Stability in the instantaneous Bethe-Salpeter formalism: reduced exact-propagator bound-state equation with harmonic interaction

TL;DR

This paper proves that a reduced instantaneous Bethe-Salpeter formalism with harmonic interactions yields stable bound-state solutions, addressing previous instabilities found in the Salpeter equation with confining interactions.

Contribution

It introduces a rigorously proven, reduced Bethe-Salpeter equation that avoids instabilities present in earlier models for harmonic confining interactions.

Findings

Bound-state energies are real and discrete.

Instabilities are absent in the reduced formalism for harmonic interactions.

The formalism generalizes the Salpeter equation with improved stability.

Abstract

Several numerical investigations of the Salpeter equation with static confining interactions of Lorentz-scalar type revealed that its solutions are plagued by instabilities of presumably Klein-paradox nature. By proving rigorously that the energies of all predicted bound states are part of real, entirely discrete spectra bounded from below, these instabilities are shown, for confining interactions of harmonic-oscillator shape, to be absent for a reduced version of an instantaneous Bethe-Salpeter formalism designed to generalize the Salpeter equation towards an approximate inclusion of the exact propagators of all bound-state constituents.