Advances in solving the two-fermion homogeneous Bethe-Salpeter equation in Minkowski space

Abstract: Actual solutions of the Bethe-Salpeter equation for a two-fermion bound system are becoming available directly in Minkowski space, by virtue of a novel technique, based on the so-called Nakanishi integral representation of the Bethe-Salpeter amplitude and improved by expressing the relevant momenta through light-front components, i.e. $k^\pm=k^0 \pm k^3$. We solve a crucial problem that widens the applicability of the method to real situations by providing an analytically exact treatment of the singularities plaguing the two-fermion problem in Minkowski space, irrespective of the complexity of the irreducible Bethe-Salpeter kernel. This paves the way for feasible numerical investigations of relativistic composite systems, with any spin degrees of freedom. We present a thorough comparison with existing numerical results, evaluated in both Minkowski and Euclidean space, fully corroborating our analytical treatment, as well as fresh light-front amplitudes illustrating the potentiality of non perturbative calculations performed directly in Minkowski space.