BPS sectors of the Skyrme model and their non-BPS extensions

Two recently found coupled Bogomol’nyi-Prasad-Sommerfield (BPS) submodels of the Skyrme model are further analyzed. First, we provide a geometrical formulation of the submodels in terms of the eigenvalues of the strain tensor. Second, we study their thermodynamical properties and show that the mean-field equations of state coincide at high pressure and read $p=\overline{\rho }/3$. We also provide evidence that matter described by the first BPS submodel has some similarity with a Bose-Einstein condensate. Moreover, we show that extending the second submodel to a non-BPS model by including certain additional terms of the full Skyrme model does not spoil the respective ansatz, leading to an ordinary differential equation for the profile of the Skymion, for any value of the topological charge. This allows for an almost analytical description of the properties of Skyrmions in this model. In particular, we analytically study the breaking and restoration of the BPS property. Finally, we provide an explanation of the success of the rational map ansatz.

Figure 1

Left: Profile function $\xi$ in the $E^{(2)}+E_0$ model with the pion mass potential and $m=1$. Right: Energy divided by $12{\pi }^2|B|$ as a function of the mass parameter for $B=1$ (there is no discernible difference for $B=2$, 3, 4).

Figure 2

Profile function $\xi$ in the ${\mathcal{L}}^{(2)}+{\mathcal{L}}_{BPS}$ in the $B=1$ sector. Here we assume the pion mass potential ${\mathcal{U}}_{\pi }$ with $m=1$.

Figure 3

Energy per baryon charge (left) and energy divided by the topological bound (5.16) (right) for $B=1$, 2, 3, 4 in the ${\mathcal{L}}^{(2)}+{\mathcal{L}}_{BPS}$ model, as a function of the coupling constant $\lambda$. Here we assume the pion mass potential ${\mathcal{U}}_{\pi }$ with $m=1$.

Figure 4

Dependence of the radius and $f(R)$ function of the $B=1$ Skyrmion in the ${\mathcal{L}}_{24}^{(1)}+{\mathcal{L}}_{\pi }$ model on $1/2m$.

Figure 5

Left: On-shell energy of the first and second BPS submodels divided by the corresponding topological bounds as a function of the topological charge. Right: Ratio between the on-shell energies of the first and second BPS submodel.