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.
Highlights
Strikingly as a tension between unitarity and crossing symmetry
We explore the S-matrices of gapped, unitary, Lorentz invariant quantum field theories with a global O(N ) symmetry in 1+1 dimensions
By looking for theories which maximize particular couplings, we will rediscover in this way the two most famous integrable models with O(N ) symmetry — the O(N ) nonlinear sigma model and the O(N ) Gross-Neveu model — whose S-matrices were found by Zamolodchikov and Zamolodchikov in their seminal 1979 paper [4]
Summary
We consider relativistic particles of mass m transforming in the O(N ) vector representation. There could be bound states showing up in the S-matrix of two fundamental particles They would transform in the singlet, anti-symmetric or symmetric traceless representation of O(N ) and show up as a pole in the corresponding channel. If there is a single bound state transforming in the anti-symmetric representation with mass mBS = 2 cos(λ/2) there is a pole in the corresponding channel as. There is no integrable theory where the particles form a single bound state in the singlet channel. (We can view this as a nice feature: the bootstrap of O(N ) symmetric theories with particles in the vector and anti-symmetric representations alone, will necessarily land us outside the integrable world It describes the scattering of the kinks and anti-kinks of the sine-Gordon model (which is dual to the massive Thirring model where these kinks correspond to the fundamental fermions)
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