Vortex Solutions for A Mixed Boundary-Value Problem in the Abelian-Higgs Model with A Neutral Scalar Field
Guange Su, Xiaosen Han
TL;DR
The paper proves the existence of radially symmetric vortex solutions in an Abelian-Higgs model with a neutral scalar field under a mixed boundary-value problem on a semi-infinite domain. It combines shooting methods with a Schauder fixed-point argument to construct the three vortex-profile functions and analyzes their monotonicity and asymptotics. A sharp phase boundary between Abelian and non-Abelian vortices is established, depending on the parameter alpha, with a rigorous demonstration that the boundary occurs at alpha = 1/√2 and precise small- and large-r behavior of the profiles is obtained. The results provide a rigorous mathematical foundation for vortex solutions in this mixed-boundary Abelian-Higgs setup and confirm numerical phase distinctions observed in prior work.
Abstract
Vortices represent a class of topological solitons arising in gauge theories coupled with complex scalar fields, holding significant importance across various domains of modern physics. In this paper we establish the existence of vortex solutions for a mixed boundary-value problem derived from the Abelian-Higgs model incorporating a neutral scalar field, a system recently investigated by Eto, Peterson et al. [7]. By synergistically combining the shooting method with the Schauder fixed-point theorem, we derive sharp analytical criteria that delineate the Abelian vortex phase from the non-Abelian one. We also rigorously establish the monotonicity, uniform boundedness, and precise asymptotic behavior of the vortex profile functions. Our results provide rigorous confirmation of numerical observations regarding the phase boundary between these distinct vortex types.