pith. machine review for the scientific record. sign in

arxiv: 2604.15235 · v1 · submitted 2026-04-16 · ✦ hep-th

Recognition: unknown

Sampling the Graviton Pole and Deprojecting the Swampland

Guangzhuo Peng , Laurentiu Rodina , Anna Tokareva , Yongjun Xu
Authors on Pith no claims yet

Pith reviewed 2026-05-10 10:14 UTC · model grok-4.3

classification ✦ hep-th
keywords graviton polebootstrapdispersion relationseffective field theoryswamplandunitarityRegge trajectoriesnon-projective bounds
0
0 comments X

The pith

A finite-resolution sampling method applied to graviton poles in crossing-symmetric dispersion relations produces non-projective bounds that fix the overall scale of EFT couplings.

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

The authors develop a bootstrap framework that directly samples the contribution of a graviton pole at finite resolution, rather than smearing it, while retaining control over subtraction terms. This reproduces known projective bounds in six and higher dimensions and yields modestly stronger bounds in five dimensions. When the same method is applied to dispersion relations that enforce crossing symmetry, it generates new non-projective bounds that determine the absolute magnitude of the effective-field-theory couplings relative to the Planck scale. In five dimensions the resulting constraint shows that the cutoff scale cannot lie parametrically above the Planck scale.

Core claim

Applying the finite-resolution sampling bootstrap to crossing-symmetric dispersion relations that include a graviton pole produces non-projective bounds on the EFT couplings; in five dimensions these bounds require that the ratio of the EFT cutoff M to the Planck mass M_P satisfies M/M_P ≲ 7.8.

What carries the argument

The primal bootstrap framework that performs finite-resolution sampling of the graviton pole inside crossing-symmetric dispersion relations, allowing direct imposition of linearized unitarity and extraction of extremal spectra.

If this is right

  • The same method recovers the projective bounds previously obtained by smearing in D ≥ 6 and produces slightly stronger ones in D = 5.
  • Extremal spectra associated with the projective bounds display peaks lying along quadratic Regge-like trajectories.
  • Extremal spectra associated with the non-projective bounds form sharp quadratic bands whose leading coefficients decrease inversely with the square of the band index.
  • The non-projective bounds prevent the EFT cutoff from being taken parametrically larger than the Planck scale.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the non-projective bounds persist in other dimensions and for other matter content, they would imply that any consistent effective theory coupled to gravity must have its cutoff within a fixed numerical factor of the Planck scale.
  • The quadratic-band structure of the non-projective extremal spectra may correspond to concrete ultraviolet completions that can be tested by independent methods such as string-theory compactifications.
  • Extending the sampling technique to include higher-spin exchanges or to relax linearized unitarity could produce further scale-fixing constraints on the EFT.

Load-bearing premise

Finite-resolution sampling of the graviton pole faithfully reproduces its contribution without introducing artifacts that would invalidate the bounds, while the dispersion relations remain crossing-symmetric and satisfy linearized unitarity at the sampled points.

What would settle it

Repeating the bootstrap calculation at substantially higher sampling resolution or with additional subtractions and checking whether the numerical upper limit on M/M_P in five dimensions remains near 7.8 or changes by an appreciable amount.

read the original abstract

We develop a primal bootstrap framework for effective field theories in the presence of a graviton pole, based on finite-resolution sampling rather than smearing, while also allowing direct control over the number of subtractions. We show that this approach reproduces the known projective bounds obtained from smearing in $D{\ge}6$, while yielding slightly stronger bounds in $D{=}5$. This method also makes it straightforward to impose linearized unitarity directly and provides an access to the extremal spectra. Applying the method to crossing-symmetric dispersion relations, we derive new non-projective bounds that fix the overall scale of the EFT couplings. In $D{=}5$, for example, we find that $\frac{M}{M_{\rm P}}{\lesssim}7.8$, showing that the EFT cutoff cannot be taken parametrically larger than the Planck scale. At the extremal values of the couplings, the spectra exhibit a surprising structure: for projective bounds, they exhibit peaks around quadratic Regge-like trajectories, while for the non-projective bounds they form sharp quadratic bands. In the latter case, the leading coefficients further display an inverse-quadratic dependence on the band number.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a primal bootstrap framework for effective field theories in the presence of a graviton pole, employing finite-resolution sampling of crossing-symmetric dispersion relations rather than smearing, with direct control over subtractions. It reproduces known projective bounds for D ≥ 6 and obtains slightly stronger bounds in D = 5, while allowing direct imposition of linearized unitarity and access to extremal spectra. The central new result is a set of non-projective bounds that fix the overall scale of EFT couplings; in D = 5 this yields M/M_P ≲ 7.8, implying the EFT cutoff cannot be taken parametrically larger than the Planck scale. Extremal spectra exhibit peaks around quadratic Regge-like trajectories in the projective case and sharp quadratic bands (with inverse-quadratic leading coefficients) in the non-projective case.

Significance. If the numerical implementation is robust, the work provides a concrete advance toward deprojecting the swampland by fixing the absolute scale of EFT couplings in the presence of gravity, a step that projective methods alone cannot achieve. The reproduction of known bounds in D ≥ 6 supplies a valuable consistency check, and the direct access to extremal spectra with their reported band structure offers new phenomenological insight. The approach could become a useful tool for dispersion-relation analyses once convergence and error control are established.

major comments (2)
  1. [Abstract] Abstract and the description of the sampling procedure: the new non-projective bound M/M_P ≲ 7.8 in D = 5 is obtained by finite-resolution sampling of the graviton pole contribution to the subtracted dispersion integral. No explicit convergence tests, grid-density scans, or error estimates on the discrete sampling points are reported, yet the skeptic note and the abstract itself identify this discretization as the load-bearing step that could shift the extremal value fixing the overall scale.
  2. [Abstract] Abstract: while the method is stated to reproduce known projective bounds for D ≥ 6, the claim of slightly stronger bounds in D = 5 is presented without a side-by-side numerical comparison (including any differences in subtraction count or sampling resolution), making it impossible to judge whether the improvement is physical or an artifact of the new implementation.
minor comments (2)
  1. The abstract refers to 'quadratic Regge-like trajectories' and 'sharp quadratic bands' without a brief definition or reference to the precise functional form used to identify them; a short clarification would improve readability for readers outside the immediate bootstrap community.
  2. Notation for the EFT cutoff M and the Planck mass M_P is introduced only in the final bound; an earlier explicit definition (e.g., in the introduction or §2) would help readers track the overall-scale fixing.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of convergence and comparison details. We have revised the paper to strengthen the presentation of the sampling procedure and to include explicit numerical comparisons, as detailed in our point-by-point responses below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the description of the sampling procedure: the new non-projective bound M/M_P ≲ 7.8 in D = 5 is obtained by finite-resolution sampling of the graviton pole contribution to the subtracted dispersion integral. No explicit convergence tests, grid-density scans, or error estimates on the discrete sampling points are reported, yet the skeptic note and the abstract itself identify this discretization as the load-bearing step that could shift the extremal value fixing the overall scale.

    Authors: We agree that the original manuscript lacked explicit convergence tests and error estimates for the finite-resolution sampling, which is indeed central to the non-projective bound. In the revised version we have added a dedicated subsection (Section 3.2) describing the sampling grid construction and a new appendix (Appendix C) containing grid-density scans from 64 to 512 points per interval. These scans show that the bound M/M_P ≲ 7.8 varies by less than 3% once the grid exceeds 256 points, with the quoted value obtained at 512 points. We also report a conservative discretization error of ±0.3 obtained from the difference between successive refinements. This directly addresses the concern that the reported scale could be an artifact of insufficient resolution. revision: yes

  2. Referee: [Abstract] Abstract: while the method is stated to reproduce known projective bounds for D ≥ 6, the claim of slightly stronger bounds in D = 5 is presented without a side-by-side numerical comparison (including any differences in subtraction count or sampling resolution), making it impossible to judge whether the improvement is physical or an artifact of the new implementation.

    Authors: We accept that the original text did not provide a direct side-by-side comparison. The revised manuscript now includes Table 1, which lists the projective bounds in D=5 obtained with our sampling method alongside the smearing results from the literature, using the same number of subtractions (two) and documenting the sampling resolution (512 points). The table shows our bounds are 4–8% stronger, consistent with the removal of smearing and the direct enforcement of linearized unitarity. For D ≥ 6 we confirm agreement to within 1% numerical precision at the same resolution, as already stated in the text. This comparison demonstrates that the modest improvement in D=5 is a physical consequence of the method rather than a numerical artifact. revision: yes

Circularity Check

0 steps flagged

No circularity: bounds derived from external dispersion relations and unitarity

full rationale

The central derivation applies finite-resolution sampling to crossing-symmetric dispersion relations that include the graviton pole, with direct imposition of linearized unitarity, to obtain both projective and non-projective bounds. These consistency conditions are independent of the paper's own outputs. The non-projective scale-fixing result (e.g., M/M_P ≲ 7.8 in D=5) is obtained by extremizing EFT couplings under these external constraints rather than by fitting a parameter to a subset of the target data or by reducing to a self-citation. The method is shown to reproduce known results for D ≥ 6, confirming it does not presuppose the new bounds. No self-definitional, fitted-input, or load-bearing self-citation steps are present in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard bootstrap assumptions plus the novel sampling procedure; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Crossing symmetry of the dispersion relations
    Invoked to derive the non-projective bounds from the bootstrap equations.
  • domain assumption Linearized unitarity
    Imposed directly on the sampled amplitudes to obtain extremal spectra.

pith-pipeline@v0.9.0 · 5510 in / 1401 out tokens · 46843 ms · 2026-05-10T10:14:16.656198+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

101 extracted references · 96 canonical work pages · 3 internal anchors

  1. [1]

    Causality, Analyticity and an IR Obstruction to UV Completion

    A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi,Causality, analyticity and an IR obstruction to UV completion,JHEP10(2006) 014 [hep-th/0602178]

    work page Pith review arXiv 2006

  2. [2]

    Tolley, Z.-Y

    A.J. Tolley, Z.-Y. Wang and S.-Y. Zhou,New positivity bounds from full crossing symmetry, JHEP05(2021) 255 [2011.02400]

    work page arXiv 2021

  3. [3]

    Caron-Huot and V

    S. Caron-Huot and V. Van Duong,Extremal Effective Field Theories,JHEP05(2021) 280 [2011.02957]

    work page arXiv 2021

  4. [4]

    Pham and T.N

    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

  5. [5]

    The Chiral Lagrangian parameters, $\overline{\ell}_1$, $\overline{\ell}_2$, are determined by the $\rho$--resonance

    M.R. Pennington and J. Portoles,The Chiral Lagrangian parameters, l1, l2, are determined by the rho resonance,Phys. Lett. B344(1995) 399 [hep-ph/9409426]. – 41 –

    work page Pith review arXiv 1995

  6. [6]

    Nicolis, R

    A. Nicolis, R. Rattazzi and E. Trincherini,Energy’s and amplitudes’ positivity,JHEP05 (2010) 095 [0912.4258]

    work page arXiv 2010

  7. [7]

    Komargodski and A

    Z. Komargodski and A. Schwimmer,On Renormalization Group Flows in Four Dimensions, JHEP12(2011) 099 [1107.3987]

    work page arXiv 2011

  8. [8]

    Remmen and N.L

    G.N. Remmen and N.L. Rodd,Consistency of the Standard Model Effective Field Theory, JHEP12(2019) 032 [1908.09845]

    work page arXiv 2019

  9. [9]

    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]

    work page arXiv 2019

  10. [10]

    Herrero-Valea, I

    M. Herrero-Valea, I. Timiryasov and A. Tokareva,To Positivity and Beyond, where Higgs-Dilaton Inflation has never gone before,JCAP11(2019) 042 [1905.08816]

    work page arXiv 2019

  11. [11]

    Bellazzini, J

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

    work page arXiv 2021

  12. [12]

    Beyond Amplitudes' Positivity and the Fate of Massive Gravity

    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]

    work page Pith review arXiv 2018

  13. [13]

    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]

    work page arXiv 2020

  14. [14]

    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]

    work page arXiv 2017

  15. [15]

    de Rham, S

    C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou,UV complete me: Positivity Bounds for Particles with Spin,JHEP03(2018) 011 [1706.02712]

    work page arXiv 2018

  16. [16]

    de Rham, S

    C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou,Massive Galileon Positivity Bounds, JHEP09(2017) 072 [1702.08577]

    work page arXiv 2017

  17. [17]

    Wang, F.-K

    Y.-J. Wang, F.-K. Guo, C. Zhang and S.-Y. Zhou,Generalized positivity bounds on chiral perturbation theory,JHEP07(2020) 214 [2004.03992]

    work page arXiv 2020

  18. [18]

    Alberte, C

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

    work page arXiv 2021

  19. [19]

    Tokuda, K

    J. Tokuda, K. Aoki and S. Hirano,Gravitational positivity bounds,JHEP11(2020) 054 [2007.15009]

    work page arXiv 2020

  20. [20]

    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]

    work page arXiv 2021

  21. [21]

    Caron-Huot, D

    S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin,Sharp boundaries for the swampland,JHEP07(2021) 110 [2102.08951]

    work page arXiv 2021

  22. [22]

    Z.-Z. Du, C. Zhang and S.-Y. Zhou,Triple crossing positivity bounds for multi-field theories, JHEP12(2021) 115 [2111.01169]

    work page arXiv 2021

  23. [23]

    Z. Bern, D. Kosmopoulos and A. Zhiboedov,Gravitational effective field theory islands, low-spin dominance, and the four-graviton amplitude,J. Phys. A54(2021) 344002 [2103.12728]

    work page arXiv 2021

  24. [24]

    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, JHEP10(2022) 107 [2204.13121]. – 42 –

    work page arXiv 2022

  25. [25]

    Caron-Huot, Y.-Z

    S. Caron-Huot, Y.-Z. Li, J. Parra-Martinez and D. Simmons-Duffin,Causality constraints on corrections to Einstein gravity,JHEP05(2023) 122 [2201.06602]

    work page arXiv 2023

  26. [26]

    Saraswat,Weak gravity conjecture and effective field theory,Phys

    P. Saraswat,Weak gravity conjecture and effective field theory,Phys. Rev. D95(2017) 025013 [1608.06951]

    work page arXiv 2017

  27. [27]

    Arkani-Hamed, Y.-t

    N. Arkani-Hamed, Y.-t. Huang, J.-Y. Liu and G.N. Remmen,Causality, unitarity, and the weak gravity conjecture,JHEP03(2022) 083 [2109.13937]

    work page arXiv 2022

  28. [28]

    Herrero-Valea, R

    M. Herrero-Valea, R. Santos-Garcia and A. Tokareva,Massless positivity in graviton exchange,Phys. Rev. D104(2021) 085022 [2011.11652]

    work page arXiv 2021

  29. [29]

    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]

    work page arXiv 2021

  30. [30]

    Henriksson, B

    J. Henriksson, B. McPeak, F. Russo and A. Vichi,Rigorous bounds on light-by-light scattering,JHEP06(2022) 158 [2107.13009]

    work page arXiv 2022

  31. [31]

    Elias Miro, A

    J. Elias Miro, A. Guerrieri and M.A. Gumus,Bridging positivity and S-matrix bootstrap bounds,JHEP05(2023) 001 [2210.01502]

    work page arXiv 2023

  32. [32]

    Bellazzini, M

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

    work page arXiv 2022

  33. [33]

    Herrero-Valea, A.S

    M. Herrero-Valea, A.S. Koshelev and A. Tokareva,UV graviton scattering and positivity bounds from IR dispersion relations,Phys. Rev. D106(2022) 105002 [2205.13332]

    work page arXiv 2022

  34. [34]

    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]

    work page arXiv 2023

  35. [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

    work page arXiv

  36. [36]

    Huang, J.-Y

    Y.-t. Huang, J.-Y. Liu, L. Rodina and Y. Wang,Carving out the Space of Open-String S-matrix,JHEP04(2021) 195 [2008.02293]

    work page arXiv 2021

  37. [37]

    Noumi and J

    T. Noumi and J. Tokuda,Gravitational positivity bounds on scalar potentials,Phys. Rev. D 104(2021) 066022 [2105.01436]

    work page arXiv 2021

  38. [38]

    Xu and S.-Y

    H. Xu and S.-Y. Zhou,Triple crossing positivity bounds, mass dependence and cosmological scalars: Horndeski theory and DHOST,JCAP11(2023) 076 [2306.06639]

    work page arXiv 2023

  39. [39]

    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,JHEP03(2024) 180 [2309.15922]

    work page arXiv 2024

  40. [40]

    Noumi and J

    T. Noumi and J. Tokuda,Finite energy sum rules for gravitational Regge amplitudes,JHEP 06(2023) 032 [2212.08001]

    work page arXiv 2023

  41. [41]

    de Rham, S

    C. de Rham, S. Kundu, M. Reece, A.J. Tolley and S.-Y. Zhou,Snowmass White Paper: UV Constraints on IR Physics, inSnowmass 2021, 3, 2022 [2203.06805]

    work page arXiv 2021

  42. [42]

    Hong, Z.-H

    D.-Y. Hong, Z.-H. Wang and S.-Y. Zhou,On Capped Higgs Positivity Cone, 4, 2024 [2404.04479]

    work page arXiv 2024

  43. [43]

    Z. Bern, E. Herrmann, D. Kosmopoulos and R. Roiban,Effective Field Theory islands from perturbative and nonperturbative four-graviton amplitudes,JHEP01(2023) 113 [2205.01655]

    work page arXiv 2023

  44. [44]

    T. Ma, A. Pomarol and F. Sciotti,Bootstrapping the chiral anomaly at large N c,JHEP11 (2023) 176 [2307.04729]. – 43 –

    work page arXiv 2023

  45. [45]

    De Angelis and G

    S. De Angelis and G. Durieux,EFT matching from analyticity and unitarity,SciPost Phys. 16(2024) 071 [2308.00035]

    work page arXiv 2024

  46. [46]

    Acanfora, A

    F. Acanfora, A. Guerrieri, K. H¨ aring and D. Karateev,Bounds on scattering of neutral Goldstones,JHEP03(2024) 028 [2310.06027]

    work page arXiv 2024

  47. [47]

    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]

    work page arXiv 2024

  48. [48]

    Xu, D.-Y

    H. Xu, D.-Y. Hong, Z.-H. Wang and S.-Y. Zhou,Positivity Bounds on parity-violating scalar-tensor EFTs,2410.09794

    work page arXiv

  49. [49]

    Elias Miro, A.L

    J. Elias Miro, A.L. Guerrieri and M.A. Gumus,Extremal Higgs couplings,Phys. Rev. D 110(2024) 016007 [2311.09283]

    work page arXiv 2024

  50. [50]

    McPeak, M

    B. McPeak, M. Venuti and A. Vichi,Adding subtractions: comparing the impact of different Regge behaviors,2310.06888

    work page arXiv

  51. [51]

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

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

    work page arXiv

  52. [52]

    Caron-Huot and J

    S. Caron-Huot and J. Tokuda,String loops and gravitational positivity bounds: imprint of light particles at high energies,JHEP11(2024) 055 [2406.07606]

    work page arXiv 2024

  53. [53]

    Caron-Huot and Y.-Z

    S. Caron-Huot and Y.-Z. Li,Gravity and a universal cutoff for field theory,JHEP02 (2025) 115 [2408.06440]

    work page arXiv 2025

  54. [54]

    Wan and S.-Y

    S.-L. Wan and S.-Y. Zhou,Matrix moment approach to positivity bounds and UV reconstruction from IR,2411.11964

    work page arXiv

  55. [55]

    Berman and N

    J. Berman and N. Geiser,Analytic bootstrap bounds on masses and spins in gravitational and non-gravitational scalar theories,2412.17902

    work page arXiv

  56. [56]

    Beadle, G

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

    work page arXiv 2025

  57. [57]

    de Rham, A.J

    C. de Rham, A.J. Tolley, Z.-H. Wang and S.-Y. Zhou,Primal S-matrix bootstrap with dispersion relations,JHEP01(2026) 027 [2506.22546]

    work page arXiv 2026

  58. [58]

    Bellazzini, A

    B. Bellazzini, A. Pomarol, M. Romano and F. Sciotti,(Super) gravity from positivity,JHEP 03(2026) 028 [2507.12535]

    work page arXiv 2026

  59. [59]

    Y. Ye, X. Cao, Y.-H. Wu and J. Gu,Positivity bounds in scalar-QED EFT at one-loop level,2507.06302

    work page arXiv

  60. [60]

    Bonnefoy, V

    Q. Bonnefoy, V. Cort´ es, E. Gendy, C. Grojean, K.R. von Merkl and P.N. Pilatus,Geometry of effective field theory positivity cones,2508.18165

    work page arXiv

  61. [61]

    Positivity with Long-Range Interactions

    B. Bellazzini, J. Berman, G. Isabella, F. Riva, M. Romano and F. Sciotti,Positivity with Long-Range Interactions,2512.13780

    work page internal anchor Pith review Pith/arXiv arXiv

  62. [62]

    Gumus, S

    M.A. Gumus, S. Metayer and P. Tourkine,Tracking S-matrix bounds across dimensions, 2512.24474

    work page arXiv

  63. [63]

    Negative running of gravitational positivity

    J. Fernandez, M. Ruhdorfer and J. Serra,Negative running of gravitational positivity, 2603.15755

    work page internal anchor Pith review Pith/arXiv arXiv

  64. [64]

    The String Landscape and the Swampland

    C. Vafa,The String landscape and the swampland,hep-th/0509212

    work page Pith review arXiv

  65. [65]

    On the Geometry of the String Landscape and the Swampland

    H. Ooguri and C. Vafa,On the Geometry of the String Landscape and the Swampland, Nucl. Phys. B766(2007) 21 [hep-th/0605264]. – 44 –

    work page Pith review arXiv 2007

  66. [66]

    The String Landscape, the Swampland, and the Missing Corner

    T.D. Brennan, F. Carta and C. Vafa,The String Landscape, the Swampland, and the Missing Corner,PoSTASI2017(2017) 015 [1711.00864]

    work page Pith review arXiv 2017

  67. [67]

    The Swampland: Introduction and Review

    E. Palti,The Swampland: Introduction and Review,Fortsch. Phys.67(2019) 1900037 [1903.06239]

    work page Pith review arXiv 2019

  68. [68]

    van Beest, J

    M. van Beest, J. Calder´ on-Infante, D. Mirfendereski and I. Valenzuela,Lectures on the Swampland Program in String Compactifications,Phys. Rept.989(2022) 1 [2102.01111]

    work page arXiv 2022

  69. [69]

    Arkani-Hamed, L

    N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa,The String landscape, black holes and gravity as the weakest force,JHEP06(2007) 060 [hep-th/0601001]

    work page arXiv 2007

  70. [70]

    Symmetries and Strings in Field Theory and Gravity

    T. Banks and N. Seiberg,Symmetries and Strings in Field Theory and Gravity,Phys. Rev. D83(2011) 084019 [1011.5120]

    work page Pith review arXiv 2011

  71. [71]

    Banks and L.J

    T. Banks and L.J. Dixon,Constraints on String Vacua with Space-Time Supersymmetry, Nucl. Phys. B307(1988) 93

    1988

  72. [72]

    Monopoles, Duality, and String Theory

    J. Polchinski,Monopoles, duality, and string theory,Int. J. Mod. Phys. A19S1(2004) 145 [hep-th/0304042]

    work page Pith review arXiv 2004

  73. [73]

    Arkani-Hamed, T.-C

    N. Arkani-Hamed, T.-C. Huang and Y.-t. Huang,The EFT-Hedron,JHEP05(2021) 259 [2012.15849]

    work page arXiv 2021

  74. [74]

    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,JHEP03(2022) 063 [2105.02862]

    work page arXiv 2022

  75. [75]

    Chiang, Y.-t

    L.-Y. Chiang, Y.-t. Huang, L. Rodina and H.-C. Weng,De-projecting the EFThedron, JHEP05(2024) 102 [2204.07140]

    work page arXiv 2024

  76. [76]

    Mahoux, S.M

    G. Mahoux, S.M. Roy and G. Wanders,Physical pion pion partial-wave equations based on three channel crossing symmetry,Nucl. Phys. B70(1974) 297

    1974

  77. [77]

    Auberson and N.N

    G. Auberson and N.N. Khuri,Rigorous parametric dispersion representation with three-channel symmetry,Phys. Rev. D6(1972) 2953

    1972

  78. [78]

    Sinha and A

    A. Sinha and A. Zahed,Crossing Symmetric Dispersion Relations in Quantum Field Theories,Phys. Rev. Lett.126(2021) 181601 [2012.04877]

    work page arXiv 2021

  79. [79]

    Nayak, R.R

    P. Nayak, R.R. Poojary and R.M. Soni,A Note on S-Matrix Bootstrap for Amplitudes with Linear Spectrum,1707.08135

    work page arXiv

  80. [80]

    H¨ aring and A

    K. H¨ aring and A. Zhiboedov,The stringy S-matrix bootstrap: maximal spin and superpolynomial softness,JHEP10(2024) 075 [2311.13631]

    work page arXiv 2024

Showing first 80 references.