The Role of Boundary Conditions in Solving Finite-Energy, Two-Body, Bound-State Bethe-Salpeter Equations

TL;DR

This paper presents a boundary condition-based basis function expansion method for numerically solving finite-energy, two-body Bethe-Salpeter equations, demonstrated on the Wick-Cutkosky model with unequal masses.

Contribution

It introduces a new approach using boundary condition-compliant basis functions to solve Bethe-Salpeter equations requiring Wick rotation and angular separation.

Findings

Successfully computed finite-energy solutions for the Wick-Cutkosky model

Method applicable to equations with separated angular variables

Demonstrated effectiveness for unequal-mass constituents

Abstract

The difficulties that typically prevent numerical solutions from being obtained to finite-energy, two-body, bound-state Bethe-Salpeter equations can often be overcome by expanding solutions in terms of basis functions that obey the boundary conditions. The method discussed here for solving the Bethe-Salpeter equation requires only that the equation can be Wick rotated and that the two angular variables associated with rotations in three-dimensional space can be separated, properties that are possessed by many Bethe-Salpeter equations including all two-body, bound-state Bethe-Salpeter equations in the ladder approximation. The efficacy of the method is demonstrated by calculating finite-energy solutions to the partially-separated Bethe-Salpeter equation describing the Wick-Cutkosky model when the constituents do not have equal masses.