Past exam of the theoretical physics major option course of the University of Oxford / 2018 / 2 / d

The breaking of the internal global symmetry in field theory $SU(n)\times SU(n)$ to $SU(n)$ removes $2(n^2-1)-(n^2-1)$ generators. The relativistic Goldstone theorem therefore requiresOther states fall into multiplets of the unbroken special unitary group, but their masses are not fixed by this counting. For the usual diagonal embedding $SU(n{)}_V\subset SU(n{)}_L\times SU(n{)}_R$, the Goldstone bosons form its adjoint multiplet.

If small explicit symmetry-breaking terms are added, the broken-generator directions generally acquire small curvature, producing light pseudo-Goldstone bosons rather than exactly massless states. Their squared masses tend to zero with those breaking parameters. Small breaking can also split multiplet degeneracies. These claims concern a small perturbation of the broken phase; approximate symmetry alone does not establish which vacuum is selected.