Abstract
We study the simplest $SO(2)$ gauged $O(5)$ Skyrme models in $4+1$ (flat) dimensions. In the gauge decoupled limit, the model supports topologically stable solitons (Skyrmions) and after gauging, the static energy of the solutions is bounded from below by a "baryon number". The studied model features both Maxwell and Maxwell--Chern-Simons dynamics. The considered configurations are subject to bi-azimuthal symmetry in the ${\mathbb R}^4$ subspace resulting in a two dimensional subsystem, as well as subject to an enhanced symmetry relating the two planes in the ${\mathbb R}^4$ subspace, which results in a one dimensional subsystem. Numerical solutions are constructed in both cases. In the purely magnetic case, fully bi-azimuthal solutions were given, while electrically charged and spinning solutions were constructed only in the radial (enhanced symmetric) case, both in the presence of a Chern-Simons term, and in its absence. We find that, in contrast with the analogous models in $2+1$ dimensions, the presence of the Chern-Simons term in the model under study here results only in quantitative effects.
Highlights
The gauging of the Skyrmion, namely of the soliton of the Oð4Þ sigma model on R3, is recognized to be of physical relevance in the study of the electrically charged nucleon
Gauging a Skyrme scalar results in the deformation of the lower bound on the energy, which prior to gauging is the topological charge, namely the winding number
The mass is maximized by the g 1⁄4 0 configurations, a limit which corresponds to the ungauged O(5) sigma model, whose solutions were discussed in [12]
Summary
The gauging of the Skyrmion, namely of the soliton of the Oð4Þ sigma model on R3, is recognized to be of physical relevance in the study of the electrically charged nucleon. This is unsatisfactory in the context of the problem at hand, where it is desirable to gauge two pairs of the Skyrme scalar with SOð2Þ, with the aim of imposing biazimuthal symmetry in R4 Such a gauging prescription together with the corresponding topological charge density is constructed in Appendix A of the present paper. Appendix A defines the “topological charge” supplying the lower bound of the energy Such a charge density is provided in [6], which is not adequate for the present application since only two of the five components of the Skyrme scalar are gauged in that case. We need to gauge two pairs of Skyrme scalars to enable the imposition of the enhanced symmetry rendering the biazimuthal system a radial one. (Appendix A stands on its own as a supplement to the corresponding result in [6].) In Appendix B, we have established the Belavin inequalities that give the Bogomol’nyi lower bounds, a task which is appreciably more involved than the corresponding one for the ungauged Oð5Þ model, studied in [12]
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