FindBounce: package for multi-field bounce actions

TL;DR

This work presents FindBounce, a Mathematica package implementing the polygonal bounce method to compute the Euclidean bounce action for metastable vacua in multi-field theories at $D=3,4$. It delivers a semi-analytical, linearly scaling solution by discretizing the potential into linear segments and, for multiple fields, deforming a path in field space with a linear system. The package offers robust performance across single and multi-field cases (up to ~20 fields) and provides extensive examples, including thermal corrections and nucleation temperature calculations, while staying competitive with existing tools. These capabilities enable fast, precise assessments of vacuum stability and first-order phase transitions in beyond-the-Standard-Model scenarios, with practical implications for cosmology and gravitational-wave phenomenology.

Abstract

We are launching FindBounce, a Mathematica package for the evaluation of the Euclidean bounce action that enters the decay rate of metastable states in quantum and thermal field theories. It is based on the idea of polygonal bounces, which is a semi-analytical approach to solving the bounce equation by discretizing the potential into piecewise linear segments. This allows for a fast and robust evaluation of arbitrary potentials with specified precision and any number of scalar fields. Time cost grows linearly with the number of fields and/or the number of segments. Computation with 20 fields takes $\sim 2$ seconds with $0.5\%$ accuracy of the action. The FindBounce function is simple to use with the native Mathematica look and feel, it is easy to install, and comes with detailed documentation and physical examples, such as the calculation of the nucleation temperature. We also provide timing benchmarks with comparisons to existing tools, where applicable.

Figures (9)

• Figure 1: Left: Linearly off-set quartic potential in gray and the polygonal approximation with $N=7$ in blue. Right: The bounce field configuration corresponding to the potential on the left, computed with the polygonal bounce approximation.
• Figure 2: A schematic overview of finding the initial radius $R_{in}$. In this particular example, the solution starts from segment $s = 2$, and is bounded by $R_2^{\min} = 0$ and $R_2^{\max}$, which can be computed from the segmentation, see Guada:2018jek for details. Note that starting from these two boundary radii, the final radius $R_{N-1}$ changes from real to complex, which happens only when starting from a segment, where a solution exists, in this example it is $s=2$.
• Figure 3: Left: The benchmark potential from Eq. \ref{['eq:BM']} for different values of $\alpha$ going from thick $\alpha = 0.6$ to thin wall $\alpha = 0.99$. Right: The bounce action $\mathcal{S}_s$ for each potential configuration and a given number of field points $s$, normalized to $s = 400$ and computed in $D=4$.
• Figure 4: Left: Evaluation time with respect to the number of field points, averaged over two intervals of $\alpha$ corresponding to thin and thick wall regimes. Right: The bounce field configuration and action with $N=31$ (default) field points for different tolerance value of the action controlled by "ActionTolerance". Reference values of the action for "FieldPoints"-> 10 and 100 field points with "ActionTolerance"->$10^{-6}$ (default) are shown on the green background.
• Figure 5: Left: The piecewise quartic potential for different values of the potential difference between the vacua, going from the thin wall $\Delta V = -0.1$ to thick wall $\Delta V = -20$ regime. Right: The bounce action $S_s$ for different number of field points, normalized to the exact result $S_E$ of the quartic-quartic potential.