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

Assume ${\mu }^2,\lambda >0$. At zero source, the vacuum manifold is the sphere of radius $v_0=\mu /\sqrt{\lambda }$. The source tilts the scalar potential by $-ϵ{\varphi }_1$. Its force should be small compared with the radial restoring-force scale ${\mu }^2{v}_0$, so a useful quantitative condition isFor nonzero $ϵ$, the exact global symmetry in field theory of the Lagrangian is $O(n-1)$, rotating ${\varphi }_2,\dots ,{\varphi }_n$. The global vacuum aligns with the source:This vacuum preserves the entire remaining $O(n-1)$ symmetry. There is therefore no exact Goldstone boson for $ϵ\ne 0$. Expanding the scalar potential to quadratic order givesThus there are $n-1$ degenerate light pseudo-Goldstone bosons and one heavy radial scalar. The small-source condition makes their squared masses parametrically smaller than the radial squared mass. Sending $ϵ$ to zero restores the original massless modes.