Bethe–Salpeter bound-state structure in Minkowski space

The quantitative investigation of the scalar Bethe–Salpeter equation in Minkowski space, within the ladder-approximation framework, is extended to include the excited states. This study has been carried out for an interacting system composed by two massive bosons exchanging a massive scalar, by adopting (i) the Nakanishi integral representation of the Bethe–Salpeter amplitude, and (ii) the formally exact projection onto the null plane. Our analysis, on one hand, confirms the reliability of the method already applied to the ground state and, on the other one, extends the investigation from the valence distribution in momentum space to the corresponding quantity in the impact-parameter space, pointing out some relevant features, like (i) the equivalence between Minkowski and Euclidean transverse-momentum amplitudes, and (ii) the leading exponential fall-off of the valence wave function in the impact-parameter space.