Study of the Homogeneous Bethe-Salpeter Equation in the 2+1 Minkowski Space

Emanuel Ydrefors,Jorge H Alvarenga Nogueira,Cristian Gutierrez,Vitor Gigante

doi:10.1142/s2010194517600552

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.

Highlights

Studies of relativistic bound systems in Minkowski space are important in order to understand the structure properties of few-body system in the non-perturbative regime
The same formalism can be used to study electroweak form factors in Minkowski space, which is of interest for applications in high-energy neutrino physics, e.g. originated from cosmic rays or remnants 8, which is measured by experiments as ICECUBE 9
We have solved the BSE in Minkowski space by using the NIR and the LF projection technique, and in this contribution we show that we can obtain precise coupling constants in comparison with the ones obtained in Euclidean space

Summary

Introduction

Studies of relativistic bound systems in Minkowski space are important in order to understand the structure properties of few-body system in the non-perturbative regime. Solving the BSE in Minkowski space constitutes a difficult task because of the singularities associated with the propagators and the ones in the interaction kernel. The first successful solution of the Bethe-Salpeter equation in Minkowski space was obtained by Kusaka and Williams 2.

Results

Conclusion