 Solution

For ${\mu }^2<0$, minimize the Higgs potential to obtain $v^2=-{\mu }^2/\lambda$. In unitary gauge, the physical scalar fluctuation is real andComparing with $m_W^2{W}_j^{\mu }{W}_{j\mu }/2$ and $-m_{\sigma }^2{\sigma }^2/2$ givesAll three gauge generators are broken by this complex doublet; this pure SU(2) model has no surviving massless photon. Three scalar angular degrees of freedom supply the longitudinal vector polarizations, leaving one real Higgs boson. The count is $3\times 2+4=3\times 3+1=10$. The corresponding fundamental-doublet example also appears in Gauge Theory by David Tong.^[ref]

The remaining interactions are $g^{2}v\sigma W_j{W}_j/4$, $g^2{\sigma }^2{W}_j{W}_j/8$, $-\lambda v{\sigma }^3$ and $-\lambda {\sigma }^4/4$. They give vertices with two equal-species vector legs and one or two scalar legs, or three or four scalar legs, as drawn above. The coefficients in the Lagrangian must be differentiated with respect to identical fields when obtaining fully normalized Feynman rules; the written coefficients alone are not all the vertex factors.