Oscillons in the broken vacuum and global vortex annihilation

Abstract: In contrast to the complex $φ^4$ model, vortex-antivortex collisions in the complex $φ^6$ theory reveal a resonant structure due to the existence of a remarkably stable, long-lived, large amplitude oscillon in the broken vacuum. Surprisingly, it persists despite the absence of a mass gap associated with the flat direction in the broken vacuum. We demonstrate that its existence is related to a far-distance modification of the potential, namely, the appearance of an unbroken (false or true) vacuum.

• The paper demonstrates that vortex-antivortex collisions in the φ⁶ model yield stable, long-lived oscillons as emergent attractors.
• It employs numerical simulations to contrast the energy dissipation in the φ⁶ model with the direct radiation seen in the φ⁴ case.
• The study reveals significant cosmological implications, suggesting that oscillon dynamics may impact axion relic abundances and dark matter models.

Oscillon Formation and Dynamics in Broken Vacuum: Insights from Global Vortex Annihilation

Introduction and Motivation

This work examines the dynamics of global vortex-antivortex (VAV) annihilation in (2+1)D complex scalar field theories with $\phi^4$ and $\phi^6$ potentials, focusing on the existence, formation, and characteristics of oscillons in the broken vacuum sector (2603.28298). The investigation is motivated by implications for axionic string evolution in the early universe and the associated axion relic abundance, since oscillons—long-lived, spatially localized, nontopological field configurations—can dramatically alter the energy budget following string decay processes.

The central result is that, contrary to expectations from the absence of a mass gap in the broken vacuum, the complex $\phi^6$ model supports robust oscillon formation as an emergent attractor after VAV collisions. This is in stark contrast to the $\phi^4$ case, where VAV annihilation is direct and no oscillon plateau is observed.

Theoretical Framework: Global Vortices in Complex Scalar Models

The Lagrangian considered is

$$\mathcal{L} = \frac{1}{2} \partial_\mu \phi\, \overline{\partial^\mu \phi} - V(|\phi|),$$

where $\phi$ is a complex scalar and $V(|\phi|)$ is parameterized to interpolate between $\phi^4$ (Mexican hat) and $\phi^6$ models:

$$V(|\phi|) = \frac{1}{8(1+\nu^2)} (\nu^2 + |\phi|^2)\left(|\phi|^2 - 2\right)^2,$$

with $\phi^6$0 controlling the potential structure.

Both cases feature a $\phi^6$1 symmetry and broken phase with $\phi^6$2, supporting global vortices with nontrivial $\phi^6$3 topology. The key distinction lies in the potential at $\phi^6$4: for $\phi^6$5 ($\phi^6$6), there is a true/unbroken vacuum, while for $\phi^6$7 ($\phi^6$8), it is a maximum.

Figure 1: Profiles of the unit charge vortices in the $\phi^6$9 (blue curve) and $\phi^6$0 models (orange curve).

Vortex-Antivortex Collision Dynamics and Oscillon Emergence

Resonant Structure in $\phi^6$1 VAV Collisions

Numerical simulations of head-on VAV collisions reveal that, in $\phi^6$2, post-collision dynamics often settle into a long-lived, spatially localized, radially symmetric oscillating structure—a genuine oscillon—centered at the origin. This oscillator has a dominant frequency $\phi^6$3, beneath the radial perturbation threshold $\phi^6$4, and displays modulated amplitude oscillations.

Figure 2: The VAV collision. Energy density at the origin as a function of time for various initial velocities $\phi^6$5. Upper: $\phi^6$6, Lower: $\phi^6$7.

This result is highly nontrivial: in conventional wisdom, the flat direction in the broken vacuum implies gapless Goldstone modes rendering oscillons immediately radiative and thus non-existent except in highly fine-tuned situations. The simulations, however, show that in the complex $\phi^6$8 potential, oscillons are generic attractors for a wide class of violent initial data, not artifacts of fine tuning.

Figure 3: Formation of a radially symmetric oscillon in the VAV collision in the $\phi^6$9 model with $\phi^4$0.

Absence of Oscillons in the $\phi^4$1 Model

By contrast, in the complex $\phi^4$2 theory, VAV collision energy is efficiently radiated through massless Goldstone channels, and intermediate oscillon-like plateaus are absent; the system quickly settles to vacuum.

Field Content of Post-collision Oscillon

Examination of the field components at the origin shows the imaginary part decays rapidly via radiation, leaving only a stable, real, breathing oscillation. The power spectrum analysis confirms the coherence and modulation structure of this oscillon.

Figure 4: The real and imaginary component of $\phi^4$3 at the origin in the VAV collision with $\phi^4$4 and $\phi^4$5. Upper: $\phi^4$6, Lower: $\phi^4$7.

Figure 5: Evolution of the normalized energy enclosed in a disc of radius $\phi^4$8 for $\phi^4$9 in $$\mathcal{L} = \frac{1}{2} \partial_\mu \phi\, \overline{\partial^\mu \phi} - V(|\phi|),$$0 (blue), $$\mathcal{L} = \frac{1}{2} \partial_\mu \phi\, \overline{\partial^\mu \phi} - V(|\phi|),$$1 (green), and perturbed Gaussian data (orange).

Origin and Robustness of the Broken Vacuum Oscillon

The oscillon observed aligns with those generated directly from real scalar Gaussian initial data in the $$\mathcal{L} = \frac{1}{2} \partial_\mu \phi\, \overline{\partial^\mu \phi} - V(|\phi|),$$2 theory, perturbed along the imaginary direction. Independent of the route (VAV collision or crafted pulse), the attractor is the same modulated real oscillon, with comparable frequencies and dynamical properties.

Figure 6: Embedded real $$\mathcal{L} = \frac{1}{2} \partial_\mu \phi\, \overline{\partial^\mu \phi} - V(|\phi|),$$3 oscillon (from Gaussian data with $$\mathcal{L} = \frac{1}{2} \partial_\mu \phi\, \overline{\partial^\mu \phi} - V(|\phi|),$$4, $$\mathcal{L} = \frac{1}{2} \partial_\mu \phi\, \overline{\partial^\mu \phi} - V(|\phi|),$$5), showing rapid decay of the imaginary component.

Figure 7: Power spectrum at the origin for the oscillon found via VAV collision ($$\mathcal{L} = \frac{1}{2} \partial_\mu \phi\, \overline{\partial^\mu \phi} - V(|\phi|),$$6, $$\mathcal{L} = \frac{1}{2} \partial_\mu \phi\, \overline{\partial^\mu \phi} - V(|\phi|),$$7, red) and embedded oscillon from perturbed Gaussian initial data ($$\mathcal{L} = \frac{1}{2} \partial_\mu \phi\, \overline{\partial^\mu \phi} - V(|\phi|),$$8, $$\mathcal{L} = \frac{1}{2} \partial_\mu \phi\, \overline{\partial^\mu \phi} - V(|\phi|),$$9, blue dashed).

Energy analysis shows these oscillons can inherit only $\phi$0 of the original VAV energy, with the rest radiated, emphasizing their nature as strong attractors. The parameter scan in the potential, varying $\phi$1, demonstrates oscillon stability only for sufficiently deep false (unbroken) vacua at $\phi$2—a marked dependence on global (not local) potential structure.

Figure 8: Oscillon existence as a function of the parameter $\phi$3 in $\phi$4 for $\phi$5.

Contrasts with Other Soliton Dynamics

• The formation mechanism and oscillation frequencies are not associated with Feshbach-like internal modes of the global vortex. Instead, the oscillon arises as a genuine collective phenomenon unrelated to single-vortex vibrational spectra.
• In Abelian-Higgs (local) vortex models, oscillons are allowed due to the mass gap; in the complex $\phi$6 theory, even embedding real $\phi$7 oscillons proves unstable because of immediate coupling to radiative massless modes.

Cosmological and Theoretical Implications

The presence of robust, long-lived, broken vacuum oscillons induced by VAV annihilation in higher-order potential models signifies a profound alteration of post-phase-transition field dynamics. For axionic string networks, this implies energy sequestration into oscillons, potentially changing axion emission rates and dark matter relic computations.

Since the existence of these oscillons depends sensitively on the remote structure of the scalar potential (e.g., the depth of the false vacuum at the origin), minor modifications to high-energy extensions of the Standard Model may yield macro-scale consequences for nonperturbative early universe dynamics. Extensions to local vortices and monopole systems with higher-order interactions are natural next directions.

Conclusion

This study conclusively demonstrates that the complex $\phi$8 model supports dynamically robust, long-lived oscillons in the broken vacuum following global vortex-antivortex collision, overturning expectations based on the absence of a mass threshold. The existence and properties of these oscillons are governed not by local vacuum structure but by the global topology of the potential, specifically the presence of a sufficiently deep, distant (false) vacuum at the origin. This result necessitates reconsideration of nonperturbative energy relaxation mechanisms in axionic and related cosmological scenarios, with implications for dark matter phenomenology and the physics of physics beyond the Standard Model.

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