Radial and angular masses of a phase-invariant scalar potential

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