= Radial and angular masses of a phase-invariant scalar potential
For a <complex scalar field> with canonical kinetic term $|\partial\phi|^2$ and potential $V=F(|\phi|^2)$, take a broken-symmetry minimum $\phi=v>0$, where $F'(v^2)=0$. Writing $\phi=v+(h+i\pi)/\sqrt2$ gives quadratic potential $F(v^2)+v^2F''(v^2)h^2$. Hence the canonical squared masses are $m_h^2=2v^2F''(v^2)$ and $m_\pi^2=0$. Coupling $D_\mu=\partial_\mu+ieA_\mu$ gives gauge-boson squared mass $m_A^2=2e^2v^2$; the factor of two depends on defining $v$ as the complex field's vacuum value rather than its canonically normalized real component.
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