= Gauge-boson mass rank from a real scalar vacuum
{title2=$(M_A^2)_{ab}=g^2(T_a\phi_0)^TT_b\phi_0$}
With positive canonical kinetic terms and nonzero coupling $g$, expansion of the covariant scalar kinetic term gives the displayed <gauge-boson mass matrix>. It is a <Gram matrix>: for real $c_a$, $c^TM_A^2c=g^2|\sum_ac_aT_a\phi_0|^2$. Its kernel is the <Lie algebra> of the <stabilizer subgroup> of the vacuum. Thus its rank is $\dim G-\dim H$, and exactly that many gauge-field combinations acquire nonzero squared masses. Removal of these gauge-orbit directions does not remove accidental zero eigenvalues of the <scalar mass matrix> on the transverse complement.
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