Near-BPS baby Skyrmions

Abstract

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 ϵ. Classical solutions of this model, to leading order in ϵ, are called restricted harmonic maps. In the BPS limit (ϵ → 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 ϵ with which we can calculate the corrections to the BPS mass. The leading order ($$ \mathcal{O}\left({\upepsilon}^1\right) $$ O ϵ 1 ) 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 ϵ to capture the relevant terms in perturbation theory, however, the binding energy contributes to the total energy at order ϵ2. 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.

Near-BPS baby Skyrmions with Gaussian tails

“…the solution for > 0 is axially symmetric. We study the axially symmetric baby Skyrmions in the perturbative scheme, put forward in our previous paper [17] and find very good agreement between the perturbative results and the full numerical ODE results. The main difference here with respect to ref.…”

“…distances nor for the binding energies and we conclude in this paper that the perturbative scheme fails for solitons with tails -as opposed to compactons [17].…”

“…The main difference here with respect to ref. [17] is that the BPS solution here is not a compacton and therefore there is no need for imposing special cusp conditions at the boundary of the would-be compacton. The most stable axially symmetric baby Skyrmion has charge Q = N = 2 and therefore it is expected that for large topological charge Q, the most stable solution is made of Q 2 almost axially symmetric N = 2 components -stitched together in some fashion.…”

“…Because the near-BPS limit of the BPS Skyrme model is very challenging, we have in ref. [17] considered the analogous problem in the toy model, known as the baby Skyrme model [18][19][20][21]. The baby Skyrme model is "smaller" than the Skyrme model in two senses: it is formulated in 2+1 dimensional spacetime instead of 3+1 and it has the target space S 2 instead of S 3 .…”

“…Furthermore, the BPS baby Skyrme model also enjoys area-preserving diffeomorphism invariance for static solutions [22], and it was in fact discovered earlier in this context, than in the full 3+1 dimensional Skyrme model. In our previous paper [17], we chose to stick with the most simplistic potential, namely the pion mass term. This had the consequence that the BPS solutions in that model are of the compacton type [23][24][25].…”