Efficient numerical solution to vacuum decay with many fields

TL;DR

This paper introduces a new efficient numerical method for solving vacuum decay bubble nucleation problems involving multiple fields, overcoming limitations of traditional shooting and minimization techniques.

Contribution

The authors develop a multiple shooting-based method that directly integrates equations of motion, significantly improving speed and accuracy for multi-field vacuum decay solutions.

Findings

Finds solutions with three fields in under a minute

Handles up to eight fields in about an hour

Provides a Mathematica package for implementation

Abstract

Finding numerical solutions describing bubble nucleation is notoriously difficult in more than one field space dimension. Traditional shooting methods fail because of the extreme non-linearity of field evolution over a macroscopic distance as a function of initial conditions. Minimization methods tend to become either slow or imprecise for larger numbers of fields due to their dependence on the high dimensionality of discretized function spaces. We present a new method for finding solutions which is both very efficient and able to cope with the non-linearities. Our method directly integrates the equations of motion except at a small number of junction points, so we do not need to introduce a discrete domain for our functions. The method, based on multiple shooting, typically finds solutions involving three fields in under a minute, and can find solutions for eight fields in about an hour. We include a numerical package for Mathematica which implements the method described here.