# Study of the homogeneous Bethe-Salpeter equation in the 2+1 Minkowski space

## TL;DR

This paper solves the Bethe-Salpeter equation for a two-scalar boson system in 2+1 dimensions using the Nakanishi integral representation and Light-Front projection, enabling Minkowski space calculations that agree with Euclidean results.

## Contribution

It demonstrates the effectiveness of the Nakanishi integral representation in solving the Bethe-Salpeter equation directly in Minkowski space in 2+1 dimensions.

## Key findings

- Good agreement between Minkowski and Euclidean calculations.
- Validation of Nakanishi integral representation in 2+1 dimensions.
- Potential applications in condensed matter physics.

## Abstract

The problem of a relativistic bound-state system consisting of two scalar bosons interacting through the exchange of another scalar boson, in 2+1 space-time dimensions, has been studied. The Bethe-Salpeter equation (BSE) was solved by adopting the Nakanishi integral representation (NIR) and the Light-Front projection. The NIR allows us to solve the BSE in Minkowski space, which is a big and important challenge, since most of non-perturbative calculations are done in Euclidean space, e.g. Lattice and Schwinger-Dyson calculations. We have in this work adopted an interaction kernel containing the ladder and cross-ladder exchanges. In order to check that the NIR is also a good representation in 2+1, the coupling constants and Wick-rotated amplitudes have been computed and compared with calculations performed in Euclidean space. Very good agreement between the calculations performed in the Minkowski and Euclidean spaces has been found. This is an important consistence test that allows Minkowski calculations with the Nakanishi representation in 2+1 dimensions. This relativistic approach will allow us to perform applications in condensed matter problems in a near future.