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. arXiv:1909.06495v2 [hep-th]

The "bootstrap determination" of the geometrical correlation functions in the two-dimensional Potts model proposed in a paper by Picco et al. [1] was later shown in [2] to be incorrect, the actual spectrum of the model being considerably more complex than initially conjectured. We provide in this paper a geometrical interpretation of the fourpoint functions built in [1], and explain why the results obtained by these authors, albeit incorrect, appeared so close to those of their numerical simulations of the Potts model. Our strategy is based on a cluster expansion of correlation functions in RSOS minimal models, and a subsequent numerical and algebraic analysis of the corresponding s-channel spectrum, in full analogy with our early work on the Potts model [2]. Remarkable properties of the lattice amplitudes are uncovered, which explain in particular the truncation of the spectrum of [2] to the much simpler one of the RSOS models, and which will be used in a forthcoming paper to finally determine the geometrical four-point functions of the Potts model itself.