Existence of Dyons in the Coupled Georgi–Glashow–Skyrme Model

Abstract

We prove the existence of a continuous family of finite-energy particle-like solutions in the coupled Georgi-Glashow-Skyrme model carrying both electric and magnetic charges, known as dyons. Due to the presence of electricity and the Minkowski spacetime signature, we need to solve a variational problem with an indefinite action functional. Our results show that, while the magnetic charge is uniquely determined by the topological monopole number, the electric charge of a solution can be arbitrarily prescribed in an open interval.

Existence of dyons in minimally gauged Skyrme model via constrained minimization

“…where S 2 r denotes the 2-sphere of radius r, centered at the origin in the 3-space. Similarly, the monopole charge Q M may be shown to be 1 as well [25].…”

“…In [7], numerical solutions are obtained which convincingly support the existence of such solutions. The purpose of this paper is to give an analytic proof for the existence of these solutions, extending the methods developed in the earlier studies [25,34,44,45] for the dyon solutions in other models in field theory described above. See also [5,16].…”

“…Since the presence of the function g gives rise to a nonvanishing temporal component of the gauge field, g may be regarded as an electric potential. With (2.12), the Skyrme charge Q S can be shown to be given by [3,7,25]…”

“…In order to tackle the problem arising from the negative terms involving the function g in the action (2.24), we use the methods developed in [25,34,44] by imposing the constraint ∞ 0 (r 2 g ′ G ′ + 2a 2 gG) dr = 0, (3.6) to 'freeze' the troublesome g-sector, where G is an arbitrary test function satisfying G(∞) = 0 and…”

“…See also [5,16]. Note that, since here we are interested in the minimally gauged Skyrme model where no Higgs field is present, we lose the control over the negative terms in the indefinite action functional which can otherwise be controlled if a Higgs field is present [8,25,34,44,45]. In order to overcome this difficulty, we need to obtain suitable uniform estimates for a minimizing sequence at singular boundary points and to achieve strong convergence results, for the sequences of the negative terms.…”