= Solution
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 gives
$$
\boxed{m_{3,8}=0,\qquad m_{1,2}=2g|v|,\qquad m_{4,5,6,7}=g|v|.}
$$
Only the two diagonal generators commute with the vacuum. Its unbroken group is the <maximal torus>
$$
\boxed{H=\{\operatorname{diag}(e^{i\alpha},e^{i\beta},e^{-i(\alpha+\beta)})\}\cong U(1)\times U(1).}
$$
The 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.
\Image[/past-exam-of-the-gauge-field-theory-course-of-the-university-of-cambridge/2014/paper-1-adjoint-gauge-spectrum.png]
{title=Gauge-boson masses for two diagonal SU(3) adjoint vacua}
Back to article page
