Solving the inhomogeneous Bethe-Salpeter Equation in Minkowski space: the zero-energy limit

TL;DR

This paper numerically investigates the inhomogeneous Bethe-Salpeter Equation for two massive scalars in Minkowski space at zero energy, calculating scattering lengths and analyzing Nakanishi weight functions to advance non-perturbative field theory methods.

Contribution

It extends the Nakanishi integral representation approach to zero-energy states in Minkowski space, providing new numerical results for scattering lengths and wave functions in a non-perturbative framework.

Findings

Calculated scattering lengths for various Yukawa couplings.

Analyzed the behavior of Nakanishi weight functions near zero energy.

Identified a non-trivial change in the support width of the Nakanishi weight function.

Abstract

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.