Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Geometry of effective field theory positivity cones

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For three flavors, the forward-limit positivity cone $\mathcal{C}_W$ has a complete classification: every extremal element is one of three explicit types, and the third type generates all inelastic bounds.

desk verdict A credible first classification of the three-flavor positivity cone whose key symbolic elimination in Prop. 4.25 is asserted rather than shown—worth refereeing, with a request for a checkable derivation. read the letter →

arxiv 2508.18165 v1 pith:HPP5QQRE submitted 2025-08-25 math-ph hep-thmath.AGmath.MP

Quentin Bonnefoy , Vicente Cortés , Emanuele Gendy , Christophe Grojean , Karim Ritter von Merkl , Paula Naomi Pilatus This is my paper · ORCID
classification math-phhep-thmath.AGmath.MP MSC 52A2015B4881T12
keywords positivityboundseffectivefieldtheoryspectrahedronextremalraysconvexconeelasticinelasticthreeflavors
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

Positivity bounds restrict the Wilson coefficients of any effective field theory that must come from a unitary, local, causal quantum theory. This paper solves the geometric core of those bounds for three particle flavors: it classifies every extremal ray of the forward-limit positivity cone $\mathcal{C}_W$, the cone of positive semidefinite four-tensors with the symmetries of a crossing-symmetric amplitude. The classification has exactly three families, and the third family produces the inelastic bounds that cannot be reached from elastic scattering alone. The same theorem yields complete positivity bounds for three-flavor amplitudes, and it shows that amplitudes with $O(3)$, $\mathbb{Z}_2^3$, or $SO(2)$ symmetry are fully constrained by elastic bounds alone.

What carries the argument

The object doing the work is the linear spectrahedron $\mathcal{C}_W = \{S\in W : S\ge 0\}$, where $W$ is the space of four-tensors in $\mathrm{Sym}^2(\mathrm{Sym}^2 V^*)\oplus \mathrm{Sym}^2(\Lambda^2 V^*)$ that are invariant under the transposition $\tau$. Extremality is tested through the minimal-face characterization: $S$ is extremal precisely when no non-zero $S'\in\mathcal{C}_W$ has kernel strictly containing $\ker S$. The proof organizes $\mathcal{C}_W$ by the rank of the curvature-type restriction $R'(S)$ and, for rank 1, by the dimension of $\ker S\cap (z\vee V)$; the surviving case $\operatorname{rank} R'(S)=1$, $\operatorname{rank} R(S)=3$, and $\ker S\cap (z\vee V)=\{0\}$ forces the explicit three-parameter kernel basis of Proposition 4.25, which is then converted into the closed form of Theorem 4.1.

What would settle it

For a fixed basis of $V$, draw many positive semidefinite $S\in\mathcal{C}_W$ by taking random sums of tensor squares $Q_i\otimes Q_i$ with $Q_i\in\mathrm{Sym}^2 V$ or $Q_i\in\Lambda^2 V$ and imposing $\tau S=S$; then filter for four-dimensional kernel and test extremality through the kernel-inclusion criterion. Any extremal $S$ outside the three families of Theorem 4.1 would refute the classification.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 4.1: when $\dim V = 3$, an element $S\in\mathcal{C}_W$ is extremal if and only if it is one of (1) $S = \alpha^4$ for $\alpha\in V^*\setminus\{0\}$; (2) $S = (\alpha_1\vee\alpha_2)^{\otimes 2} + (\alpha_1\wedge\alpha_2)^{\otimes 2}$ for linearly independent $\alpha_1,\alpha_2$; or (3) $S = S_{\mathrm{tot}} + 2(g^2+d^2-1-dh)\big((\alpha_2\otimes\alpha_3)^{\otimes 2} + (\alpha_3\otimes\alpha_2)^{\otimes 2}\big)$ for a basis $\{\alpha_1,\alpha_2,\alpha_3\}$ and parameters satisfying $g^2 > 1-d^2+dh$, with $S_{\mathrm{tot}}$ the displayed totally symmetric quartic. Types 1 and 2 are exactly the elastic bounds $M(\alpha\otimes\beta,\alpha\otimes\beta)\ge 0$; type 3 is the inelastic family, so every inelastic bound for three flavors comes from this family. The proof also establishes that in the symmetric cases $O(3)$, $\mathbb{Z}_2^3$, and $SO(2)$, every extremal ray of the invariant cone is the projection of an elastic extremal ray, so elastic bounds alone cut out the full invariant cone.

Load-bearing premise

The load-bearing premise is that the four-tensor extracted from forward two-to-two amplitudes lies in $\mathcal{C}_W^*$ whenever the UV completion is unitary, local, and causal—an identification imported from earlier work and only sketched in Appendix A.1—together with the paper's explicit restriction to tree-level, quartic-in-momentum, non-truncated cones without full crossing constraints.

Editorial extensions

If this is right

  • For three flavors, the complete set of positivity bounds is now explicit: elastic bounds from types 1 and 2 plus the inelastic inequalities from type 3, so no further extremal rays are missing.
  • Any amplitude with $O(3)$, $\mathbb{Z}_2^3$, or $SO(2)$ symmetry is fully constrained by elastic bounds, so inelastic bounds add nothing in those symmetric sectors.
  • In chiral perturbation theory for pions, the positivity cone reduces to the elastic conditions $\ell_2\ge -\ell_1$ and $\ell_2\ge 0$, reproducing the bounds known from earlier dispersion-relation analyses.
  • Without symmetry, the elastic cone is strictly smaller than the full cone, so there exist three-flavor Wilson-coefficient tensors that pass every elastic bound yet are excluded by an inelastic bound.
  • The paper recovers the known two-flavor result as a special case: there the third family degenerates and all extremal rays are elastic.

Reading between the lines

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

  • A natural stress test is to push the same rank-of-$R'$ strategy to $n=4$: nothing in the kernel-dimensionality arguments is obviously four-flavor-specific, and the growth of $W$ suggests additional inelastic families beyond the three.
  • The $O(3)/\mathbb{Z}_2^3/SO(2)$ results suggest a pattern: when the symmetry group is large enough that projecting an inelastic extremal ray lowers its rank, elastic bounds become sufficient; finding the threshold symmetry group would be a clean follow-up.
  • The explicit type-3 family gives a concrete target for numerical searches: sample the 21-dimensional three-flavor tensor space near tensors that annihilate an inelastic ray, since the paper notes uninformed random directions rarely expose the elastic/inelastic gap.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the forward-limit positivity cone C_W for n flavors, viewed as a linear spectrahedron of four-tensors, and determines all its extremal elements for dim V = 3. The main result, Theorem 4.1, gives an if-and-only-if classification into three explicit families: rank-one totally symmetric tensors; tensors of the form (α1∨α2)^2 + (α1∧α2)^2; and a three-parameter family parameterized by (g,d,h) with g^2 > 1 - d^2 + dh. The proof splits according to the rank of R'(S), using a kernel-inclusion criterion for extremality in spectrahedra (Cor. 2.14) and a sequence of auxiliary propositions. The paper then derives the resulting elastic and inelastic positivity bounds, proves that for amplitudes with O(3), Z_2^3, or SO(2) symmetry the elastic bounds suffice, and applies the O(3) result to pion scattering in chiral perturbation theory.

Significance. If correct, the classification solves the extremal representation problem for the three-flavor forward-limit positivity cone, thereby completing the two-flavor result of Ref. [98] and providing the full set of positivity bounds for three-flavor EFTs, including the inelastic bounds that are absent in the two-flavor case. The general structural results—the kernel-based extremality criterion, the two-dimensional recovery in Theorem 3.17, and the necessary kernel-dimension bounds in Proposition 3.12—are clean and potentially reusable. The paper also delivers concrete, falsifiable inequalities for the symmetric cases and reproduces the known pion bounds. These are significant contributions if the algebraic core of the proof is fully secured.

major comments (3)
  1. [§4.4, Proposition 4.25] The load-bearing step of the classification is asserted rather than demonstrated. After setting up the 6×6 matrix in terms of parameters a,b,c,d,e,f,g,h,q, the proof states without derivation that requiring the fourth principal minors to vanish, the first three principal minors to be nonnegative, and q>0 yields a=1+d^2, b=1+dh, c=dg, e=1+g^2+h^2, f=g(d+h), b+q=d^2+g^2 and g^2>1-d^2+dh. This elimination is the only step connecting the abstract extremality criterion (Propositions 4.15 and 4.24) to the explicit third family in Theorem 4.1, and the inelastic bounds in §5.1.2 depend on it directly. The authors should display the elimination, provide a verifiable symbolic computation, or give a rigorous argument showing that no other branch and no additional inequality can arise.
  2. [§4.4, Proposition 4.25 (matrix and kernel basis)] The preliminary matrix displayed in the proof and the final matrix in the proposition are not reconciled. With the stated basis {e1^2, e2^2, e3^2, e1∨e2, e1∨e3, e2∨e3}, the preliminary matrix has entries (4,5)=0, (4,6)=d, (5,6)=g, (6,6)=b+q, whereas the displayed final matrix has (4,5)=d, (4,6)=g, (5,6)=0, (6,6)=1. Unless a column/row reordering or a rescaling of basis vectors is being performed, the two matrices are not the same. A reader cannot reproduce the claimed kernel basis without knowing which convention is used. The three cases of the kernel basis (d,g≠0; d=0; g=0) are also presented without derivation; since these vectors are used to check the no-nontrivial-solution conditions in Proposition 4.15, each case needs an explicit verification that the displayed vectors do indeed span the kernel.
  3. [§4.2, Proposition 4.15] The proof of extremality for the rank-3, rank-R'=1 case relies on the claim that each of the three displayed systems of quadratic equations has no non-trivial real solution. This claim is stated without proof. Because Corollary 2.14 requires exactly this vanishing condition, a short case-by-case justification (for instance, by sign analysis or by reducing to a contradiction with g^2>1-d^2+dh) should be included. Without it, the iff direction of Theorem 4.1 for the third family is incomplete.
minor comments (5)
  1. [§5.2.2, Lemma 5.5] There is a typo: 'Secrion 3' should be 'Section 3'.
  2. [§5.1.1, Proposition 5.2] The proof's case analysis for vectors with four or more non-vanishing components is concise and somewhat informal; the statement that 'there will always remain a non-trivial constraint' for four non-vanishing components is asserted after only one example. Since Proposition 5.2 is used to characterize elastic bounds, a more systematic enumeration or a reference to the full verification would improve readability.
  3. [Figure 1] The caption says the regions were approximated by numerically sampling extremal rays, but no details on the sampling algorithm or accuracy are given. A brief description in the text or a supplementary file would make the figure reproducible.
  4. [Footnote 3] The list of restrictions in footnote 3 is dense and would benefit from being expanded in the main text, since it delimits the physical scope of the otherwise purely mathematical classification.
  5. [§5.2.2, Remark 5.8] The reference to 'the appendix of [98]' for the classification of extremal rays of the Z_2^3-invariant cone should be more specific, since the appendix is long and the relevant result is used for an alternative proof of Theorem 5.6.

Circularity Check

0 steps flagged · score 1.0 of 10

The classification is derived from the definition of CW as a spectrahedron; no predicted quantity is fitted or defined in terms of the result, and self-citations are not load-bearing.

full rationale

The paper's central result, Theorem 4.1, is obtained by a convex-geometric analysis of the cone CW defined as positive semidefinite tensors in W, using standard spectrahedron facts (Corollaries 2.13 and 2.14) and a case-by-case study of the ranks of R(S) and R'(S). There is no quantity that is fitted and then called a prediction: the apparent parameters d, g, h in the third family arise from an explicit basis choice and from the principal-minor conditions in Proposition 4.25, not from any external data. The identification of the physical four-tensor M with an element of C*W is imported from Refs. [94,98] and sketched in Appendix A.1; those are external references, and the present authors' own citations ([15,67]) appear only in the literature survey and in physical applications, not as premises of the classification. The elastic-bounds-sufficiency results in Section 5 are proved from Theorem 4.1 and explicit projection arguments, and the inelastic bounds are defined directly as dual inequalities of the third extremal family. The load-bearing algebraic step in Proposition 4.25, where the kernel basis is stated after a summarized determinant elimination, is a verification/completeness concern about exposition rather than circularity, since the statement is not equivalent to its assumptions by construction. No self-definitional, fitted-input, or self-citation circularity was found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to data; d, g, and h parameterize the third extremal family rather than being adjustable inputs. The central derivation relies on standard convex-geometry theorems and on the physical cone identification imported from Refs. [94,98]. No new particles, forces, dimensions, or other invented entities are introduced; the third extremal family is a new classification element within the already-defined cone.

assumptions (5)
  • standard math A closed convex cone with no lines is generated by its extremal rays (Klee/Rockafellar).
    Used in Section 2.2 and Theorem 2.8 to justify that every positivity bound can be derived from extremal rays of C_W.
  • standard math For a spectrahedron, the minimal face of x is {z in C : ker z contains ker x} (Ramana-Goldman).
    This is the kernel-inclusion extremality test applied throughout Sections 3 and 4, especially Corollary 2.14.
  • domain assumption Positive semidefinite symmetric four-tensors in W correspond to elements of the dual positivity cone C_W^* from physical dispersion relations.
    Imported from Refs. [94,98] and sketched in Appendix A.1; it connects the mathematical classification to scattering-amplitude positivity bounds.
  • domain assumption Physical scope is restricted to tree-level amplitudes, quartic-in-momentum positivity, non-truncated cones, forward kinematics, and no complete crossing constraints.
    Explicitly declared in footnote 3; the physical applications, including chiral perturbation theory, inherit these restrictions.
  • standard math O(3)-invariant four-tensors on R^3 have the three-parameter Jeffreys form used in Theorem 5.3.
    Cited to Ref. [103] and used to reduce O(3)-invariant amplitudes to two independent coefficients.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Geometry of effective field theory positivity cones." pith.science (2026). https://pith.science/paper/HPP5QQRE

@misc{pith2026250818165,
  author       = {Pith},
  title        = {Pith review of: Geometry of effective field theory positivity cones},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HPP5QQRE}},
  note         = {Machine review of arXiv:2508.18165}
}
abstract

Positivity bounds are theoretical constraints on the Wilson coefficients of an effective field theory. These bounds emerge from the requirement that a given effective field theory must be the low-energy limit of a relativistic quantum theory that satisfies the fundamental principles of unitarity, locality, and causality. The task of deriving these bounds can be reformulated as the geometric problem of finding the extremal representation of a closed convex cone~$\mathcal C_W$. More precisely, in the presence of multiple particle flavors, the forward-limit positivity cone $\mathcal C_W$ consists of all positive semi-definite tensors in $W =\left\{ S \in \mathrm{Sym}^2 (\mathrm{Sym}^2\, V^*)\oplus \mathrm{Sym}^2 \left({\Lambda}^2 V^*\right) : \tau S = S \right\} \subset \mathrm{Sym}^2(V^*\otimes V^*)$, where $\tau$ denotes transposition in the second and fourth tensor factor and $V\cong\mathbb{R}^n$, where $n$ is the number of flavors. In this work, we solve this question up to three flavors, i.e.~$n=3$, proving a full classification of all extremal elements in these cases. We furthermore study the implications of our findings, deriving the full positivity bounds for amplitudes with and without additional symmetries. In the cases with additional symmetries that we consider, we find that the so-called elastic bounds are sufficient to give rise to the full positivity bounds.

Figures

Figures reproduced from arXiv: 2508.18165 by the authors.

Figure 1
Figure 1. Regions of parameters consistent with positivity bounds. The union of the blue and orange regions corresponds to points which abide by the elastic bounds and the orange region corresponds to imposing all (elastic and inelastic) bounds. The 21-dimensional parameter space has been restricted to M1111 and M1112, more pre￾cisely to the perturbations in their direction around the following reference point : M1111 = 0.332… view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Positivity and partial wave unitarity bounds on ALP theories via amplitude methods

    hep-ph 2025-10 conditional novelty 6.0 of 10

    Complete partial-wave unitarity and positivity bounds are derived for ALP effective interactions up to dimension 8, with new SMEFT positivity constraints as a byproduct.

Reference graph

Works this paper leans on

108 extracted references · 4 canonical work pages · cited by 1 Pith paper

  1. [98]

    X. Li, H. Xu, C. Yang, C. Zhang and S.-Y. Zhou,Positivity in Multifield Effective Field Theories, Phys. Rev. Lett.127 (2021) 121601, [2101.01191]

    arXiv 2021

  2. [1]

    T. N. Pham and T. N. Truong,Evaluation of the Derivative Quartic Terms of the Meson Chiral Lagrangian From Forward Dispersion Relation, Phys. Rev. D31 (1985) 3027

    1985

  3. [2]

    M. R. Pennington and J. Portoles,The Chiral Lagrangian parameters,ℓ1, ℓ2, are determined by theρ-resonance, Phys. Lett. B344 (1995) 399–406, [hep-ph/9409426]

    arXiv 1995

  4. [3]

    Ananthanarayan, D

    B. Ananthanarayan, D. Toublan and G. Wanders,Consistency of the chiral pion pion scattering amplitudes with axiomatic constraints, Phys. Rev. D51 (1995) 1093–1100, [hep-ph/9410302]

    arXiv 1995

  5. [4]

    Adams, N

    A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi,Causality, analyticity and an IR obstruction to UV completion, Journal of High Energy Physics 2006 (oct, 2006) 014

    2006

  6. [5]

    Bellazzini, C

    B. Bellazzini, C. Cheung and G. N. Remmen,Quantum Gravity Constraints from Unitarity and Analyticity, Phys. Rev. D93 (2016) 064076, [1509.00851]

    arXiv 2016

  7. [6]

    Guerrieri, J

    A. Guerrieri, J. Penedones and P. Vieira,Where Is String Theory in the Space of Scattering Amplitudes?, Phys. Rev. Lett.127 (2021) 081601, [2102.02847]

    arXiv 2021

  8. [7]

    Z. Bern, E. Herrmann, D. Kosmopoulos and R. Roiban,Effective Field Theory islands from perturbative and nonperturbative four-graviton amplitudes, JHEP 01 (2023) 113, [2205.01655]

    arXiv 2023

Show all 108 references
  1. [8]

    Bellazzini and F

    B. Bellazzini and F. Riva,New phenomenological and theoretical perspective on anomalous ZZ and Zγ processes, Phys. Rev. D98 (2018) 095021, [1806.09640]

    2018 arXiv

  2. [9]

    Q. Bi, C. Zhang and S.-Y. Zhou,Positivity constraints on aQGC: carving out the physical parameter space, JHEP 06 (2019) 137, [1902.08977]

    2019 arXiv

  3. [10]

    G. N. Remmen and N. L. Rodd,Consistency of the Standard Model Effective Field Theory, JHEP 12 (2019) 032, [1908.09845]

    2019 arXiv

  4. [11]

    G. N. Remmen and N. L. Rodd,Flavor Constraints from Unitarity and Analyticity, Phys. Rev. Lett.125 (2020) 081601, [2004.02885]

    2020 arXiv

  5. [12]

    B. Fuks, Y. Liu, C. Zhang and S.-Y. Zhou,Positivity in electron-positron scattering: testing the axiomatic quantum field theory principles and probing the existence of UV states, Chin. Phys. C 45 (2021) 023108, [2009.02212]

    2021 arXiv

  6. [13]

    Yamashita, C

    K. Yamashita, C. Zhang and S.-Y. Zhou,Elastic positivity vs extremal positivity bounds in SMEFT: a case study in transversal electroweak gauge-boson scatterings, JHEP 01 (2021) 095, [2009.04490]

    2021 arXiv

  7. [14]

    Gu, L.-T

    J. Gu, L.-T. Wang and C. Zhang,Unambiguously Testing Positivity at Lepton Colliders, Phys. Rev. Lett.129 (2022) 011805, [2011.03055]

    2022 arXiv

  8. [15]

    Bonnefoy, E

    Q. Bonnefoy, E. Gendy and C. Grojean,Positivity bounds on Minimal Flavor Violation, JHEP 04 (2021) 115, [2011.12855]

    2021 arXiv

  9. [16]

    Henriksson, B

    J. Henriksson, B. McPeak, F. Russo and A. Vichi,Rigorous bounds on light-by-light scattering, JHEP 06 (2022) 158, [2107.13009]

    2022 arXiv

  10. [17]

    Zhang,SMEFTs living on the edge: determining the UV theories from positivity and extremality, JHEP 12 (2022) 096, [2112.11665]

    C. Zhang,SMEFTs living on the edge: determining the UV theories from positivity and extremality, JHEP 12 (2022) 096, [2112.11665]

    2022 arXiv

  11. [18]

    X. Li, K. Mimasu, K. Yamashita, C. Yang, C. Zhang and S.-Y. Zhou,Moments for positivity: using Drell-Yan data to test positivity bounds and reverse-engineer new physics, JHEP 10 (2022) 107, [2204.13121]

    2022 arXiv

  12. [19]

    Ghosh, R

    D. Ghosh, R. Sharma and F. Ullah,Amplitude’s positivity vs. subluminality: causality and unitarity constraints on dimension 6 & 8 gluonic operators in the SMEFT, JHEP 02 (2023) 199, [2211.01322]. References 51

    2023 arXiv

  13. [20]

    Li,Positivity bounds at one-loop level: the Higgs sector, JHEP 05 (2023) 230, [2212.12227]

    X. Li,Positivity bounds at one-loop level: the Higgs sector, JHEP 05 (2023) 230, [2212.12227]

    2023 arXiv

  14. [21]

    Gu and C

    J. Gu and C. Shu,Probing positivity at the LHC with exclusive photon-fusion processes, JHEP 05 (2024) 183, [2311.07663]

    2024 arXiv

  15. [22]

    Q. Chen, K. Mimasu, T. A. Wu, G.-D. Zhang and S.-Y. Zhou,Capping the positivity cone: dimension-8 Higgs operators in the SMEFT, JHEP 03 (2024) 180, [2309.15922]

    2024 arXiv

  16. [23]

    Davighi, S

    J. Davighi, S. Melville, K. Mimasu and T. You,Positivity and the electroweak hierarchy, Phys. Rev. D109 (2024) 033009, [2308.06226]

    2024 arXiv

  17. [24]

    G. N. Remmen and N. L. Rodd,Positively Identifying HEFT or SMEFT, 2412.07827

  18. [25]

    Chakraborty, S

    D. Chakraborty, S. Chattopadhyay and R. S. Gupta,Towards the HEFT-hedron: the complete set of positivity constraints at NLO, 2412.14155

    arXiv

  19. [26]

    Cheung and G

    C. Cheung and G. N. Remmen,Positive Signs in Massive Gravity, JHEP 04 (2016) 002, [1601.04068]

    2016 arXiv

  20. [27]

    Bellazzini, F

    B. Bellazzini, F. Riva, J. Serra and F. Sgarlata,Beyond Positivity Bounds and the Fate of Massive Gravity, Phys. Rev. Lett.120 (2018) 161101, [1710.02539]

    2018 arXiv

  21. [28]

    de Rham, S

    C. de Rham, S. Melville and A. J. Tolley,Improved Positivity Bounds and Massive Gravity, JHEP 04 (2018) 083, [1710.09611]

    2018 arXiv

  22. [29]

    Alberte, C

    L. Alberte, C. de Rham, S. Jaitly and A. J. Tolley,Positivity Bounds and the Massless Spin-2 Pole, Phys. Rev. D102 (2020) 125023, [2007.12667]

    2020 arXiv

  23. [30]

    Tokuda, K

    J. Tokuda, K. Aoki and S. Hirano,Gravitational positivity bounds, JHEP 11 (2020) 054, [2007.15009]

    2020 arXiv

  24. [31]

    Alberte, C

    L. Alberte, C. de Rham, S. Jaitly and A. J. Tolley,QED positivity bounds, Phys. Rev. D 103 (2021) 125020, [2012.05798]

    2021 arXiv

  25. [32]

    Caron-Huot, D

    S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin,Sharp boundaries for the swampland, JHEP 07 (2021) 110, [2102.08951]

    2021 arXiv

  26. [33]

    Alberte, C

    L. Alberte, C. de Rham, S. Jaitly and A. J. Tolley,Reverse Bootstrapping: IR Lessons for UV Physics, Phys. Rev. Lett.128 (2022) 051602, [2111.09226]

    2022 arXiv

  27. [34]

    Caron-Huot, Y.-Z

    S. Caron-Huot, Y.-Z. Li, J. Parra-Martinez and D. Simmons-Duffin,Causality constraints on corrections to Einstein gravity, JHEP 05 (2023) 122, [2201.06602]

    2023 arXiv

  28. [35]

    Chiang, Y.-t

    L.-Y. Chiang, Y.-t. Huang, W. Li, L. Rodina and H.-C. Weng,(Non)-projective bounds on gravitational EFT, 2201.07177

    arXiv

  29. [36]

    de Rham, S

    C. de Rham, S. Jaitly and A. J. Tolley,Constraints on Regge behavior from IR physics, Phys. Rev. D108 (2023) 046011, [2212.04975]

    2023 arXiv

  30. [37]

    Hong, Z.-H

    D.-Y. Hong, Z.-H. Wang and S.-Y. Zhou,Causality bounds on scalar-tensor EFTs, JHEP 10 (2023) 135, [2304.01259]

    2023 arXiv

  31. [38]

    Alviani and A

    E. Alviani and A. Falkowski,Matching and positivity beyond minimal coupling in effective theories of photons and gravitons, Eur. Phys. J. C85 (2025) 154, [2408.03439]

    2025 arXiv

  32. [39]

    Cheung and G

    C. Cheung and G. N. Remmen,Infrared Consistency and the Weak Gravity Conjecture, JHEP 12 (2014) 087, [1407.7865]

    2014 arXiv

  33. [40]

    Andriolo, D

    S. Andriolo, D. Junghans, T. Noumi and G. Shiu,A Tower Weak Gravity Conjecture from Infrared Consistency, Fortsch. Phys. 66 (2018) 1800020, [1802.04287]

    2018 arXiv

  34. [41]

    De Rham, L

    C. De Rham, L. Heisenberg and A. J. Tolley,Spin-2 fields and the weak gravity conjecture, Phys. Rev. D100 (2019) 104033, [1812.01012]

    2019 arXiv

  35. [42]

    Bellazzini, M

    B. Bellazzini, M. Lewandowski and J. Serra,Positivity of Amplitudes, Weak Gravity Conjecture, and Modified Gravity, Phys. Rev. Lett.123 (2019) 251103, [1902.03250]

    2019 arXiv

  36. [43]

    Arkani-Hamed, Y.-t

    N. Arkani-Hamed, Y.-t. Huang, J.-Y. Liu and G. N. Remmen,Causality, unitarity, and the weak gravity conjecture, JHEP 03 (2022) 083, [2109.13937]. References 52

    2022 arXiv

  37. [44]

    Melville and J

    S. Melville and J. Noller,Positivity in the Sky: Constraining dark energy and modified gravity from the UV, Phys. Rev. D101 (2020) 021502, [1904.05874]

    2020 arXiv

  38. [45]

    Grall and S

    T. Grall and S. Melville,Positivity bounds without boosts: New constraints on low energy effective field theories from the UV, Phys. Rev. D105 (2022) L121301, [2102.05683]

    2022 arXiv

  39. [46]

    de Rham, S

    C. de Rham, S. Melville and J. Noller,Positivity bounds on dark energy: when matter matters, JCAP 08 (2021) 018, [2103.06855]

    2021 arXiv

  40. [47]

    Freytsis, S

    M. Freytsis, S. Kumar, G. N. Remmen and N. L. Rodd,Multifield positivity bounds for inflation, JHEP 09 (2023) 041, [2210.10791]

    2023 arXiv

  41. [48]

    Xu and S.-Y

    H. Xu and S.-Y. Zhou,Triple crossing positivity bounds, mass dependence and cosmological scalars: Horndeski theory and DHOST, JCAP 11 (2023) 076, [2306.06639]

    2023 arXiv

  42. [49]

    S.-S. Kim, H. M. Lee and K. Yamashita,Positivity bounds on Higgs-portal freeze-in dark matter, JHEP 11 (2023) 119, [2308.14629]

    2023 arXiv

  43. [50]

    de Rham, S

    C. de Rham, S. Melville, A. J. Tolley and S.-Y. Zhou,Massive Galileon Positivity Bounds, JHEP 09 (2017) 072, [1702.08577]

    2017 arXiv

  44. [51]

    Wang, F.-K

    Y.-J. Wang, F.-K. Guo, C. Zhang and S.-Y. Zhou,Generalized positivity bounds on chiral perturbation theory, JHEP 07 (2020) 214, [2004.03992]

    2020 arXiv

  45. [52]

    A. L. Guerrieri, J. Penedones and P. Vieira,S-matrix bootstrap for effective field theories: massless pions, JHEP 06 (2021) 088, [2011.02802]

    2021 arXiv

  46. [53]

    Fernandez, A

    C. Fernandez, A. Pomarol, F. Riva and F. Sciotti,Cornering large-Nc QCD with positivity bounds, JHEP 06 (2023) 094, [2211.12488]

    2023 arXiv

  47. [54]

    Acanfora, A

    F. Acanfora, A. Guerrieri, K. Häring and D. Karateev,Bounds on scattering of neutral Goldstones, JHEP 03 (2024) 028, [2310.06027]

    2024 arXiv

  48. [55]

    Chala,Constraints on anomalous dimensions from the positivity of the S matrix, Phys

    M. Chala,Constraints on anomalous dimensions from the positivity of the S matrix, Phys. Rev. D108 (2023) 015031, [2301.09995]

    2023 arXiv

  49. [56]

    Y.-P. Liao, J. Roosmale Nepveu and C.-H. Shen,Positivity in Perturbative Renormalization: an EFTa-theorem, 2505.02910

    arXiv

  50. [57]

    Chala and X

    M. Chala and X. Li,Positivity restrictions on the mixing of dimension-eight SMEFT operators, Phys. Rev. D109 (2024) 065015, [2309.16611]

    2024 arXiv

  51. [58]

    Bonifacio and K

    J. Bonifacio and K. Hinterbichler,Bounds on Amplitudes in Effective Theories with Massive Spinning Particles, Phys. Rev. D98 (2018) 045003, [1804.08686]

    2018 arXiv

  52. [59]

    de Rham, S

    C. de Rham, S. Melville, A. J. Tolley and S.-Y. Zhou,Positivity Bounds for Massive Spin-1 and Spin-2 Fields, JHEP 03 (2019) 182, [1804.10624]

    2019 arXiv

  53. [60]

    Bellazzini, F

    B. Bellazzini, F. Riva, J. Serra and F. Sgarlata,Massive Higher Spins: Effective Theory and Consistency, JHEP 10 (2019) 189, [1903.08664]

    2019 arXiv

  54. [61]

    Alberte, C

    L. Alberte, C. de Rham, A. Momeni, J. Rumbutis and A. J. Tolley,Positivity Constraints on Interacting Spin-2 Fields, JHEP 03 (2020) 097, [1910.11799]

    2020 arXiv

  55. [62]

    Z.-Y. Wang, C. Zhang and S.-Y. Zhou,Generalized elastic positivity bounds on interacting massive spin-2 theories, JHEP 04 (2021) 217, [2011.05190]

    2021 arXiv

  56. [63]

    de Rham, L

    C. de Rham, L. Engelbrecht, L. Heisenberg and A. Lüscher,Positivity bounds in vector theories, JHEP 12 (2022) 086, [2208.12631]

    2022 arXiv

  57. [64]

    Bellazzini, G

    B. Bellazzini, G. Isabella, S. Ricossa and F. Riva,Massive gravity is not positive, Phys. Rev. D 109 (2024) 024051, [2304.02550]

    2024 arXiv

  58. [65]

    K. Aoki, T. Noumi, R. Saito, S. Sato, S. Shirai, J. Tokuda et al.,Gravitational positivity for phenomenologists: Dark gauge boson in the swampland, Phys. Rev. D110 (2024) 016002, [2305.10058]

    2024 arXiv

  59. [66]

    Melville, D

    S. Melville, D. Roest and D. Stefanyszyn,UV Constraints on Massive Spinning Particles: Lessons from the Gravitino, JHEP 02 (2020) 185, [1911.03126]. References 53

    2020 arXiv

  60. [67]

    Bonnefoy, G

    Q. Bonnefoy, G. Casagrande and E. Dudas,Causality constraints on nonlinear supersymmetry, JHEP 11 (2022) 113, [2206.13451]

    2022 arXiv

  61. [68]

    Chandrasekaran, G

    V. Chandrasekaran, G. N. Remmen and A. Shahbazi-Moghaddam,Higher-Point Positivity, JHEP 11 (2018) 015, [1804.03153]

    2018 arXiv

  62. [69]

    Arkani-Hamed, C

    N. Arkani-Hamed, C. Cheung, C. Figueiredo and G. N. Remmen,Multiparticle Factorization and the Rigidity of String Theory, Phys. Rev. Lett.132 (2024) 091601, [2312.07652]

    2024 arXiv

  63. [70]

    L. C. Bresciani, G. Levati and P. Paradisi,Amplitudes and partial wave unitarity bounds, 2504.12855

    arXiv

  64. [71]

    Cheung and G

    C. Cheung and G. N. Remmen,Multipositivity Bounds, 2505.05553

    arXiv

  65. [72]

    de Rham, S

    C. de Rham, S. Melville, A. J. Tolley and S.-Y. Zhou,Positivity bounds for scalar field theories, Phys. Rev. D96 (2017) 081702, [1702.06134]

    2017 arXiv

  66. [73]

    Davighi, S

    J. Davighi, S. Melville and T. You,Natural selection rules: new positivity bounds for massive spinning particles, JHEP 02 (2022) 167, [2108.06334]

    2022 arXiv

  67. [74]

    Beadle, G

    C. Beadle, G. Isabella, D. Perrone, S. Ricossa, F. Riva and F. Serra,Non-Forward UV/IR Relations, 2407.02346

  68. [75]

    Chala and J

    M. Chala and J. Santiago,Positivity bounds in the standard model effective field theory beyond tree level, Phys. Rev. D105 (2022) L111901, [2110.01624]

    2022 arXiv

  69. [76]

    Bellazzini, M

    B. Bellazzini, M. Riembau and F. Riva,IR side of positivity bounds, Phys. Rev. D106 (2022) 105008, [2112.12561]

    2022 arXiv

  70. [77]

    Hamada, R

    Y. Hamada, R. Kuramochi, G. J. Loges and S. Nakajima,On (scalar QED) gravitational positivity bounds, JHEP 05 (2023) 076, [2301.01999]

    2023 arXiv

  71. [78]

    Caron-Huot and J

    S. Caron-Huot and J. Tokuda,String loops and gravitational positivity bounds: imprint of light particles at high energies, JHEP 11 (2024) 055, [2406.07606]

    2024 arXiv

  72. [79]

    Y. Ye, B. He and J. Gu,Positivity bounds in scalar Effective Field Theories at one-loop level, JHEP 12 (2024) 046, [2408.10318]

    2024 arXiv

  73. [80]

    Peng, L.-Q

    G. Peng, L.-Q. Shao, A. Tokareva and Y. Xu,Running EFT-hedron with null constraints at loop level, 2501.09717

    arXiv

  74. [81]

    Chang and J

    C.-H. Chang and J. Parra-Martinez,Graviton loops and negativity, 2501.17949

    arXiv

  75. [82]

    Beadle, G

    C. Beadle, G. Isabella, D. Perrone, S. Ricossa, F. Riva and F. Serra,The EFT bootstrap at finite MPL, JHEP 06 (2025) 209, [2501.18465]

    2025 arXiv

  76. [83]

    Bellazzini, J

    B. Bellazzini, J. Elias Miró, R. Rattazzi, M. Riembau and F. Riva,Positive moments for scattering amplitudes, Phys. Rev. D104 (2021) 036006, [2011.00037]

    2021 arXiv

  77. [84]

    Arkani-Hamed, T.-C

    N. Arkani-Hamed, T.-C. Huang and Y.-t. Huang,The EFT-Hedron, JHEP 05 (2021) 259, [2012.15849]

    2021 arXiv

  78. [85]

    Chiang, Y.-t

    L.-Y. Chiang, Y.-t. Huang, W. Li, L. Rodina and H.-C. Weng,Into the EFThedron and UV constraints from IR consistency, JHEP 03 (2022) 063, [2105.02862]

    2022 arXiv

  79. [86]

    Caron-Huot and V

    S. Caron-Huot and V. Van Duong,Extremal Effective Field Theories, JHEP 05 (2021) 280, [2011.02957]

    2021 arXiv

  80. [87]

    Chiang, Y.-t

    L.-Y. Chiang, Y.-t. Huang, L. Rodina and H.-C. Weng,De-projecting the EFThedron, JHEP 05 (2024) 102, [2204.07140]

    2024 arXiv

  81. [88]

    Elias Miro, A

    J. Elias Miro, A. Guerrieri and M. A. Gumus,Bridging positivity and S-matrix bootstrap bounds, JHEP 05 (2023) 001, [2210.01502]

    2023 arXiv

  82. [89]

    Riembau,Full Unitarity and the Moments of Scattering Amplitudes, 2212.14056

    M. Riembau,Full Unitarity and the Moments of Scattering Amplitudes, 2212.14056

    arXiv

  83. [90]

    Wan and S.-Y

    S.-L. Wan and S.-Y. Zhou,Matrix moment approach to positivity bounds and UV reconstruction from IR, JHEP 02 (2025) 168, [2411.11964]. References 54

    2025 arXiv

  84. [91]

    A. J. Tolley, Z.-Y. Wang and S.-Y. Zhou,New positivity bounds from full crossing symmetry, JHEP 05 (2021) 255, [2011.02400]

    2021 arXiv

  85. [92]

    S. D. Chowdhury, K. Ghosh, P. Haldar, P. Raman and A. Sinha,Crossing Symmetric Spinning S-matrix Bootstrap: EFT bounds, SciPost Phys. 13 (2022) 051, [2112.11755]

    2022 arXiv

  86. [93]

    Z.-Z. Du, C. Zhang and S.-Y. Zhou,Triple crossing positivity bounds for multi-field theories, JHEP 12 (2021) 115, [2111.01169]

    2021

  87. [94]

    Zhang and S.-Y

    C. Zhang and S.-Y. Zhou,Convex geometry perspective on the (standard model) effective field theory space, Phys. Rev. Lett.125 (Nov, 2020) 201601, [2005.03047]

    2020 arXiv

  88. [95]

    Bellazzini, L

    B. Bellazzini, L. Martucci and R. Torre,Symmetries, Sum Rules and Constraints on Effective Field Theories, JHEP 09 (2014) 100, [1405.2960]

    2014 arXiv

  89. [96]

    Trott,Causality, unitarity and symmetry in effective field theory, JHEP 07 (2021) 143, [2011.10058]

    T. Trott,Causality, unitarity and symmetry in effective field theory, JHEP 07 (2021) 143, [2011.10058]

    2021 arXiv

  90. [97]

    C. Yang, Z. Ren and J.-H. Yu,Positivity from J-Basis operators in the standard model effective Field Theory, JHEP 05 (2024) 221, [2312.04663]

    2024 arXiv

  91. [99]

    R. T. Rockafellar,Convex Analysis. Princeton landmarks in mathematics and physics. Princeton University Press, 1970

    1970

  92. [100]

    V. L. Klee Jr,Extremal structure of convex sets, Archiv der Mathematik8 (1957) 234–240

    1957

  93. [101]

    Ramana and A

    M. Ramana and A. J. Goldman,Some geometric results in semidefinite programming, Journal of Global Optimization7 (1995) 33–50

    1995

  94. [102]

    Vinzant,The geometry of spectrahedra, inSum of squares: theory and applications, vol

    C. Vinzant,The geometry of spectrahedra, inSum of squares: theory and applications, vol. 77 ofProc. Sympos. Appl. Math., pp. 11–35. Amer. Math. Soc., Providence, RI, 2020

    2020

  95. [103]

    Jeffreys,Cartesian tensors

    H. Jeffreys,Cartesian tensors. Cambridge University Press, New York, 1962

    1962

  96. [104]

    R. A. Horn and C. R. Johnson,Matrix analysis. Cambridge University Press, Cambridge, second ed., 2013

    2013

  97. [105]

    Bellazzini,Softness and amplitudes’ positivity for spinning particles, JHEP 02 (2017) 034, [1605.06111]

    B. Bellazzini,Softness and amplitudes’ positivity for spinning particles, JHEP 02 (2017) 034, [1605.06111]

    2017 arXiv

  98. [106]

    de Rham, S

    C. de Rham, S. Melville, A. J. Tolley and S.-Y. Zhou,UV complete me: Positivity Bounds for Particles with Spin, JHEP 03 (2018) 011, [1706.02712]

    2018 arXiv

  99. [107]

    Froissart,Asymptotic behavior and subtractions in the Mandelstam representation, Phys

    M. Froissart,Asymptotic behavior and subtractions in the Mandelstam representation, Phys. Rev. 123 (1961) 1053–1057

    1961

  100. [108]

    Martin,Unitarity and high-energy behavior of scattering amplitudes, Phys

    A. Martin,Unitarity and high-energy behavior of scattering amplitudes, Phys. Rev. 129 (1963) 1432–1436

    1963

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.