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Scattering amplitudes can be extracted from time-ordered $n$-point functions by means of the well known LSZ reduction formula, even in non-perturbative Quantum Field Theories, such as Quantum Chromodynamics (QCD). However, in the context of Lattice QCD, one can access only Euclidean $n$-point functions sampled at discrete points and with finite (but systematically improvable) precision and accuracy. This makes the problem of analytically continuing back to Minkowski space-time ill-posed. I will present here one particular strategy which allows to extract scattering amplitudes from Euclidean correlators, while avoiding analytic continuation, technically turning an ill-posed problem into a merely ill-conditioned one.
Working in the axiomatic framework of the Haag-Ruelle scattering theory, we show that scattering amplitudes can be approximated arbitrarily well in terms of linear combinations of Euclidean correlators at discrete time separations. The essential feature of the proposed approximants is that one can calculate them, at least in principle, from Lattice-QCD data. In this talk, after reviewing the basic ideas behind Haag-Ruelle scattering theory, I will sketch the derivation of the approximations formulae, and discuss extensively how they can be used in practical numerical calculations. Also, similarities and differences with other methods, e.g. Lüscher's formalism, will be reviewed.