Past exam of the gauge field theory course of the University of Cambridge / 2014 / Paper 1 / 1

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.

For eigenvalues $(v,v,-2v)$, the first two entries coincide. Consequently the $12$ generators $t^1,t^2$, as well as $t^3,t^8$, commute with the vacuum. The $13$ and $23$ eigenvalue differences both have magnitude $3|v|$, givingThere are four massless and four massive gauge bosons. The centralizer of a Hermitian matrix in SU(n) preserves the degenerate two-dimensional eigenspace, so the unbroken group isAt the Lie algebra level this is $\mathfrak{su}(2)\oplus \mathfrak{u}(1)$, generated by $t^{1,2,3}$ and $t^8$. The finite quotient specifies the actual subgroup inside SU(3); simply writing $SU(2)\times U(1)$ suppresses that global identification. The four real massive fields combine into a complex SU(2) doublet and its conjugate. Unbroken SU(2) rotations relate its two components, explaining the equal masses of the $13$ and $23$ sectors in this SU(3) gauge theory with an adjoint Higgs field.

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.