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Towards the HEFT-hedron: the complete set of positivity constraints at NLO

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2412.14155 v1 pith:UXEXXMB2 submitted 2024-12-18 hep-ph hep-exhep-th

Debsubhra Chakraborty , Susobhan Chattopadhyay , Rick S. Gupta This is my paper · ORCID
classification hep-phhep-exhep-th
keywords positivityboundsHiggseffectivefieldtheoryHEFTlongitudinalgauge-HiggsscatteringWilsoncoefficientsSMEFTunitaritycrossingsymmetry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

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.

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 (3)
  1. [Secs. 2.1 and 4, eqs. (2.7), (3.2), (4.17)] 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.
  2. [Eq. (2.45) vs Table 4 and Appendix B, eq. (B.12)] 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.
  3. [Sec. 4, Table 5] 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.
minor comments (3)
  1. [Secs. 5.1 and 6] 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.
  2. [Table 6 and text] 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)).
  3. [Table 5] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the constraints follow from dispersion relations, unitarity and crossing symmetry; the only self-citation is a non-load-bearing talk notice.

full rationale

The central derivation is self-contained. The analytical bounds of Table 4 follow from the twice-subtracted dispersion relation (3.1)-(3.2), the optical theorem, and the positive-definiteness of the matrix gamma_beta in (3.7); the matrix elements are the same c^{2,0}_{ijkl} that the dispersion relation renders positive, so the argument is the standard positivity logic rather than an input-output identification. The capping bounds in Table 6 are produced by linear programming over discretized spectral densities subject to unitarity bounds (4.10)-(4.13), null constraints from st-crossing (4.8), and U(1)_em symmetry constraints, with no experimental data fitted. The list of 15 operators and the a_i parametrization are definitions and algebraic mappings (Section 2), not predictions; eq. (2.45) is a dictionary between c^{2,0}_{ijkl} and HEFT WCs, not a circular derivation of the bounds. The only self-citation is Ref. [25], a talk by the same authors reporting these results, cited in a Note added; no load-bearing claim in the derivation depends on it. The stated assumptions about single insertions and neglected EFT loops (Sections 2.1 and 4) are explicit power-counting limitations on the word 'complete'; they are correctness/regime concerns, not cases where a prediction reduces by construction to its input.

Assumptions & free parameters 2 free parameters · 8 assumptions · 0 invented entities

The derivation rests on standard dispersion-relation and EFT assumptions rather than on new fitted parameters or postulated particles. The only hand-chosen inputs are the cutoff scale Lambda and the numerical truncation parameters, which control the capping bounds.

free parameters (2)
  • New physics cutoff scale Lambda = 1.8 TeV and 2.4 TeV
    The numerical capping bounds in Table 6 and figures are computed for these two choices, selected to match the lower bounds on Lambda derived from current experimental limits. The final numbers scale with Lambda and would change for other values.
  • Spectral density discretization truncation orders N and l_M = not stated
    The linear programming model in Table 5 discretizes the UV integral with N points and truncates angular momentum at l_M. The paper says these are chosen to be sufficiently large but does not report the values, making the numerical bounds irreproducible as published.
assumptions (8)
  • domain assumption The twice-subtracted dispersion relation of eq. (3.1) holds for the IR-subtracted amplitude Mtilde, with only s- and u-channel singularities above the cutoff Lambda^2.
    Invoked in Sec. 3 to derive positivity; requires a mass gap and that all t-channel poles are removed by the M_sing subtraction term defined in eq. (2.37).
  • standard math The Froissart-Martin bound applies to the subtracted HEFT amplitude.
    Used in Sec. 3 to justify the twice-subtracted dispersion relation; relies on Refs. [46,47] and on the claim that t-channel pole residues are at most linear in s.
  • domain assumption Longitudinal gauge boson scattering is mapped to goldstone scattering by the equivalence theorem.
    Used throughout to compute the s^2 growth from the HEFT operators in Table 1 via the goldstone amplitudes (Secs. 2.1 and 2.4).
  • domain assumption The 15 operators of Table 1 from Ref. [36] are the complete set of NLO HEFT operators that generate s^2 growth in 2-to-2 longitudinal gauge-Higgs scattering.
    The 'complete set' claim inherits the completeness of the operator basis of Ref. [36]; the paper does not rederive the basis.
  • ad hoc to paper Only single insertions of NLO operators matter; two insertions of lower-order operators are suppressed by the power counting of eq. (2.7).
    Stated in Sec. 2.1 as a justification for ignoring contributions from other operator categories. If this suppression fails, the stated completeness of the constraints is not established.
  • domain assumption EFT loop contributions to the s^2 coefficient can be neglected.
    Stated explicitly in Sec. 4 ('with our assumption that EFT loops can be neglected'). The paper cites Refs. [44,45] on loop effects but does not include them in the bounds.
  • standard math The spectral density unitarity constraints of eqs. (4.10)-(4.13) are valid.
    Taken from Ref. [18] and used as the unitarity input to the linear programming bounds without rederivation.
  • domain assumption The low-energy amplitude is invariant only under U(1)em, not the full SU(2)L x U(1)Y gauge symmetry.
    Deliberate choice in Secs. 2.4 and 4 to keep the treatment general for HEFT; the resulting constraints are weaker than those of Refs. [18,49], which impose the full SM symmetry.

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Pith. "Pith review of Towards the HEFT-hedron: the complete set of positivity constraints at NLO." pith.science (2026). https://pith.science/paper/UXEXXMB2

@misc{pith2026241214155,
  author       = {Pith},
  title        = {Pith review of: Towards the HEFT-hedron: the complete set of positivity constraints at NLO},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UXEXXMB2}},
  note         = {Machine review of arXiv:2412.14155}
}
abstract

We present the complete set of positivity bounds on the Higgs Effective Field Theory (HEFT) at next-to-leading order (NLO). We identify the 15 operators that can be constrained by positivity, as they contribute to $s^2$-growth in the amplitude for longitudinal gauge-Higgs scattering, that is to all possible 2-to-2 scattering processes involving longitudinal gauge bosons, $V_L = W_L^\pm, Z_L$, and the Higgs boson, $h$. We find two sets of constraints: (i) specific linear combinations of CP-even Wilson coefficients (WCs) must be positive, and (ii) the magnitudes of some WCs -- including all CP-odd ones -- must be smaller than products of other CP-even WCs. We present our final constraints on the 15 dimensional HEFT space and show how known positivity bounds on the 3 dimensional space of dimension 8 SMEFT can be recovered from them. We find that only about $5\%$ of the parameter space for WCs of HEFT operators at NLO complies with these positivity constraints. Additionally, we obtain double-sided bounds on these WCs by fully exploiting the implications of unitarity and $st$-crossing symmetry. For WCs contributing to the vector boson scattering process our final constraints are in most cases significantly stronger than the experimental ones. For the $V_L V_L, hh \to hh$ and $V_LV_L, hh \to V_Lh$ process, there are no reported experimental limits and our theoretical constraints provide the first bounds.

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Forward citations

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Reviewed August 11, 2026 · model on record in the stance chip above.