Derrick dimension test for the Skyrme model ($E(L)=L^{d-2}{E}_2+L^{d-4}{E}_4$)

For a spatially dilated Skyrme model field $U_L(x)=U(x/L)$, the quadratic and quartic derivative energies scale as shown. A static stationary field must satisfy $(d-2)E_2+(d-4)E_4=0$. In three spatial dimensions this permits $E_2=E_4$ and a positive second variation along scale. In two dimensions it requires $E_4=0$, and in four or more it forces a constant finite-energy smooth field. In one dimension the commutator term vanishes identically. Derrick scaling tests a necessary variational condition, not existence or stability under all deformations.

For a smooth finite-energy static planar Skyrme model field, Derrick scaling gives zero quartic energy, hence $[R_1,R_2]=0$ pointwise. The quartic first variation then vanishes, leaving the sigma-model equation ${\partial }_i{R}_i=0$. The right-current Maurer-Cartan equation gives ${\partial }_1{R}_2-{\partial }_2{R}_1=[R_1,R_2]=0$. Each current component is therefore harmonic on the plane and square-integrable. The mean-value estimate on expanding disks makes each component zero, so $U$ is constant. This excludes smooth finite-energy planar static solutions of this potential-free SU(2) model, without misapplying the same conclusion to a pure curved-target sigma model.