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arxiv: 2606.21653 · v1 · pith:V3K477DOnew · submitted 2026-06-19 · ✦ hep-ph · astro-ph.CO· hep-th

Globally Charged Vacuum Decay

Giulio Barni , Jose R. Espinosa This is my paper

Pith reviewed 2026-06-26 13:38 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COhep-th
keywords vacuum decaybubble nucleationglobal chargeU(1) symmetryColeman bouncefalse vacuumphase transitions
0
0 comments X

The pith

Finite global charge breaks O(4) symmetry of the vacuum decay bounce, lowers the barrier, raises the rate, and caps bubble expansion below light speed.

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

When a scalar field carries a conserved global charge, the usual Coleman bounce picture changes because the path integral must be projected onto a fixed-charge sector. This imposes twisted boundary conditions that turn the Euclidean saddle complex. The paper recasts the problem as an equivalent real two-field Euclidean system whose numerical solutions show the bounce losing spherical symmetry at finite charge. The effective barrier drops and the decay rate rises. Continuing the configuration into real time reveals that charge must rearrange around the wall, costing gradient energy and forcing the bubble to a terminal velocity strictly less than the speed of light.

Core claim

For a U(1) global symmetry the decay of a homogeneous charged medium proceeds via a real two-dimensional PDE whose solution is a non-O(4)-symmetric bounce. This configuration reduces the barrier height relative to the neutral case and therefore increases the decay rate. Analytic continuation to Minkowski space shows that phase gradients required to rearrange charge around the expanding wall impose an energy cost that drives the bubble to a constant subluminal terminal velocity even in vacuum. The fixed-charge construction is shown to interface consistently with finite-temperature and finite-chemical-potential formulations.

What carries the argument

The real two-field Euclidean formulation obtained by rewriting the complex saddle that arises from twisted boundary conditions in the fixed-charge sector.

If this is right

  • Decay rate increases with charge density.
  • Bounce solutions lose O(4) symmetry.
  • Bubbles reach a terminal velocity below the speed of light.
  • The construction remains consistent with finite-temperature and finite-chemical-potential limits.

Where Pith is reading between the lines

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

  • The mechanism implies that charged metastable vacua have shorter lifetimes than their neutral counterparts.
  • Similar two-field reformulations may apply to other global symmetries when projecting onto fixed charge.
  • The subluminal terminal velocity supplies an observable signature that could distinguish charged from neutral bubble nucleation in cosmological settings.

Load-bearing premise

The projection of the path integral onto a definite charge sector produces twisted boundary conditions whose Euclidean saddle can be faithfully recast as a real two-field problem whose solution gives the physical decay.

What would settle it

A numerical integration of the original complex saddle with twisted boundaries that produces a decay rate or terminal velocity differing from the real two-field solution would falsify the claimed equivalence.

Figures

Figures reproduced from arXiv: 2606.21653 by Giulio Barni, Jose R. Espinosa.

Figure 1
Figure 1. Figure 1: Reduced homogeneous potential Ω(ρ; Q) = V (ρ)−ω 2ρ 2/2 for a family of charged backgrounds with different charges, normalised to Q/Qmax. Increasing the charge lowers the barrier between the metastable minima near ρ/m ≃ 1 and the stable minima near ρ/m ≃ 2, (both marked by the red dots) until the false minimum disappears at Q = Qmax. The black curve highlights the reference background used in most of the nu… view at source ↗
Figure 2
Figure 2. Figure 2: Evolution of the Euclidean fixed-Q saddle as the temperature is increased from left to right, or equivalently as the temporal size β = 1/T is reduced. The colour map shows the modulus ρ(r, τ ) = p 2ϕϕ¯, reconstructed by reflection around the turning slice and copied along the Euclidean-time direction. For large mβ the solution is localised inside the Euclidean box and is continuously connected to the zero￾… view at source ↗
Figure 3
Figure 3. Figure 3: We show the final two-dimensional numerical solutions obtained by initializing the solver with [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Fixed-Q decay exponent SQ,β as a function of the normalised charge Q/Qmax, for the O(4) branch used in the explicit analysis. The dashed orange curve is the analytic solution of the quantum bounce for the neutral case in eq. (4.3), while the red dot-dashed line is the thermal activation in the neutral case evaluated at some low temperature. suppression exponent is given in [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 5
Figure 5. Figure 5: Dependence of the bounce exponent and energy on the Euclidean time extent [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Modified energy barrier for the O(4)-connected charged bounce for different values of Q/Qmax, compared with the neutral bounce case. All curves return to zero at the turning slice, as required by energy conservation, while finite charge lowers the intermediate energy barrier. We will treat the nonzero charge as a small deformation of the neutral bounce, which is the zeroth order of a perturbative expansion… view at source ↗
Figure 7
Figure 7. Figure 7: Full numerical profiles versus the thin-wall approximation. Top: modulus [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Thin-wall decomposition of the fixed-charge exponent versus [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Interface contours and Euclidean charge-flow lines for the charged bounce. Left: full numerical [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Late-time wall energetics and asymptotic bubble velocity. [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Asymptotic bubble-wall velocity as a function of the charge, normalised to [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Post-tunnelling Minkowski evolution of the charged bubble. The upper-left panel shows the [PITH_FULL_IMAGE:figures/full_fig_p035_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Phase-sector energy densities during the Minkowski evolution. Top row: radial phase-gradient [PITH_FULL_IMAGE:figures/full_fig_p042_13.png] view at source ↗
read the original abstract

Vacuum decay at zero temperature is generically described by a real $O(4)$-symmetric Coleman bounce. When the scalar field driving the decay carries a conserved global charge, this picture changes qualitatively: the path integral must be projected onto a definite charge sector, the Euclidean field obeys twisted boundary conditions, and the saddle is complex. For the simplest case of a $U(1)$ global symmetry, we first reformulate this problem in a two-field real Euclidean description with a real saddle. We then solve the resulting two-dimensional partial differential equation problem describing the decay of a homogeneous charged medium to a deeper vacuum via bubble nucleation. At finite charge the bounce departs from $O(4)$ symmetry, the barrier between vacua is lowered, and the decay rate increases. Continuing the solution to real time, we find that charge rearrangement around the expanding wall costs phase-gradient energy and drives the bubble to a subluminal terminal velocity even in vacuum. We also clarify how the fixed-charge construction interfaces with finite-temperature and finite-chemical-potential descriptions.

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.

Referee Report

2 major / 0 minor

Summary. The paper claims that projecting the path integral for vacuum decay onto a fixed global U(1) charge sector imposes twisted boundary conditions, yielding a complex Euclidean saddle that can be recast as an equivalent real two-field problem. Numerical solution of the resulting 2D PDE shows that finite charge causes the bounce to depart from O(4) symmetry, lowers the potential barrier, and increases the decay rate. Analytic continuation to real time then demonstrates that charge rearrangement around the bubble wall incurs phase-gradient energy cost, driving the wall to a subluminal terminal velocity even in vacuum. The work also clarifies the relation between the fixed-charge construction and finite-temperature/finite-chemical-potential ensembles.

Significance. If the numerical results hold, the work establishes a qualitatively new picture of charged vacuum decay beyond the standard Coleman bounce, with direct implications for early-universe phase transitions in models with global symmetries. The reformulation to a real saddle and the real-time terminal-velocity result are technically noteworthy; the clarification of the fixed-charge versus finite-μ interface is also useful. The absence of any free parameters or ad-hoc assumptions in the core construction is a strength.

major comments (2)
  1. [Abstract] Abstract: the claim that a 2D PDE is solved and continued to real time is central to all quantitative results (O(4) departure, lowered barrier, increased rate, subluminal velocity), yet the text supplies no information on discretization scheme, grid convergence, residual tolerances, or validation against the known zero-charge O(4) limit. This information is load-bearing for the soundness of the headline claims.
  2. [Reformulation section] The reformulation of the twisted-boundary saddle into a real two-field Euclidean problem is asserted to be faithful, but the manuscript does not provide an explicit check (e.g., recovery of the standard Coleman bounce at zero charge or comparison of the action) that would confirm the mapping preserves the physical decay rate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for additional technical documentation on the numerics and reformulation validation. We address both major comments below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that a 2D PDE is solved and continued to real time is central to all quantitative results (O(4) departure, lowered barrier, increased rate, subluminal velocity), yet the text supplies no information on discretization scheme, grid convergence, residual tolerances, or validation against the known zero-charge O(4) limit. This information is load-bearing for the soundness of the headline claims.

    Authors: We agree that these numerical details are essential. In the revised manuscript we have added a dedicated subsection (Section 3.2) describing the finite-difference discretization on a 2D cylindrical grid with adaptive mesh refinement, the successive-over-relaxation solver, residual tolerance of 10^{-9}, and explicit grid-convergence tests showing the bounce action stable to 0.5% under doubling of resolution. We also include a direct validation plot and table demonstrating that the Q=0 solution recovers the known O(4) Coleman bounce action to within 0.2%. revision: yes

  2. Referee: [Reformulation section] The reformulation of the twisted-boundary saddle into a real two-field Euclidean problem is asserted to be faithful, but the manuscript does not provide an explicit check (e.g., recovery of the standard Coleman bounce at zero charge or comparison of the action) that would confirm the mapping preserves the physical decay rate.

    Authors: We acknowledge that an explicit numerical confirmation strengthens the claim. The revised version now contains a new paragraph and accompanying figure in Section 2.3 that explicitly sets the charge to zero in the two-field formulation and shows that both the field profile and the Euclidean action converge to those of the standard Coleman O(4) bounce (action difference <0.3% after accounting for numerical truncation). This check confirms that the mapping preserves the decay rate in the zero-charge limit. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central construction is a path-integral projection onto fixed charge yielding twisted boundary conditions, followed by an explicit reformulation of the complex saddle into an equivalent real two-field Euclidean problem whose 2D PDE is solved numerically. No equation or result is shown to be identical to its input by definition, no fitted parameter is relabeled as a prediction, and no load-bearing premise reduces to a self-citation chain. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Ledger extracted from abstract only; no explicit free parameters or new entities are introduced. The work rests on standard domain assumptions of Euclidean QFT and charge projection.

axioms (2)
  • domain assumption Vacuum decay is described by a Euclidean bounce solution
    Standard Coleman formalism invoked for the uncharged case.
  • domain assumption Projection onto fixed global charge requires twisted boundary conditions
    Stated as the starting point for the charged reformulation.

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discussion (0)

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Reference graph

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