Abstract
We use the numerical S-matrix bootstrap method to obtain bounds on the two leading Wilson coefficients (or low energy constants) of the chiral lagrangian controlling the low-energy dynamics of massless pions. This provides a proof of concept that the numerical S-matrix bootstrap can be used to derive non-perturbative bounds on massless EFTs in more than two spacetime dimensions.
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A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE].
B. Bellazzini, Softness and amplitudes’ positivity for spinning particles, JHEP 02 (2017) 034 [arXiv:1605.06111] [INSPIRE].
L. Vecchi, Causal versus analytic constraints on anomalous quartic gauge couplings, JHEP 11 (2007) 054 [arXiv:0704.1900] [INSPIRE].
A. Nicolis, R. Rattazzi and E. Trincherini, Energy’s and amplitudes’ positivity, JHEP 05 (2010) 095 [Erratum ibid. 11 (2011) 128] [arXiv:0912.4258] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, Positivity bounds for scalar field theories, Phys. Rev. D 96 (2017) 081702 [arXiv:1702.06134] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, UV complete me: positivity bounds for particles with spin, JHEP 03 (2018) 011 [arXiv:1706.02712] [INSPIRE].
N. Arkani-Hamed, Positive geometry of effective field theory 1, lecture at the CERN winter school on supergravity, strings and gauge theory, CERN, Geneva, Switzerland (2019).
N. Arkani-Hamed, Positive geometry of effective field theory 2, lecture at the CERN winter school on supergravity, strings and gauge theory, CERN, Geneva, Switzerland (2019).
N. Arkani-Hamed, Positive geometry of effective field theory 3, lecture at the CERN winter school on supergravity, strings and gauge theory, CERN, Geneva, Switzerland (2019).
N. Arkani-Hamed, Positive geometry of effective field theory 4, lecture at the CERN winter school on supergravity, strings and gauge theory, CERN, Geneva, Switzerland (2019).
B. Bellazzini, J. Elias Miró, R. Rattazzi, M. Riembau and F. Riva, Positive moments for scattering amplitudes, arXiv:2011.00037 [INSPIRE].
A.V. Manohar and V. Mateu, Dispersion relation bounds for ππ scattering, Phys. Rev. D 77 (2008) 094019 [arXiv:0801.3222] [INSPIRE].
Y.-J. Wang, F.-K. Guo, C. Zhang and S.-Y. Zhou, Generalized positivity bounds on chiral perturbation theory, JHEP 07 (2020) 214 [arXiv:2004.03992] [INSPIRE].
A.J. Tolley, Z.-Y. Wang and S.-Y. Zhou, New positivity bounds from full crossing symmetry, JHEP 05 (2021) 255 [arXiv:2011.02400] [INSPIRE].
S. Weinberg, Phenomenological Lagrangians, Physica A 96 (1979) 327 [INSPIRE].
J. Gasser and H. Leutwyler, Chiral perturbation theory to one loop, Annals Phys. 158 (1984) 142 [INSPIRE].
J. Bijnens, G. Colangelo, G. Ecker, J. Gasser and M.E. Sainio, Elastic ππ scattering to two loops, Phys. Lett. B 374 (1996) 210 [hep-ph/9511397] [INSPIRE].
J. Bijnens, G. Colangelo, G. Ecker, J. Gasser and M.E. Sainio, Pion-pion scattering at low energy, Nucl. Phys. B 508 (1997) 263 [Erratum ibid. 517 (1998) 639] [hep-ph/9707291] [INSPIRE].
L. Girlanda, M. Knecht, B. Moussallam and J. Stern, Comment on the prediction of two loop standard chiral perturbation theory for low-energy ππ scattering, Phys. Lett. B 409 (1997) 461 [hep-ph/9703448] [INSPIRE].
G. Amoros, J. Bijnens and P. Talavera, Kℓ4 form-factors and π − π scattering, Nucl. Phys. B 585 (2000) 293 [Erratum ibid. 598 (2001) 665] [hep-ph/0003258] [INSPIRE].
G. Colangelo, J. Gasser and H. Leutwyler, ππ scattering, Nucl. Phys. B 603 (2001) 125 [hep-ph/0103088] [INSPIRE].
T.N. Pham and T.N. Truong, Evaluation of the derivative quartic terms of the meson chiral Lagrangian from forward dispersion relation, Phys. Rev. D 31 (1985) 3027 [INSPIRE].
J. Elias Miró, A.L. Guerrieri, A. Hebbar, J. Penedones and P. Vieira, Flux tube S-matrix bootstrap, Phys. Rev. Lett. 123 (2019) 221602 [arXiv:1906.08098] [INSPIRE].
A.L. Guerrieri, J. Penedones and P. Vieira, Where is string theory?, work in progress.
A.L. Guerrieri, J. Penedones and P. Vieira, Bootstrapping QCD using pion scattering amplitudes, Phys. Rev. Lett. 122 (2019) 241604 [arXiv:1810.12849] [INSPIRE].
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap. Part III. Higher dimensional amplitudes, JHEP 12 (2019) 040 [arXiv:1708.06765] [INSPIRE].
S.O. Aks, Proof that scattering implies production in quantum field theory, J. Math. Phys. 6 (1965) 516.
A.J. Dragt, Amount of four-particle production required in S-matrix theory, Phys. Rev. 156 (1967) 1588.
M. Correia, A. Sever and A. Zhiboedov, An analytical toolkit for the S-matrix bootstrap, arXiv:2006.08221 [INSPIRE].
S.D. Protopopescu et al., ππ partial wave analysis from reactions π+p → π+π−∆++ and π+p → K+K−∆++ at 7.1 GeV/c, Phys. Rev. D 7 (1973) 1279 [INSPIRE].
M.J. Losty et al., A study of π−π− scattering from π−p interactions at 3.93 GeV/c, Nucl. Phys. B 69 (1974) 185 [INSPIRE].
G. Grayer et al., High statistics study of the reaction π−p → π−π+n: apparatus, method of analysis, and general features of results at 17 GeV/c, Nucl. Phys. B 75 (1974) 189 [INSPIRE].
P. Estabrooks and A.D. Martin, ππ phase shift analysis below the \( K\overline{K} \) threshold, Nucl. Phys. B 79 (1974) 301 [INSPIRE].
W. Hoogland et al., Measurement and analysis of the π+π+ system produced at small momentum transfer in the reaction π+p → π+π+n at 12.5 GeV, Nucl. Phys. B 126 (1977) 109 [INSPIRE].
NA48/2 collaboration, Precise tests of low energy QCD from Ke4 decay properties, Eur. Phys. J. C 70 (2010) 635 [INSPIRE].
R. Garcia-Martin, R. Kaminski, J.R. Pelaez, J. Ruiz de Elvira and F.J. Yndurain, The pion-pion scattering amplitude. IV: improved analysis with once subtracted Roy-like equations up to 1100 MeV, Phys. Rev. D 83 (2011) 074004 [arXiv:1102.2183] [INSPIRE].
W. Landry and D. Simmons-Duffin, Scaling the semidefinite program solver SDPB, arXiv:1909.09745 [INSPIRE].
L. Córdova, Y. He, M. Kruczenski and P. Vieira, The O(N) S-matrix monolith, JHEP 04 (2020) 142 [arXiv:1909.06495] [INSPIRE].
A.L. Guerrieri, A. Homrich and P. Vieira, Dual S-matrix bootstrap. Part I. 2D theory, JHEP 11 (2020) 084 [arXiv:2008.02770] [INSPIRE].
Y. He and M. Kruczenski, S-matrix bootstrap in 3 + 1 dimensions: regularization and dual convex problem, to appear.
M. Kruczenski, Loop equations and bootstrap methods in the lattice, talk at the Bootstrap 2020 annual conference, via Zoom, Boston, MA, U.S.A., June 2020.
A. Bose, P. Haldar, A. Sinha, P. Sinha and S.S. Tiwari, Relative entropy in scattering and the S-matrix bootstrap, SciPost Phys. 9 (2020) 081 [arXiv:2006.12213] [INSPIRE].
J. Koschinski, M.V. Polyakov and A.A. Vladimirov, Leading infrared logarithms from unitarity, analyticity and crossing, Phys. Rev. D 82 (2010) 014014 [arXiv:1004.2197] [INSPIRE].
J. Gasser and H. Leutwyler, On the low-energy structure of QCD, Phys. Lett. B 125 (1983) 321 [INSPIRE].
M. Froissart, Asymptotic behavior and subtractions in the Mandelstam representation, Phys. Rev. 123 (1961) 1053 [INSPIRE].
A. Martin, Unitarity and high-energy behavior of scattering amplitudes, Phys. Rev. 129 (1963) 1432 [INSPIRE].
S.M. Roy, Exact integral equation for pion pion scattering involving only physical region partial waves, Phys. Lett. B 36 (1971) 353 [INSPIRE].
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ArXiv ePrint: 2011.02802
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Guerrieri, A.L., Penedones, J. & Vieira, P. S-matrix bootstrap for effective field theories: massless pions. J. High Energ. Phys. 2021, 88 (2021). https://doi.org/10.1007/JHEP06(2021)088
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DOI: https://doi.org/10.1007/JHEP06(2021)088