For the first time, the inhomogeneous Bethe-Salpeter Equation for an
interacting system, composed by two massive scalars exchanging a massive
scalar, is numerically investigated in ladder approximation, directly in
Minkowski space, by using an approach based on the Nakanishi integral
representation. In this paper, the limiting case of zero-energy states is
considered, extending the approach successfully applied to bound states. The
numerical values of scattering lengths, are calculated for several values of
the Yukawa coupling constant, by using two different integral equations that
stem within the Nakanishi framework. Those low-energy observables are compared
with (i) the analogous quantities recently obtained in literature, within a
totally different framework and (ii) the non relativistic evaluations, for
illustrating the relevance of a non perturbative, genuine field theoretical
treatment in Minkowski space, even in the low-energy regime. Moreover,
dynamical functions, like the Nakanishi weight functions and the distorted part
of the zero-energy Light-front wave functions are also presented.
Interestingly, a highly non trivial issue related to the abrupt change in the
width of the support of the Nakanishi weight function, when the zero-energy
limit is approached, is elucidated, ensuring a sound basis to the forthcoming
evaluation of phase-shifts.Comment: 23 pages and 4 figures. Minor changes in the abstract, typos fixed
and added a figure. Submitted for publicatio
