Derrick scaling of Yang-Mills-Higgs energy ($V_{\lambda }=\lambda V_B+{\lambda }^{-1}{V}_D+{\lambda }^{-3}{V}_U$)

In three space dimensions, rescale $A_i(x)$ to $\lambda A_i(\lambda x)$ and $\Phi (x)$ to $\Phi (\lambda x)$. The gauge curvature then scales with two derivatives, and the adjoint covariant derivative of the Higgs field with one. Magnetic, Higgs-gradient and potential energies give the displayed formula. Stationarity requires $V_B=V_D+3V_U$, which permits nonzero energies. In the Bogomolny-Prasad-Sommerfield monopole limit $V_U=0$, the Bogomolny equations give $V_B=V_D$.