Nontopological Electromagnetic Hedgehogs

TL;DR

This work demonstrates the existence of nonrelativistic, non-gravitating solitons in a theory of a self-interacting complex vector field with a global $U(1)$ symmetry, showing that, when coupled to the electromagnetic tensor via a parity-even term, these solitons can confine radially oriented electric or magnetic fields within their bulk. Using a spherically symmetric ansatz and a dimensionless reformulation, the authors identify condensate states, construct soliton profiles, and analyze their stability by enforcing $E< MQ$, thereby revealing two regimes: a proper thin-wall soliton and a non-thin-wall family, with a cusp signaling EFT breakdown at certain $ oldsymbol{ hoparam}$. The resulting electromagnetic hedgehogs are not monopoles but localized field configurations supported by the vector soliton, and exist for parameter ranges $-1<oldsymbol{ hoparam}<0$, $oldsymbol{ mg} eq 0$, offering potential applications in dark matter phenomenology and condensed-matter analogs; future work includes gravity, non-Abelian couplings, and stability analyses. The study provides a concrete framework for embedding radial electromagnetic structures inside nontopological vector solitons and highlights the interplay between nonrelativistic dynamics, EFT validity, and localized field trapping.

Abstract

We study classical localised configurations - solitons - in a theory of self-interacting complex Proca field with the global $U(1)$ symmetry. We focus on spherically-symmetric solitons near the nonrelativistic limit, which are supported by the quartic interactions of the neutral Proca field. Such solitons can source the radial electric (magnetic) field if one introduces a parity-even (parity-odd) coupling of the Proca field to the electromagnetic field tensor. We discuss the conditions of existence of such nontopological ''electromagnetic hedgehogs'' and their properties.

Figures (5)

• Figure 1: Vector soliton (\ref{['ansatz']}) at $\kappa = - 0.9$ and $\mathrm{w} = 0.998$. This solution is kinematically stable, $E < MQ$.
• Figure 2: Kinematically stable vector soliton (\ref{['ansatz']}) at $\kappa = - 0.9$ and $\mathrm{w} = 0.99$. Also shown is the condensate solution (\ref{['condensate2']}) ($u_c$, $v_c$) for the same parameters.
• Figure 3: Kinematically stable vector soliton (\ref{['ansatz']}) at $\kappa = - 0.55$ and $\mathrm{w} = 0.96$. We see that the cusp is approached in the transverse components of the field.
• Figure 4: $E/M - Q$ as a function of $Q$ for vector solitons (\ref{['ansatz']}) in the theory (\ref{['eff_model']}) with $\tilde{\alpha} = 1$ and $\kappa = \tilde{\beta}/|\tilde{\alpha}|= -0.55$.
• Figure 5: The equatorial cross-section of the electric hedgehog---radial spherically-symmetric nontopological vector solition---in the theory (\ref{['L']}). We take $\kappa = \beta / \alpha = -0.9$, $\mathrm{w} = 0.998$, and $\gamma \ll \alpha$. The arrows indicate the value $\vert \vec{E}\vert/M^2$ and the direction of the electric field.