Polar coordinates for the electroweak Higgs doublet ($\varphi =\frac{v+H}{\sqrt{2}}U\left(\genfrac{}{}{0ex}{}{0}{1}\right)$)

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 $U=\left(\begin{array}{cc}{z}_2^{\ast }& z_1\\ -z_1^{\ast }& z_2\end{array}\right)$ satisfies $U(0,1{)}^T=z$. Near the vacuum, the three broken generators can equivalently be parameterized by $e^{-i\mathbf{n}\cdot \mathbf{\sigma }+i{n}_3}$. 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.