts without an explicit enumeration that no L4 operator contributes at the relevant order. This should be fixable either by supplying the missing argument or by softening the 'leading HEFT' claim in the abstract and conclusions. If the authors choose the latter, the paper would still be a useful NLO QCD calculation for the L2 Lagrangian with anomalous couplings, but the scope would need to be stated more carefully. I would be supportive of publication after a revision that addresses this point."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper gives the first full NLO QCD prediction for pp→hhjj at O(α_ew^4) in HEFT with the leading couplings cλ, cV, c2V, going beyond the structure-function approximation to include non-factorising and Higgs-strahlung topologies. That is the real content and it is genuinely new. The validation is the strongest part: LO agreement with MadGraph and Whizard+OpenLoops, amplitude-level agreement with OpenLoops at 10^-15 for Born and real amplitudes and 10^-8 for the virtual finite part over 10^6 phase-space points, and K-factors around 0.90 with reduced scale dependence. The cross-section parametrisation (4.6) is a monomial decomposition of the squared matrix element fitted to computed benchmark points, not a fit of the target physics. The UFO model and run files are promised on a repository.\n\nThe soft spot is the completeness argument in Sec 2.2. The paper asserts without showing an enumeration that no L4 operator contributes at O(g_ew^4 g_s^2). The text only discusses the bosonic X^2 U h and X U h D^2 classes; fermionic L4 operators such as ψ^2 X h or four-fermion contact terms are not addressed. In my reading this is a presentation gap rather than a fatal flaw: for the Lagrangian (2.9), the NLO QCD computation is correct, and plausible L4 effects sit at the level of NLO EW or are Yukawa suppressed, not at the level of NLO QCD. But 'one can check' is doing real work here, and the leading-HEFT interpretation depends on it. The referee should ask the authors to spell out the check or cite a complete enumeration.\n\nMinor: the Whizard-GoSam interface is not yet public, and reproducibility depends on the release. The comparison to the concurrent work [98] is deferred, which is fine. The shape effects in Δη(h,h) and ΔR(h,h) are physically plausible and clearly explained.\n\nThis is for precision Higgs phenomenologists and experimental analysts looking for VBF shape observables. It deserves a serious referee; send it out, with the L4 enumeration and tool release as the main requests.","headline":"Solid, well-validated NLO QCD prediction for VBF double-Higgs in HEFT; the completeness argument for dropping L4 operators is under-explained but does not sink the calculation.","tokens_in":20324,"tokens_out":12098,"would_cite":true,"duration_ms":125158,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["12.38.Bx","13.85.Lg","14.80.Bn"],"model":"deepseek-v4-flash","headline":"This paper establishes that the leading HEFT couplings $c_\\lambda$, $c_V$, and $c_{2V}$ control both the rate and the shape of $pp\\to hhjj$ at NLO QCD, and that combinations within current experimental bounds can strongly distort…","keywords":["Higgs pair production","vector boson fusion","NLO QCD","Higgs Effective Field Theory","anomalous Higgs couplings","LHC phenomenology","kappa framework"],"falsifier":"A concrete check is to enumerate the $X^2 U h$ and $X U h D^2$ classes of the complete $d_\\chi=4$ HEFT basis and test whether any operator contracts to a $q q \\to q q h h$ tree topology with two electroweak emissions at $O(g_{\\rm ew}^4 g_s^2)$. If such an operator exists, the restricted Lagrangian (2.9) is incomplete and the shape predictions would need revision.","tokens_in":68,"feed_emoji":"⚛️","tokens_out":10329,"duration_ms":105589,"temperature":0.7,"pith_summary":"The paper presents the full next-to-leading-order QCD prediction for Higgs-boson pair production in vector boson fusion, $pp \\to hhjj$ at order $\\alpha_{\\rm ew}^4$, within Higgs Effective Field Theory. It keeps the three leading anomalous couplings—$c_\\lambda$ for the trilinear Higgs coupling, $c_V$ for the single $hVV$ coupling, and $c_{2V}$ for the double $hhVV$ coupling—and includes non-factorisable QCD diagrams and Higgs-strahlung topologies, so it goes beyond the structure-function approximation used by earlier tools. A sympathetic reader should care because these three couplings are exactly the ones the HL-LHC can hope to constrain through double-Higgs VBF, and the paper shows that combinations of them can change not just the total rate but the shapes of $m_{hh}$, $p_T^h$, $\\Delta\\eta(h,h)$, and $\\Delta R(h,h)$ in experimentally accessible regions.","feed_headline":"HEFT anomalies flip VBF di-Higgs shapes at NLO QCD","feed_subtitle":"Full NLO QCD beyond structure-function approximation: LHC-allowed coupling shifts change rates 13-fold and flip shapes.","key_machinery":"The load-bearing object is the effective Lagrangian (2.9), which parametrises the leading HEFT effects through three coupling multipliers $c_\\lambda$, $c_V$, and $c_{2V}$ attached to the $h^3$, $hVV$, and $hhVV$ vertices. Around this Lagrangian the calculation is organised by chiral dimension: the paper assigns $d_\\chi=0$ to gauge and scalar fields and $d_\\chi=1$ to each weak coupling, derivative, and fermion bilinear, which fixes the operator counting and justifies dropping $d_\\chi=4$ operators at this order. The technical carrier is an automated NLO interface between a Monte Carlo event generator and a one-loop amplitude generator using a UFO model file, which delivers real and virtual corrections including non-factorising diagrams and $s$-channel Higgs-strahlung topologies rather than factorised DIS-like structures.","core_discovery":"The central claim is that the leading HEFT Lagrangian, restricted to the three custodial-symmetric operators of equation (2.9), is sufficient for a complete NLO QCD description of electroweak $hhjj$ production, and that this calculation can be done without the structure-function approximation. The authors argue that at $O(\\alpha_{\\rm ew}^4 \\alpha_s)$ no $d_\\chi=4$ HEFT operator can form the required tree topology with the correct coupling powers, so the anomalous vertices coincide with the $\\kappa$-framework modifiers. Numerically, the NLO QCD $K$-factor is about $0.90$ and varies by only a few percent across the coupling parameter space, while the total cross section can range from roughly half to more than an order of magnitude above the SM value. The more distinctive result is shape-level: combinations of anomalous couplings that are individually within current experimental bounds can turn the dip in the pseudorapidity separation $\\Delta\\eta(h,h)$ into a peak, create a peak-dip structure in $m_{hh}$, and enhance the high-$p_T$ tails, patterns that would not appear when varying any single coupling alone.","pith_inferences":["Editorial inference: because the anomalous couplings enter only the electroweak subdiagram, the near-universal K-factor suggests the NLO QCD correction could be transferred to any $\\kappa$-framework model as a process-level factor, but the paper does not claim this beyond the investigated parameter space.","Editorial inference: the shape flips in the central $\\Delta\\eta(h,h)$ region are promising discriminants for an experimental search, since they appear at moderate separations rather than only in high-energy tails, which is a testable strategy the paper leaves implicit.","Editorial inference: when NLO electroweak corrections are added, $d_\\chi=4$ operators will introduce vertex structures beyond the $\\kappa$-framework, so the simple coupling-multiplier picture of this paper is expected to break down at that order.","Editorial inference: the same automated chain could be used to produce NLO QCD predictions for single-Higgs VBF or for $hhjj$ in SMEFT, where the operator counting and the pattern of shape distortions would differ."],"forward_implications":["Within the stated HEFT operator set, the NLO QCD prediction for $pp\\to hhjj$ is now known without the structure-function approximation, so the non-factorisable and Higgs-strahlung contributions are included at perturbative order $\\alpha_{\\rm ew}^4 \\alpha_s$.","The K-factor is nearly flat and almost independent of the anomalous couplings, meaning the QCD correction factor can be treated as approximately universal across the HEFT parameter space.","The polynomial parametrisation of the total cross section in $c_\\lambda$, $c_V$, and $c_{2V}$ gives a fast continuous interpolation that can be evaluated for arbitrary coupling values without rerunning the full simulation.","Coupling combinations that are individually allowed by current experimental constraints can change the shape of $\\Delta\\eta(h,h)$ and $m_{hh}$ in ways that a single-coupling analysis would miss.","The new NLO setup is usable for arbitrary user-defined processes beyond this one, so the same interface can be applied to related VBF and EFT observables."],"supporting_citations":[{"why":"supplies the complete NLO HEFT operator basis used to identify and exclude $d_\\chi=4$ contributions.","marker":"[38]"},{"why":"defines the chiral-dimension power counting that fixes which operators enter at the targeted coupling order.","marker":"[56]"},{"why":"provides the previous NNLO-plus-EW structure-function prediction that this calculation goes beyond.","marker":"[14]"},{"why":"analyses unitarity in $VV\\to hh$, used to explain why $c_V$ and $c_{2V}$ distortions grow in the high-$p_T$ tails.","marker":"[23]"},{"why":"the one-loop amplitude generator whose automated approach is the core of the NLO virtual amplitudes.","marker":"[40]"},{"why":"the Monte Carlo event generator that provides phase-space integration and real-emission handling.","marker":"[44]"},{"why":"defines the standard interface connecting the event generator to the one-loop provider, enabling the automated NLO chain.","marker":"[45]"},{"why":"the simultaneous complementary NLO+PS calculation in the structure-function approximation, used to position the present full-topology result.","marker":"[98]"}],"fun_headline_variants":["Anomalous HEFT terms reshape VBF di-Higgs observables at NLO","NLO QCD beyond structure functions alters VBF hh shapes","Allowed coupling shifts flip di-Higgs shapes in VBF at NLO","Full NLO QCD shows HEFT shape changes in VBF di-Higgs","HEFT coupling shifts within LHC bounds flip VBF di-Higgs shapes"],"cache_read_input_tokens":22400,"weakest_assumption_plain":"The calculation's HEFT interpretation rests on the claim that no higher-order effective operator can contribute to this process at $O(\\alpha_{\\rm ew}^4 \\alpha_s)$; the paper asserts this is checkable by power counting but does not show the full enumeration.","fun_headline_variants_meta":{"raw":{"variants":["Anomalous HEFT terms reshape VBF di-Higgs observables at NLO","NLO QCD beyond structure functions alters VBF hh shapes","Allowed coupling shifts flip di-Higgs shapes in VBF at NLO","Full NLO QCD shows HEFT shape changes in VBF di-Higgs","HEFT coupling shifts within LHC bounds flip VBF di-Higgs shapes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000853,"raw_usage":{"total_tokens":3680,"prompt_tokens":888,"completion_tokens":2792,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":504,"completion_tokens_details":{"reasoning_tokens":2687}},"tokens_in":504,"tokens_out":2792,"duration_ms":18576,"temperature":1.0,"reasoning_tokens":2687,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T22:32:52.746569+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check is to enumerate the $X^2 U h$ and $X U h D^2$ classes of the complete $d_\\chi=4$ HEFT basis and test whether any operator contracts to a $q q \\to q q h h$ tree topology with two electroweak emissions at $O(g_{\\rm ew}^4 g_s^2)$. If such an operator exists, the restricted Lagrangian (2.9) is incomplete and the shape predictions would need revision.","supporting_citations":[],"review_version":1}