Solutions of Bethe-Salpter equations in QED3

TL;DR

This paper investigates solutions to the Bethe-Salpeter equation in QED3, demonstrating consistency with Ward-Takahashi identities and deriving eigenvalues for scalar-vector fermion-antifermion systems through approximate methods.

Contribution

It provides an approximate method to solve the Bethe-Salpeter equation in QED3 and analyzes the existence of bound states in scalar-vector fermion-antifermion systems.

Findings

Solutions for axial-scalar are consistent with Ward-Takahashi identities.

Eigenvalues for the coupling constant are derived from boundary conditions.

Massless scalar-vector fermion-antifermion systems do not seem to exist in this approximation.

Abstract

To understand the mechanism of the fermion pair and fermion-antifermion pair condensation,the solutions of Bethe-Salpeter equation in QED$_{3}$ is examined.In the ladder appoximation our solution for the axial-scalar is consistent with Ward-Takahashi-identity for the axial vector currents.Since the massless scalar-vector sector is described by a coupled integral equation,it is difficult to solve explicitly.We approximate the equation for large and small momentum region separately and convert them into differential equations in position space.These equation can be solved easily.Boundary condition at the origin leads the eigenvalue for dimensionless coupling constant $\lambda=e^{2}/m$.There exists solutions for massless scalar-vector fermion-antifermion (fa) system with discrete spectrum. In our approximation massless-scalar-vector ff systemes does not seem to exist.