SO(4) vector-Higgs gauge-boson spectrum ($SO(4)\to SO(3),m_{+}=0,m_{-}=|gv|$)

A real four-vector vacuum ${\Phi }_0=v{e}_4$ leaves the rotations of its perpendicular three-space unbroken. In a Hermitian generator basis split into two commuting SU(2) triples with $(T^a+T^{a+3})e_4=0$, define $A^{a,\pm }=(A^a\pm A^{a+3})/\sqrt{2}$. For scalar kinetic term $|D\Phi {|}^2$ and canonical vector kinetic terms, its vacuum value is $g^2{v}^2{\sum }_a(A^{a,-}{)}^2/2$. Comparison with $m^2{A}^2/2$ gives three massless and three mass-$|gv|$ gauge bosons.

The factor multiplying the scalar kinetic term matters: replacing it by $|D\Phi {|}^2/2$ at fixed numerical $v$ changes the massive value to $|gv|/\sqrt{2}$. The unbroken global subgroup is SO(3), rather than its simply connected SU(2) cover. The Spin(4) double cover accounts for that central identification.