Minkowski space approach for the Bethe-Salpeter equation

Lauro Tomio,Vitor Gigante,Cristian Gutierrez,S Bondarenko,A Malakhov,V Burov

doi:10.1051/epjconf/201713801013

Abstract

The bound-state energy spectrum for two-boson system with a fixed exchanged massive boson is investigated by considering the scalar Bethe-Salpeter (BS) equation in Minkowski space within the ladder-approximation framework. The approach is extending to excited states a previous study performed for the ground state. The number of states in the spectrum is restricted by the coupling strength of the two-body interaction, as well as the mass of the exchanged boson. In the present approach, we use the Nakanishi Integral Representation and the formally exact null-plane projection. As usual a comparison between eigenvalues obtained in the Minkowski and Euclidean space is presented. Within such scheme, the light-front momentum-space valence wave functions are obtained for the ground and excited states, together with the corresponding quantities in the impactparameter space. Among relevant features emerging from this study, we can point out the node structure of light-front wave function and the leading exponential fall-off of the valence wave function in the impact-parameter space.

Highlights

The bound-state energy spectrum for two-boson system with a fixed exchanged massive boson is investigated by considering the scalar Bethe-Salpeter (BS) equation in Minkowski space within the ladder-approximation framework
In order to widen the applicability of the calculations in Minkowski space, we explore the structure of the valence wave function in the impact parameter (IP) space
We are reporting some recent investigations on the spectrum of excited states of the Bethe-Salpeter equation in 3 + 1 space-time dimensions, considering the s-wave state in the ladder approximation, using the Nakanishi integral representation

Summary

Introduction

As a natural extension of the previous analysis, which was limited only for ground states, within a recent work done in [13], the s−wave spectrum of excited states for the scalar theory was explored in the ladder kernel approximation. This problem, as far as we know, was for the first time investigated in the Minkowski space, with relevant details appearing in [14].

Minkowski space Bethe-Salpeter equation and bound-state spectrum

Comparison between the Minkowski and Euclidean eigenvalues

Momentum space valence wave function

Valence wave function in the impact-parameter space

Conclusions