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We propose a non-perturbative formulation of planar scattering amplitudes inN=4 SYM or, equivalently, polygonal Wilson loops. The construction is based onthe OPE approach and introduces a new decomposition of the Wilson loop in termsof fundamental building blocks named Pentagon transitions. These transitionssatisfy a simple relation to the worldsheet S-matrix on top of the so calledGubser-Klebanov-Polyakov vacuum which allows us to bootstrap them at any valueof the coupling. In this letter we present a subsector of the full solution toscattering amplitudes which we call the gluonic part. We match our results withboth weak and strong coupling data available in the literature. For example,the strong coupling Y-system can be understood in this approach.

Space-time S-matrix and Flux-tube S-matrix at Finite Coupling