Linear confinement in momentum space: singularity-free bound-state equations

TL;DR

This paper introduces a subtraction method to eliminate principal value singularities in momentum-space bound-state equations with linear confinement, improving numerical stability and accuracy, demonstrated through bottomonium spectrum fitting.

Contribution

A novel subtraction technique is developed to remove singularities in momentum-space bound-state equations, facilitating more accurate and stable numerical solutions for confining interactions.

Findings

Singularity can be effectively removed using the subtraction method.

The method enables stable solutions for linear confining potentials.

Successful fit of bottomonium spectrum demonstrates practical utility.

Abstract

Relativistic equations of Bethe-Salpeter type for hadron structure are most conveniently formulated in momentum space. The presence of confining interactions causes complications because the corresponding kernels are singular. This occurs not only in the relativistic case but also in the nonrelativistic Schr\"odinger equation where this problem can be studied more easily. For the linear confining interaction the singularity reduces to one of Cauchy principal value form. Although this singularity is integrable, it still makes accurate numerical solutions difficult. We show that this principal value singularity can be eliminated by means of a subtraction method. The resulting equation is much easier to solve and yields accurate and stable solutions. To test the method's numerical efficiency, we performed a three-parameter least-squares fit of a simple linear-plus-Coulomb potential to the bottomonium spectrum.