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The Derrick scaling of the Abelian Higgs energy must rescale the gauge potential as well as the Higgs field. SetThen $B_{\lambda }(x)={\lambda }^{2}B(\lambda x)$ andLet $E_B=\int B^2$, $E_D=\int |D\Phi {|}^2$ and $E_P=\frac{1}{4}\int (1-|\Phi {|}^2{)}^2$, with all integrals over the plane. Changing variables givesAt a static solution, $V^{\prime }(1)=2E_B-2E_P=0$. ConsequentlyThe covariant-gradient contribution is scale invariant, while the magnetic and potential terms balance. This is why the scalar-only nonexistence conclusion does not extend to the Abelian Higgs model.