Abstract
We explore the S-matrices of gapped, unitary, Lorentz invariant quantum field theories with a global O(N) symmetry in 1+1 dimensions. We extremize various cubic and quartic couplings in the two-to-two scattering amplitudes of vector particles. Saturating these bounds, we encounter known integrable models with O(N) symmetry such as the O(N) Gross-Neveu and non-linear sigma models and the scattering of kinks in the sine-Gordon model. We also considered more general mass spectra for which we move away from the integrable realm. In this regime we find (numerically, through a large N analysis and sometimes even analytically) that the S-matrices saturating the various coupling bounds have an extremely rich structure exhibiting infinite resonances and virtual states in the various kinematical sheets. They are rather exotic in that they admit no particle production yet they do not obey Yang-Baxter equations. We discuss their physical (ir)relevance and speculate, based on some preliminary numerics, that they might be close to more realistic theories with particle production.
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References
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap. Part I: QFT in AdS, JHEP 11 (2017) 133 [arXiv:1607.06109] [INSPIRE].
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap II: two dimensional amplitudes, JHEP 11 (2017) 143 [arXiv:1607.06110] [INSPIRE].
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix Bootstrap III: Higher Dimensional Amplitudes, arXiv:1708.06765 [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized s Matrices in Two-Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Models, Annals Phys. 120 (1979) 253 [INSPIRE].
A.L. Guerrieri, J. Penedones and P. Vieira, Bootstrapping QCD: the Lake, the Peninsula and the Kink, arXiv:1810.12849 [INSPIRE].
M. Hortacsu, B. Schroer and H.J. Thun, A Two-dimensional σ Model With Particle Production, Nucl. Phys. B 154 (1979) 120 [INSPIRE].
M.F. Paulos and Z. Zheng, Bounding scattering of charged particles in 1 + 1 dimensions, arXiv:1805.11429 [INSPIRE].
Y. He, A. Irrgang and M. Kruczenski, A note on the S-matrix bootstrap for the 2d O(N) bosonic model, JHEP 11 (2018) 093 [arXiv:1805.02812] [INSPIRE].
E. Witten, Some Properties of the \( {\left(\overline{\psi}\psi \right)}^2 \) Model in Two-Dimensions, Nucl. Phys. B 142 (1978) 285 [INSPIRE].
R. Shankar and E. Witten, The S Matrix of the Kinks of the \( {\left(\overline{\psi}\psi \right)}^2 \) Model, Nucl. Phys. B 141 (1978) 349 [Erratum ibid. B 148 (1979) 538] [INSPIRE].
M. Karowski and H.J. Thun, Complete S Matrix of the O(2N) Gross-Neveu Model, Nucl. Phys. B 190 (1981) 61 [INSPIRE].
N. Doroud and J. Elias Miró, S-matrix bootstrap for resonances, JHEP 09 (2018) 052 [arXiv:1804.04376] [INSPIRE].
B. Gabai, D. Mazáč, A. Shieber, P. Vieira and Y. Zhou, No Particle Production in Two Dimensions: Recursion Relations and Multi-Regge Limit, arXiv:1803.03578 [INSPIRE].
A. Zamolodchikov and I. Ziyatdinov, Inelastic scattering and elastic amplitude in Ising field theory in a weak magnetic field at T > T c : Perturbative analysis, Nucl. Phys. B 849 (2011) 654 [arXiv:1102.0767] [INSPIRE].
A.B. Zamolodchikov, Z 4 Symmetric factorized s matrix in two space-time dimensions, Commun. Math. Phys. 69 (1979) 165 [INSPIRE].
G. Mussardo and S. Penati, A Quantum field theory with infinite resonance states, Nucl. Phys. B 567 (2000) 454 [hep-th/9907039] [INSPIRE].
A.B. Zamolodchikov, Exact S matrix associated with selfavoiding polymer problem in two-dimensions, Mod. Phys. Lett. A 6 (1991) 1807 [INSPIRE].
F.A. Smirnov, A Comment on A. Zamolodchikov’s paper concerning selfavoiding polymers, Phys. Lett. B 275 (1992) 109 [INSPIRE].
P. Fendley, Taking N → O with S matrices, cond-mat/0111582.
A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
D. Iagolnitzer, Factorization of the Multiparticle s Matrix in Two-Dimensional Space-Time Models, Phys. Rev. D 18 (1978) 1275 [INSPIRE].
D. Iagolnitzer, Scattering in quantum field theories: The Axiomatic and constructive approaches, Princeton University Press, Princeton, U.S.A. (1993) [INSPIRE].
F.A. Smirnov and A.B. Zamolodchikov, On space of integrable quantum field theories, Nucl. Phys. B 915 (2017) 363 [arXiv:1608.05499] [INSPIRE].
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ArXiv ePrint: 1805.11143v3
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Córdova, L., Vieira, P. Adding flavour to the S-matrix bootstrap. J. High Energ. Phys. 2018, 63 (2018). https://doi.org/10.1007/JHEP12(2018)063
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DOI: https://doi.org/10.1007/JHEP12(2018)063