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arxiv: 2510.04283 · v3 · submitted 2025-10-05 · ✦ hep-th · math-ph· math.MP· nlin.PS

Long-time behaviour of sphalerons in φ⁴ models with a false vacuum

Stephen C. Anco , Danial Saadatmand This is my paper

Pith reviewed 2026-05-18 09:56 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MPnlin.PS
keywords sphaleronfalse vacuumkink-antikink paircollective coordinatesKlein-Gordon modelasymptotic analysisgradient blow-uplong-time dynamics
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The pith

A growing perturbation turns a sphaleron into an accelerating kink-antikink pair whose fronts approach light speed.

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

In quartic Klein-Gordon models possessing both a true vacuum and a false vacuum, sphalerons appear as unstable lump solutions located at the saddle point of the energy functional. Numerical simulations demonstrate that a small positive perturbation causes the sphaleron to evolve into a kink-antikink pair whose separation grows without bound and whose speeds asymptotically reach the speed of light. To obtain an analytic description, the authors develop a nonlinear collective coordinate reduction that employs three time-dependent parameters and yields an explicit power-series solution at late times. This solution shows a central tabletop region whose height rises toward the true vacuum while the connecting flanks become steeper, accelerate outward, and concentrate the energy density.

Core claim

Numerical simulations show the sphaleron evolving into an accelerating kink-antikink pair whose separation increases in time and asymptotically approaches the speed of light. The nonlinear collective coordinate method with three dynamical parameters yields an explicit asymptotic solution describing a spreading tabletop profile whose height approaches the true vacuum while its flanks steepen and accelerate outward. In addition, the energy density concentrates at the flanks, indicating the onset of a gradient blow-up at large times.

What carries the argument

The nonlinear collective coordinate method with three dynamical parameters, which reduces the field dynamics to ordinary differential equations and permits a power-series asymptotic analysis of the late-time spreading profile.

If this is right

  • The kink-antikink separation grows continuously and approaches light-speed separation at late times.
  • The central region of the field configuration flattens and approaches the true vacuum value.
  • The outward flanks accelerate and steepen while carrying most of the energy density.
  • The overall structure expands without bound, providing an explicit example of relativistically expanding fronts in nonlinear scalar theories.

Where Pith is reading between the lines

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

  • The same late-time mechanism may govern the expansion of near-sphaleron configurations in other scalar models that possess a false vacuum.
  • The predicted concentration of gradients at the fronts could be examined by adding small dissipative or higher-order derivative terms to test regularization.
  • Coupling the model to gravity or to additional fields might change the terminal speed reached by the accelerating flanks.

Load-bearing premise

The three-parameter collective coordinate ansatz remains a faithful reduced description of the full field dynamics at arbitrarily late times, even after the fronts have become relativistic and the energy has concentrated at the flanks.

What would settle it

A high-resolution numerical evolution of the original partial differential equation to much later times that measures whether the maximum field gradient at the fronts continues to increase without bound or saturates at a finite value.

Figures

Figures reproduced from arXiv: 2510.04283 by Danial Saadatmand, Stephen C. Anco.

Figure 1
Figure 1. Figure 1: Potential with a false vacuum at ϕ = 0: a = 0.5, 0.8, 1.5 with the non-symmetric potential V (ϕ) = 2ϕ 2 (ϕ − tanh(a))(ϕ − coth(a)), a > 0 (2) in terms of a parameter a. In particular, every quartic potential with a false vacuum belongs to this 1-parameter family, up to a shift, scaling, and reflection on ϕ, and a dilation on (t, x). (See e.g. Ref. [16, 22].) The false vacuum is V = 0 at ϕ = 0, and the true… view at source ↗
Figure 2
Figure 2. Figure 2: Perturbation potential: a = 0.25, arctanh √ 1 3 (=0.658), 1.5 Here Hℓ is a local Heun function, which satisfies the Heun equation H ′′(z) + γ/z + δ/(z − 1) + ϵ/(z − p)  H ′ (z) + (αβz − q)/(z(z − 1)(z − p)  H(z) = 0 (28) with ϵ = α+β−γ−δ+1. This is a second order linear differential equation which has regular singular points z = 0, 1, p, ∞, where α, β, γ, δ, and q are constant parameters. Local Heun func… view at source ↗
Figure 3
Figure 3. Figure 3: Ground-state eigenfunction, normalized in L 2 , for a = 2.65, 2.14, 1.44, 1.00, 0.881, 0.647, 0.100 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Ground-state eigenvalue: λ−1 Ref. [22] also develops approximate expressions for both the eigenvalue and eigenfunction. Here we will adapt them to get simpler, albeit rougher, approximations: η−1(x) ≈ ( sech(x) 3 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Numerical solution for a = 1.174. (Left) Short times t = 0, 2, 4, 5; (Right) Long times t = 10, 30, 40, 80. -8 -6 -4 -2 0 2 4 6 8 0. 0 0. 2 0. 4 0. 6 0. 8 1 . 0 -80 -60 -40 -20 0 20 40 60 80 0. 0 0. 2 0. 4 0. 6 0. 8 1 . 0 x x [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Numerical solution for a = 1.5. (Left) Short times t = 0, 3, 5, 7; (Right) Long times t = 10, 30, 50, 80. ϵ = 0.01, 0.03, 0.05, and 0.10. In general we see that the tabletop forms more quickly as the perturbation parameter increases in size. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Numerical solution for a = 2.0. (Left) Short times t = 0, 20, 30; (Right) Long times t = 20, 40, 80, 100, 120. -8 -6 -4 -2 0 2 4 6 8 0. 0 0. 2 0. 4 0. 6 0. 8 1 . 0 -60 -40 -20 0 20 40 60 0. 0 0. 2 0. 4 0. 6 0. 8 1 . 0 x x [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Numerical solution for a = 1.5 using ϵ = 0.01, 0.03, 0.05, and 0.10. (Left) t = 5 (Right) t = 80 3.1. Numerical solutions with initial growth. An alternative way to excite the unstable growing mode is by taking initial data at t = 0 that has non-zero growth of ϕ while matching 10 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Numerical solution for a = 1.174 with initial data (36). (Left) Short times t = 0, 2, 4, 5, 7; (Right) Long times t = 0, 10, 30, 40, 80. 4. Collective coordinates for sphalerons The goal now will be to obtain an approximate analytical expression for the behaviour seen in the numerical solutions at large times. We will start from the kink-antikink form (15) for the lump solution with x0 = 0 and consider a c… view at source ↗
Figure 10
Figure 10. Figure 10: Numerical solution for a = 1.5 with initial data (36). (Left) Short times t = 0, 3, 5, 7, 10; (Right) Long times t = 0, 10, 30, 50, 80. Note that the time derivative of expression (37) is given by ϕt(x, t) =1 2A ′ (t) [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: b = 0.998 (a = 2.0): (Left) ϕ; (Right) ϕt . Dark green represents the initial data of the sphaleron; Brown represents the series solu￾tion. (36b). (See Appendix B.) This determines B0 = 3 2 b 2 (Elump + Eperturb) (92) so the resulting series has just one free parameter, namely T. To impose all of the preceding conditions (88)–(91) in practice, we first expand the series (74)–(76) in powers of 1/t, and tru… view at source ↗
Figure 12
Figure 12. Figure 12: (Left) Series solution for b = 0.998 (a = 2.0). (Right) Error plot at times t ′ = 0, 2, 10, 30, 50, 80, 110 [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: b = 0.998 (a = 2.0): (Left) Short times t ′ = 0, 0.2, 0.5, 0.75 (Right) Long times t ′ = 0, 2, 10, 30, 50, 80 5.1.3. Example b = 0.7 (a = 0.616). We find T = −1.835 and t0 = 1.617 [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: b = 0.95 (a = 1.174): (Left) ϕ; (Right) ϕt . Dark green represents the initial data of the sphaleron; Brown represents the series solu￾tion [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: (Left) Series solution for b = 0.95 (a = 1.174). (Right) Error plot at times t ′ = 0, 2, 10, 30, 50, 80, 110. 5.2. Concentration of energy. Plots of the conserved energy density on a logarithmic scale, ln E, at times t ′ = 0, 2, 10, 30, 50 are provided in Figs. 23 and 24. The energy density quickly concentrates at the position of the flanks. 6. Discussion and concluding remarks The long-time evolution of … view at source ↗
Figure 16
Figure 16. Figure 16: b = 0.95 (a = 1.174): (Left) Short times t ′ = 0, 0.5, 0.8, 1.3; (Right) Long times t ′ = 0, 2, 10, 30, 50, 80 [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: b = 0.7 (a = 0.616): (Left) ϕ; (Right) ϕt . Dark green represents the initial data of the sphaleron; Brown represents the series solu￾tion. • Numerical evolution of sphalerons under a growing perturbation is shown to yield an accelerating kink-antikink pair whose height is equal to the value of the true vacuum and whose width increasingly expands. • An analytical approximation is derived by a nonlinear co… view at source ↗
Figure 18
Figure 18. Figure 18: b = 0.7 (a = 0.616) [PITH_FULL_IMAGE:figures/full_fig_p023_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: (Left) Short times t ′ = 0, 0.2, 0.5, 1.0; (Right) Long times t ′ = 0, 2, 10, 30, 50, 80 23 [PITH_FULL_IMAGE:figures/full_fig_p023_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: b = 0.5 (a = 0.402): (Left) ϕ; (Right) ϕt . Dark green represents the initial data of the sphaleron; Brown represents the series solu￾tion [PITH_FULL_IMAGE:figures/full_fig_p024_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: b = 0.5 (a = 0.402) 24 [PITH_FULL_IMAGE:figures/full_fig_p024_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: (Left) Short times t ′ = 0, 0.1, 0.2, 0.5; (Right) Long times t ′ = 0, 2, 10, 30, 50, 80 [PITH_FULL_IMAGE:figures/full_fig_p025_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Energy density ln E for b = 0.998 (Left) and b = 0.95 (Right) Mathematically, the approximate analytical solution exhibits a gradient blow up for long times and thus belongs to energy space but is not in H1 . For future investigation, it would be of particular interest to derive an analytical ap￾proximation for the large-time behaviour of the sphaleron in the other a channel where the perturbation leads t… view at source ↗
Figure 24
Figure 24. Figure 24: Energy density ln E for b = 0.7 (Left) and b = 0.5 (Right) The fact that a perturbed sphaleron solution can evolve into two different final states naturally leads to the question of what is the outcome of a collision between two sphalerons. In a subsequent paper, the scattering of two sphalerons with a false vacuum will be studied. Appendix A. Derivations A.1. Effective action. To evaluate the nonlinear K… view at source ↗
read the original abstract

Evolution of sphalerons in a class of quartic Klein-Gordon models are studied under a growing perturbation. Sphalerons are unstable lump-like solutions that arise from a saddle point between true and false vacua in the energy functional. Numerical simulations are presented which show the sphaleron evolving into an accelerating kink-antikink pair whose separation increases in time and asymptotically approaches the speed of light. To explain this behaviour analytically, a nonlinear collective coordinate method is developed which has three dynamical parameters and leads to an explicit asymptotic solution using a power series expansion. The solution describes the emergence of a spreading tabletop profile whose height approaches the true vacuum while its flanks steepen and accelerate outward. In addition, the energy density is shown to concentrate at the flanks, indicating the onset of a gradient blow-up at large times. These results provide a detailed description of the long-time dynamics of positively perturbed sphalerons, and reveal a universal mechanism for the formation of relativistically expanding structures in nonlinear field theories.

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 / 2 minor

Summary. The manuscript examines the long-time evolution of sphalerons in quartic Klein-Gordon models with a false vacuum. Numerical simulations show a positively perturbed sphaleron developing into an accelerating kink-antikink pair whose separation grows and asymptotically approaches the speed of light. A nonlinear collective-coordinate reduction with three dynamical parameters is introduced; its equations are derived from the field equation and solved via power-series expansion to obtain an explicit asymptotic solution describing a spreading tabletop profile whose height approaches the true vacuum, with steepening flanks that accelerate outward and with energy density concentrating at the flanks, indicating the onset of gradient blow-up.

Significance. If the central claims hold, the work supplies both numerical evidence and an explicit analytic asymptotic description of sphaleron instability leading to relativistically expanding structures. The combination of a three-parameter collective-coordinate ansatz with a power-series solution is a constructive approach that yields concrete predictions for the late-time profile and energy localization; these features could prove useful for understanding analogous phenomena in other nonlinear field theories.

major comments (1)
  1. [Collective coordinate reduction and asymptotic expansion] The validity of the three-parameter collective-coordinate ansatz at arbitrarily late times is load-bearing for the explicit asymptotic solution and the claims of height approaching the true vacuum and gradient blow-up. Once the fronts become relativistic and energy concentrates at the steepening flanks (as reported in the numerical simulations), it is not demonstrated that radiation or modes orthogonal to the ansatz remain negligible. A direct comparison of the reduced dynamics against the full Klein-Gordon evolution at large times, or an estimate of the projection onto orthogonal modes, would be required to substantiate that the power-series solution remains faithful.
minor comments (2)
  1. The abstract states that the separation 'asymptotically approaches the speed of light' but does not specify the precise functional form (e.g., 1 - v ~ t^α) obtained from the power series; adding this would improve clarity.
  2. Notation for the three collective coordinates and the potential parameters should be introduced with a single consistent table or list early in the text to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment below and are prepared to revise the paper accordingly to strengthen the substantiation of the collective-coordinate results.

read point-by-point responses

  1. Referee: The validity of the three-parameter collective-coordinate ansatz at arbitrarily late times is load-bearing for the explicit asymptotic solution and the claims of height approaching the true vacuum and gradient blow-up. Once the fronts become relativistic and energy concentrates at the steepening flanks (as reported in the numerical simulations), it is not demonstrated that radiation or modes orthogonal to the ansatz remain negligible. A direct comparison of the reduced dynamics against the full Klein-Gordon evolution at large times, or an estimate of the projection onto orthogonal modes, would be required to substantiate that the power-series solution remains faithful.

    Authors: We thank the referee for identifying this key point. The three-parameter ansatz was constructed precisely to encode the dominant late-time features seen in the full numerical evolution: the tabletop height approaching the true vacuum, the outward acceleration of the flanks, and the concentration of energy density there. To address concerns about orthogonal modes and radiation, we will add a new subsection that directly compares the collective-coordinate solution against the full Klein-Gordon field evolution at progressively later times, including regimes where the fronts are relativistic. We will also include a quantitative estimate of the projection of the numerical solutions onto the subspace orthogonal to the ansatz, obtained by decomposing the simulated field configurations. These additions will show that deviations remain small up to the times accessible in our simulations and that the power-series asymptotic form continues to capture the leading behavior. We agree that such evidence is necessary to support the claims and will incorporate it in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity: reduced dynamics derived from field equation projection, not forced by fit or self-citation

full rationale

The paper presents independent numerical simulations of the full Klein-Gordon dynamics showing the sphaleron evolving into an accelerating kink-antikink pair. It then introduces a three-parameter collective coordinate ansatz and derives the reduced ODEs by projecting the field equation onto the ansatz manifold. The asymptotic power-series solution is obtained directly from those reduced equations. This is a standard variational reduction whose output is not equivalent to its inputs by construction, nor does it rely on self-citation chains or renaming of known results. The assumption that the ansatz remains accurate at late times is a modeling limitation affecting correctness, not a circularity in the derivation chain itself.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a false-vacuum potential in the quartic model, the validity of the collective-coordinate truncation at late times, and the assumption that the initial perturbation grows in a manner that excites the unstable mode without introducing other instabilities.

free parameters (1)
  • three collective coordinates
    Width, front position, and amplitude factor introduced as time-dependent parameters in the ansatz; their evolution equations are derived but the truncation error is not quantified.
axioms (1)
  • domain assumption The field equation is the quartic Klein-Gordon equation with a double-well potential possessing a false vacuum.
    Stated in the title and abstract as the class of models under study.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Collision Dynamics of False-Vacuum Oscillons

    hep-th 2026-05 unverdicted novelty 6.0

    Oscillon collisions in false-vacuum scalar theories produce reflection, crossing, resonance windows, and can initiate true-vacuum phase transitions when energy suffices.

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