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For eigenvalues $(v,-v,0)$, all entries are distinct. The $12$ difference has magnitude $2|v|$, while both the $13$ and $23$ differences have magnitude $|v|$. The adjoint scalar gauge-boson mass matrix therefore givesOnly the two diagonal generators commute with the vacuum. Its unbroken group is the maximal torusThe two real fields belonging to each $ij$ pair combine into oppositely charged conjugate vector states with equal mass. The further equality between the $13$ and $23$ masses comes from the equally spaced vacuum eigenvalues; the surviving Abelian group alone does not force distinct root-charge sectors to have equal masses. Thus a fourfold massive degeneracy occurs in both examples, but its relation to the unbroken continuous symmetry differs. The scalar potential has not been specified, so no additional symmetry of that potential or physical scalar mass spectrum can be inferred.

In spontaneous symmetry breaking, the action respects a symmetry while a selected vacuum state is invariant only under a proper subgroup. A nonzero vacuum expectation value can identify that choice. For a local gauge theory, gauge transformations are redundancies rather than transformations between physically different states; a gauge-fixed scalar expectation value describes the Higgs mechanism and its massive excitations. It should not be interpreted as breaking an observable local redundancy.

Write the adjoint scalar field as $\Phi ={\Phi }_0+\phi$, with Hermitian gauge generators normalized by $\mathrm{tr}(t^a{t}^b)={\delta }^{ab}/2$. Since ${\Phi }_0$ is constant, the part of the covariant derivative linear in the gauge field is $D_{\mu }{\Phi }_0=ig{A}_{\mu }^a[t^a,{\Phi }_0]$. Both $t^a$ and ${\Phi }_0$ are Hermitian, so their commutator is anti-Hermitian. The scalar kinetic term therefore contributesMatching coefficients gives the adjoint scalar gauge-boson mass matrixIt is positive semidefinite: for real coefficients $x_a$, its quadratic form is $2g^2\mathrm{tr}(C^{†}C)\ge 0$, where $C=[x_a{t}^a,{\Phi }_0]$. Its null space consists of generators commuting with the vacuum, the Lie algebra of its centralizer.

For a diagonal vacuum ${\Phi }_0=\mathrm{diag}({\lambda }_1,{\lambda }_2,{\lambda }_3)$, the matrix unit $E_{ij}$ obeys $[E_{ij},{\Phi }_0]=({\lambda }_j-{\lambda }_i)E_{ij}$. The two normalized Hermitian generators built from $E_{ij},E_{ji}$ each acquireThe diagonal generators $t^3,t^8$ commute with every such vacuum and remain massless. Orthogonality of these root generators also shows that the mass matrix is diagonal in the supplied Gell-Mann matrices basis. Take $g>0$ and $v\ne 0$ below; $v=0$ restores the full SU(3) group.

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.