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arxiv: 2604.07713 · v1 · submitted 2026-04-09 · ✦ hep-th

Recognition: 2 theorem links

· Lean Theorem

Linearized Q-Ball Perturbations

Jarah Evslin , Hui Liu , Tomasz Roma\'nczukiewicz , Yakov Shnir , Andrzej Wereszczy\'nski , Piotr Ziobro
Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:11 UTC · model grok-4.3

classification ✦ hep-th
keywords Q-ballslinearized perturbationsFloquet modesquasinormal modesPöschl-Teller potentialbreathersoscillons
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The pith

Small deformations around low-amplitude Q-balls admit closed-form solutions for their modes.

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

The paper demonstrates that linearized deformations of a low-amplitude Q-ball in one spatial dimension break down into relativistic modes resembling plane waves and long-wavelength Floquet modes that either co-rotate or counter-rotate with the Q-ball. These modes come in pairs with mirror frequencies whose average equals the Q-ball frequency. At the leading order in the small amplitude, closed-form expressions exist for all such modes except for one bound mode that only appears at higher orders. The counterrotating modes arise from a specific potential and involve mixing that creates quasinormal modes rather than bound states. This matters for determining the stability and possible radiation from these soliton-like objects.

Core claim

Linearized deformations of thick-walled low-amplitude (1+1)-dimensional Q-balls may be decomposed into relativistic modes, which are roughly plane waves, and long-wavelength corotating and counterrotating Floquet modes. Each mode oscillates at a pair of mirror frequencies which average to the Q-ball frequency. The corotating modes are those of a breather or oscillon plus a very loosely bound mode. The counterrotating modes are described by an irrational-level Pöschl-Teller potential, with two discrete modes which mix with their unbound mirrors, unbinding them and turning them into Feshbach-type quasinormal modes. Expanding to leading order in the Q-ball amplitude, we present all of these in

What carries the argument

Leading order expansion in Q-ball amplitude for solving the linearized perturbation equations, yielding closed forms for relativistic and Floquet modes.

Load-bearing premise

The deformations of the Q-ball can be separated into relativistic and long-wavelength Floquet modes with the small-amplitude approximation holding throughout.

What would settle it

A direct numerical solution of the linearized equations at small but finite Q-ball amplitude that fails to match the closed-form mode expressions would falsify the leading-order results.

Figures

Figures reproduced from arXiv: 2604.07713 by Andrzej Wereszczy\'nski, Hui Liu, Jarah Evslin, Piotr Ziobro, Tomasz Roma\'nczukiewicz, Yakov Shnir.

Figure 1
Figure 1. Figure 1: The solutions G(x) (left) and H(x) (right) of Eqs. (2.19) and (2.20) at the con￾tinuum threshold Ω + ω = m for ϵ equal to 0.1 to 0.7 in even steps shown in red, orange, yellow, green, blue, purple and ultraviolet respectively. While the red curve in the G(x) plot is well-approximated by -tanh2 (ϵx) as in Eq. (3.4), at large |x| our ϵ expansion misses the linear rise and the inevitable x-intercept. But what… view at source ↗
Figure 2
Figure 2. Figure 2: The solutions G(x) (left) and H(x) (right) of Eqs. (2.19) and (2.20) at ϵ = 0.6 at frequencies ω ranging from the continuum threshold ω = m − Ω to the bound state at ω = m − Ω − 0.0139 in even steps shown in red, orange, yellow, green, blue and purple respectively. The red curve is the threshold mode while the purple curve is the bound mode. For all solutions in between, G(x) is exponentially divergent. sm… view at source ↗
Figure 3
Figure 3. Figure 3: We set m = 1. (a),(b) Power spectrum of the field at the center for the squashed Q-ball Ω = 0.97 in the inverted ϕ 4 model together with (c) the bound mode profile and (d) the even quasinormal mode profile. Note that the bound mode extends far beyond the Q-ball itself [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The odd quasinormal mode, evaluated numerically at Ω = 0 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Four linearly independent solutions for Ω = 0 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
read the original abstract

Linearized deformations of the thick-walled (low-amplitude) (1+1)-dimensional Q-ball may be decomposed into relativistic modes, which are roughly plane waves, and also long-wavelength corotating and counterrotating Floquet modes. Each mode oscillates at a pair of mirror frequencies which average to the Q-ball frequency. The corotating modes are those of a breather or oscillon plus a very loosely bound mode. The counterrotating modes are described by an irrational-level P\"oschl-Teller potential, with two discrete modes which mix with their unbound mirrors, unbinding them and turning them into Feshbach-type quasinormal modes. Expanding to leading order in the Q-ball amplitude, we present all of these modes in closed form, except for the bound mode which does not exist at leading order.

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

0 major / 3 minor

Summary. The manuscript develops a controlled low-amplitude expansion for linearized perturbations of thick-walled (1+1)-dimensional Q-balls. It decomposes the deformations into relativistic (roughly plane-wave) modes and long-wavelength corotating/counterrotating Floquet modes, each pair oscillating at mirror frequencies whose average equals the Q-ball frequency. Closed-form expressions are obtained for all modes at leading order in the amplitude, except the bound mode (which is absent at this order); the counterrotating sector is mapped to an irrational-level Pöschl-Teller potential whose discrete levels mix with continuum mirrors to produce Feshbach-type quasinormal modes.

Significance. If the derivations are correct, the work supplies explicit analytical expressions for the leading-order perturbation spectrum of low-amplitude Q-balls, a technically useful result in soliton and nonlinear-field-theory studies. The closed-form mode functions (especially the Pöschl-Teller treatment of counterrotating modes) constitute a concrete advance that can be used for further analytic or numerical investigations of stability and radiation.

minor comments (3)
  1. [Section 4] The statement that the bound mode 'does not exist at leading order' is clear in the abstract but would benefit from an explicit sentence in the main text (near the discussion of the effective potential) confirming that the corresponding eigenvalue vanishes identically at O(ε).
  2. [Section 5.2] The definition of the 'irrational-level' Pöschl-Teller potential should include the explicit value of the depth parameter (or its leading-order expression in the amplitude) so that readers can immediately reproduce the discrete spectrum.
  3. [Section 6] A short table or list summarizing the frequencies and decay rates of the Feshbach quasinormal modes would improve readability and allow quick comparison with the relativistic continuum.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript on linearized Q-ball perturbations and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives the linearized modes by expanding the field equations to leading order in the small Q-ball amplitude, obtaining an effective linear operator whose potentials (including the Pöschl-Teller form) and solutions follow directly from that expansion. The decomposition into relativistic plane-wave modes and long-wavelength Floquet modes is a structural consequence of the time-dependent background at this order, not an imposed ansatz. The explicit absence of the bound mode is a direct result of the leading-order potential having no discrete eigenvalue for it, rather than a fitted or self-defined input. No load-bearing step reduces to a self-citation, a renamed known result, or a prediction that is the input by construction; the closed-form expressions are obtained by solving the linearized system under the stated low-amplitude approximation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on the standard framework of classical scalar field theory in 1+1 dimensions and linear perturbation theory around known soliton solutions; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Existence and stability properties of thick-walled Q-ball solutions in the underlying scalar field model
    Invoked to define the background around which perturbations are linearized.
  • domain assumption Validity of the low-amplitude expansion for decomposing modes into relativistic and Floquet types
    Central to the closed-form derivations described.

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Reference graph

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