Techniques for Solving Static Klein-Gordon Equation with Self-Interaction $\lambda\phi^4$ and Arbitrary Spherical Source Terms

TL;DR

This paper develops numerical techniques to solve the static Klein-Gordon equation with self-interaction and arbitrary spherical sources, providing adaptable code for various density profiles in physics applications.

Contribution

It introduces new numerical methods for solving the Klein-Gordon equation with self-interaction and supplies adaptable code for diverse source profiles.

Findings

Effective numerical techniques for $ abla^2 o abla^2 + m^2$ equations

Solutions for scalar fields with self-interaction in spherical symmetry

Open-source code for arbitrary density profiles

Abstract

The Klein-Gordon equation for a scalar field sourced by a static spherically symmetric background is an interesting second-order differential equation with applications in particle physics, astrophysics, and elsewhere. Here we present static solutions for generic source density profiles in the case where the scalar field has no interactions or a mass term. For a $\lambda\phi^4$ self-interaction term, we develop the techniques that are necessary numerical computation. We also provide code to perform the numerical calculations that can be adapted for arbitrary density profiles.