Electroweak doublet gauge-boson mass matrix ($m_W^2=g^2{v}^2/4$)

For a Higgs doublet with hypercharge $1/2$ and vacuum expectation value $(0,v/\sqrt{2}{)}^T$, its gauge-covariant kinetic term gives the neutral gauge-boson mass matrixin the $(W^3,B)$ basis. Its eigenvalues are zero and $(g^2+g^{\prime 2})v^2/4$, and the Weinberg angle rotation gives the photon and Z boson. The two charged electroweak gauge bosons have $m_W^2=g^2{v}^2/4$. The zero eigenvalue expresses the unbroken electric-charge gauge symmetry.