= Polar coordinates for the electroweak Higgs doublet
{title2=$\phi=\frac{v+H}{\sqrt2}U\binom01$}
A nonzero <Higgs doublet> has a real radial magnitude and three real angular coordinates. For a normalized vector $z=(z_1,z_2)^T$, the <SU(2) matrix> $U=\left(\begin{smallmatrix}z_2^*&z_1\\-z_1^*&z_2\end{smallmatrix}\right)$ satisfies $U(0,1)^T=z$. Near the vacuum, the three broken generators can equivalently be parameterized by $e^{-i\boldsymbol n\cdot\boldsymbol\sigma+in_3}$. A local <gauge transformation> removes these angular fields in <unitary gauge>. This coordinate choice is local about a nonzero vacuum and need not extend through zeros or topological defects.
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