arxiv: 2604.26795 · v1 · submitted 2026-04-29 · ✦ hep-th · astro-ph.CO· gr-qc· hep-ph

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A positive definite formulation of vacuum decay with reduced symmetry

Jos\'e R. Espinosa , Ryusuke Jinno , Thomas Konstandin , Shogo Matake , Taiga Miyachi

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Pith reviewed 2026-05-07 13:01 UTC · model grok-4.3

classification ✦ hep-th astro-ph.COgr-qchep-ph

keywords vacuum decaytunneling actionO(3) symmetrypositive definitebounce solutionimpuritiesfalse vacuumtunneling potential

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The pith

An explicitly positive definite action calculates vacuum decay rates when symmetry drops to O(3) due to impurities.

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

The authors construct a method to compute the Euclidean tunneling action for vacuum decay that remains positive definite even when spherically symmetric impurities reduce the usual O(4) symmetry of the bounce to O(3). This generalizes the tunneling potential approach, which had assumed full O(4) symmetry. A reader would care because the positivity guarantees that numerical evaluations stay stable and that decay rates in the presence of defects can be calculated without unphysical negative contributions. The construction recovers the standard tunneling potential result whenever O(4) symmetry is restored and supplies explicit analytic solutions for O(3) bounces of any wall thickness.

Core claim

We present a formulation for the calculation of the tunneling decay action, that is explicitly positive definite, for impurities whose effects are spherically symmetric so that the bounce symmetry is reduced to O(3). The action constructed can be regarded as a generalization of the tunneling potential method, which implicitly assumed O(4) symmetry. We show that the action obtained reduces to the tunneling potential for O(4)-symmetric cases and provide analytic examples with O(3) symmetry and arbitrary wall thickness.

What carries the argument

The generalized positive definite tunneling potential action, built to remain non-negative for O(3)-symmetric bounces sourced by spherically symmetric impurities.

If this is right

The new action reduces exactly to the standard tunneling potential whenever O(4) symmetry holds.
Analytic O(3) solutions exist for arbitrary wall thickness.
Decay rates can be computed for impurity-catalyzed processes while preserving positivity of the action.
The formulation applies to any scalar potential whose impurity perturbations preserve spherical symmetry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same construction could serve as a starting point for perturbative treatments of weakly non-spherical impurities.
It supplies a controlled benchmark against which full 3+1 dimensional numerical bounce codes can be tested.
Applications to vacuum decay on cosmic strings or domain walls become more tractable if the method is adapted to those geometries.

Load-bearing premise

A positive definite expression for the action can be constructed by generalizing the tunneling potential method to the reduced O(3) symmetry case.

What would settle it

Direct numerical integration of the Euclidean equations of motion for an O(3) symmetric potential that yields an action value smaller than the result from the new formula.

Figures

Figures reproduced from arXiv: 2604.26795 by Jos\'e R. Espinosa, Ryusuke Jinno, Shogo Matake, Taiga Miyachi, Thomas Konstandin.

**Figure 1.** Figure 1: Potential V (ϕ, r) and tunneling potential Vt(ϕ, r) for the deformed thinwall example of subsection 4.2 with α = 1 and ϕ0 chosen to give a bubble radius R ≃ 6 view at source ↗

**Figure 2.** Figure 2: Action density s(ϕ, r) for the deformed thin-wall example of subsection 4.2 with α = 1 and ϕ0 chosen to give a bubble radius R ≃ 6. The red line shows ϕB(r, 0). chosen very close to 1 so as to give a bubble radius R ≃ 6 (a thin-wall case) view at source ↗

**Figure 3.** Figure 3: Action (normalized to the O(4) value) for the deformed thin-wall example of subsection 4.2 with ϕ0 chosen to give a bubble radius R ≃ 6, α = αV in the potential and α = αV + δαϕ in the profile of the deformed bounce. The green line shows the case with same α value in both functions. field profiles view at source ↗

read the original abstract

The Euclidean bounce for vacuum decay enjoys an $O(4)$ symmetry that is lost in the presence of impurities than can catalyze the decay. We present a formulation for the calculation of the tunneling decay action, that is explicitly positive definite, for impurities whose effects are spherically symmetric so that the bounce symmetry is reduced to $O(3)$. The action constructed can be regarded as a generalization of the tunneling potential method, which implicitly assumed $O(4)$ symmetry. We show that the action obtained reduces to the tunneling potential for $O(4)$-symmetric cases and provide analytic examples with $O(3)$ symmetry and arbitrary wall thickness.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They've built a positive-definite action for O(3)-symmetric vacuum decay that generalizes the tunneling potential method and supplies analytic examples with arbitrary wall thickness.

read the letter

The core advance is a formulation of the Euclidean bounce action that stays explicitly positive definite when impurities break the usual O(4) symmetry down to O(3). It is presented as a direct generalization of the tunneling potential approach, and the authors show that the new action recovers the older O(4) result exactly when spherical symmetry is restored. They also give analytic O(3) solutions that hold for any wall thickness, which is a concrete step beyond thin-wall or numerical-only treatments.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a positive-definite formulation for the tunneling action in vacuum decay when the O(4) symmetry of the Euclidean bounce is reduced to O(3) due to spherically symmetric impurities. This is achieved by generalizing the tunneling potential method, with the new action shown to reduce exactly to the O(4) case and illustrated with analytic examples for arbitrary wall thicknesses.

Significance. Should the derivation hold, this provides a valuable extension for computing decay rates in the presence of impurities that break the full rotational symmetry, potentially enabling more realistic models of catalyzed vacuum decay. The explicit positivity and the reduction property are key strengths, as is the inclusion of analytic O(3) examples which allow direct verification.

minor comments (2)

[Abstract] The abstract mentions analytic examples but does not specify the form of the potential or the impurity; including a brief indication would help readers assess the generality.
[Section 5] The transition from the general O(3) formulation to the specific analytic examples could be expanded with an intermediate step showing how the wall-thickness parameter enters the reduced action.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recommending minor revision. The referee's description accurately captures our development of a positive-definite action for O(3)-symmetric vacuum decay that reduces to the standard tunneling potential method under restored O(4) symmetry.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper constructs an explicitly positive-definite action for O(3)-symmetric vacuum decay by generalizing the independent tunneling-potential method (which assumed O(4)). It asserts and demonstrates reduction to the known O(4) case plus supplies analytic O(3) examples with arbitrary wall thickness. No load-bearing step reduces by definition, fitted input, or self-citation chain to the target result itself; the central claim is a new formulation whose correctness is externally verifiable against the O(4) limit and explicit solutions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable from the given text.

pith-pipeline@v0.9.0 · 5423 in / 1026 out tokens · 64382 ms · 2026-05-07T13:01:48.389631+00:00 · methodology

discussion (0)

Reference graph

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