= Derrick scaling of the Abelian Higgs energy
{c}
{title2=$E(\lambda)=\lambda^2E_B+E_D+\lambda^{-2}E_P$}
In two dimensions, scale $\Phi_\lambda(x)=\Phi(\lambda x)$ and $A_{j,\lambda}(x)=\lambda A_j(\lambda x)$. The <magnetic field> has scaling weight two, the <gauge covariant derivative> of the <Higgs field> has weight one, and the potential has weight zero. Their integrated energies therefore have the powers shown. Stationarity gives $E_B=E_P$, a <Derrick virial identity>. This balance allows a planar <Abelian Higgs vortex> to have finite size even though a scalar-gradient-plus-nonnegative-potential model has no comparable magnetic term.
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