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arxiv: 2606.19420 · v1 · pith:U4WIFMDTnew · submitted 2026-06-17 · ✦ hep-th · hep-lat· hep-ph

Bootstrapping Pion Form Factors at Large N

Jan Albert , Dilara Kosva , Leonardo Rastelli This is my paper

Pith reviewed 2026-06-26 19:39 UTC · model grok-4.3

classification ✦ hep-th hep-lathep-ph
keywords large N QCDpion form factorbootstrapchiral Lagrangianasymptotic freedomvector currentpion decay constantcharge radius
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The pith

Large-N bootstrap from analyticity and unitarity constrains low-energy pion form factor coefficients and yields bounds on the decay constant and charge radius.

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

The paper applies bootstrap techniques to a mixed system of the vector-current two-point function, the pion vector form factor, and pion scattering in the chiral limit of large-N QCD. These observables are meromorphic, with spectral data fixed by unitarity, crossing symmetry, and Regge boundedness. From analyticity, unitarity, and Brodsky-Farrar scaling, universal bounds are derived on the low-energy coefficients of the form factor. Adding perturbative ultraviolet behavior at a finite scale produces concrete limits on the pion decay constant, a lower bound on the scale of asymptotic freedom extracted from lattice data, and a constraint on the pion charge radius, shrinking the space of allowed chiral Lagrangians toward the region occupied by large-N QCD.

Core claim

At large N the vector-current two-point function, pion vector form factor, and pion scattering amplitude are meromorphic, with spectral data constrained by unitarity, crossing symmetry, and Regge boundedness. Rigorous bounds on low-energy form-factor coefficients follow from analyticity, unitarity, and Brodsky-Farrar scaling. Imposing perturbative ultraviolet behavior at a finite scale bounds the pion decay constant, converts a large-N lattice measurement into a lower bound on the onset of asymptotic freedom, and constrains the pion charge radius, thereby restricting the allowed parameter space of chiral Lagrangians.

What carries the argument

The mixed bootstrap system of the vector-current two-point function, pion vector form factor, and pion scattering amplitude in the chiral limit, required to be meromorphic with spectral data fixed by unitarity, crossing symmetry, and Regge boundedness.

If this is right

  • Low-energy coefficients in the expansion of the pion vector form factor satisfy rigorous upper and lower bounds.
  • The pion decay constant is bounded from above when perturbative QCD input is supplied at a finite scale.
  • A large-N lattice value for a low-energy observable can be converted into a lower bound on the scale where asymptotic freedom begins.
  • The pion charge radius is constrained by the same inputs.
  • The allowed region of parameter space for chiral Lagrangians is reduced toward the values realized by large-N QCD.

Where Pith is reading between the lines

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

  • The same mixed-system bootstrap could be applied to other mesons or currents in large-N gauge theories.
  • Higher-precision lattice runs at increasing N could directly test the numerical bounds on the decay constant.
  • Gauge-invariant local probes of this type may serve as canonical bridges between bootstrap methods and microscopic Lagrangians in other confining theories.
  • The derived constraints could be used to estimate the radius of convergence of the chiral expansion at large N.

Load-bearing premise

The observables are meromorphic at large N, with all spectral data fixed by unitarity, crossing symmetry, and Regge boundedness.

What would settle it

A lattice computation at sufficiently large but finite N that places the pion decay constant outside the derived numerical bound would falsify the phenomenological use of the bounds.

read the original abstract

We initiate a bootstrap study of pion form factors in large $N$ QCD. We consider the mixed system of the vector-current two-point function, the pion vector form factor, and the pion scattering amplitude in the chiral limit. At large $N$ these observables are meromorphic, with spectral data constrained by unitarity, crossing symmetry, and Regge boundedness. We obtain bounds of two kinds. The first are rigorous and universal: from analyticity, unitarity and the asymptotic Brodsky-Farrar scaling, we constrain low-energy form-factor coefficients. The second are more phenomenological, of the Shifman-Vainshtein-Zakharov type: feeding in the perturbative ultraviolet behavior at a finite scale lets us bound the pion decay constant, convert a large $N$ lattice measurement into a lower bound on the scale at which asymptotic freedom sets in, and constrain the pion charge radius. Combining these inputs, the space of allowed chiral Lagrangians shrinks toward the region where large $N$ QCD is expected to sit. Our results illustrate how local gauge-invariant probes provide a canonical bridge between the hadronic bootstrap and the microscopic QCD Lagrangian.

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 paper initiates a bootstrap analysis of pion form factors in large-N QCD by studying the mixed system of the vector-current two-point function, the pion vector form factor, and the pion scattering amplitude in the chiral limit. At large N these observables are taken to be meromorphic, with spectral data fixed by unitarity, crossing symmetry, and Regge boundedness. From analyticity, unitarity, and Brodsky-Farrar scaling the authors derive rigorous universal bounds on low-energy form-factor coefficients; incorporating perturbative UV behavior at a finite scale then yields SVZ-style bounds on f_π, a lower bound on the onset of asymptotic freedom from lattice data, and a constraint on the pion charge radius, thereby shrinking the allowed space of chiral Lagrangians.

Significance. If the central claims hold, the work supplies a concrete bridge between large-N bootstrap methods and microscopic QCD parameters, extending analyticity-based techniques to mixed correlators and producing falsifiable constraints on low-energy constants. The combination of universal bounds with phenomenological UV matching is a strength, though its impact depends on the rigor with which the mixed-system constraints are shown to close.

major comments (2)
  1. [abstract / mixed-system paragraph] Abstract and the paragraph describing the mixed system: the assertion that spectral data are 'fully constrained by unitarity, crossing symmetry, and Regge boundedness' is load-bearing for the universal bounds; the manuscript must demonstrate explicitly that the coupled functional equations for the three channels admit no residual continuous parameters (e.g., from multi-particle cuts or non-linear trajectories) once these principles are imposed, otherwise the reported bounds on low-energy coefficients reduce to conditional statements rather than universal results.
  2. [phenomenological section (SVZ matching)] The transition from rigorous bounds to the SVZ-style extractions of f_π and the AF scale relies on feeding in perturbative UV behavior at a finite matching scale; the manuscript should quantify the sensitivity of the extracted bounds to the choice of this scale and provide a clear error estimate or stability analysis, as this choice directly affects the phenomenological claims.
minor comments (2)
  1. Notation for the mixed system (vector two-point function, form factor, scattering amplitude) should be introduced with explicit definitions of the relevant residues and pole locations to facilitate verification of the unitarity and crossing constraints.
  2. [abstract] The abstract mentions 'asymptotic Brodsky-Farrar scaling' without a reference or brief derivation; a short footnote or citation would clarify the precise asymptotic form used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points regarding the rigor of our claims on spectral constraints and the stability of phenomenological extractions. We address each major comment below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: [abstract / mixed-system paragraph] Abstract and the paragraph describing the mixed system: the assertion that spectral data are 'fully constrained by unitarity, crossing symmetry, and Regge boundedness' is load-bearing for the universal bounds; the manuscript must demonstrate explicitly that the coupled functional equations for the three channels admit no residual continuous parameters (e.g., from multi-particle cuts or non-linear trajectories) once these principles are imposed, otherwise the reported bounds on low-energy coefficients reduce to conditional statements rather than universal results.

    Authors: We agree that an explicit demonstration is necessary to establish the universality of the bounds. While the large-N meromorphicity assumption together with unitarity, crossing symmetry, and Regge boundedness is used to fix the spectral data in the mixed system, the manuscript does not contain a dedicated derivation showing the absence of residual continuous parameters in the coupled functional equations. In the revised version we will add a subsection (likely in Section 2) that writes the coupled dispersion relations for the vector two-point function, the pion form factor, and the scattering amplitude, and argues that the only remaining freedom consists of discrete choices of pole locations and residues consistent with the imposed principles, with no continuous parameters surviving from multi-particle cuts or trajectory nonlinearities. This will make the universal bounds unconditional as stated. revision: yes

  2. Referee: [phenomenological section (SVZ matching)] The transition from rigorous bounds to the SVZ-style extractions of f_π and the AF scale relies on feeding in perturbative UV behavior at a finite matching scale; the manuscript should quantify the sensitivity of the extracted bounds to the choice of this scale and provide a clear error estimate or stability analysis, as this choice directly affects the phenomenological claims.

    Authors: We concur that the choice of matching scale introduces a systematic uncertainty that should be quantified. The current manuscript presents results for a single representative scale without a stability analysis. In the revised version we will add a dedicated paragraph (in the phenomenological section) that varies the matching scale over a physically motivated interval, recomputes the extracted bounds on f_π, the AF onset scale, and the charge radius, and reports the resulting variation as an error estimate. This will make the sensitivity explicit and strengthen the phenomenological conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from stated principles

full rationale

The paper states that rigorous universal bounds follow from analyticity, unitarity, crossing symmetry, Regge boundedness and Brodsky-Farrar scaling applied to the mixed system of vector two-point function, pion form factor and pion scattering (abstract). Phenomenological SVZ-style bounds are obtained by additionally feeding in perturbative UV behavior at a finite scale. No equations or steps are quoted that reduce a claimed prediction to a fitted parameter by construction, rename a known result, or make a load-bearing claim rest solely on a self-citation whose content is unverified. The meromorphicity and spectral constraints are presented as following directly from the listed standard large-N properties without additional internal fitting or self-referential closure. This is the normal case of a bootstrap analysis whose central claims remain independent of the outputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; ledger entries are therefore minimal and provisional. The central claim rests on the assumption that large-N observables are meromorphic and that perturbative UV input can be matched at a finite scale without introducing uncontrolled errors.

free parameters (1)
  • UV matching scale
    The scale at which perturbative QCD behavior is inserted to obtain SVZ-type bounds; its value is chosen by hand and affects the numerical limits on f_pi and charge radius.
axioms (2)
  • domain assumption Observables are meromorphic at large N with poles fixed by unitarity, crossing, and Regge boundedness
    Stated in the abstract as the starting point for the mixed system analysis.
  • domain assumption Brodsky-Farrar scaling holds asymptotically
    Used to constrain low-energy coefficients from high-energy behavior.

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discussion (0)

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Reference graph

Works this paper leans on

126 extracted references · 1 canonical work pages

  1. [1]

    Albert and L

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

    arXiv 2022

  2. [2]

    Albert and L

    J. Albert and L. Rastelli,Bootstrapping pions at large N. Part II. Background gauge fields and the chiral anomaly,JHEP09(2024) 039, arXiv:2307.01246 [hep-th]

    arXiv 2024

  3. [3]

    Albert, J

    J. Albert, J. Henriksson, L. Rastelli, and A. Vichi,Bootstrapping mesons at large N: Regge trajectory from spin-two maximization,JHEP09(2024) 172, arXiv:2312.15013 [hep-th]

    arXiv 2024

  4. [4]

    Fernandez, A

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

    arXiv 2023

  5. [5]

    T. Ma, A. Pomarol, and F. Sciotti,Bootstrapping the chiral anomaly at large Nc,JHEP11 (2023) 176, arXiv:2307.04729 [hep-th]

    arXiv 2023

  6. [6]

    Li,Effective field theory bootstrap, large-NχPT and holographic QCD,JHEP01 (2024) 072, arXiv:2310.09698 [hep-th]

    Y.-Z. Li,Effective field theory bootstrap, large-NχPT and holographic QCD,JHEP01 (2024) 072, arXiv:2310.09698 [hep-th]

    arXiv 2024

  7. [7]

    Eckner, F

    C. Eckner, F. Figueroa, and P. Tourkine,On the number of Regge trajectories for dual amplitudes,JHEP02(2025) 103, arXiv:2405.21057 [hep-th]

    arXiv 2025

  8. [8]

    Berman,Analytic bounds on the spectrum of crossing symmetric S-matrices,JHEP08 (2025) 066, arXiv:2410.01914 [hep-th]

    J. Berman,Analytic bounds on the spectrum of crossing symmetric S-matrices,JHEP08 (2025) 066, arXiv:2410.01914 [hep-th]

    arXiv 2025

  9. [9]

    Martin,Scattering Theory: Unitarity, Analyticity and Crossing, pp

    A. Martin,Scattering Theory: Unitarity, Analyticity and Crossing, pp. 1–117. Springer Berlin Heidelberg, Berlin, Heidelberg, 1969.https://doi.org/10.1007/BFb0101044

    work page doi:10.1007/bfb0101044 1969

  10. [10]

    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

  11. [11]

    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, arXiv:hep-ph/9410302

    Pith/arXiv arXiv 1995

  12. [12]

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

    Pith/arXiv arXiv 1995

  13. [13]

    Comellas, J

    J. Comellas, J. I. Latorre, and J. Taron,Constraints on chiral perturbation theory parameters from QCD inequalities,Phys. Lett. B360(1995) 109–116, arXiv:hep-ph/9507258

    Pith/arXiv arXiv 1995

  14. [14]

    Dita,Positivity constraints on chiral perturbation theory pion pion scattering amplitudes,Phys

    P. Dita,Positivity constraints on chiral perturbation theory pion pion scattering amplitudes,Phys. Rev. D59(1999) 094007, arXiv:hep-ph/9809568

    Pith/arXiv arXiv 1999

  15. [15]

    Adams, N

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

    Pith/arXiv arXiv 2006

  16. [16]

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

    arXiv 2021

  17. [17]

    Caron-Huot and V

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

    arXiv 2021

  18. [18]

    Arkani-Hamed, T.-C

    N. Arkani-Hamed, T.-C. Huang, and Y.-T. Huang,The EFT-Hedron,JHEP05(2021) 259, arXiv:2012.15849 [hep-th]. – 38 –

    arXiv 2021

  19. [19]

    Caron-Huot, D

    S. Caron-Huot, D. Mazáč, L. Rastelli, and D. Simmons-Duffin,Sharp boundaries for the swampland,Journal of High Energy Physics2021(July, 2021)

    2021

  20. [20]

    Caron-Huot, D

    S. Caron-Huot, D. Mazac, L. Rastelli, and D. Simmons-Duffin,AdS bulk locality from sharp CFT bounds,JHEP11(2021) 164, arXiv:2106.10274 [hep-th]

    arXiv 2021

  21. [21]

    Z. Bern, D. Kosmopoulos, and A. Zhiboedov,Gravitational effective field theory islands, low-spin dominance, and the four-graviton amplitude,Journal of Physics A: Mathematical and Theoretical54(Aug., 2021) 344002

    2021

  22. [22]

    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, arXiv:2201.06602 [hep-th]

    arXiv 2023

  23. [23]

    Caron-Huot, Y.-Z

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

    arXiv 2023

  24. [24]

    Henriksson, B

    J. Henriksson, B. McPeak, F. Russo, and A. Vichi,Rigorous bounds on light-by-light scattering,Journal of High Energy Physics2022(June, 2022)

    2022

  25. [25]

    Henriksson, B

    J. Henriksson, B. McPeak, F. Russo, and A. Vichi,Bounding violations of the weak gravity conjecture,Journal of High Energy Physics2022(Aug., 2022)

    2022

  26. [26]

    Berman, H

    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]

    arXiv 2024

  27. [27]

    McPeak, M

    B. McPeak, M. Venuti, and A. Vichi,Adding subtractions: comparing the impact of different Regge behaviors,SciPost Phys.20(2026) 085, arXiv:2310.06888 [hep-th]

    arXiv 2026

  28. [28]

    Eckner, F

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

    arXiv 2025

  29. [29]

    Bertucci, J

    F. Bertucci, J. Henriksson, B. McPeak, S. Ricossa, F. Riva, and A. Vichi,Positivity bounds on massive vectors,JHEP12(2024) 051, arXiv:2402.13327 [hep-th]

    arXiv 2024

  30. [30]

    Berman and H

    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]

    arXiv 2024

  31. [31]

    Berman, H

    J. Berman, H. Elvang, N. Geiser, and L. L. Lin,Bootstrapping extremal scalar amplitudes with and without supersymmetry,JHEP05(2026) 149, arXiv:2412.13368 [hep-th]

    arXiv 2026

  32. [32]

    Albert, W

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

    arXiv 2025

  33. [33]

    Häring and A

    K. Häring and A. Zhiboedov,The stringy s-matrix bootstrap: maximal spin and superpolynomial softness,Journal of High Energy Physics2024(Oct., 2024)

    2024

  34. [34]

    Z.-Y. Dong, T. Ma, A. Pomarol, and F. Sciotti,Bootstrapping the chiral-gravitational anomaly,JHEP05(2025) 114, arXiv:2411.14422 [hep-th]

    arXiv 2025

  35. [35]

    Berman, H

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

    arXiv 2025

  36. [36]

    Bellazzini, A

    B. Bellazzini, A. Pomarol, M. Romano, and F. Sciotti,(Super) gravity from positivity, JHEP03(2026) 028, arXiv:2507.12535 [hep-th]

    arXiv 2026

  37. [37]

    Bellazzini, J

    B. Bellazzini, J. Berman, G. Isabella, F. Riva, M. Romano, and F. Sciotti,Positivity with Long-Range Interactions,arXiv:2512.13780 [hep-th]. – 39 –

    Pith/arXiv arXiv

  38. [38]

    Elvang, A

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

    Pith/arXiv arXiv

  39. [39]

    Boisvert, W

    M. Boisvert, W. Knop, and L. Rastelli,Where is tree-level heterotic string theory?, arXiv:2606.09980 [hep-th]

    Pith/arXiv arXiv

  40. [40]

    Albert, J

    J. Albert, J. Henriksson, L. Rastelli, and A. Vichi. To appear

  41. [41]

    S. J. Brodsky and G. R. Farrar,Scaling Laws at Large Transverse Momentum,Phys. Rev. Lett.31(1973) 1153–1156

    1973

  42. [42]

    G. P. Lepage and S. J. Brodsky,Exclusive Processes in Perturbative Quantum Chromodynamics,Phys. Rev. D22(1980) 2157

    1980

  43. [43]

    Bocchia and A

    S. Bocchia and A. Vichi,Primal Bootstrap for Pion Scattering at Large-N, arXiv:2606.14676 [hep-th]

    arXiv

  44. [44]

    K. G. Wilson,Nonlagrangian models of current algebra,Phys. Rev.179(1969) 1499–1512

    1969

  45. [45]

    D. J. Gross and F. Wilczek,Ultraviolet Behavior of Nonabelian Gauge Theories,Phys. Rev. Lett.30(1973) 1343–1346

    1973

  46. [46]

    H. D. Politzer,Reliable Perturbative Results for Strong Interactions?,Phys. Rev. Lett.30 (1973) 1346–1349

    1973

  47. [47]

    Albert and A

    J. Albert and A. Homrich,Imprints of asymptotic freedom on confining strings, arXiv:2602.15097 [hep-th]

    arXiv

  48. [48]

    Karateev, S

    D. Karateev, S. Kuhn, and J. Penedones,Bootstrapping Massive Quantum Field Theories, JHEP07(2020) 035, arXiv:1912.08940 [hep-th]

    arXiv 2020

  49. [49]

    M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov,QCD and Resonance Physics. Theoretical Foundations,Nucl. Phys. B147(1979) 385–447

    1979

  50. [50]

    M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov,QCD and Resonance Physics: Applications,Nucl. Phys. B147(1979) 448–518

    1979

  51. [51]

    Davier, A

    M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang,A new evaluation of the hadronic vacuum polarisation contributions to the muon anomalous magnetic moment and to α(m2 Z),Eur. Phys. J. C80(2020) 241, arXiv:1908.00921 [hep-ph]. [Erratum: Eur.Phys.J.C 80, 410 (2020)]

    arXiv 2020

  52. [52]

    L. J. Reinders, H. Rubinstein, and S. Yazaki,Hadron Properties from QCD Sum Rules, Phys. Rept.127(1985) 1

    1985

  53. [53]

    Colangelo and A

    P. Colangelo and A. Khodjamirian,QCD sum rules, a modern perspective, arXiv:hep-ph/0010175

    Pith/arXiv arXiv

  54. [54]

    Gubler and D

    P. Gubler and D. Satow,Recent progress in qcd condensate evaluations and sum rules, Progress in Particle and Nuclear Physics106(May, 2019) 1–67

    2019

  55. [55]

    Caron-Huot, A

    S. Caron-Huot, A. Pokraka, and Z. Zahraee,Two-point sum-rules in three-dimensional Yang-Mills theory,JHEP01(2024) 195, arXiv:2309.04472 [hep-th]

    arXiv 2024

  56. [56]

    He and M

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

    arXiv 2024

  57. [57]

    He and M

    Y. He and M. Kruczenski,Gauge Theory Bootstrap: Pion amplitudes and low energy parameters,arXiv:2403.10772 [hep-th]. – 40 –

    arXiv

  58. [58]

    He and M

    Y. He and M. Kruczenski,The Gauge Theory Bootstrap: Predicting pion dynamics from QCD,arXiv:2505.19332 [hep-th]

    arXiv

  59. [59]

    H. Chen, A. L. Fitzpatrick, and D. Karateev,Form factors and spectral densities from Lightcone Conformal Truncation,JHEP04(2022) 109, arXiv:2107.10285 [hep-th]

    arXiv 2022

  60. [60]

    H. Chen, A. L. Fitzpatrick, and D. Karateev,Bootstrapping 2dϕ4 theory with Hamiltonian truncation data,JHEP02(2022) 146, arXiv:2107.10286 [hep-th]

    arXiv 2022

  61. [61]

    Correia, J

    M. Correia, J. Penedones, and A. Vuignier,Injecting the UV into the bootstrap: Ising Field Theory,JHEP08(2023) 108, arXiv:2212.03917 [hep-th]

    arXiv 2023

  62. [62]

    Cordova, M

    L. Cordova, M. Correia, A. Georgoudis, and A. Vuignier,The O(N) monolith reloaded: sum rules and Form Factor Bootstrap,JHEP01(2024) 093, arXiv:2311.03031 [hep-th]

    arXiv 2024

  63. [63]

    M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira,The S-matrix bootstrap. Part I: QFT in AdS,JHEP11(2017) 133, arXiv:1607.06109 [hep-th]

    Pith/arXiv arXiv 2017

  64. [64]

    M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira,The S-matrix bootstrap II: two dimensional amplitudes,JHEP11(2017) 143, arXiv:1607.06110 [hep-th]

    Pith/arXiv arXiv 2017

  65. [65]

    M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira,The S-matrix bootstrap. Part III: higher dimensional amplitudes,JHEP12(2019) 040, arXiv:1708.06765 [hep-th]

    Pith/arXiv arXiv 2019

  66. [66]

    Homrich, J

    A. Homrich, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira,The S-matrix Bootstrap IV: Multiple Amplitudes,JHEP11(2019) 076, arXiv:1905.06905 [hep-th]

    arXiv 2019

  67. [67]

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

    Pith/arXiv arXiv 2019

  68. [68]

    A. L. Guerrieri, J. Penedones, and P. Vieira,S-matrix bootstrap for effective field theories: massless pions,JHEP06(2021) 088, arXiv:2011.02802 [hep-th]

    arXiv 2021

  69. [69]

    Guerrieri, K

    A. Guerrieri, K. Häring, and N. Su,From data to the analytic S-matrix: A Bootstrap fit of the pion scattering amplitude,SciPost Phys.20(2026) 034, arXiv:2410.23333 [hep-th]

    arXiv 2026

  70. [70]

    Bonanno, M

    C. Bonanno, M. García Pérez, A. González-Arroyo, K.-I. Ishikawa, and M. Okawa, Non-perturbative determination of meson masses and low-energy constants in large-N QCD,JHEP12(2025) 096, arXiv:2508.05446 [hep-lat]

    arXiv 2025

  71. [71]

    S. R. Coleman and E. Witten,Chiral Symmetry Breakdown in Large N Chromodynamics, Phys. Rev. Lett.45(1980) 100

    1980

  72. [72]

    Weinberg,The Quantum theory of fields

    S. Weinberg,The Quantum theory of fields. Vol. 1: Foundations. Cambridge University Press, 6, 2005

    2005

  73. [73]

    ’t Hooft,A Planar Diagram Theory for Strong Interactions,Nucl

    G. ’t Hooft,A Planar Diagram Theory for Strong Interactions,Nucl. Phys. B72(1974) 461

    1974

  74. [74]

    Witten,Baryons in the 1/n Expansion,Nucl

    E. Witten,Baryons in the 1/n Expansion,Nucl. Phys. B160(1979) 57–115

    1979

  75. [75]

    Okubo,Phi meson and unitary symmetry model,Phys

    S. Okubo,Phi meson and unitary symmetry model,Phys. Lett.5(1963) 165–168

    1963

  76. [76]

    Zweig,An SU(3) model for strong interaction symmetry and its breaking

    G. Zweig,An SU(3) model for strong interaction symmetry and its breaking. Version 2, pp. 22–101. Hadronic Press, Nonantum, MA, 2, 1964

    1964

  77. [77]

    Iizuka,Systematics and phenomenology of meson family,Prog

    J. Iizuka,Systematics and phenomenology of meson family,Prog. Theor. Phys. Suppl.37 (1966) 21–34

    1966

  78. [78]

    Gasser and H

    J. Gasser and H. Leutwyler,Chiral Perturbation Theory to One Loop,Annals Phys.158 (1984) 142. – 41 –

    1984

  79. [79]

    Gasser and H

    J. Gasser and H. Leutwyler,Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark,Nucl. Phys. B250(1985) 465–516

    1985

  80. [80]

    Kaiser and H

    R. Kaiser and H. Leutwyler,Large N(c) in chiral perturbation theory,Eur. Phys. J. C17 (2000) 623–649, arXiv:hep-ph/0007101

    Pith/arXiv arXiv 2000

Showing first 80 references.