Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 303 / 3 / c / iv / Solution

If ${\mu }^2={\lambda }^2<0$, the enhanced $O(3)$-symmetric potential has the sphere of minimaChoosing a ground state breaks $O(3)$ to $O(2)$. The vacuum manifold $O(3)/O(2)\simeq S^2$ has dimension two, so there are two Goldstone modes.