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FindBounce: package for multi-field bounce actions

Victor Guada, Miha Nemevšek, Matevž Pintar

arXiv:2002.00881v2hep-phastro-ph.CO

TL;DR

Metastable-state decay requires the Euclidean bounce action, while numerical bounce calculations can be challenging in multifield and thin-wall cases. FindBounce implements a semi-analytical polygonal method in Mathematica and computes actions and field configurations, including finite-temperature examples such as nucleation temperature.

  • Problem

    Evaluating the Euclidean bounce action that controls metastable-state decay rates is challenging, particularly for multifield calculations involving coupled equations.

    Source: Program Summary · p. 2, §1 · p. 4, §2.1 · p. 8

  • Method

    FindBounce implements a semi-analytical polygonal bounce method in Mathematica, returning the bounce action and associated field solution.

    Source: §1 · p. 5, §2.1 · p. 8

  • Results

    Runtime is under 1 second for 10 fields and about 2 seconds for 20 fields, with runtime increasing linearly with field count.

    Source: Program Summary · p. 2

  • Takeaways & Limitations

    The output supports finite-temperature nucleation-temperature calculations below the critical temperature and can inform gravitational-wave-spectrum parameters α and β.

    Source: §6 · p. 25

    Randomly chosen starting coefficients may lead FindRoot to a saddle rather than the potential minimum, in which case FindBounce returns $Failed.

    Source: §5.7 · p. 23

Abstract

from arXiv · show

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.

Program Summary

FindBounce is a Mathematica package for evaluating the Euclidean bounce action in thermal and quantum field theories. It supports D = 3, 4 calculations and scales linearly with field count.

Program Summary

  • Runtime is under 1 second for 10 fields and about 2 seconds for 20 fields, increasing linearly with field count.It was tested on Mac OS X and Windows and should work wherever Mathematica is installed.Source: Program Summary · p. 2
  • The package evaluates the Euclidean bounce action controlling metastable-minimum decay rates in thermal and quantum field theories.Its solution method is semi-analytical and based on the polygonal bounce idea [47].Source: Program Summary · p. 2
  • FindBounce requires Mathematica 10 or later and supports spacetime dimensions D = 3, 4.It is distributed as a .paclet under GPLv3.Source: Program Summary · p. 2

1. Introduction

False-vacuum decay calculations seek the dominant semiclassical contribution that estimates the decay-rate factor B, but numerical methods face precision challenges, especially for thin-wall and multifield cases. FindBounce implements a polygonal bounce method in Mathematica for arbitrary-field calculations.

1. Introduction

  • The bounce solution supplies the dominant semiclassical contribution estimating the exponential factor B in false-vacuum decay.Metastable states can transition to lower-energy states through quantum or thermal fluctuations.Source: §1 · p. 4
  • Numerical bounce calculations require careful initial values and integration control, particularly for nearly degenerate vacua in the thin-wall regime.In multifield theories, the path through field space is not fixed and finding the initial value is highly nonlinear.Source: §1 · p. 4
  • The polygonal bounce method extends triangular potentials to arbitrary polygonal shapes, spacetime dimensions D = 3, 4, and field counts, enabling arbitrarily precise evaluation [47].Its piecewise-linear potential yields a simple analytical solution on each segment.Source: §1 · p. 4
  • FindBounce implements the method as a documented, publicly available Mathematica package that returns the bounce action and a manipulable field-solution object.The associated BounceFunction stores field-solution parameters for further manipulation.Source: §1 · p. 5
  • Finite-temperature bounce action is needed to extract parameters related to gravitational-wave production from a first-order phase transition, and is important for electroweak baryogenesis and primordial inter-galactic magnetic-field production [59] [60] [61].Source: §1 · p. 5

2. Basics of false vacuum decay

The polygonal bounce approach replaces the infinite radial domain with a finite one and constructs the solution segment by segment. Its multifield extension deforms a starting path through an iterated semi-analytical procedure.

2. Basics of false vacuum decay

  • Shooting methods need precise initial values and sustained integration accuracy, especially for large-radius thin-wall bubbles, and must terminate before divergence or oscillation.The polygonal approach instead confines field evolution to a finite radial interval and holds the field constant outside it, regularizing the solution.Source: §2 · p. 6

2.1. Polygonal bounce method

  • The potential is segmented into linear pieces, whose solutions are matched in value and derivative to construct a continuously differentiable bounce.The segmentation points sample the potential; FindBounce defaults to 31 points, corresponding to 30 segments.Source: §2.1 · p. 6
  • The solution is determined by an initial radius, found by locating a segment with a solution and bracketing the radius until the final-boundary condition is met.The initial radius can be rescaled to test the solution using Derrick’s theorem.Source: §2.1 · pp. 7–8
  • The default second-order expansion improves polygonal precision through matching linearized in small radius shifts; higher orders require costly numerical root-finding at each segment.For greater practical precision, the paper recommends increasing FieldPoints rather than including still higher orders.Source: §2.1 · p. 8
  • For multiple fields, the method perturbs an initial field-space path and solves a coupled linear system for path deformation, scaling linearly with fields and segments.The starting path may be a straight line, a triangle through an intermediate point, or a user-supplied path.Source: §2.1 · p. 9
  • Each multifield iteration accounts for kinetic and potential minimization together, converges quickly, and retains the compactified radial formulation that works in thin-wall cases.The resulting solution has closed analytical form for further manipulation and parameter extraction.Source: §2.1 · p. 9

3. Quick installation and running guide

FindBounce is installed from a .paclet and run through Mathematica’s native interface. Its function accepts standard potentials, segmented potential points, or multifield potentials.

3.1. Download and installation

  • The .paclet contains the package code, documentation, and resources; installation and update information is available through Mathematica’s paclet tools.The documentation may require a Mathematica restart to update.Source: §3.1 · p. 10

3.2. Running

  • After loading the package with Needs, FindBounce accepts a potential and variables with the two minima, and can also accept piecewise-linear field-value and potential-value pairs.The function is overloaded for multifield potentials, and its output can be accessed through the BounceFunction object.Source: §3.2 · pp. 10–11

3.3. Contributing

  • Bug reports and feature ideas go through the repository’s issues page, while pull requests are welcome and major changes should be discussed in an issue first.Developer instructions cover tests, building the .paclet, and creating documentation.Source: §3.3 · p. 11

4. Using the FindBounce and its output

FindBounce exposes options for controlling potential derivatives, field-space segmentation, convergence tolerances, and the calculation dimension. Its BounceFunction output stores the solution and related parameters for inspection, reuse, and plotting.

4.1. FindBounce options

  • ActionTolerance sets the relative action-change threshold, while MaxRadiusIterations caps radius iterations and may be exceeded to meet the tolerance.The default ActionTolerance is 10^-6; radius iteration usually takes O(1) steps at that tolerance.Source: §4.1 · pp. 11–12
  • Dimension selects the spacetime dimension: D = 4 by default for zero-temperature tunneling and D = 3 for thermal field theory.BottomlessPotential specifies use of the polygonal-quartic combination for single-field potentials unbounded from below.Source: §4.1 · pp. 11–12
  • FieldPoints controls the segmentation, while MidFieldPoint can place an intermediate point, including the potential's local maximum on the straight path.The default multi-field trajectory is a straight line between minima; arbitrary field coordinates can instead define the segmentation.Source: §4.1 · pp. 12–13
  • Gradient and Hessian settings select analytic derivatives, finite differences, supplied derivatives, or—in the gradient case—no second-order extension.Finite differences are suitable for non-analytic potentials or when Mathematica's default derivative is slow; disabling the gradient can be necessary for discontinuous derivatives.Source: §4.1 · p. 12
  • PathTolerance limits the maximum normalized field-point displacement between path iterations, and FindBounce stops when it or either iteration limit or ActionTolerance is satisfied.The default PathTolerance is 10^-2.Source: §4.1 · pp. 12–13

4.2. Bounce function output and manipulation

  • BounceFunction stores the action, bounce solution, path, radii, coefficients, and calculation dimension; BouncePlot plots field configurations, including multiple fields together.The stored path can be reused as an initial ansatz for similar potentials.Source: §4.2 · pp. 13–14

5. Examples

The examples demonstrate FindBounce’s use in single- and multi-field bounce calculations, including thermal nucleation, and explore methods for several potential and instanton cases. They report action accuracy, runtime scaling, and timing comparisons with other tools.

5.1. Single field benchmark

  • With N=30 field points, action error falls below 1% in both wall regimes; evaluation time scales linearly with field points and is ~10% faster for thin walls.The default second-order potential correction speeds convergence; action estimates remain within the requested tolerance even when the first-segment boundary conditions are not exact.Source: §5.1 · pp. 15–16

5.2. Quartic-quartic potential

  • For this example, FindBounce reaches the solution within 1% accuracy using 50 field points in the thin-wall regime and 100 in the thick-wall regime, with somewhat slower convergence.The first derivative is undefined at ϕ = 0, so gradient evaluation and automatic polygonal extension were turned off. Setting MidFieldPoint to ϕ = 0 stabilizes and improves bounce-action accuracy, especially with few field points.Source: §5.2 · p. 17

5.3. Intermediate minima and disappearing instantons

  • In the five-point intermediate-minimum example, direct ABC tunneling disappears below a threshold depth, while the two-step AB and BC decay channels remain nonzero.As the intermediate minimum approaches the lower vacuum, the AB radius grows until the required ordering of bounce radii for the direct solution fails.Source: §5.3 · p. 18

5.4. Potentials unbounded from below

  • For an unbounded potential, FindBounce joins a polygonal segment to an exact unstable-quartic solution; in the example the bounce materializes at ϕ0≈90 despite matching at ϕ=5.The quartic is matched by boundary conditions at the connecting field value, with the matching radius determined numerically.Source: §5.4 · p. 19

5.5. Two field benchmark

  • A two-field benchmark demonstrates bounce trajectories and supports straight, midpoint-constrained, or arbitrary initial paths for path deformation.The arbitrary-path option can encode a known field-space trajectory, such as a parabola used in the example.Source: §5.5 · p. 20

5.6. Thermal corrections and nucleation temperature

  • For a singlet-scalar thermal benchmark, the nucleation criterion is B=S3/TN≈140 under radiation domination, using high-temperature corrections valid up to O(T^4).The example neglects contributions from the potential’s quartic coupling in the thermal corrections and introduces a FindBounce calculation of B versus temperature.Source: §5.6 · pp. 21–22
  • The estimated nucleation temperature is approximately 81 GeV, and 10 field points already give a reasonably accurate, computationally inexpensive estimate using the stated leading contributions.Precision can be improved by increasing FieldPoints and ActionTolerance; the example notes that quantum corrections, daisy resummation, the prefactor A, and contributions from lighter fermions are among the omitted considerations.Source: §5.6 · p. 22

5.7. Beyond two fields

  • Because random parameters can lead FindRoot to a saddle rather than a minimum, FindBounce returns $Failed in that case; Vevacious can find tree-level extrema of a generic one-loop effective potential [114].The example notes that Vevacious can be used to compute extrema for more general potentials.Source: §5.7 · p. 23
  • Actions with N ≲20 field points are accurate to roughly 1%, while N = 100 reproduces values from other methods; FindBounce handles both thin- and thick-wall cases.For nϕ ≤8, the selected parameters yield thick-wall bounces; for larger field counts, the random parameters include thin-wall cases at nϕ = 15 and 19.Source: Table 1 · p. 24, Figure 9 · p. 24, §5.7 · p. 24
  • Runtime scales linearly with the number of fields and field points, independently of field-point count and across thin- and thick-wall cases.The linear scaling with field count is shown for different field-point settings in Figure 9.Source: Figure 9 · p. 24, §5.7 · p. 24
  • With 31 field points, FindBounce takes less than 1 second for 10 fields and less than 2 seconds for 20; its speed is comparable to other tools.At N = 10 field points, its time demand is comparable to SimpleBounce and CosmoTransition; those tools use C++ and Python, while FindBounce uses Mathematica without significant numerical optimization.Source: Figure 9 · p. 24, §5.7 · pp. 24–25

6. Conclusions and outlook

FindBounce computes bounce field configurations and Euclidean actions for false-vacuum decay, including calculations with up to 20 fields in seconds. The conclusion highlights finite-temperature nucleation-temperature estimates and identifies extensions involving bubble growth, multi-field barrierless tunneling, and decay-rate prefactors.

6. Conclusions and outlook

  • Up to 20 fields can be handled efficiently, with the package computing bounce configurations and associated Euclidean actions for false-vacuum decay.The authors describe calculations ranging up to 20 fields in a matter of seconds.Source: §6 · p. 25
  • Bubble growth after nucleation requires bounce equations in Minkowski spacetime, and matching those solutions to later tunneling may be needed for potentials with multiple minima.The authors leave these directions for future work.Source: §6 · p. 25
  • The demonstrated barrierless-tunneling treatment matches a single-field polygonal potential to an unstable quartic; extending it to multiple fields requires additional care.The multi-field extension is deferred to a future package version.Source: §6 · p. 25
  • A direct polygonal-potential calculation of the decay-rate prefactor is proposed as a possible extension beyond thin-wall cases and toward multiple fields.The authors suggest calculating eigenfunctions of the potential's second derivative.Source: §6 · p. 25
  • FindBounce output can be used to determine the nucleation temperature T_N below the critical temperature from finite-temperature bounce calculations.The authors also mention computing α and β parameters for insight into gravitational-wave spectra, with further study deferred.Source: §6 · p. 25

A. One-Loop effective potential

This appendix presents one-loop effective-potential results for quantum and thermal settings and introduces the thermal functions used in the calculation.

A. One-Loop effective potential

  • The appendix gives quantum and thermal effective-potential results at one-loop order in the MS scheme.It situates the calculation alongside earlier zero-temperature and finite-temperature work.Source: A. One-Loop effective potential · p. 26
  • The effective-potential sum runs over species coupled to ϕ, using their degrees of freedom and field-dependent squared masses; the sign distinguishes bosons from fermions.The renormalization scale is µ_R, and C_i is 3/2 for scalars and fermions and 5/6 for gauge bosons.Source: A. One-Loop effective potential · p. 26
  • The appendix gives relevant thermal functions and notes that they can be expanded at high temperatures.Source: A. One-Loop effective potential · p. 26

B. Multi field potential parameters

The appendix specifies multi-field potential parameters used in comparisons and explains how the final coefficient controls vacuum degeneracy and thin-wall behavior.

B. Multi field potential parameters

  • The multi-field potential used for comparisons with other packages, including Table 1, is taken from and defined for each field count n_ϕ.Source: B. Multi field potential parameters · p. 26
  • The final coefficient c_(n_ϕ+1) controls vacuum degeneracy; tiny values yield thin-wall cases for 15 and 19 fields, where the action is notably larger.Despite the larger actions, bounce-action computation time is practically independent of whether the regime is thin- or thick-wall.Source: B. Multi field potential parameters · p. 26
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