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VacuumTunneling: A package to solve bounce equation with renormalization factor

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Pith's one-line read The paper presents a Mathematica package, VacuumTunneling, that computes the Euclidean bounce action for first-order phase transitions with the wavefunction renormalization factor $Z$ included, and demonstrates that $Z$ can significantly…

desk verdict First public bounce solver with field-dependent Z; single-field LO-validated, but multi-field NLO path-deformation equation has a dimensional inconsistency that must be fixed. read the letter →

arxiv 2501.15236 v1 pith:AAPPEGPA submitted 2025-01-25 hep-ph astro-ph.CO

Banghui Hua , Jiang Zhu This is my paper · ORCID
classification hep-phastro-ph.CO
keywords vacuumdecayratebounceactionwavefunctionrenormalizationfactorfirst-orderphasetransitionshootingmethodpathdeformationsupercoolingMathematicapackage
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that computing vacuum decay rates at next-to-leading order requires solving the bounce equation that comes from the derivative-expanded effective action with the wavefunction renormalization factor $Z$ retained, not just the leading-order equation with $Z=1$. To make this possible, the authors build a Mathematica package, VacuumTunneling, which solves the modified single-field bounce equation by shooting and extends it to multi-field tunneling by path deformation, including the extra friction-like term $-\frac{1}{2}\frac{d\log Z}{d\sigma}\left(\frac{d\sigma}{dr}\right)^2$. They show that the presence of $Z$ changes both the tunneling path in field space and the value of the bounce action; in one two-field example the action changes by more than two orders of magnitude, while at $Z=1$ the package reproduces the actions of existing packages within about one percent. A sympathetic reader would care because a consistent next-to-leading-order treatment is needed to address the gauge dependence of tunneling rates, and because some phase transitions have kinetic terms that vanish at leading order, making the $Z$ factor the leading contribution to the bounce.

What carries the argument

The load-bearing object is the modified Euclidean bounce equation for an $O(D)$-symmetric bubble, Eq. (2.6): $\frac{d^2\sigma}{dr^2}+\frac{D-1}{r}\frac{d\sigma}{dr}-\frac{1}{2}\frac{\partial\log Z_\sigma}{\partial\sigma}\left(\frac{d\sigma}{dr}\right)^2 = Z_\sigma\frac{\partial V_{\rm eff}}{\partial \sigma}$, with boundary conditions $\sigma'(0)=0$ and $\sigma(\infty)=\sigma_F$. This is the usual Coleman bubble equation with two alterations: the effective force is multiplied by $Z_\sigma$, and the friction receives a contribution proportional to the logarithmic derivative of $Z_\sigma$, which can either increase or decrease friction depending on its sign. It is solved by bisection on weighted averages of the true-vacuum and barrier-crossing points, with a rescaling $V_{\rm eff}=\alpha^2 U$, $U(\sigma_b)=1$, and $R=\alpha r$ to keep the numbers moderate. For multiple fields the same equation is solved along a path of unit speed, while the perpendicular component of the equation defines a perpendicular force $N = \left(\frac{dx}{dr}\right)^2\frac{d^2\sigma}{dx^2} - \frac{1}{2}\frac{d^2x}{dr^2}\nabla_\perp \log Z_\sigma - Z_\sigma\nabla_\perp V_{\rm eff}$ that drives path deformation until it vanishes. The package interpolates purely numerical potentials and renormalization factors, so no analytic expressions are required.

What would settle it

Directly minimize the next-to-leading-order action functional $S=\int dr\,r^{D-1}\left(\frac{Z^{-1}}{2}\sigma'^2 + V_{\rm eff}\right)$ for a single-field model with a non-constant $Z$ using an independent global method, such as fine-grid relaxation or an analytic solution where one exists, and compare the action and profile with the package's output; agreement to the claimed one-percent accuracy would confirm that Eq. (2.6) is being solved, while a larger mismatch would falsify the solver's central claim.

Watch

Extended reading notes

Core claim

The central discovery is that the next-to-leading-order tunneling problem can be solved numerically with two small modifications of the standard machinery. In single-field shooting, the particle-in-a-potential picture is retained but the driving force becomes $-Z_\sigma \partial V_{\rm eff}/\partial \sigma$ and the friction is augmented by the term $-\frac{1}{2}\frac{\partial \log Z_\sigma}{\partial \sigma}\left(\frac{d\sigma}{dr}\right)^2$. In multi-field tunneling, the perpendicular force that drives path deformation acquires an extra term involving the transverse gradient of $\log Z_\sigma$, so the renormalization factor modifies not only the bubble profile but the tunneling path itself. The package rescales the potential by its barrier height to control numerical error, handles potentials and renormalization factors that are only known numerically, and includes a supercooling-specific narrowing of the shooting search interval. With these ingredients, the paper argues, the bounce action at next-to-leading order can be computed in seconds to minutes, and the results show that $Z$ affects the action substantially even when its average value between the two vacua is normalized to 1.

Load-bearing premise

The calculation assumes the wavefunction renormalization factor $Z$ is strictly positive between the false and true vacua; when $Z$ crosses zero, as it does in electroweak phase transitions, the coefficient in Eq. (2.6) blows up and the solver cannot run without a regularization the paper does not provide.

Editorial extensions

If this is right

  • Including $Z$ can change the bounce action by more than an order of magnitude; in the two-field example of Sec. IV.B the action shifts from 171.2 to 71283.7, so the next-to-leading-order correction is not automatically small.
  • The tunneling path in field space is modified by the renormalization factor, not just the field profile along the path; the $Z\neq 1$ path differs visibly from the $Z=1$ straight-line path in the same example.
  • At leading order ($Z=1$), the package reproduces the actions of CosmoTransitions, FindBounce, and AnyBubble within about one percent, so the code passes the leading-order validation benchmark.
  • For supercooling transitions, where the barrier sits extremely close to the false vacuum, the narrowed search interval allows the solver to work at temperatures where FindBounce and other packages fail, including a test model whose vacuum separation is $10^8$ times the barrier distance.
  • For dark chiral phase transitions with no leading-order kinetic term, the package computes the next-to-leading-order action; full path deformation changes the action by only 0.6% relative to the lowest-potential one-dimensional path.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the solver cannot run when $Z$ changes sign between the vacua, the electroweak case the Introduction singles out remains unsolved; a regularization that handles the $Z=0$ crossing is a natural next step the paper leaves open.
  • The large action shifts shown in the examples imply that predicted gravitational-wave spectra and nucleation temperatures could move substantially once realistic next-to-leading-order $Z$-terms are included; this is a concrete consequence worth testing in specific beyond-the-Standard-Model scenarios.
  • A direct test of the central claim would be to run the package on a single-field model with a non-constant $Z$ for which Eq. (2.6) has an independent solution (for example, a potential where the equation is integrable or can be solved by a completely different minimizer) and to compare the resulting action and profile; the paper validates only the $Z=1$ limit against other packages.
  • The path-deformation claim could be checked by comparing the package's deformed path with the path obtained by directly minimizing the next-to-leading-order action functional over paths with an independent optimizer; agreement would confirm the $Z$-dependent perpendicular force is correctly implemented.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper presents a Mathematica package, VacuumTunneling, for computing Euclidean bounce actions in first-order phase transitions with a field-dependent wavefunction renormalization factor Z in the derivative-expanded effective action. The single-field solver is a modified shooting method that includes the extra velocity-squared term in Eq. (2.6); the multi-field solver is a path-deformation method that adjusts a path in field space until a 'perpendicular force' N in Eq. (2.35) vanishes. The paper validates the Z=1 limit against CosmoTransitions, FindBounce, and AnyBubble (action agreement within 1%), and applies the package to a two-field model, random multi-field potentials with 2-8 fields, a supercooled conformal model, and a PNJL chiral phase transition. The paper also documents optimizations for supercooled transitions and for numerically defined potentials and renormalizations.

Significance. The package fills a genuine gap: existing public codes solve the Z=1 bounce equation, whereas NLO effective actions contain a field-dependent wavefunction renormalization. The single-field NLO equation (2.6) is plausibly derived, and the Z=1 validation against three independent packages is real external support; the supercooling optimization and support for numerical input are useful features. The paper is also explicit about the restriction to positive Z and about the need for regularization in electroweak applications. However, the multi-field Z-neq-1 results are not currently supported, because the perpendicular projection equation (2.34) is dimensionally inconsistent and the multi-field equation of motion (2.27) is not the variation of (2.5). Since the headline claim that the renormalization factor changes the tunneling path rests on this multi-field algorithm, the central advertised capability is unverified as written.

major comments (3)
  1. [II.C, Eq. (2.27)] The multi-field equation of motion stated in Eq. (2.27) is not the Euler-Lagrange equation of the action in Eq. (2.5). Varying S = integral r^{D-1}[Z^{-1}|sigma'|^2/2 + V] gives sigma'' + (D-1)/r sigma' - (nabla log Z dot sigma') sigma' + (1/2) nabla log Z |sigma'|^2 = Z nabla V. Equation (2.27) instead contains only the term -(1/2) nabla log Z |sigma'|^2 and omits the projection term -(nabla log Z dot sigma') sigma'. For a single field the two terms combine into -(1/2) partial_logZ/partial_sigma sigma'^2 and Eq. (2.6) is recovered, which is why single-field tests cannot detect the problem. In multi-field tunneling the omitted term is generally nonzero, so the NLO multi-field paths and actions reported in Table I and Fig. 5 are not computed from the stated action.
  2. [II.C, Eq. (2.34)] The perpendicular projection is dimensionally inconsistent. Writing sigma(r) = sigma(x(r)) with |d sigma/dx| = 1 gives sigma' = T dx/dr and sigma'' = T d^2x/dr^2 + (dT/dx)(dx/dr)^2, with T = d sigma/dx. Projecting the correct equation perpendicular to T yields (dx/dr)^2 d^2 sigma/dx^2 + (1/2)(dx/dr)^2 nabla_perp log Z = Z nabla_perp V. Equation (2.34) instead has -(1/2)(d^2x/dr^2) nabla_perp log Z; this term has dimension 1/(L^2 [field]) while all other terms have dimension [field]/L^2. Consequently the perpendicular force N in Eq. (2.35) and the path deformation (2.38) do not enforce Eq. (2.27) when Z is not equal to 1. The Z=1 validation against FindBounce cannot detect this because the offending term vanishes for constant Z.
  3. [Table I and Fig. 5] The multi-field Z-neq-1 numbers in Table I and the modified path in Fig. 5 are the paper's main evidence for the claim that Z significantly changes the tunneling path and action. Because the algorithm's dynamical equations are the ones criticized above, these numbers cannot currently be interpreted as solutions of the stated NLO problem. A revision should correct Eqs. (2.27) and (2.34), re-run the multi-field examples, and add an independent check of the corrected multi-field solver, for example by comparing a two-field Z-neq-1 solution against a direct numerical solution of the full boundary-value problem.
minor comments (5)
  1. [Abstract] The abstract contains a typo: 'center rule' should be 'central role', and 'as well on the tunneling path' should be 'as well as on the tunneling path'.
  2. [IV.B and Fig. 5] The heading of Section IV.B reads 'tow-field tunneling' and should be 'two-field tunneling'; the caption of Fig. 5 contains the stray text 'renormalizat]on'.
  3. [IV.A, Fig. 4] In the right panel of Fig. 4 the table header reads 'packge action'; it should be 'package action'.
  4. [II.B, Eq. (2.5)] The sentence 'we would like to redefine the renormalization factor Z by 1/Z' is ambiguous; clarify whether Z_sigma in Eqs. (2.5)-(2.6) is the inverse of the function called Z in Eq. (2.1), since the same symbol is used for both.
  5. [I and Program Summary] The Introduction states that the package works 'at any dimension', while the Program Summary restricts the implementation to D = 3, 4; please reconcile these statements.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the Z=1 limit is benchmarked against independent packages, and the Z≠1 examples solve the stated bounce equations rather than being fitted to target actions.

full rationale

The paper's central derivation is self-contained. The single-field algorithm starts from the action Eq. (2.5), derives the equation of motion Eq. (2.6), and uses bisection on the initial field value to satisfy the boundary conditions; the reported action is then obtained by integrating the resulting profile, so no parameter fitted to a target action is renamed as a prediction. The multi-field path-deformation method similarly solves Eq. (2.27) by computing the perpendicular-force residual N in Eq. (2.35) and deforming the path until N=0; this is an iterative equation solver, not a construction that assumes the desired action. The Z=1 limit is checked against CosmoTransitions, FindBounce, and AnyBubble in Fig. 4 and Table I with agreement within 1%, providing genuine external grounding. There is one self-citation, Ref. [10] by Kang and Zhu, used for the supercooling model and the S3/T∼70 nucleation condition; that reference motivates an example and does not force any package output. The chiral-phase-transition example instead uses the external model [90] and matches it to 0.1%. The manuscript explicitly limits itself to Z>0 and acknowledges in Section V and the footnote on p. 24 that electroweak transitions where Z crosses zero require regularization; this is an admitted limitation rather than a circular step. The dimensional concern about Eq. (2.34) is a possible correctness error in the multi-field NLO projection, but a wrong equation is not an equivalence between input and output, so it does not constitute circularity under the review rules.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central computation rests on the derivative-expanded effective action, O(D)-symmetric bounces, positivity of Z, and several numerical convergence assumptions. No physics parameters are fitted to data; the listed free parameters are hand-chosen algorithm settings. The multi-field perpendicular-force equation is the least secure input.

free parameters (3)
  • SuperCoolingSearchWindow = 10
    Section II.D searches for the initial point in an interval of 10 times the false-vacuum-to-barrier distance. This hand-chosen factor determines which region of field space is probed in thick-wall transitions and is not derived from the action.
  • DrDivisor = 2000
    Appendix B sets the integration step to dr = DeltaR/2000, justified only by the statement that the action differs by less than 1 percent from a 1/100 finer step in most tested cases. This is an unverified numerical convergence choice.
  • RelativeAccuracy = 0.01
    The default endpoint tolerance in Eq. (2.9) and Eq. (2.10) decides when the shooting endpoint is accepted as the false vacuum and affects the reported action values. It is an adjustable algorithmic parameter, not a physical input.
assumptions (6)
  • domain assumption The effective action can be truncated to S = integral [Z(phi)/2 (d phi)^2 + V_eff(phi)], dropping higher-derivative terms.
    Used throughout and derived in Appendix A via the zero-momentum expansion. Section V notes the expansion may fail for gauge-field cases, so the bounce equation solved is an approximation.
  • standard math The bounce solution has O(D) spherical symmetry, reducing the field equation to a radial ODE.
    Used in Eq. (2.5) and Eq. (2.6). This is standard for flat-space vacuum decay, but the paper does not test its validity for the NLO kinetic term.
  • domain assumption Z(phi) is positive between the true and false vacua.
    Stated in Section II.B; required for a well-defined kinetic term and nonsingular coefficient in Eq. (2.6). Section V explicitly says this fails for electroweak phase transitions.
  • domain assumption The multi-field tunneling path can be found by iterated deformation driven by the perpendicular force N of Eq. (2.35).
    Section II.C assumes convergence of the path deformation. Because Eq. (2.34) as printed is dimensionally inconsistent, this assumption is not currently established by the paper.
  • ad hoc to paper Integration step dr = DeltaR/2000 gives action values accurate to about 1 percent.
    Appendix B states this only for 'most of our tested cases'; no complete convergence study is provided, and it is not applied to the multi-field NLO runs.
  • domain assumption In supercooled transitions, the bounce is controlled by the potential and Z near the false vacuum and the barrier.
    Section II.D restricts the search interval based on this claim. It is plausible for thick-wall bounces but is not demonstrated in the paper.

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Cite this review

Pith. "Pith review of VacuumTunneling: A package to solve bounce equation with renormalization factor." pith.science (2026). https://pith.science/paper/AAPPEGPA

@misc{pith2026250115236,
  author       = {Pith},
  title        = {Pith review of: VacuumTunneling: A package to solve bounce equation with renormalization factor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AAPPEGPA}},
  note         = {Machine review of arXiv:2501.15236}
}
abstract

The Vacuum tunneling rate $\Gamma$ from the effective action is a key to studying the cosmological first-order phase transition(FOPT). One solid way to compute the $\Gamma$ is to start with the derivative expansion of the effective action and solve the bounce equation numerically. In this process, the renormalization factor $Z$ of the tunneling field may play a center rule, which is not considered in existing packages. Therefore, we present a \texttt{Mathematica} package \vt to compute the bounce action with or without the renormalization factor. Applying the \vt package, we find that the presence of $Z$ has a significant impact on the action, as well on the tunneling path. We provide some concrete examples to demonstrate the difference between the solution with and without the renormalization factor, both in the action and tunneling path. This package is based on the modified shooting and path deformation method. We also made some optimizations for the super-cooling phase transition(thick wall scenario), in which other numerical package works poorly. This package works as long as the expressions can give values of the potential and the renormalization at a certain field point. This means the input potential and the renormalization can be merely numerical quantities without analytical expressions. The computation time can be as short as 1 second in single-field tunneling and several seconds in multi-field cases.

Figures

Figures reproduced from arXiv: 2501.15236 by the authors.

Figure 1
Figure 1. FIG. 1. The blue solid line is the effective potential, in which shooting without renomalization [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Multi-Field Algorithm Program Flowchart: The main structure consists of a large loop [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The left panel shows the curve of the action varying with [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The left panel is the comparison between the field profile given by the program in [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The left figure is the tunneling path in the field space. The blue solid line is the path [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Upper left and lower left: Figure of potential scaled to including the true and false vacuum [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The left panel shows bounce solution of effective potential described by Eq.(4.11) and [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]

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Forward citations

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