Numerical Approach to Multi Dimensional Phase Transitions

TL;DR

This paper develops a robust two-stage numerical strategy to compute bounce solutions for first-order phase transitions in theories with multiple scalar fields. It first solves the undamped (energy-conserving) problem by transforming the potential to guide a stable Newton minimization, then smoothly continues to the physically damped case relevant for thermal or vacuum transitions. The method yields highly accurate bounce configurations and actions, demonstrated in both one- and two-field examples, and remains stable and scalable for multi-dimensional field spaces. The approach is positioned as a versatile tool for analyzing phase transitions in beyond-Standard-Model scenarios, with potential applications to electroweak baryogenesis and related cosmological phenomena.

Abstract

We present an algorithm to analyze numerically the bounce solution of first-order phase transitions. Our approach is well suited to treat phase transitions with several fields. The algorithm consists of two parts. In the first part the bounce solution without damping is determined, in which case energy is conserved. In the second part the continuation to the physically relevant case with damping is performed. The presented approach is numerically stable and easily implemented.

Figures (7)

• Figure 1: The potential for a theory with a false vacuum.
• Figure 2: The bounce configuration in the one-dimensional degenerate case. The deviations from the exact analytic result are $O(10^{-4})$.
• Figure 3: Potential $V$ and kinetic energy $E_\textrm{kin}$ of the one-dimensional configuration. The total energy $E=E_{\textrm{kin}}-V$ is conserved up to $0.1\%$.
• Figure 4: The bounce configuration in the two-dimensional non-degenerate case. The exponentially decreasing tail $[0,250]$ has been cut off.
• Figure 5: Potential $V$ and kinetic energy $E_\textrm{kin}$ of the two-dimensional configuration. The total energy $E=E_\textrm{kin} - V$ is conserved up to $0.1\%$. The exponentially decreasing tail $[0,250]$ has again been cut off.