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We consider the scattering matrices of massive quantum field theories with no bound states and a global O(N) symmetry in two spacetime dimensions. In particular we explore the space of two-to-two S-matrices of particles of mass m transforming in the vector representation as restricted by the general conditions of unitarity, crossing, analyticity and O(N) symmetry. We found a rich structure in that space by using convex maximization and in particular its convex dual minimization problem. At the boundary of the allowed space special geometric points such as vertices were found to correspond to integrable models. The dual convex minimization problem provides a novel and useful approach to the problem allowing, for example, to prove that generically the S-matrices so obtained saturate unitarity and, in some cases, that they are at vertices of the allowed space.

Perimeter Institute for Theoretical Physics

Department of Physics and Astronomy & Guelph-Waterloo Physics Institute University of Waterloo

Institut de Physique Théorique, CEA Saclay

Department of Physics and Astronomy Purdue University

Purdue Quantum Science and Engineering Institute (PQSEI) Purdue University

Instituto de Física Teórica UNESP ICTP South American Institute for Fundamental Research

Philippe Meyer Institute Physics Department École Normale Suṕerieure PSL Research University, 24 rue Lhomond

Instituto de Física Teórica UNESP ICTP South American Institute for Fundamental Research