SU(3) gauge theory ($SU(3)$ gauge theory)

An SU(3) gauge theory has eight gauge fields, one for each generator of $SU(3)$. A vacuum expectation value of one scalar in the fundamental representation generically leaves an $SU(2)$ subgroup unbroken.

A traceless Hermitian $3\times 3$ adjoint scalar field transforms by $\Phi \mapsto U\Phi U^{†}$. A diagonal vacuum with eigenvalues $({\lambda }_1,{\lambda }_2,{\lambda }_3)$, summing to zero, leaves its centralizer unbroken. The adjoint scalar gauge-boson mass matrix assigns mass $g|{\lambda }_i-{\lambda }_j|$ to the vector fields linking the $i$th and $j$th eigenspaces. Coincident eigenvalues preserve non-Abelian rotations of their eigenspace; distinct eigenvalues preserve only a maximal torus.

For a complex fundamental scalar with vacuum expectation value proportional to $(0,0,v)$, the stabilizer is the $SU(2)$ acting on the first two components. Five broken generators produce five massive vector bosons, while three gauge bosons remain massless and one radial scalar remains physical.