Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 303 / 3 / b / ii / Solution

Choose the vacuum $\varphi =(v+\sigma ,{\pi }_1,\dots ,{\pi }_{N-1})$. The quadratic free energy contains a positive mass term for the radial field $\sigma$ but no mass term for any transverse field ${\pi }_a$:Thus the momentum-space correlation function is $⟨{\pi }_a(k){\pi }_b(-k)⟩={\delta }_{ab}/k^2$. Since a massive correlator has denominator $k^2+{\xi }^{-2}$, these Goldstone modes have ${\xi }^{-1}=0$ and hence infinite correlation length.