Two-body scattering states in Minkowski space and the Nakanishi integral representation onto the null plane

TL;DR

This paper develops a Minkowski space approach using the Nakanishi integral representation to analyze two-body scattering states via the Bethe-Salpeter equation, with explicit equations and applications to scalar interactions.

Contribution

It introduces a detailed formalism for scattering states in Minkowski space using the Nakanishi representation and null-plane projection, extending previous bound state studies.

Findings

Derived explicit scattering integral equations in ladder approximation.

Applied formalism to zero-energy limit and Wick-Cutkosky model.

Provided a workable Minkowski space treatment for scattering states.

Abstract

The Nakanishi perturbative integral representation of the four-dimensional T-matrix is investigated in order to get a workable treatment for scattering states, solutions of the inhomogeneous Bethe-Salpeter Equation, in Minkowski space. The projection onto the null-plane of the four-dimensional inhomogeneous Bethe-Salpeter Equation plays a key role for devising an equation for the Nakanishi weight function (a real function), as in the homogeneous case that corresponds to bound states and it has been already studied within different frameworks. In this paper, the whole formal development is illustrated in detail and applied to a system, composed by two massive scalars interacting through the exchange of a massive scalar. The explicit expression of the scattering integral equations are also obtained in ladder approximation, and, as simple applications of our formalism, some limiting cases, like the zero-energy limit and the Wick-Cutkosky model in the continuum, are presented.