= Tree-level electroweak gauge-boson masses
{title2=$m_W=gv/2,\quad m_Z=v\sqrt{g^2+g'^2}/2,\quad m_A=0$}
For the <Higgs doublet> vacuum $(0,v)^T/\sqrt2$, the neutral <gauge-boson mass matrix> in $(W^3,B)$ is $(v^2/4)\begin{pmatrix}g^2&-gg'\\-gg'&g'^2\end{pmatrix}$. Its zero <eigenvector> is proportional to $(g',g)$, the <photon>; the perpendicular vector is the <Z boson>. The charged fields $W^\pm=(W^1\mp iW^2)/\sqrt2$ have the stated <W boson> mass. Thus $m_W=m_Z\cos\theta_W$, where $\theta_W$ is the <Weinberg angle>. The unbroken charge generator annihilates the vacuum and explains the zero eigenvalue.
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