Partial Waves for Multipositivity

Jaehoon Jeong

REVIEW 2 major objections 7 minor 1 cited by

Partial Waves for Multipositivity

T0 review · 2 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper constructs spin-resolved partial wave expansions for four-, five-, and six-point massless planar amplitudes on complex-forward kinematic charts, claiming this establishes the missing spin-resolved framework for multipositivity.

desk verdict Useful construction of kinematic charts and explicit partial waves for higher-point multipositivity, but the expansion's completeness and analyticity are asserted rather than proved. read the letter →

arxiv 2608.02719 v1 pith:YIBIXAE5 submitted 2026-08-03 hep-th hep-ph

Jaehoon Jeong This is my paper · ORCID

classification hep-thhep-ph

keywords partialwavesmultipositivitycomplex-forwardlimitmasslessplanaramplitudesspinresolutionWess-Zumino-Witteneffectivefieldtheoryscattering

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

The paper develops a partial-wave formalism for multipositivity, a set of positivity inequalities among massless planar scattering amplitudes with different numbers of external particles. Until now these higher-point positivity bounds had no spin-resolved partial-wave framework like the one used in two-to-two scattering, so the residues at heavy-state poles could not be identified with definite UV angular momentum. The author constructs kinematic charts satisfying two conditions—the complex-forward matching of momentum sums and the independence of the factorization invariant from the other Mandelstams—and uses complexified rotations to build the one-to-two and one-to-three wavefunctions. The result is explicit spin-resolved partial wave expansions for four-, five-, and six-point amplitudes, equations (37)–(39), which directly provide the ingredients for the multipositivity bounds. If correct, this lets higher-point positivity bounds be read as statements about which spins appear in the UV spectrum.

What carries the argument

The load-bearing object is the wavefunction $\langle JM,\tau|p_1^{h_1},\ldots,p_n^{h_n}\rangle$ defined on the kinematic charts. The mechanism that produces it is the complexified rotation $R(\vec z)=R_+(z_1)R_-(z_2)R_0(z_3)$, Gauss-decomposed into unipotent and diagonal pieces, acting on reference momenta built from the $\ell_\pm$ basis matrices; this turns the Mandelstam parameters into rotation parameters. The two kinematic conditions $\langle 1\rangle$ and $\langle 2\rangle$ are what guarantee the combined charts realize the complex-forward limit and isolate $s$ as the factorization invariant, so that the residues at heavy poles factor into products of definite-spin couplings. The rotation matrix elements $F^J_{MM'}(\vec z)$ and their unipotent limits $f^J_{MM'}(z)$ carry the angular dependence of the final partial waves.

What would settle it

Take a known five-point amplitude on the chart (17), such as the WZW amplitude, compute its residues at a heavy pole $s=m_k^2$, and check whether the spin-resolved sum rule (44) reproduces the low-energy coefficient from the partial wave expansion (38); a mismatch would show the wavefunctions do not span the chart.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that multipositivity bounds—positivity inequalities relating massless planar amplitudes with different numbers of external particles—can be carried over to a spin-resolved partial wave expansion. The author writes the general $(n+m)$-point amplitude on a kinematic chart satisfying the conditions $\langle 1\rangle$ (the complex-forward momentum-conservation condition) and $\langle 2\rangle$ (the factorization invariant $s$ is independent of the other Mandelstams) as a sum over angular momentum $J$ and magnetic quantum numbers of products of wavefunctions $\langle JM,\tau|p_1,\ldots,p_n\rangle$ and $\langle JM,\tau'|-\bar p_{n+m},\ldots,-\bar p_{n+1}\rangle^*$, with Mandelstam dependence carried by $H_J$. For four-, five-, and six-point kinematics the paper gives explicit charts and constructs the wavefunctions from complexified rotations; the resulting expansions (37)–(39) provide the spin-resolved ingredients for the multipositivity inequalities, and the WZW example at five points resolves the low-energy coefficient into definite-spin UV couplings. The author's claim is that this establishes the spin-resolved framework for higher-point multipositivity, placing it on the same footing as conventional two-to-two partial wave analyses.

Load-bearing premise

The partial-wave expansion (22) is assumed to be complete on the complex-forward kinematic charts: every amplitude on the chart is representable as a sum over the constructed wavefunctions $\langle JM,\tau|p_1,\ldots,p_n\rangle$, but no completeness proof is supplied.

Editorial extensions

If this is right

The multipositivity bounds $|A^{(4)}|^2 \le (4pt)(4pt)$, $|A^{(5)}|^2 \le (4pt)(6pt)$, and $|A^{(6)}|^2 \le (6pt)(6pt)$ can be decomposed by spin, so the positivity statement can be refined sector by sector in exchanged angular momentum.
The five-point WZW coefficient is written as a sum over UV resonance data with definite spin (Eq. (44)), and similar sum rules follow for other odd-point amplitudes.
The same chart-building procedure works for charts where some two-particle Mandelstams vanish, which is relevant for theories without massless three-point amplitudes such as chiral perturbation theory.
Because the partial wave construction itself does not use planarity, the same wavefunctions can expand nonplanar amplitudes as well.

Reading between the lines

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

A natural extension is to recursive one-to-many wavefunctions: the six-point state is a one-to-three block containing a one-to-two block, so one can test whether a one-to-four wavefunction for seven-point kinematics factorizes in the same iterative way.
The collinear reorganization into reduced coefficients $H^{J_1}_{M_2;mnl}$ suggests a general rule: in the $s_{45}\to 0$ limit the UV spin tower of a three-particle block is repackaged into projections of a two-particle subsystem, which could be checked against a model with a few explicit resonances.
For nonplanar multipositivity, the same wavefunctions may apply, but the kinematic conditions $\langle 1\rangle$ and $\langle 2\rangle$ would need re-deriving because the factorization invariants differ; the paper leaves that case open.
A massive kinematics extension is announced as separate future work; a concrete check would be to take the massive limit of these complexified-rotation wavefunctions and compare with standard massive helicity partial waves.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper builds real machinery: kinematic charts for four-, five-, and six-point multipositivity, explicit one-to-three wavefunctions, and a worked WZW sum rule. What is genuinely new is the systematic chart construction under conditions ⟨1⟩ and ⟨2⟩, the one-to-three wavefunctions of Eq. (36), and the explicit partial waves of Eqs. (37)–(39). The WZW application is honest: the coefficient is fixed by the QCD anomaly, not fitted, and the sum rule in Eq. (44) is a concrete spin-resolved statement. The algebra with complexified rotations is internally consistent, and the charts do satisfy on-shell and momentum-conservation conditions as stated.

The largest soft spot is completeness. Eq. (22) is asserted to be a complete partial-wave expansion, but no convergence or basis-completeness argument is given. The stress-test observation is concrete: the five-point basis functions contain factors like F(b1 sqrt(s45)/(s-s45), b2/sqrt(s45), 0), which have branch cuts in s45. Physical tree amplitudes on the chart are rational—the WZW amplitude itself is polynomial. So either the J-sum cancels the cuts, which would require non-analytic coefficient functions, or the basis cannot represent generic analytic amplitudes. The appendix’s resummation around Eq. (43) is a workaround for the s45 → 0 limit, not a completeness proof. The paper should either prove completeness or explicitly rescope the claim; the comparison to 2-to-2 partial waves is an overstatement until this is settled.

A second, smaller issue: five-point partial waves already exist in Ref. [103], and the paper does not clearly delineate what is new relative to that work. That distinction matters. Minor: a few statements that charts “automatically” satisfy the kinematic conditions are too terse; I would want the Gram determinant checks spelled out at least once.

Who this is for: amplitudes and EFT bootstrap people working on higher-point positivity. They will want to read it and likely use the chart technology, but they should treat the completeness claim with caution. The paper deserves peer review—it is serious, well-constructed work—but the referee should demand a rigorous treatment of completeness and analyticity. If the author can supply that, or honestly rescope the result, this would be a solid contribution. My recommendation: send it out, and make the completeness/analyticity issue the major point of the report.

Referee Report

2 major / 7 minor

Summary. This paper develops a partial-wave (PW) formalism adapted to the 'multipositivity' constraints on massless planar amplitudes in the complex-forward limit. The author formulates two kinematic conditions — ⟨1⟩ (complex-forward completion) and ⟨2⟩ (independence of the factorization invariant) — constructs explicit momentum charts for (2|2), (2|3), and (3|3) kinematics, derives one-to-two and one-to-three wavefunctions as matrix elements of complexified SL(2,C) rotations acting on reference configurations with definite J0 eigenvalues, and assembles the partial-wave expansions in Eqs. (37)–(39). An appendix applies the five-point expansion to the WZW amplitude and derives a spin-resolved sum rule, Eq. (44), for the anomaly coefficient κ̃. The Discussion claims that this places higher-point multipositivity on the same footing as conventional two-to-two partial-wave analyses.

Significance. If the claimed expansions are complete, this is a useful and timely technical contribution to a fast-moving area. The momentum charts are explicit and checkable; on-shell conditions, momentum conservation, and condition ⟨1⟩ are satisfied by direct inspection, and the algebraic construction of the wavefunctions via complexified rotations is internally consistent. The WZW application is genuinely non-circular: κ̃ is fixed by the QCD chiral anomaly, not fitted, and the derivation leads to a concrete, testable sum rule (44) expressing a low-energy coefficient through definite-spin UV couplings. I also verified that the five-point basis has the right exponent-space support to match Mandelstam functions on chart (17), which is why the WZW monomials in (42) are reachable. However, the central gap — completeness of the expansion (22) on the charts and the analyticity of the coefficient functions in s45 — is load-bearing for the advertised claim, and the paper does not provide an inversion or convergence argument. As it stands, the formal structure is sound but the 'same footing as two-to-two' conclusion is not yet established.

major comments (2)

[Partial Waves for Multipositivity (Eqs. 38, 40, 43)] The completeness of the expansion (22) on the complex-forward charts is asserted, not proved, and as written the five-point basis is not manifestly analytic in the Mandelstam variable s45. Substituting z1 = −b1√s45/(s−s45) and z2 = b2/√s45 into Eq. (40), each k-term of F^{J2}_{M2 h54} carries the overall factor s45^{(M2−h54)/2} (the k-dependence cancels in the s45 exponent), so for odd M2−h54 every basis element in Eq. (38) has a square-root branch point in s45, whereas the physical amplitude on the chart — e.g., the WZW amplitude in Eq. (42) — is polynomial. The equality (38) can hold only if the coefficient functions H^{J1|J2M2} absorb compensating non-analyticities or if the J2/M2 sums cancel the half-integer powers; neither mechanism is demonstrated, and no inversion or orthogonality relation fixing H is provided. The sentence 'By identifying these wavefunctions on our kinematic charts, we complete the PW expansions' is an assertion, and the Appendix's Eq. (43) postulates the required resummation ('where we take the full amplitude to be analytic in the collinear limit'), which is an assumption about the amplitude, not a consequence of basis completeness; the same issue affects the six-point expansion (39) through its F^{J3} factor. On the positive side, the exponent-space support of the five-point basis (for scalars, m_{b1} = m_a + m_{b2}) matches the scaling of the planar Mandelstams on chart (17), so the representable class plausibly includes all Mandelstam functions; what is missing is a proof that the even-M2 sector suffices, or a canonical projection that removes the odd-M2 branch-cut sector.
[One-to-Three Wavefunctions (Eqs. 33–36)] The label J1 in the one-to-three wavefunction is not identified with the total angular momentum of the three-particle block. The reference product |K12;J2M2⟩ ⊗ |k3^{h3}⟩ has a definite J0 eigenvalue M2−h3 but is a superposition of several total-J1 components, so the step 'in a fixed-J1 partial wave, we project it onto |J1, M2−h3⟩' suppresses the recoupling overlap between the two-particle subsystem and the angular-momentum content of the third particle. That overlap — a Clebsch-Gordan-type coefficient depending on J1, J2, M2, and h3 — must be exhibited in Eq. (36) or absorbed by an explicit, canonical normalization of the basis states. Without this, the summation label J1 in Eqs. (38)–(39) is not shown to be the spin of the exchanged state that factorizes in the residue formula (8), which is exactly what is needed for the 'spin-resolved' interpretation claimed in the abstract and Discussion.

minor comments (7)

[One-to-Two Wavefunction (Eq. 28)] The rotation parameter in Eq. (28) is printed as b, but the momenta p1, p2 in Eq. (14) carry the parameter a and Eq. (29) correctly uses a; conjugation with R+(a/√s) reproduces Eq. (14) (R+ℓ+R+^{-1} = ℓ+ + (a/√s)n+ and R+ℓ−R+^{-1} = ℓ− − (a/√s)n+), so Eq. (28) should be corrected and the p → RpR^{-1} convention stated explicitly.
[Kinematic Conditions for Multipositivity (Eq. 24)] With the explicit generators in Eq. (23) the commutators are [J−,ℓ−] = −n− and [J−,ℓ+] = n− − n+, so the printed relation [J−,ℓ±] = ±n− is not consistent with the matrix algebra; Eq. (25) is correct.
[(p1,2|p3,4,5) Kinematics] The word 'vainishing' should read 'vanishing'.
[Momentum parametrizations (Eqs. 17 and 32)] The matrix displays in Eqs. (17) and (32) are garbled (missing brackets and misplaced entries) and should be typeset clearly, since these charts underlie all subsequent formulas.
[Introduction (comparison with Ref. [103])] The novelty claim relative to Ref. [103] (five-point partial waves) rests on a single sentence; the authors should state explicitly what the complex-forward chart construction adds to the general multiparticle partial-wave frameworks of Refs. [103–105].
[Note [115] and Eq. (37)] To pin down the normalization of the coefficient functions H, the authors should show explicitly how Eq. (37) reduces to the standard two-to-two helicity partial wave, including the (2J+1) factor and the identification of the M-sum of f-factors with the Wigner d-function of the scattering angle.
[Kinematic Conditions (condition ⟨2⟩)] The statements that the constructions 'automatically' satisfy condition ⟨2⟩ (e.g., for (−p̄4,3|p3,4)) are checkable but not shown; one-line verifications for each chart would help the reader confirm that condition ⟨2⟩ indeed holds.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the partial-wave formalism is built on independently specified multipositivity charts, the WZW coefficient is fixed by the QCD anomaly rather than fitted, and the self-citations are not load-bearing.

full rationale

The derivation chain is not circular. The kinematic conditions ⟨1⟩ and ⟨2⟩ are adopted from Ref. [101] (Cheung and Remmen), which is external to the present author, and the paper's central output—the spin-resolved partial waves in Eqs. (37)–(39)—is a new construction built from complexified rotations on those charts rather than a restatement of the input conditions. The wavefunctions in Eqs. (30) and (36) are derived by standard angular-momentum projection, and the expansion coefficients H_J are left as arbitrary coefficient functions, not fitted to the target amplitudes. In the Appendix, the WZW coefficient κ is fixed by the QCD chiral anomaly ('with κ=Nc/(4√2 Fπ^5) fixed by the QCD chiral anomaly'), so the sum rule (44) is a genuine dispersion-relation prediction rather than a fitted parameter renamed as a prediction. The self-citations that appear are not load-bearing: [102] is cited for the tetrad basis and for the statement that multipositivity has found an application in the chiral Lagrangian, while [104] is listed among general partial-wave references; neither supports the central derivation. The one substantive weakness is that the completeness of the expansion (22) on the kinematic charts is asserted ('By identifying these wavefunctions on our kinematic charts, we complete the PW expansions') but not proved, and the five- and six-point basis functions contain √s45 factors. This is a completeness/justification gap, not circularity: the expansion does not reduce to the chart conditions by construction, nor is any predicted quantity defined in terms of the data it is supposed to constrain. Hence the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; the variables a, b, a1, a2, b1, b2, s, s12, s45, s56 are chart coordinates, not fitted constants. The central claim rests on standard angular momentum algebra and on the physical and analytic assumptions of the multipositivity framework. No new entities are introduced.

assumptions (4)

standard math Standard angular momentum representation theory, including |JM⟩ states, the J±, J0 algebra, and Wigner matrix elements (Eqs. (23)-(26) and (40)).
Used throughout the wavefunction construction; no proof is given because it is standard background.
domain assumption The partial wave expansion (22) is complete and convergent on the complex-forward kinematic charts; the wavefunctions ⟨JM|p1,...,pn⟩ form a basis for amplitudes on the chart.
Assumed in the section 'One-to-Many Wavefunctions'; no convergence or completeness argument is provided.
domain assumption Conditions ⟨1⟩ and ⟨2⟩, adopted from Ref [101], are the correct kinematic requirements for multipositivity, particularly the independence of the factorization invariant s from all other kinematic invariants.
The charts are constructed to satisfy these conditions; if the conditions are not sufficient, the spin-resolved bounds do not follow.
domain assumption In the WZW application, the five-point amplitude obeys A^(5)/s → 0 as s → ∞ with other Mandelstams fixed (no pomeron, α_i < 1), and is analytic in the collinear limit s45 → 0.
Used in the Appendix to drop boundary terms in the dispersion relation and to define reduced coefficients via Eq. (43).

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Cite this review

Pith. "Pith review of Partial Waves for Multipositivity." pith.science (2026). https://pith.science/paper/YIBIXAE5

@misc{pith2026260802719,
  author       = {Pith},
  title        = {Pith review of: Partial Waves for Multipositivity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YIBIXAE5}},
  note         = {Machine review of arXiv:2608.02719}
}

read the original abstract

We develop a partial-wave formalism providing the missing spin-resolved framework for multipositivity among massless planar amplitudes. We present a systematic procedure for constructing kinematic charts realizing the complex-forward limit. Applying it to obtain the four-, five-, and six-point charts, we show how to derive the one-to-two and one-to-three wavefunctions and construct the associated partial waves. The resulting formalism places higher-point multipositivity on the same footing as conventional partial-wave analyses of two-to-two scattering.

Figures

Figures reproduced from arXiv: 2608.02719 by the authors.

**Figure 1.** Diagrammatic illustration of Eq. (11). kinematics required for the Cauchy–Schwarz inequality, we define p¯i(s, c) ≡ p † i (s ∗ , c), (9) so that p¯i = p † i for real s. With this definition, the amplitudes of two combined kinematics (p1,...,n| − p¯n,...,1) and (pn+1,...,n+m| − p¯n+m,...,n+1) reproduce the residues |A (1+n) k | 2 and |A (1+m) k | 2 , respectively. Here, the definition p¯i automatically satisfies th… view at source ↗

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B. Bellazzini, C. Cheung, and G. N. Remmen, “Quantum Gravity Constraints from Unitarity and Analyticity,” Phys. Rev. D93(2016) 064076, arXiv:1509.00851 [hep- th]

2016 arXiv
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Positivity of Curvature- Squared Corrections in Gravity,

C. Cheung and G. N. Remmen, “Positivity of Curvature- Squared Corrections in Gravity,”Phys. Rev. Lett.118 (2017) 051601,arXiv:1608.02942 [hep-th]

2017 arXiv
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Causality Constraints on Corrections to the Graviton Three-Point Coupling,

X. O. Camanho, J. D. Edelstein, J. Maldacena, and A. Zhiboedov, “Causality Constraints on Corrections to the Graviton Three-Point Coupling,”JHEP02(2016) 020,arXiv:1407.5597 [hep-th]

2016 arXiv
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Causality Constrains Higher Curvature Corrections to Gravity,

A. Gruzinov and M. Kleban, “Causality Constrains Higher Curvature Corrections to Gravity,”Class. Quant. Grav.24(2007) 3521,arXiv:hep-th/0612015

2007 arXiv
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Infrared Consistency and the Weak Gravity Conjecture,

C. Cheung and G. N. Remmen, “Infrared Consistency and the Weak Gravity Conjecture,”JHEP12(2014) 087,arXiv:1407.7865 [hep-th]

2014 arXiv
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Posi- tivity of Amplitudes, Weak Gravity Conjecture, and Modified Gravity,

B. Bellazzini, M. Lewandowski, and J. Serra, “Posi- tivity of Amplitudes, Weak Gravity Conjecture, and Modified Gravity,”Phys. Rev. Lett.123(2019) 251103, arXiv:1902.03250 [hep-th]

2019 arXiv
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Causality constraints on cor- rections to Einstein gravity,

S. Caron-Huot, Y.-Z. Li, J. Parra-Martinez, and D. Simmons-Duffin, “Causality constraints on cor- rections to Einstein gravity,”JHEP05(2023) 122, arXiv:2201.06602 [hep-th]

2023 arXiv
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Graviton partial waves and causal- ity in higher dimensions,

S. Caron-Huot, Y.-Z. Li, J. Parra-Martinez, and D. Simmons-Duffin, “Graviton partial waves and causal- ity in higher dimensions,”Phys. Rev. D108(2023) 026007,arXiv:2205.01495 [hep-th]

2023 arXiv
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Positive Signs in Mas- sive Gravity,

C. Cheung and G. N. Remmen, “Positive Signs in Mas- sive Gravity,”JHEP04(2016) 002, arXiv:1601.04068 [hep-th]

2016 arXiv
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Improved Pos- itivity Bounds and Massive Gravity,

C. de Rham, S. Melville, and A. J. Tolley, “Improved Pos- itivity Bounds and Massive Gravity,”JHEP04(2018) 083,arXiv:1710.09611 [hep-th]

2018 arXiv
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Mas- sive gravity is not positive,

B. Bellazzini, G. Isabella, S. Ricossa, and F. Riva, “Mas- sive gravity is not positive,”Phys. Rev. D109(2024) 024051,arXiv:2304.02550 [hep-th]

2024 arXiv
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Gravita- tional effective field theory islands, low-spin dominance, and the four-graviton amplitude,

Z. Bern, D. Kosmopoulos, and A. Zhiboedov, “Gravita- tional effective field theory islands, low-spin dominance, and the four-graviton amplitude,”J. Phys. A54(2021) 344002,arXiv:2103.12728 [hep-th]

2021 arXiv
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Flattening of the EFT-hedron: supersymmetric positivity bounds and the search for string theory,

J. Berman, H. Elvang, and A. Herderschee, “Flattening of the EFT-hedron: supersymmetric positivity bounds and the search for string theory,”JHEP03(2024) 021, arXiv:2310.10729 [hep-th]

2024 arXiv
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Multifield positivity bounds for inflation,

M. Freytsis, S. Kumar, G. N. Remmen, and N. L. Rodd, “Multifield positivity bounds for inflation,”JHEP09 (2023) 041,arXiv:2210.10791 [hep-th]

2023 arXiv
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Consistency of the Stan- dard Model Effective Field Theory,

G. N. Remmen and N. L. Rodd, “Consistency of the Stan- dard Model Effective Field Theory,”JHEP12(2019) 032,arXiv:1908.09845 [hep-ph]

2019 arXiv
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FlavorConstraintsfrom Unitarity and Analyticity,

G.N.RemmenandN.L.Rodd, “FlavorConstraintsfrom Unitarity and Analyticity,”Phys. Rev. Lett.125(2020) 081601, arXiv:2004.02885 [hep-ph] . [Erratum:Phys. Rev. Lett.127, 149901 (2021)]

2020 arXiv
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Signs, spin, SMEFT: Sum rules at dimension six,

G. N. Remmen and N. L. Rodd, “Signs, spin, SMEFT: Sum rules at dimension six,”Phys. Rev. D105(2022) 036006,arXiv:2010.04723 [hep-ph]

2022 arXiv
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Softness and amplitudes’ positiv- ity for spinning particles,

B. Bellazzini, “Softness and amplitudes’ positiv- ity for spinning particles,”JHEP02(2017) 034, arXiv:1605.06111 [hep-th]

2017 arXiv
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Spinning sum rules for the dimension-six SMEFT,

G. N. Remmen and N. L. Rodd, “Spinning sum rules for the dimension-six SMEFT,”JHEP09(2022) 030, arXiv:2206.13524 [hep-ph]

2022 arXiv
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Positively identify- ing Higgs effective field theory or standard model ef- fective field theory,

G. N. Remmen and N. L. Rodd, “Positively identify- ing Higgs effective field theory or standard model ef- fective field theory,”Phys. Rev. D113(2026) 036027, arXiv:2412.07827 [hep-ph]

2026
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Positivity constraints on aQGC: carving out the physical parameter space,

Q. Bi, C. Zhang, and S.-Y. Zhou, “Positivity constraints on aQGC: carving out the physical parameter space,” JHEP06(2019) 137,arXiv:1902.08977 [hep-ph]

2019 arXiv
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Positivity bounds on vector boson scattering at the LHC,

C. Zhang and S.-Y. Zhou, “Positivity bounds on vector boson scattering at the LHC,”Phys. Rev. D100(2019) 095003,arXiv:1808.00010 [hep-ph]

2019 arXiv
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Corners and islands in the S-matrix bootstrap of the open superstring,

J. Berman and H. Elvang, “Corners and islands in the S-matrix bootstrap of the open superstring,”JHEP09 (2024) 076,arXiv:2406.03543 [hep-th]

2024 arXiv
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Amplitudes and the Riemann Zeta Function,

G. N. Remmen, “Amplitudes and the Riemann Zeta Function,”Phys. Rev. Lett.127(2021) 241602, arXiv:2108.07820 [hep-th]

2021 arXiv
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Stringy Completions of the StandardModelfromtheBottomUp,

B. Bachu and A. Hillman, “Stringy Completions of the StandardModelfromtheBottomUp,” arXiv:2212.03871 [hep-th]

arXiv
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UV-complete gravity amplitudes and the triple product,

Y.-t. Huang and G. N. Remmen, “UV-complete gravity amplitudes and the triple product,”Phys. Rev. D106 (2022) L021902,arXiv:2203.00696 [hep-th]

2022 arXiv
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Veneziano variations: how unique are string amplitudes?,

C. Cheung and G. N. Remmen, “Veneziano variations: how unique are string amplitudes?,”JHEP01(2023) 122,arXiv:2210.12163 [hep-th]

2023 arXiv
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Stringy dynamics from an amplitudes bootstrap,

C. Cheung and G. N. Remmen, “Stringy dynamics from an amplitudes bootstrap,”Phys. Rev. D108(2023) 026011,arXiv:2302.12263 [hep-th]

2023 arXiv
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Bespokedualresonance,

C.CheungandG.N.Remmen, “Bespokedualresonance,” Phys. Rev. D108(2023) 086009, arXiv:2308.03833 [hep- th]

2023 arXiv
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The stringy S-matrix bootstrap: maximal spin and superpolynomial softness,

K. Häring and A. Zhiboedov, “The stringy S-matrix bootstrap: maximal spin and superpolynomial softness,” JHEP10(2024) 075,arXiv:2311.13631 [hep-th]

2024 arXiv
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What is the graviton pole made of?,

K. Häring and A. Zhiboedov, “What is the graviton pole made of?,”arXiv:2410.21499 [hep-th]

arXiv
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On unitarity of tree-level string amplitudes,

N. Arkani-Hamed, L. Eberhardt, Y.-t. Huang, and 7 S. Mizera, “On unitarity of tree-level string amplitudes,” JHEP02(2022) 197,arXiv:2201.11575 [hep-th]

2022 arXiv
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Unitarity of bespoke amplitudes,

R. Bhardwaj, M. Spradlin, A. Volovich, and H.-C. Weng, “Unitarity of bespoke amplitudes,”Phys. Rev. D110 (2024) 106016,arXiv:2406.04410 [hep-th]

2024 arXiv
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Positivity in Amplitudes from Quantum Entanglement,

R. Aoude, G. Elor, G. N. Remmen, and O. Sumensari, “Positivity in Amplitudes from Quantum Entanglement,” arXiv:2402.16956 [hep-th]

arXiv
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Strings from Almost Nothing,

C. Cheung, G. N. Remmen, F. Sciotti, and M. Tarquini, “Strings from Almost Nothing,”arXiv:2508.09246 [hep- th]

arXiv
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Where Is String Theory in the Space of Scattering Amplitudes?,

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

2021 arXiv
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Where is tree-level string theory?,

J. Albert, W. Knop, and L. Rastelli, “Where is tree-level string theory?,”JHEP02(2025) 157, arXiv:2406.12959 [hep-th]

2025 arXiv
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Boot- strapping Extremal Scalar Amplitudes With and With- out Supersymmetry,

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String The- ory from Maximal Supersymmetry,

H. Elvang, A. Herderschee, and R. Morales, “String The- ory from Maximal Supersymmetry,”arXiv:2601.11705 [hep-th]

arXiv
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Bootstrap Principle for the Spectrum and Scattering of Strings,

C. Cheung, A. Hillman, and G. N. Remmen, “Bootstrap Principle for the Spectrum and Scattering of Strings,” Phys. Rev. Lett.133(2024) 251601, arXiv:2406.02665 [hep-th]

2024 arXiv
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Uniqueness criteria for the Virasoro-Shapiro amplitude,

C. Cheung, A. Hillman, and G. N. Remmen, “Uniqueness criteria for the Virasoro-Shapiro amplitude,”Phys. Rev. D111(2025) 086034,arXiv:2408.03362 [hep-th]

2025 arXiv
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Spectral Constraints on Theories of Colored Par- ticles and Gravity,

A. Hillman, Y.-t. Huang, L. Rodina, and J. Rum- butis, “Spectral Constraints on Theories of Colored Par- ticles and Gravity,”Phys. Rev. Lett.135(2025) 061604, arXiv:2411.04857 [hep-th]

2025
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Strings from Massive Higher Spins: The Asymp- totic Uniqueness of the Veneziano Amplitude,

S.Caron-Huot, Z.Komargodski, A.Sever, andA.Zhiboe- dov, “Strings from Massive Higher Spins: The Asymp- totic Uniqueness of the Veneziano Amplitude,”JHEP 10(2017) 026,arXiv:1607.04253 [hep-th]

2017 arXiv
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Positivity with Long-Range In- teractions,

B. Bellazzini, J. Berman, G. Isabella, F. Riva, M. Ro- mano, and F. Sciotti, “Positivity with Long-Range In- teractions,”arXiv:2512.13780 [hep-th]

arXiv
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The Rise of Linear Trajectories,

Y.-t. Huang, S. Ricossa, F. Riva, and J.-D. Tsai, “The Rise of Linear Trajectories,”arXiv:2510.07991 [hep-th]

arXiv
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The EFT bootstrap at finiteMP L,

C. Beadle, G. Isabella, D. Perrone, S. Ricossa, F. Riva, and F. Serra, “The EFT bootstrap at finiteMP L,”JHEP 06(2025) 209,arXiv:2501.18465 [hep-th]

2025 arXiv
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Non-forward UV/IR relations,

C. Beadle, G. Isabella, D. Perrone, S. Ricossa, F. Riva, and F. Serra, “Non-forward UV/IR relations,”JHEP08 (2025) 188,arXiv:2407.02346 [hep-th]

2025
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Bootstrapping Gravity with Cross- ing Symmetric Dispersion Relations,

C. Pasiecznik, “Bootstrapping Gravity with Cross- ing Symmetric Dispersion Relations,”arXiv:2506.09884 [hep-th]

arXiv
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Gravity and a uni- versal cutoff for field theory,

S. Caron-Huot and Y.-Z. Li, “Gravity and a uni- versal cutoff for field theory,”JHEP02(2025) 115, arXiv:2408.06440 [hep-th]

2025 arXiv
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Bootstrapping gauge theories,

Y. He and M. Kruczenski, “Bootstrapping gauge theories,”Phys. Rev. Lett.133(2024) 191601, arXiv:2309.12402 [hep-th]

2024 arXiv
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The O(N) S-matrix Monolith,

L. Córdova, Y. He, M. Kruczenski, and P. Vieira, “The O(N) S-matrix Monolith,”JHEP04(2020) 142, arXiv:1909.06495 [hep-th]

2020 arXiv
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Bootstrap- ping QCD Using Pion Scattering Amplitudes,

A. L. Guerrieri, J. Penedones, and P. Vieira, “Bootstrap- ping QCD Using Pion Scattering Amplitudes,”Phys. Rev. Lett.122(2019) 241604, arXiv:1810.12849 [hep- th]

2019 arXiv
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S-matrix bootstrap for effective field theories: massless pions,

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2021 arXiv
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The Phases of the Scalar S-Matrix Island,

J. Elias Miró, A. Guerrieri, and M. A. Gümüş, “The Phases of the Scalar S-Matrix Island,”arXiv:2605.06613 [hep-th]

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Extremal Higgs couplings,

J. Elias Miró, A. Guerrieri, and M. A. Gümüş, “Extremal Higgs couplings,”Phys. Rev. D110(2024) 016007, arXiv:2311.09283 [hep-ph]

2024 arXiv
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Regge boot- strap: From linear to nonlinear trajectories,

C. Eckner, F. Figueroa, and P. Tourkine, “Regge boot- strap: From linear to nonlinear trajectories,”Phys. Rev. D111(2025) 126005,arXiv:2401.08736 [hep-th]

2025 arXiv
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Onthenumber of Regge trajectories for dual amplitudes,

C.Eckner, F.Figueroa, andP.Tourkine, “Onthenumber of Regge trajectories for dual amplitudes,”JHEP02 (2025) 103,arXiv:2405.21057 [hep-th]

2025 arXiv
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Thermal Positivity,

C. Cheung and R. A. Rosen, “Thermal Positivity,” arXiv:2606.05136 [hep-th]

arXiv
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Bootstrapping Pion Form Factors at LargeN,

J. Albert, D. Kosva, and L. Rastelli, “Bootstrapping Pion Form Factors at LargeN,” arXiv:2606.19420 [hep- th]

arXiv
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S-matrix bootstrap bounds on self-interacting dark matter,

Q. Chen, Z.-H. Wang, and S.-Y. Zhou, “S-matrix bootstrap bounds on self-interacting dark matter,” arXiv:2607.13141 [hep-ph]

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Unitary Dual-Resonance S-matrices,

C. de Rham, A. J. Tolley, Z.-H. Wang, and S.-Y. Zhou, “Unitary Dual-Resonance S-matrices,”arXiv:2607.24922 [hep-th]

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Analytic Boundaries of Infinite-Spin-Tower Amplitudes from Hidden Zero,

L.-Q. Shao and A. Vichi, “Analytic Boundaries of Infinite-Spin-Tower Amplitudes from Hidden Zero,” arXiv:2607.27300 [hep-th]

arXiv
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Causality and the Equivalence Principle for Higher Energy Scattering,

Y.-t. Huang and L. W. Lindwasser, “Causality and the Equivalence Principle for Higher Energy Scattering,” arXiv:2606.27222 [hep-th]

arXiv
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Theoretical and Obser- vational Bounds on Dynamical Chern-Simons Gravity as an Effective Field Theory,

A. Cassem and M. P. Hertzberg, “Theoretical and Obser- vational Bounds on Dynamical Chern-Simons Gravity as an Effective Field Theory,”arXiv:2604.07332 [hep-th]

arXiv
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A Dispersive Bootstrap for the Virasoro-Shapiro Amplitude,

Y. Xu, “A Dispersive Bootstrap for the Virasoro-Shapiro Amplitude,”arXiv:2606.19283 [hep-th]

arXiv
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Multiparticle Factorization and the Rigidity of String Theory,

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, arXiv:2312.07652 [hep-th]

2024 arXiv
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Higher-Spin and Higher-Point Constraints on Stringy Amplitudes,

I. Basile, G. N. Remmen, and G. Staudt, “Higher-Spin and Higher-Point Constraints on Stringy Amplitudes,” arXiv:2603.04485 [hep-th]

arXiv
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Higher-Point Positivity,

V. Chandrasekaran, G. N. Remmen, and A. Shahbazi- Moghaddam, “Higher-Point Positivity,”JHEP11(2018) 015,arXiv:1804.03153 [hep-th]

2018 arXiv
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Splitting regions and shrinking islands from higher point con- straints,

J. Berman, H. Elvang, and C. Figueiredo, “Splitting regions and shrinking islands from higher point con- straints,”JHEP10(2025) 226, arXiv:2506.22538 [hep- th]

2025 arXiv
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Amplitudes and partial wave unitarity bounds,

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2026 arXiv
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Multipositivity bounds for scattering amplitudes,

C. Cheung and G. N. Remmen, “Multipositivity bounds for scattering amplitudes,”Phys. Rev. D112(2025) 016017,arXiv:2505.05553 [hep-th]

2025 arXiv
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Five-point partial waves, split- ting constraints and hidden zeros,

A. P. Saha and A. Sinha, “Five-point partial waves, split- ting constraints and hidden zeros,” arXiv:2601.15088 [hep-th]
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Nondiagonal pair of Ma- jorana particles ate+e− colliders,

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Throughout this work, we use the shorthand notation p1,...,N≡(p 1,...,p N )for arbitraryN
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In the all-incoming convention, the conjugation of an N-point amplitude isA (N)∗ (p1,...,N ) =A (N) (−p† N,...,1)
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A shift with theℓ± matrices cannot turn on the mixed Mandelstam because of diagonality
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Thus, nonplanar amplitudes can also be expanded in the PWs derived here

Note that the PW construction itself is independent of planarity. Thus, nonplanar amplitudes can also be expanded in the PWs derived here
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For example, for a four-point am- plitude one may writeH J (s) =⟨JM|M (s)|JM⟩, with the normalization factors between the momentum and angular-momentum bases absorbed intoHJ

The coefficients in our PWs are the usual ones in the conventional PW expansion, defined in the angular- momentum basis. For example, for a four-point am- plitude one may writeH J (s) =⟨JM|M (s)|JM⟩, with the normalization factors between the momentum and angular-momentum base...
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1977

Pith tools

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

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Multipositivity Constrains the Chiral Lagrangian,

C. Cheung, J. Jeong, P. Ko, A. Pomarol, G. N. Remmen, and F. Sciotti, “Multipositivity Constrains the Chiral Lagrangian,”arXiv:2605.21582 [hep-th]. 8

arXiv

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Causality, analyticity and an IR obstruction to UV completion,

A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nico- lis, and R. Rattazzi, “Causality, analyticity and an IR obstruction to UV completion,”JHEP10(2006) 014, arXiv:hep-th/0602178

arXiv 2006

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To turn on this Mandelstam under the conditions in Eqs

= 0. To turn on this Mandelstam under the conditions in Eqs. (4) and (5), we shiftp1,2 =q1,2±an + [109], which follows the shift introduced in Ref. [101], itself inspired by on-shell recursion relations [110]. This gives s23 = det(an+−a∗n−) =|a| 2 in the chart(p1,2|−¯p2,1). For the other side, to satisfy the condition⟨2⟩, we choose p3,4 =−pT 2,1(a→b ), wh...

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Positive moments for scatter- ing amplitudes,

B. Bellazzini, J. Elias Miró, R. Rattazzi, M. Riem- bau, and F. Riva, “Positive moments for scatter- ing amplitudes,”Phys. Rev. D104(2021) 036006, arXiv:2011.00037 [hep-th]

arXiv 2021

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The EFT-Hedron,

N. Arkani-Hamed, T.-C. Huang, and Y.-t. Huang, “The EFT-Hedron,”JHEP05(2021) 259, arXiv:2012.15849 [hep-th]

arXiv 2021

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Extremal Effective Field Theories,

S. Caron-Huot and V. Van Duong, “Extremal Effective Field Theories,”JHEP05(2021) 280, arXiv:2011.02957 [hep-th]

arXiv 2021

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Bootstrapping pions at large N,

J. Albert and L. Rastelli, “Bootstrapping pions at large N,”JHEP08(2022) 151,arXiv:2203.11950 [hep-th]

arXiv 2022

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Cor- nering large-Nc QCD with positivity bounds,

C. Fernandez, A. Pomarol, F. Riva, and F. Sciotti, “Cor- nering large-Nc QCD with positivity bounds,”JHEP06 (2023) 094,arXiv:2211.12488 [hep-th]

arXiv 2023