= Solution
Choose the vacuum $\phi=(v+\sigma,\pi_1,\ldots,\pi_{N-1})$. The quadratic free energy contains a positive mass term for the radial field $\sigma$ but no mass term for any transverse field $\pi_a$:
$$
F^{(2)}_\pi=\frac12\sum_{a=1}^{N-1}\int d^dx,(\nabla\pi_a)^2.
$$
Thus the momentum-space <correlation function> is $\langle\pi_a(k)\pi_b(-k)\rangle=\delta_{ab}/k^2$. Since a massive correlator has denominator $k^2+\xi^{-2}$, these Goldstone modes have $\xi^{-1}=0$ and hence infinite <correlation length>.
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