Solution of the Scalar Coulomb Bethe-Salpeter Equation

TL;DR

This paper derives and analytically solves a relativistic two-body wave equation for scalar particles with Coulomb interaction, providing precise energy level calculations up to order alpha^6, aligning with Bethe-Salpeter results.

Contribution

It introduces a local two-body wave equation from the Bethe-Salpeter framework and solves it analytically for scalar Coulomb systems, extending energy calculations to higher order corrections.

Findings

Analytical solution matches Bethe-Salpeter energies to order alpha^4.

Extended energy level calculations to order alpha^6.

Derived a two-body Bohr-Sommerfeld formula for scalar particles.

Abstract

A relativistic two-body wave equation, local in configuration space, is derived from the Bethe-Salpeter equation for two scalar particles bound by a scalar Coulomb interaction. The two-body bound-state wave equation is solved analytically, giving a two-body Bohr-Sommerfeld formula whose energies agree with the Bethe-Salpeter equation to order alpha^4 for all quantum states. From the Bohr-Sommerfeld formula, along with the expectation values of two remaining small corrections, the energy levels of the scalar Coulomb Bethe-Salpeter equation are worked out to order alpha^6 for all states.