Near-BPS baby Skyrmions

Abstract

<jats:title>A<jats:sc>bstract</jats:sc> </jats:title><jats:p>We consider the baby-Skyrme model in the regime close to the so-called restricted baby-Skyrme model, which is a BPS model with area-preserving diffeomorphism invariance. The perturbation takes the form of the standard kinetic Dirichlet term with a small coefficient <jats:italic>ϵ</jats:italic>. Classical solutions of this model, to leading order in <jats:italic>ϵ</jats:italic>, are called restricted harmonic maps. In the BPS limit (<jats:italic>ϵ →</jats:italic> 0) of the model with the potential being the standard pion-mass term, the solution with unit topological charge is a compacton. Using analytical and numerical arguments we obtain solutions to the problem for topological sectors greater than one. We develop a perturbative scheme in <jats:italic>ϵ</jats:italic> with which we can calculate the corrections to the BPS mass. The leading order (<jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathcal{O}\left({\upepsilon}^1\right) $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msup> <mml:mi>ϵ</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mfenced> </mml:math></jats:alternatives></jats:inline-formula>) corrections show that the baby Skyrmion with topological charge two is energetically preferred. The binding energy requires us to go to the third order in <jats:italic>ϵ</jats:italic> to capture the relevant terms in perturbation theory, however, the binding energy contributes to the total energy at order <jats:italic>ϵ</jats:italic><jats:sup>2</jats:sup>. We find that the baby Skyrmions — in the near-BPS regime — are compactons of topological charge two, that touch each other on their periphery at a single point and with orientations in the attractive channel.</jats:p>