 Solution

Take $r>0$ and $m>0$, the positive-energy sigma-model regime, and keep $\theta =0$ for the energy calculation. For a static field the energy isFor either sign $s=\pm 1$, complete the square:Here $\int {\varphi }_x\sin \varphi dx=-[\cos \varphi {]}_{-\infty }^{+\infty }=2T$. Choose $s$ to have the sign of $T$. The Bogomolny bound is thereforeSaturation requires ${\varphi }_x=sm\sin \varphi$ and $\sin \varphi {\alpha }_x=0$. Finite-energy vacua have $\sin \varphi =0$, so their endpoint cosine values are $\pm 1$ and this particular topological charge is $T\in \{-1,0,1\}$. This is the static kink bound in a mass-deformed spherical sigma model. Negative $r$ would instead make the derivative energy unbounded below, so positivity is an essential physical convention.