{"id":"1dd70e40-4925-46be-9086-e687992299f7","arxiv_id":"2412.14155","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Complete positivity constraints from causality and unitarity are derived for the 15 NLO HEFT operators contributing to longitudinal gauge-Higgs scattering, shrinking the allowed space to about 5 percent.","lead":"This paper works out the full set of mathematical limits that causality and unitarity place on 15 coupling constants in the Higgs Effective Field Theory, the general low-energy description of the Higgs boson and W/Z particles. The limits rule out most of the parameter space and give the first theoretical bounds on several scattering processes that experiments have not yet measured.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Completeness claim in 'complete set of positivity constraints' is not established: the s^2 coefficient used is tree-level single-insertion only, while one-loop LO-HEFT contributions are O(Λ^2/(16π^2 v^2)) relative to NLO tree and are not negligible for v/Λ ≥ 1/(4π).","rationale":"The reader's verdict (CONDITIONAL) and weakest assumption point to exactly this. I agree: the single-insertion and no-EFT-loop assumptions are the load-bearing premises behind 'complete.' The paper itself flags the assumption in footnote 6 and Sec. 4, so this is not manufactured. The numerical capping procedure and the 5% volume-fraction definition are also not fully reproducible, but they are secondary: if the tree-level coefficient is not the full NLO coefficient, even a perfectly reproducible capping computation would determine the wrong object. I checked the apparent discrepancy between the a_i definitions in Eq. (2.45) and Table 4/Appendix B; it is a relabelling of the same linear combinations, not a mathematical inconsistency, so it is not the main concern. The proposed test is feasible: the one-loop coefficient can be computed in the SM/HEFT LO sector and compared numerically; this would settle whether the omitted contributions are negligible at the quoted cutoffs. Until then, the central claim should be read as 'tree-level, single-insertion positivity constraints,' not the complete NLO statement advertised.","tokens_in":30542,"tokens_out":20397,"duration_ms":180709,"concrete_test":"Compute the one-loop contribution to c^{2,0}_{ijkl} in the LO HEFT (SM Higgs-goldstone sector) for a representative channel, e.g. W_L W_L -> W_L W_L or Z_L h -> Z_L h, with the same IR subtraction as Eq. (2.37), and compare it with the smallest entries of Table 6 at Λ=1.8 TeV (for example c2 ~ 0.44 x 10^-2 or c15 upper bound ~ 0.021). If the loop coefficient is of order s^2/(16π^2v^2) and comparable to these bounds, the tables do not bound the 15 WCs as stated. A cleaner check: re-run the dispersion/linear-programming analysis including the one-loop spectral density in Eq. (4.3); if the allowed ranges shift by more than ~30% (the estimated ratio Λ^2/(16π^2v^2) at 1.8 TeV), the completeness claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that Tables 4 and 6 give the complete set of NLO HEFT positivity constraints on the 15 WCs. This requires that the c^{2,0}_{ijkl} entering the dispersion relation (3.2) and the linear program of Sec. 4 are exactly the single-insertion tree-level contributions of the 15 operators in Table 1. The paper asserts this twice: Sec. 2.1 ('If we consider more insertions, operators from other categories... might also result in s^2 growth... can therefore be safely ignored') and Sec. 4 ('with our assumption that EFT loops can be neglected'). The stated power counting (2.7), however, suppresses extra derivatives and h factors, not loop factors. A one-loop diagram built from the two-derivative LO HEFT vertices has four derivatives and contributes to c^{2,0}_{ijkl} at order s^2/(16π^2 v^2). The ratio to a single NLO insertion, c_i s^2/v^4 with c_i ~ v^2/Λ^2, is Λ^2/(16π^2v^2). With the paper's own regime v/Λ ≥ 1/(4π), this ratio is O(1): ~0.34 at Λ=1.8 TeV and ~0.6 at Λ=2.4 TeV. Thus the neglected EFT-loop term is not parametrically subleading exactly when HEFT is the relevant framework. If the loop term shifts the effective coefficient, the bounds of Table 4/6 constrain c_tree + c_loop, not the 15 WCs as labelled. Multi-insertion NLO terms are suppressed by v^2/Λ^2 and are less dangerous, but the loop contribution is not. The abstract's 'complete' is therefore not supported without either computing these loop contributions or explicitly restricting the claim to tree-level single-insertion NLO amplitude.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives positivity constraints on 15 Wilson coefficients of NLO HEFT operators that contribute to the s^2 piece of forward longitudinal gauge-Higgs scattering amplitudes. Using a U(1)_em invariant parametrization, the authors show that the s^2 coefficient has 15 independent components, map them to the HEFT basis of Ref. [36], derive analytical cone-shaped constraints (Table 4) from positive semidefiniteness of the gamma_beta matrix, and then obtain double-sided numerical 'capping' bounds via linear programming over discretized spectral densities subject to unitarity, st-crossing null constraints, and U(1)_em symmetry (Table 6). The paper recovers the known SMEFT positivity bounds as a special case and compares its results with experimental vector-boson-scattering bounds.","tokens_in":30983,"tokens_out":14212,"duration_ms":117764,"significance":"If the results hold as stated, this is a useful contribution: it extends positivity constraints to the full NLO HEFT operator space, provides first bounds for several processes without experimental limits, and connects the HEFT positivity cone to the SMEFT positivity cone in a transparent way. The analytical derivation is coherent and follows standard dispersion-relation logic, and the explicit mapping between HEFT WCs, amplitude parameters, and anomalous couplings is a valuable phenomenological tool. The main caveats concern the 'complete set' claim: the derivation is tree-level and neglects EFT loops, and the numerical capping procedure lacks reproducibility details. These issues are fixable in revision but currently limit the strength of the central claim.","major_comments":[{"comment":"The claim that Tables 4 and 6 give the 'complete set' of NLO positivity constraints is not established because the low-energy coefficient c^{2,0}_{ijkl} is computed at tree level with a single insertion of the 15 NLO operators, while EFT loops are neglected. The power counting in eq. (2.7) controls insertions of derivatives and Higgs fields, not loop factors. A one-loop diagram built from the two-derivative LO HEFT vertices contributes to c^{2,0} at order s^2/(16π^2 v^4), while a single NLO tree insertion contributes c_i s^2/v^4 with c_i ~ v^2/Λ^2; the ratio is Λ^2/(16π^2 v^2), which is 0.34 at Λ=1.8 TeV and 0.60 at Λ=2.4 TeV. Thus the neglected contribution is O(1) precisely in the regime v/Λ ≥ 1/(4π) adopted in eq. (2.7). As a result, the inequalities of Tables 4 and 6 constrain the sum c_i^{tree} + Δc_i^{loop}, not the bare Wilson coefficients as labelled, and the word 'complete' in the abstract and Sec. 1 is not supported. Please include the one-loop HEFT contributions or explicitly restrict the completeness claim to the tree-level single-insertion approximation.","section":"Secs. 2.1 and 4, eqs. (2.7), (3.2), (4.17)"},{"comment":"The definitions of the amplitude parameters a_i are inconsistent between eq. (2.45) and the Table 4 caption / Appendix B. For example, eq. (2.45) gives a7 = 2c6 + 2c7 + c9 and a9 = c6 + c7/2, whereas Table 4 and eq. (B.12) give a7 = c6 + c7/2 and a9 = c7; the assignments of a8, a12, a14, a15, and a16 also differ. Since Table 4 is the main analytical result and all constraints are written in terms of a_i, this mismatch prevents the reader from verifying the final bounds from the stated amplitude mapping. Please reconcile eq. (2.45) with eq. (2.43), Table 4, and Appendix B, and state which mapping was used in the numerical analysis.","section":"Eq. (2.45) vs Table 4 and Appendix B, eq. (B.12)"},{"comment":"The discretized linear program is not fully reproducible because the truncation orders N and l_M are not stated and no convergence checks are reported. The final numerical bounds in Table 6 depend on these choices; please report the values used and demonstrate that the bounds stabilize as N and l_M are increased.","section":"Sec. 4, Table 5"}],"minor_comments":[{"comment":"The volume fractions 'about 95%' and 'about 74%' are quoted without specifying the measure on the unbounded HEFT cone; please define the bounding box or normalization used to compute these fractions.","section":"Secs. 5.1 and 6"},{"comment":"Several typographical issues should be corrected: the c5 row in Table 6 has a stray double bracket '[−4.31, 4.78]]', and the text contains 'Fog. 5' (Sec. 5.2) and 'whre' (before eq. (2.43)).","section":"Table 6 and text"},{"comment":"The coefficients C^{ijkl}_{r,ir}(l) in Table 5 are not defined in the text; either define them explicitly or provide a precise pointer to the equations in Ref. [18] where they appear.","section":"Table 5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically competent and the analytical derivation is largely standard; the main issue is the overreach in the word 'complete'. I believe the authors can address this by computing or estimating the one-loop HEFT contribution to the s^2 coefficient, or by carefully scoping the claim to tree-level single-insertion bounds. The a_i inconsistency is also fixable but must be resolved before the paper can be used reliably. The overlap with Ref. [26] is openly disclosed, and the comparison with that work is handled fairly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read of 2412.14155. The paper is a solid, useful piece of work on positivity constraints in HEFT at NLO. It identifies the 15 operators that can contribute to s^2 growth in longitudinal gauge-Higgs scattering, derives analytical constraints (Table 4) and numerical capping bounds (Table 6), and recovers the known SMEFT bounds as a consistency check. It also agrees with Remmen-Rodd's custodial results where they overlap, which is a good sign. The mapping to anomalous couplings in Table 2 will be handy for phenomenology. For the VBS and Higgs channels with no experimental limits, these are genuinely new bounds.\n\nThe main issue is the 'complete set' claim. The paper explicitly assumes single insertions and tree-level amplitudes (Secs. 2.1 and 4). The stress-test note quantifies the loop contribution: a one-loop diagram from LO HEFT vertices contributes to the s^2 coefficient at order s^2/(16π^2 v^4), while a tree-level NLO insertion is of order (v^2/Λ^2) s^2/v^4. The ratio is Λ^2/(16π^2 v^2). In the paper's own regime v/Λ ≥ 1/(4π), this ratio is O(1) — about 0.34 at Λ=1.8 TeV and 0.6 at Λ=2.4 TeV. So the neglected loops are not parametrically suppressed; they shift the coefficients that the positivity bounds actually constrain. The bounds in Tables 4 and 6 therefore constrain tree-level-plus-loop combinations, not cleanly the 15 WCs as labelled. Removing 'complete' or computing those loop corrections would fix this. It's a real gap, but not a fatal one — the analytical machinery is sound and the results are likely correct as constraints on the effective s^2 coefficients.\n\nOther soft spots are minor: the numerical capping procedure lacks reproducibility details (no code, no N and l_M values), and the '5% of parameter space' statement is not backed by a clear measure on the unbounded cone. The experimental comparison in Fig. 1 is more meaningful because that region is bounded.\n\nOverall: this is a serious paper worth sending to referees. The main derivation is coherent, the literature engagement is honest, and the new bounds are valuable. A careful referee should push on the loop assumption and the volume claim, but neither is a showstopper.\n\nRecommendation: accept for peer review, with the expectation that the 'complete' claim gets qualified and the numerics get documented.","headline":"Solid tree-level positivity analysis for HEFT with an overreaching 'complete' claim: the EFT-loop contribution is O(1) in the stated regime and is not accounted for.","tokens_in":31515,"tokens_out":4998,"would_cite":true,"duration_ms":44626,"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":"The complete set of positivity constraints on the Higgs Effective Field Theory at next-to-leading order is given by two tables of inequalities: analytical constraints on 15 Wilson coefficients plus numerical capping bounds, together…","keywords":["positivity bounds","Higgs effective field theory","HEFT","longitudinal gauge-Higgs scattering","Wilson coefficients","SMEFT","unitarity","crossing symmetry"],"falsifier":"Find a causal, unitary UV completion whose matched NLO HEFT Wilson coefficients violate one inequality in Table 4 (for example, a renormalizable theory that yields $c_2<0$); equivalently, compute the $s^2$ coefficient at NLO including two insertions of lower-order operators and EFT loops and show that the allowed region shrinks or the inequalities are modified.","tokens_in":30352,"feed_emoji":"📐","tokens_out":5261,"duration_ms":44255,"temperature":0.7,"pith_summary":"This paper tries to establish the complete set of theoretical constraints that causality, analyticity, unitarity, and crossing symmetry place on the Higgs Effective Field Theory at next-to-leading order. For the 15 Wilson coefficients that can produce an $s^2$ growth in longitudinal gauge-Higgs scattering, it derives two kinds of bounds: linear inequalities forcing some CP-even combinations to be positive, and product inequalities bounding CP-odd and inelastic coefficients in terms of elastic ones. The constraints define a convex 'HEFT-hedron' that leaves only about 5 percent of the 15-dimensional coefficient space, and numerical double-sided bounds cap this cone. If correct, the results give the first positivity bounds for several Higgs-associated scattering channels and are stronger than current LHC limits for most vector boson scattering coefficients.","feed_headline":"Positivity bounds cut HEFT parameter space to 5%","feed_subtitle":"Complete NLO constraints on 15 Wilson coefficients for longitudinal gauge-Higgs scattering, mostly stronger than LHC bounds.","key_machinery":"The central object is the $4\\times4$ matrix $\\gamma_\\beta$ built from the $s^2$ coefficients of the forward amplitude for superposed states $|\\alpha\\rangle=\\alpha_i|i\\rangle$ and $|\\beta\\rangle=\\beta_j|j\\rangle$ of the four Goldstone/Higgs states. Positivity of the second derivative of the forward amplitude forces $\\gamma_\\beta$ to be positive definite for every $\\beta$, which yields the analytical constraints in Table 4. The paper then re-expresses the low-energy coefficients as integrals over UV spectral densities through a twice-subtracted dispersion relation, and uses linear programming with unitarity bounds, $st$-crossing null constraints, and $U(1)_{\\rm em}$ symmetry to compute double-sided bounds that cap the cone.","core_discovery":"The central claim is that the complete set of positivity bounds at NLO is provided by Table 4 and Table 6. Table 4 gives analytical constraints: positive linear combinations of CP-even Wilson coefficients, such as $c_2>0$ and $c_1+c_2>0$, together with inequalities of the form $A^2<BC$ that bound CP-odd and inelastic coefficients by products of elastic ones. Table 6 gives numerical capping bounds obtained by imposing $st$-crossing and full unitarity, which turn the open cone into a bounded region for a chosen cutoff. The paper also claims these 15-dimensional constraints reproduce known SMEFT dimension-8 positivity bounds when intersected with the three-dimensional SMEFT plane, and that the projection of the HEFT positivity cone onto the SMEFT plane is larger than that intersection, leaving room for positivity to distinguish HEFT from SMEFT in future measurements.","pith_inferences":["The linear-programming capping procedure could be reapplied with a measured cutoff scale from global fits, turning the two $\\Lambda$ benchmarks into a continuous bound on each Wilson coefficient.","A natural next test is to compute two-insertion and EFT-loop corrections to the $s^2$ coefficient for one of the 15 operators; this would show how much of the 'complete' claim survives when the weakest assumption is relaxed.","The same positivity-matrix construction applies to any EFT with Goldstone-type scattering, such as chiral Lagrangians or composite Higgs models, so the HEFT-hedron method is not specific to electroweak symmetry breaking.","If future experiments measure only a subset of Wilson coefficients, the projection argument suggests that finding a point outside the SMEFT-consistent region can falsify SMEFT as the low-energy description even when no single coefficient measurement does."],"forward_implications":["The allowed region of the 15 Wilson coefficients shrinks to roughly 5 percent of the unconstrained space, so global HEFT fits can treat the HEFT-hedron as a sharp theoretical prior.","Wilson coefficients contributing to $V_L V_L, hh \\to hh$ and $V_L V_L, hh \\to V_L h$ receive their first reported bounds, since no experimental limits exist for those processes.","For most Wilson coefficients contributing to $V_L V_L \\to V_L V_L$, the positivity bounds are tighter than the current LHC bounds from vector boson scattering.","The known three-parameter SMEFT positivity region is recovered as the intersection of the three-dimensional SMEFT plane with the 15-dimensional HEFT-hedron.","A future measurement falling inside the HEFT positivity cone but outside its SMEFT projection could be a first sign that the low-energy theory is HEFT rather than SMEFT."],"supporting_citations":[{"why":"Supplies the foundational dispersion-relation argument that $s^{2n}$ forward-amplitude coefficients are positive.","marker":"[3]"},{"why":"Develops the full-crossing null constraints and numerical methods that the paper adapts for the capping bounds.","marker":"[8, 9]"},{"why":"Derives the SMEFT dimension-8 positivity bounds on vector boson scattering that the paper recovers from its 15-dimensional cone.","marker":"[13, 14]"},{"why":"Provides the 'capping the positivity cone' linear-programming framework with unitarity and st-crossing used for Table 6.","marker":"[18]"},{"why":"Overlapping HEFT positivity work whose projection-versus-intersection observation the paper confirms and extends.","marker":"[26]"},{"why":"Supplies the complete NLO HEFT operator basis from which the 15 $U h D^4$ operators are drawn.","marker":"[36]"},{"why":"Provides the experimental LHC bounds on HEFT Wilson coefficients that the paper compares against.","marker":"[51]"}],"fun_headline_variants":["Positivity bounds on HEFT at NLO: only 5% space survives","NLO positivity cuts HEFT Wilson coefficients to 5% of space","HEFT positivity: complete NLO bounds reduce parameter space to 5%","Positivity constraints leave only 5% of HEFT parameter space","Complete NLO positivity bounds for HEFT: 5% allowed region"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The assumption that only a single insertion of one of the 15 NLO operators contributes to the $s^2$ term, so that two-insertion contributions from lower-order HEFT operators and EFT loop corrections are negligible; if these are not suppressed, 'complete' is not established.","fun_headline_variants_meta":{"raw":{"variants":["Positivity bounds on HEFT at NLO: only 5% space survives","NLO positivity cuts HEFT Wilson coefficients to 5% of space","HEFT positivity: complete NLO bounds reduce parameter space to 5%","Positivity constraints leave only 5% of HEFT parameter space","Complete NLO positivity bounds for HEFT: 5% allowed region"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000262,"raw_usage":{"total_tokens":1638,"prompt_tokens":1025,"completion_tokens":613,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":641,"completion_tokens_details":{"reasoning_tokens":515}},"tokens_in":641,"tokens_out":613,"duration_ms":5102,"temperature":1.0,"reasoning_tokens":515,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:25:59.779419+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a causal, unitary UV completion whose matched NLO HEFT Wilson coefficients violate one inequality in Table 4 (for example, a renormalizable theory that yields $c_2<0$); equivalently, compute the $s^2$ coefficient at NLO including two insertions of lower-order operators and EFT loops and show that the allowed region shrinks or the inequalities are modified.","supporting_citations":[{"cited_title":"Adams, N","cited_arxiv_id":null,"evidence_quote":"Supplies the foundational dispersion-relation argument that $s^{2n}$ forward-amplitude coefficients are positive."},{"cited_title":"Sun, M.-L","cited_arxiv_id":null,"evidence_quote":"Supplies the complete NLO HEFT operator basis from which the 15 $U h D^4$ operators are drawn."}],"review_version":1}