Scattering of wobbling vortices

Abstract: We investigate the dynamical role of internal vibrational modes in the Abelian Higgs model, focusing on how Derrick-type excitations modify vortex dynamics and scattering processes. We study the scattering of excited vortices and show that the interplay between spectral flow and mode excitation generates effective forces and enables resonant energy transfer between translational and internal degrees of freedom. As a result, vortex dynamics become strongly non-adiabatic, exhibiting super-elastic collisions, oscillatory dependence of the final state on initial conditions, and the emergence of fractal structures in scattering diagrams. Our results demonstrate that internal vibrational modes play a fundamental role in vortex interactions, going beyond the standard moduli space approximation and revealing a rich phenomenology driven by mode dynamics.

• The paper reveals that internal vibrational modes trigger non-adiabatic effects, leading to resonant energy transfer and chaotic vortex scattering.
• Numerical simulations map critical velocity thresholds and fractal, multi-bounce windows across Type-I, BPS, and Type-II regimes.
• Phase-driven energy exchange between translational and internal modes suggests modifications to the effective equation of state for vortex gases.

Scattering Phenomena of Wobbling Vortices in the Abelian Higgs Model

Introduction

This work addresses the dynamics of vortex scattering in the Abelian Higgs model, with emphasis on the role of internal vibrational (Derrick-type) modes. Focusing on configurations where these modes are excited, the study elucidates how the coupling between translational and internal degrees of freedom produces pronounced deviations from adiabatic or moduli-space dynamics, manifesting as resonant energy transfer, super-elastic collisions, and fractal scattering patterns. The analysis is framed across the three canonical regimes: Type-I (attractive, $\lambda<1$), self-dual/BPS ($\lambda=1$), and Type-II (repulsive, $\lambda>1$).

Theoretical Framework and Internal Modes

The Abelian Higgs model in $2+1$D supports topologically stable vortex solutions, with their dynamics traditionally described by the moduli space approximation, valid in the low-velocity, near-static regime. However, moduli space dynamics does not incorporate excitations of the vortex's internal structure. Fluctuation analysis reveals the existence of localized shape modes for low winding-number vortices and moderate coupling $\lambda$. The dominant internal excitation is the radial (Derrick-type) breathing mode, present for $\lambda < 1.5$ for the fundamental 1-vortex. These modes introduce oscillatory dynamics in the vortex core, parametrized by an excitation amplitude $\eta$ and frequency $\omega$, and induce an energy exchange mechanism between translation and internal vibration.

Dynamically, excitation of these modes produces effective forces between vortices, modifying scattering outcomes beyond the force-free geodesic picture. The spectrum of these modes—and hence their influence—is $\lambda$-dependent, with the spectral flow phenomenon governing how mode frequencies evolve as vortices separate or approach.

Numerical Study of Excited Vortex Scattering

The authors systematically investigate head-on collisions of Derrick-excited 1-vortices by high-precision numerical simulation with explicit Lorentz gauge implementation and careful analysis of initial configurations via Lorentz-boosted superpositions.

Key findings:

• Internal modes act as dynamical degrees of freedom, facilitating strong non-adiabatic effects even at modest excitation amplitudes.
• The internal mode-induced spectral flow leads to effective inter-vortex forces that either enhance (attractive) or oppose (repulsive) the static interaction, contingent on the mode phase at collision.
• Collisions display a chart of possible final states (outgoing velocities, final excitations) that depends sharply on initial parameters, with intricate multi-bounce windows and fractal features analogous to kink-antikink resonances in $1+1$D scalar field theory.

Type-I Regime ($\lambda=1$0)

Intrinsic attraction dominates, generating metastable bound states and multi-bounce resonant windows even in absence of vibrational excitation. Introduction of internal mode excitation dramatically enhances chaotic scattering, producing a complex hierarchy of resonance windows and fractal dependence of observables on initial velocity. Notably, there are super-elastic events where the final kinetic energy exceeds the initial, enabled by energy transfer from vibrational modes.

Self-dual BPS Regime ($\lambda=1$1)

Static forces vanish, isolating the impact of mode-driven dynamics. For unexcited vortices, collisions are strictly $\lambda=1$2 with elastic kinematics at low velocities. With mode excitation, effective attraction emerges solely from spectral flow, and at large amplitudes the system replicates the resonant, multi-bounce structure of the Type-I regime. Oscillatory dependence of final observables on initial velocity is pronounced, with clear phase-controlled energy exchange between internal and translational channels.

Type-II Regime ($\lambda=1$3)

Repulsion prevents collisions below a critical velocity $\lambda=1$4, which decreases as excitation amplitude increases due to mode-induced attraction. Scattering remains quasi-elastic at low velocities for unexcited vortices, but once excitation is present, the system exhibits windows of multi-bounce behavior and emergent fractal structure, as seen in attractive scenarios. Importantly, the presence of sufficiently strong internal excitation can temporarily endow Type-II vortices with effective Type-I-like behavior.

Quantitative and Phenomenological Results

• Critical velocity thresholds and resonance windows are mapped with high numerical resolution across a broad parameter grid in $\lambda=1$5.
• Analytical phase-based conditions accurately predict the locations of resonance peaks in velocity/amplitude output, validating the mechanism of phase-tuned energy transfer.
• Scattering diagrams display hallmarks of deterministic chaos, with self-similar (fractal) replication of window structures at progressively smaller velocity scales.

Physical Implications and Future Directions

These results indicate that internal mode excitation is generically unavoidable in multi-vortex dynamics due to inevitable energy transfer in collisions. The persistence and efficiency of this mechanism imply that in realistic vortex ensembles, vortices will seldom remain purely moduli-locked entities; excited states, slow decay, and continual vibrational energy exchange should be considered in statistical and thermodynamic treatments.

A striking implication is the modification of the effective equation of state for a vortex gas, particularly in the BPS regime, from the classical Clausius/Boltzmann form to a potentially van der Waals-type behavior, as kinetic energy is dynamically redistributed into internal modes. The slow decay of excited modes ensures long-lived departures from pure geodesic dynamics.

Investigations are suggested for:

• The collective behavior and thermodynamics of vortex gases incorporating vibrational degrees of freedom.
• The role of higher $\lambda=1$6 "Feshbach-like" resonances in strongly Type-II vortices, with connections to recent studies of vortex-antivortex annihilation channels.
• The universality of the chaos/resonance framework across other soliton-bearing gauge field theories.
• The interplay of external excitation mechanisms (thermal, radiative, phase transition-induced) with dynamical excitation during collisions.

Conclusion

This study conclusively demonstrates that internal vibrational modes, and their associated spectral flow, fundamentally alter vortex scattering in the Abelian Higgs model across all regimes of the coupling parameter $\lambda=1$7. The non-adiabatic coupling between translational and vibrational motion drives rich dynamical phenomena, including super-elastic resonance, chaotic multi-bounce windows, and fractal dependence of observables on initial conditions. These effects necessitate an extension of standard moduli-space picture to properly describe vortex dynamics and have far-reaching theoretical and practical implications for the evolution and statistical mechanics of topological soliton ensembles.