A cluster-separable Born approximation for the 3D reduction of the three-fermion Bethe-Salpeter equation

TL;DR

This paper introduces a covariant 3D reduction method for the three-fermion Bethe-Salpeter equation, ensuring covariance and stability against continuum dissolution, applicable in external potentials and multi-fermion interactions.

Contribution

It develops a novel covariant 3D reduction technique for the three-fermion Bethe-Salpeter equation using a cluster-separable Born approximation.

Findings

Ensures covariance of two-fermion equations in external potentials.

Prevents continuum dissolution while maintaining covariance.

Applicable to truncated series of the 3D potential.

Abstract

We perform a 3D reduction of the two-fermion Bethe-Salpeter equation based on Sazdjian's explicitly covariant propagator, combined with a covariant substitute of the projector on the positive-energy free states. We use this combination in the two fermions in an external potential and in the three-fermion problems. The covariance of the two-fermion propagators insures the covariance of the two-body equations obtained by switching off the external potential, or by switching off all interactions between any pair of two fermions and the third one, even if the series giving the 3D potential is limited to the Born term or more generally truncated. The covariant substitute of the positive energy projector preserves the equations against continuum dissolution without breaking the covariance.