= Compact-core variational estimate for an Abelian Higgs vortex
{title2=$E(R)=\pi(2/R^2+2/3+R^2/24)$}
For a unit <Abelian Higgs vortex>, the piecewise profiles $h=r/R$ and $a=(r/R)^2$ inside $r=R$, with both equal to one outside, are continuous finite-energy trial fields. In the normalization $E=\int[B^2/2+|D\phi|^2/2+(1-|\phi|^2)^2/8]$, their radial energy is the displayed function. Its minimum has $R^4=48$ and $E=\pi(2/3+1/\sqrt3)$, an upper-bound estimate above the exact <Bogomolny bound> $\pi$. Derivative jumps at the matching radius are allowed in the variational energy space; the trial profile is not an exact solution.
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