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arxiv: 2606.19917 · v1 · pith:AKQCPWKWnew · submitted 2026-06-18 · ✦ hep-th · hep-ph

Metastable and critical-bubble branches of Coleman--Weinberg monopoles

Sumit Shaw This is my paper

Pith reviewed 2026-06-26 16:13 UTC · model grok-4.3

classification ✦ hep-th hep-ph
keywords Coleman-Weinberg monopolesmetastable monopolescritical bubbleHiggs-gauge systemradial Hessian spectrumradiative symmetry breakingstatic energy functional
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The pith

The metastable Coleman-Weinberg monopole remains locally stable until its lowest radial Hessian eigenvalue reaches zero at the critical rescaled scalar mass μ_c=0.064352(1), where a saddle-point monopole-critical-bubble branch appears.

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

This paper constructs the static monopole-critical-bubble configuration in the full coupled radial Higgs-gauge system and shows that it is a saddle of the static energy functional. It characterizes the metastable monopole and monopole-critical-bubble branches by their profiles, energies, and radial Hessian spectra. The monopole-bubble solution carries a negative radial mode, while the metastable monopole remains locally stable until its lowest radial Hessian eigenvalue approaches zero. The resulting branch structure supplies a direct static picture of how Coleman-Weinberg monopoles lose metastability when radiative symmetry breaking renders the broken vacuum metastable.

Core claim

The monopole-critical-bubble configuration in the full coupled radial Higgs-gauge system is a saddle of the static energy functional. The monopole-bubble solution carries a negative radial mode, while the metastable monopole remains locally stable until its lowest radial Hessian eigenvalue approaches zero at the critical rescaled scalar mass parameter μ_c=0.064352(1). This branch structure supplies a direct static picture of how Coleman-Weinberg monopoles lose metastability.

What carries the argument

The radial Hessian spectrum of the static energy functional evaluated on the monopole and monopole-bubble profiles in the coupled Higgs-gauge system.

If this is right

  • The monopole-bubble branch connects to the metastable monopole at the point where the lowest radial eigenvalue reaches zero.
  • The monopole-bubble configuration has exactly one negative radial mode and is therefore a saddle.
  • The two branches are distinguished by their radial profiles, total energies, and radial Hessian spectra.
  • The critical value is given numerically as μ_c=0.064352(1).

Where Pith is reading between the lines

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

  • The static saddle picture implies that the decay of the metastable monopole proceeds through the critical-bubble configuration, which could be checked by constructing the corresponding instanton or by evolving the fields in real time.
  • Extending the Hessian analysis to include angular dependence might reveal whether additional negative modes appear at the same critical value or at a shifted value.
  • The numerical value of μ_c could serve as a benchmark for other numerical or analytic approximations to the same coupled system in different gauges or truncations.

Load-bearing premise

That the radial Hessian spectrum alone determines the stability threshold of the monopole branch, without angular modes or full time-dependent dynamics altering the conclusion that the eigenvalue zero-crossing marks loss of metastability.

What would settle it

A computation of the full Hessian spectrum that includes angular modes and finds a different zero-crossing value for the lowest eigenvalue, or a time-dependent simulation showing continued stability past μ_c=0.064352(1), would falsify the claim that the radial eigenvalue crossing alone marks the loss of metastability.

Figures

Figures reproduced from arXiv: 2606.19917 by Sumit Shaw.

Figure 1
Figure 1. Figure 1: Coleman–Weinberg potential used in the radial monopole equations. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Radial profiles for the metastable monopole (MM), the monopole– [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Energy branch structure of the Coleman–Weinberg monopole sys [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Softening of the lowest radial Hessian eigenvalue near the branch [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
read the original abstract

We revisit the Coleman--Weinberg monopole problem introduced by Kiselev, where radiative symmetry breaking makes the broken vacuum metastable. We construct the associated static monopole--critical-bubble configuration in the full coupled radial Higgs--gauge system and show that it is a saddle of the static energy functional. The metastable monopole and monopole--critical-bubble branches are characterized by their profiles, energies, and radial Hessian spectra. The monopole--bubble solution carries a negative radial mode, while the metastable monopole remains locally stable until its lowest radial Hessian eigenvalue approaches zero. The resulting branch structure gives a direct static picture of how Coleman--Weinberg monopoles lose metastability, with critical rescaled scalar mass parameter \(\mu_c=0.064352(1)\).

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

1 major / 0 minor

Summary. The paper constructs the static monopole-critical-bubble solution in the coupled radial Higgs-gauge system for the Coleman-Weinberg monopole with a metastable broken vacuum. It characterizes the metastable monopole and monopole-bubble branches through profiles, energies, and radial Hessian spectra, showing that the bubble is a saddle with a negative radial mode while the monopole remains locally stable until its lowest radial Hessian eigenvalue reaches zero at the critical value μ_c=0.064352(1). This supplies a direct static picture of the loss of metastability along the monopole branch.

Significance. If the result holds, the work supplies a concrete numerical realization of the branch structure connecting a metastable monopole to a critical bubble in a radiatively broken gauge theory, together with a high-precision determination of the critical parameter. The explicit solution of the full coupled boundary-value problem and the reported radial spectra represent a technical advance in the study of non-perturbative instabilities.

major comments (1)
  1. [Abstract] Abstract and branch-structure description: the stability threshold is identified exclusively with the zero-crossing of the lowest eigenvalue in the radial (l=0) sector of the Hessian at μ_c=0.064352(1). For a spherically symmetric background the second-variation operator decomposes into independent angular-momentum sectors l=0,1,2,…. The manuscript supplies no evidence that the lowest eigenvalue in any l≥1 sector remains positive up to this μ; an earlier crossing in an l≥1 sector would alter the actual point at which the monopole ceases to be a local minimum, changing the claimed branch structure.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to address all angular-momentum sectors in the Hessian. We respond to the single major comment below and will make the corresponding changes.

read point-by-point responses

  1. Referee: [Abstract] Abstract and branch-structure description: the stability threshold is identified exclusively with the zero-crossing of the lowest eigenvalue in the radial (l=0) sector of the Hessian at μ_c=0.064352(1). For a spherically symmetric background the second-variation operator decomposes into independent angular-momentum sectors l=0,1,2,…. The manuscript supplies no evidence that the lowest eigenvalue in any l≥1 sector remains positive up to this μ; an earlier crossing in an l≥1 sector would alter the actual point at which the monopole ceases to be a local minimum, changing the claimed branch structure.

    Authors: We agree that the present manuscript restricts the Hessian analysis to the radial (l=0) sector and supplies no data on l≥1 sectors. The claim that the monopole remains locally stable up to μ_c therefore requires additional verification. We will revise the manuscript by extending the fluctuation operator to include angular dependence and by computing the lowest eigenvalues in the l=1 and l=2 sectors as functions of μ. The revised version will present these results (in a new subsection or appendix) and confirm that the eigenvalues remain positive through μ_c, so that the reported critical value is indeed the point at which local stability is lost. The abstract and the discussion of the branch structure will be updated accordingly. revision: yes

Circularity Check

0 steps flagged

No circularity: μ_c obtained by direct numerical solution of ODE boundary-value problem

full rationale

The paper constructs the monopole and monopole-bubble solutions by solving the coupled radial ODEs from the static energy functional, then computes the radial (l=0) Hessian spectrum on those backgrounds. The value μ_c=0.064352(1) is the parameter at which the lowest radial eigenvalue crosses zero; this is a standard numerical root-finding procedure on the linearized operator and does not reduce by the paper's own equations to a quantity defined in terms of itself or to a fitted parameter that is then relabeled a prediction. No self-citation chains, ansatzes smuggled via prior work, or uniqueness theorems imported from the same authors appear in the derivation. The central claim therefore remains self-contained against external benchmarks (the equations of motion and the definition of the second-variation operator).

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard Coleman-Weinberg potential and the assumption that static radial solutions plus radial Hessian capture the relevant stability physics; no free parameters or invented entities are introduced beyond the model definition.

axioms (1)
  • domain assumption The Coleman-Weinberg potential is taken as the effective potential generated by radiative corrections in the model.
    Standard starting point for the revisited Kiselev problem.

pith-pipeline@v0.9.1-grok · 5644 in / 1226 out tokens · 19962 ms · 2026-06-26T16:13:39.572295+00:00 · methodology

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Reference graph

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