The decay of the W bosons into the full family of leptons can be modeled by an unbounded self-adjoint operator acting on a Hilbert space of Fock type. The allowed energy values of the physical system are in the spectrum of this operator which is divided in three parts: the absolutely continuous spectrum, the singular continuous spectrum and the eigenvalues. The subspace related to the absolutely continuous part gathers the scattered states which are located far from the experiment center. As time goes to inﬁnity, it would be expected that they behave like if there were not any interaction. Mathematically, this intuition is expressed through the notion of asymptotic completeness of the wave operators. Moreover, the idea behind a particle physics experiment is to detect an out-coming scattered states knowing the in-coming scattered states. The mathematical object which relates these particles is the so-called scattering matrix. One of the goals of a scattering theory is to prove the existence of the scattering matrix and the property of asymptotic completeness. In this presentation we will give a pedagogical introduction to all these notions through the example of the weak decay of the W ± bosons into the full family of leptons.