By Skyrmion, here is meant the soliton of the $O(D+1)$ Skyrme scalar on ${\mathrm{I}\mathrm{R}}^D$: $${\varphi }^A=({\varphi }^a,{\varphi }^{D+1});A=a,D+1;a=1,2,\dots ,D$$with $|{\varphi }^A{|}^2=|{\varphi }^a{|}^2+({\varphi }^{D+1}{)}^2=1$, which in $D=3$ is parametrized as $U={\varphi }^{4}1\mathrm{I}+{\varphi }^a{\sigma }_a$ in the original work [3] of Skyrme.

Alternatively, one can invoke the inequality $${|{\varphi }_{\mu \nu }^{AB}-{\tilde{\varphi }}_{\mu \nu }^{AB}|}^2\ge 0,$$in which ${\tilde{\varphi }}_{\mu \nu }^{AB}$ is defined as $${\tilde{\varphi }}_{\mu \nu }^{AB}=\frac{1}{2^2}{ϵ}_{\mu \nu \rho \sigma }{ϵ}^{ABCDE}{\varphi }_{\rho \sigma }^{CD}{\varphi }^E,$$such that ${\varphi }_{\mu \nu }^{AB}{\tilde{\varphi }}_{\mu \nu }^{AB}=2{\varrho }_G^{(4)}$. We have eschewed this option of introducing ${\varrho }_G^{(4)}$, since in that case, the resulting energy density does not feature a quadratic Skyrme kinetic term $|{\varphi }_{\mu }^A{|}^2$. Also, in that case, Derrick scaling is not satisfied in the gauge decoupling limit.

This is not the term $|{\varphi }_{\mu \nu }^{AB}{|}^2$, which is invariant against the global $SO(5)$ rotations of the $O(5)$ model, which is given by the simpler expression $$|{\varphi }_{\mu \nu }^{AB}{|}^2=2\times 3!{\left(\frac{w\sin f}{r}\right)}^2[(f^{\prime }{)}^2+{\left(\frac{w\sin f}{r}\right)}^2]$$instead.

Should such a potential be added by hand, the situation may change, but this would go beyond the minimal models implied by the Bogomol’nyi-like inequalities.

The chiral function $f^{(I)}(r)$ here parametrizes the $O(5)$ Skyrme scalar ${\varphi }^{\overline{a}}=({\varphi }^a,{\varphi }^5)$, with $${\varphi }^a={\widehat{x}}^a\sin f^{(I)}\text{and}{\varphi }^5=\cos f^{(I)}$$satisfying the double–self-duality equation $${\varphi }_{\mu \nu }^{\overline{a}\overline{b}}=\pm \frac{1}{2^2}{ϵ}_{\mu \nu \rho \sigma }{ϵ}^{\overline{a}\overline{b}\overline{c}\overline{d}\overline{e}}{\varphi }_{\rho \sigma }^{\overline{c}\overline{d}}{\varphi }^{\overline{e}},$$where ${\varphi }_{\mu \nu }^{\overline{a}\overline{b}}={\partial }_{[\mu }{\varphi }^{\overline{a}}{\partial }_{\nu ]}{\varphi }^{\overline{b}}$.